diff --git a/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf b/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..36af06cd6d4161546cdd238b5859432dfb57975e --- /dev/null +++ b/-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:03d08b4e9476069e073997aa320a0fd1d0568f8f16c14f7fdc7678fa04234b48 +size 1564617 diff --git a/-9AzT4oBgHgl3EQfhPxU/vector_store/index.faiss b/-9AzT4oBgHgl3EQfhPxU/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8306d558984f125ac8cb8de687c0ab0f34a36318 --- /dev/null +++ b/-9AzT4oBgHgl3EQfhPxU/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2b1b9fdbf9c41a900369f168351bebd702c488928368687438e15322c56c799d +size 3080237 diff --git a/-9AzT4oBgHgl3EQfhPxU/vector_store/index.pkl b/-9AzT4oBgHgl3EQfhPxU/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..9d02e1d59ae6416a20d0eb42df73dfd10d9b9655 --- /dev/null +++ b/-9AzT4oBgHgl3EQfhPxU/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3f74d7bc7722810a8b8b73897e94d7f80c07fafae0a0510c2c3e4be9d05edbd4 +size 104199 diff --git a/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/2301.00543v1.pdf.txt b/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/2301.00543v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b689f03c6f6a670f2dc05795f749d1bc3a07b069 --- /dev/null +++ b/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/2301.00543v1.pdf.txt @@ -0,0 +1,904 @@ +arXiv:2301.00543v1 [math.GR] 2 Jan 2023 +ON PSEUDO-REAL FINITE SUBGROUPS OF PGL3(C) +E. BADR AND A. ELGUINDY +Abstract. Let G be a finite subgroup of PGL3(C), and let σ be the generator +of Gal(C/R). +We say that G has a real field of moduli if σG and G are +PGL3(C)-conjugates, that is, if ∃ φ ∈ PGL3(C) such that φ−1 G φ = +σG. +Furthermore, we say that R is a field of definition for G or that G is definable +over R if G is PGL3(C)-conjugate to some G′ ⊂ PGL3(R). In this situation, +we call G′ a model for G over R. If G has R as a field of definition but is not +definable over R, then we call G pseudo-real. +In this paper, we first show that any finite cyclic subgroup G = Z/nZ in +PGL3(C) has a real field of moduli and we provide a necessary and sufficient +condition for G = Z/nZ to be definable over R; see Theorems 2.1, 2.2, and +2.3. We also prove that any dihedral group D2n with n ≥ 3 in PGL3(C) is +definable over R; see Theorem 2.4. Furthermore, we study all six classes of +finite primitive subgroups of PGL3(C), and show that all of them except the +icosahedral group A5 are pseudo-real; see Theorem 2.5, whereas A5 is definable +over R. Finally, we explore the connection of these notions in group theory +with their analogues in arithmetic geometry; see Theorem 2.6 and Example +2.7. +1. Introduction +The projective general linear group over the complex numbers PGL3(C) is widely +studied in several branches of mathematics for many reasons. Some of these mo- +tivations come from algebraic geometry, arithmetic geometry, and also from group +theory. We give some examples of such motivations. +(1) In complex algebraic geometry, PGL3(C) can be viewed as the automorphism +group Aut(P2(C)) of the complex projective plane P2(C), see [11, Example 7.1.1] for +example. Moreover, any isomorphism between two smooth complex plane curves +C and C′ of a fixed degree d ≥ 4 is induced by an element of PGL3(C), see [8, +Theorem 1]. For such a curve we have the finiteness result | Aut(C)| < +∞ due to +Hurwitz [19], hence we can view Aut(C) as a finite subgroup of PGL3(C) acting on +a non-singular plane model F(X, Y, Z) = 0 for C inside P2(C). It is thus natural to +classify finite subgroups G in PGL3(C). Based on geometrical methods, Mitchell +[23] achieved such classification. Recently, Harui [12] made Mitchell’s classification +more precise under the assumption that G = Aut(C) for some smooth plane curves +C. +However, some of these groups live in a short exact sequence, hence group +extension problems arise, which can sometimes be hard to solve. +Another parallel line of research is to obtain the stratification of C-isomorphism +classes of smooth plane curves of a fixed degree d by their automorphism groups. +Henn in his PhD dissertation [13] and Komiya-Kuribayashi [22] accomplished this +task for smooth quartic curves (d = 4), Badr-Bars [3, 4, 5] for smooth quinitcs +(d = 5) and for smooth sextics (d = 6). +2020 Mathematics Subject Classification. 20G20, 14L35, 14H37, 22F50. +Key words and phrases. Projective linear groups; Field of moduli; Fields of definitions; Pseudo- +real; Smooth plane curves; Automorphism groups. +1 + +2 +E. BADR AND A. ELGUINDY +(2) In complex arithmetic geometry, the problem of studying fields of definition +versus fields of moduli for a Riemann surface S has attracted a lot of recent research. +For example, we refer to [1, 2, 7, 9, 14, 15, 16, 18, 21]. +More precisely, a subfield K of C is called a field of definition for S if there exists +a model of S defined by polynomials with coefficients in K. The field of moduli +for S is the intersection of all fields of definition for S. The work of Koizumi [20] +guarantee the existence of a model for S over a finite extension of its field of moduli. +In this direction, the surface S is said to be pseudo-real if its field of moduli is a +subfield of R, but S does not have R as a field of definition. +The above aspects from algebraic geometry and arithmetic geometry are the +main motivation for us to extend the notions of fields of definition, fields of moduli, +pseudo-real, to the study of arithmetic groups. Indeed, there has been other in- +stances in which it has been fruitful to translate concepts from arithmetic geometry +to group theory, as we illustrate next. +(3) In group theory, we can measure to which extent an infinite group Γ is +similar to an abelian group by computing its Jordan constant, denoted by J(Γ). +It is defined to be the smallest positive integer such that any finite subgroup of Γ +has an abelian normal subgroup with index not exceeding J(Γ). This definition +originated from the theory of abelian varieties, more specifically, [24, Definition +2.1]. +Concerning the Jordan constant J(PGL3(K)), where K is a field of characteristic +0, Hu [17] showed that it assumes only one of the values: 360, 168, 60, 24, 12, 6, +depending on whether +√ +5 or ζ3 belongs to K or not. Here ζ3 denotes a primitive 3rd +root of unity in K, a fixed algebraic closure of K. In particular, J(PGL3(C)) = 360, +see [17, Theorem 1.2] for full details. +Notations. Throughout the paper, we use the following notations. +• Norm(G, PGL3(C)) is the normalizer of G inside PGL3(C), +• ζn = e +2πi +n , a fixed primitive nth root of unity in C. +• We shall view C× as a subgroup of GL3(C) by identifying 0 ̸= c ∈ C with +diag(c, c, c). If A is in GL3(C), we let π(A) denote its image under the +canonical projection onto PGL3(C), namely π(A) is the coset (or equiva- +lence class) C×A. To ease notation, we occasionally continue to use A in +place of π(A) when the context is clear. +• If A = (ai,j) ∈ GL3(C), then the projective linear transformation π(A) ∈ +PGL3(K) is sometimes written as +[a1,1X + a1,2Y + a1,3Z : a2,1X + a2,2Y + a2,3Z : a3,1X + a3,2Y + a3,3Z]. +• The Galois group Gal(C/R)- action on PGL3(C) is a left action, denoted +by σφ for any φ ∈ PGL3(C). +• For c ∈ C, ℜ(c) and ℑ(c) denote the real and the imaginary parts of c +respectively, and |c| denotes the absolute value of c. +2. Main results +Let G ⊂ PGL3(C) be cyclic of order 1 < n < +∞. Up to PGL3(C)-conjugation, +such G is generated by a diagonal element A := diag(1, ζa +n, ζb +n), for some 0 ≤ a < +b ≤ n − 1 such that gcd(a, b) = 1. +Theorem 2.1. Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above. +Then, we have that +(1) G always has a real field of moduli. + +ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) +3 +(2) R is a field of definition for G if and only if A and A−1 are conjugates via a +transformation of the shape φ σφ−1 for some φ ∈ PGL3(C). In this situation, +φ−1 G φ would give a model for G over R. +An homology of period n is a projective linear transformation of the plane P2(C), +which is PGL3(C)-conjugate to diag(1, 1, ζn). Such a transformation fixes point- +wise a projective line L, its axis, and a point P ∈ P2(C) − L, its center. In its +canonical form, the line is L : Z = 0 and the point is P = (0 : 0 : 1). Otherwise, it +is a non-homology. +In particular, we have: +Theorem 2.2. Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above. +Then, there exists a model for G over R if and only if n = 2 or n > 2 such that +a + b, a − 2b or 2a − b equals 0 mod n. In particular, any cyclic group generated by +a homology of period n ≥ 3 is pseudo-real. +Furthermore, we can get a model for G over R generated by +φ−1 A φ = + + +2ℑ(α β) +0 +0 +0 +2ℑ(α β ζa +n) +2|β|2 sin(2πa/n) +0 +−2|α|2 sin(2πa/n) +2ℑ(α β ζ−a +n ) + + +for some α, β ∈ C∗ +The above results can be reformulated using characteristic polynomials of lifts +to B ∈ GL3(C). If we denote the characteristic polynomial of such B by fB(t), +then it is straightforward to see that for c ∈ C∗ +fcB(t) = c3fB(t/c). +(2.1) +So while we can not attach a single polynomial as a characteristic polynomial +to an element A ∈ PGL3(C), we can attach to such an A an equivalence class +of polynomials in C[t] coming from the action given by (2.1). +Such classes are +preserved under conjugation in PGL3(C), and we can prove the following result. +Corollary 2.3. A finite cyclic group G of order n ≥ 3 is definable over R if there +exists A ∈ GL3(C) such that π(A) (the image of A in PGL3(C) under the natural +projection) generates G in PGL3(C) and the characteristic polynomial fA(t) ∈ R[t]. +The converse is not necessarily true. +For G = D2n, a dihedral group in PGL3(C), we prove: +Theorem 2.4. Any dihedral group D2n of order 2n with n ≥ 3 in PGL3(C) is +conjugate to ⟨B, π(A)⟩, where B = [X : Z : Y ] and A = diag(1, ζa +n, ζ−a +n ) for some +integer a such that gcd(n, a) = 1. Moreover, we always can descend it to R as +⟨ φ−1 B φ, φ−1 A φ⟩, where φ−1 A φ is as given in Theorem 2.2 and +φ−1 B φ = + + +2ℑ(α β) +0 +0 +0 +−2ℑ(α β) +−2ℑ(β2) +0 +2ℑ(α2) +2ℑ(α β) + + +for some α, β ∈ C∗. +When G is one of the finite primitive subgroup of PGL3(C), we show the follow- +ing. +Theorem 2.5. Any of the finite primitive subgroups namely, the Hessian groups +Hess∗, for ∗ = 216, 72 and 36, the Klein group PSL(2, 7) of order 168, the icosa- +hedral group A5 of order 60 and the alternating group A6 of order 360, has a real +field of moduli. Moreover, none of them descends to R except A5. More concretely, + +4 +E. BADR AND A. ELGUINDY +we always can descend A5 to R as φ−1 ⟨ A, B, C⟩ φ, such that φ−1 A φ and φ−1 B φ +are as given in Theorem 2.4 with n = 5 and a = 4, and φ−1 C φ equals + + +4ℑ(α β) +8ℑ(α β) ℜ(α) +8ℑ(α β) ℜ(β) +2ℑ(β) +2 +� +cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ) +� +−2 cos(2π/5)ℑ(β2) +2ℑ(α) +2 cos(2π/5)ℑ(α2) +2 +� +cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ) +� + + , +for some α, β ∈ C∗. +A connection with these notions in arithmetic geometry is described by the next +result. +Theorem 2.6. Let C : F(X, Y, Z) = 0 be a smooth plane curve over C. If C has a +real field of moduli in the Arithmetic Geometry sense, then its automorphism group +Aut(C) has a real field of moduli in the Group Theory sense. +The converse of Theorem 2.6 is not necessarily true. Below is a counter example. +Example 2.7. There are infinitely many smooth plane quintic curves defined over +C by an equation of the form +Cα,β : X5 + Y 5 + Z5 + αX(Y Z)2 + βX3(Y Z) = 0, +such that the automorphism group Aut(Cα,β) = D10 has a real field of moduli, but +Cα,β does not have a real field of moduli as its field of moduli. +3. The case when G is cyclic +Suppose that G = ⟨diag(1, ζa +n, ζb +n)⟩ in PGL3(C) such that 0 ≤ a < b ≤ n − 1 and +gcd(a, b) = 1. +Since the complex conjugation automorphism σ : C → C sends ζn �→ ζ−1 +n , then +σG = ⟨diag(1, ζ−a +n , ζ−b +n )⟩ = G. In particular, G has a real field of moduli. This +proves Theorem 2.1-(1). +To prove Theorem 2.1-(2), we assume that G descends to R. That is, there exists +φ ∈ PGL3(C) satisfying φ−1 A φ ∈ PGL3(R), where A = diag(1, ζa +n, ζb +n). This holds +if and only if +φ−1 A φ = σ � +φ−1 A φ +� += σφ−1 A−1 σφ, +which we can read in two different ways. First as +� +φ σφ−1�−1 A +� +φ σφ−1� += A−1, +which shows that A and A−1 are conjugates via φ σφ−1. Second as +φ−1 A φ = σ � +φ−1 A φ +� +, +which shows that φ−1 A φ ∈ PGL3(R) as claimed. +We need the following lemma to discuss Theorem 2.2. +Lemma 3.1. Assume A and B are matrices in GL3(C) such that π(A) and π(B) +are PGL3(C)-conjugates (where π denotes the natural projection from GL3(C) to +PGL3(C)), then there is a constant c ∈ C∗ such that the eigenvalues of B are +precisely cν1, cν2, cν3, where ν1, ν2, ν3 are the eigenvalues of A. +Proof. Suppose that there is an ψ ∈ PGL3(C) such that ψ−1 π(A) ψ = π(B) in +PGL3(C). Then, this equation corresponds to ψ−1 A ψ = (1/c)B in GL3(C) for +some c ∈ C∗. Hence, A and (1/c)B are similar matrices in GL3(C), so by elementary +linear algebra, we guarantee that their characteristic polynomials have the same +roots, say ν1, ν2, ν3 . Therefore, the eigenvalues of B are cν1, cν2, cν3. +□ +We now present the proof of Theorem 2.2. + +ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) +5 +Proof. (of the necessity direction) First, assume that G is generated by a homology +A = diag(1, 1, ζn). Since {c, c, c ζn} ̸= {1, 1, ζ−1 +n } for any c ∈ C∗ unless n = 2, then +A and A−1 are never PGL3(C)-conjugates for n ≥ 3 by Lemma 3.1. In particular, +G does not have a model over R by Theorem 2.1. +Secondly, assume that G is generated by a non-homology A = diag(1, ζa +n, ζb +n) +such that {c, c ζa +n, c ζb +n} = {1, ζ−a +n , ζ−b +n } for some c ∈ C∗. Then, c is either 1, ζ−a +n +or +ζ−b +n . Moreover, +- if c = 1, then ζa +n = ζ−a +n , ζb +n = ζ−b +n +or ζa +n = ζ−b +n . That is, 2a = 2b = 0 mod n or +a + b = 0 mod n. We discard the case 2a = 2b = 0 mod n as it implies that n or +n/2 would divide gcd(a, b) = 1, a contradiction because n ≥ 3. This leaves us with +a + b = 0 mod n. +- if c = ζ−a +n , then ζb−a +n += ζ−b +n , and n | a − 2b = 0 mod n. +- if c = ζ−b +n , then ζa−b +n += ζ−a +n , and 2a − b = 0 mod n. +This completes the necessity part. +□ +Proof. (of the sufficiency direction) If G is cyclic generated by a homology of period +2, then G is PGL3(C)-conjugate to ⟨diag(1, 1, −1)⟩ in PGL3(R), and we are done. +Otherwise, G is generated by a non-homology A = diag(1, ζa +n, ζb +n) of order n ≥ 3 +such that a + b, a − 2b or 2a − b equals 0 mod n. First, we show that any of the +last two situation can be reduced to the first one. Indeed, if A = diag(1, ζ2b +n , ζb +n), +then one can take ψ = [Y : Z : X] so that +ψ−1 A ψ = diag(ζb +n, 1, ζ2b +n ) = diag(1, ζ−b +n , ζb +n) = diag(1, ζa′ +n , ζ−a′ +n +) in PGL3(C), +where a′ := −b. Similarly, if A = diag(1, ζa +n, ζ2a +n ), then take ψ = [Z : X : Y ] to get +ψ−1 A ψ = diag(ζa +n, ζ2a +n , 1) = diag(1, ζa +n, ζ−a +n ) in PGL3(C). +Now we are going to handle the situation when n divides a + b. Take +φ = + + +1 +0 +0 +0 +α +β +0 +α +β + + ∈ PGL3(C). +One easily verifies that φ σφ−1 = [X : Z : Y ] ∈ Norm(G, PGL3(C)) such that +[X : Z : Y ] A [X : Z : Y ] = A−1. In particular, we deduce by Theorem 2.1 that +φ−1 G φ ≤ PGL3(R) is a model of G over R. More specifically, +φ−1 A φ += + + +2ℑ(α β) i +0 +0 +0 +β +−β +0 +−α +α + + diag(1, ζa +n, ζ−a +n ) + + +1 +0 +0 +0 +α +β +0 +α +β + + += + + +2ℑ(α β) i +0 +0 +0 +ζa +n β +−ζ−a +n +β +0 +−ζa +n α +ζ−a +n +α + + + + +1 +0 +0 +0 +α +β +0 +α +β + + += + + +2ℑ(α β) i +0 +0 +0 +2ℑ(α β ζa +n) i +2|β|2 sin(2πa/n) i +0 +−2|α|2 sin(2πa/n) i +2ℑ(α β ζ−a +n ) i + + += + + +2ℑ(α β) +0 +0 +0 +2ℑ(α β ζa +n) +2|β|2 sin(2πa/n) +0 +−2|α|2 sin(2πa/n) +2ℑ(α β ζ−a +n ) + + ∈ PGL3(R). +This completes the proof of Theorem 2.2. +□ +Next, assume that G is generated by a non-homology π(A) ∈ PGL3(C) of order +n ≥ 3. As a consequence Theorem 2.2, we can say that fA(t) ∈ R[t] is a sufficient +(rather than necessary) condition for G to descend to R. + +6 +E. BADR AND A. ELGUINDY +Proof. (of Corollary 2.3) By Lemma 3.1, there exists c ∈ C∗ such that +fA(t) = (t − c)(t − cζa +n)(t − cζb +n) ∈ R[t]. +Moreover, the roots c, c ζa +n, c ζb +n of fA(t) are pairwise distinct, since π(A) is a non- +homology in PGL3(C) by assumption. +Now, the coefficients c3ζa+b +n +, c(1+ζa +n +ζb +n), c2(ζa+b +n ++ζa +n +ζb +n) belong to R. Thus +there are r, r′ ∈ R such that ζa+b +n += r/c3 and ζa +n + ζb +n = r′/c− 1. Consequently, the +last condition becomes c2(r/c3+r′/c−1) ∈ R, in other words, c3−r′c2+r′′c−r = 0 +for some r, r′, r′′ ∈ R. This means that c ∈ C is algebraic over R of degree dividing +3. Since C/R is a field extension of degree 2, then c must be algebraic over R of +degree 1. Therefore, c ∈ R, which in turns implies that ζa+b +n +, ζa +n + ζb +n ∈ R. +Clearly, ζa+b +n +∈ R only if a+b = k( n +2 ) with k = 1, 2 or 3, since 3 ≤ a+b ≤ 2n−3. +If k = 1 or 3, then ζa+b +n += −1 and ζa +n + ζb +n = ζa +n − ζ−a +n += 2 sin(2π a/n) i /∈ R, a +contradiction. Hence k = 1 and a + b = 0 mod n. By Theorem 2.2 we deduce that +G descends to R, which was to be shown. +To see that the converse does not hold in general, take A = diag(ζ3 +5, ζ4 +5, ζ2 +5) +in GL3(C). Clearly, fA(t) /∈ R[t]. However, G = ⟨π(A)⟩ is definable over R by +Theorem 2.2, since π(A) = diag(1, ζ5, ζ−1 +5 ) = diag(1, ζa +n, ζb +n) with n | a + b. +□ +4. The case when G is a Dihedral group +Suppose that G = ⟨A, B : An = B2 = 1, BAB = A−1⟩ is a dihedral group D2n +in PGL3(C) with n ≥ 3. There is no loss of generality to take A = diag(1, ζa +n, ζb +n) +up to conjugation and projective equivalence. +Since A and A−1 are PGL3(C)- +conjugates via B, then, by Theorem 2.2, A must be a non-homology. Moreover, +we can always reduce to the case b = −a modulo n. Furthermore, we can assume +by [18, Lemma 2.3.7] that B belongs to PBD(2, 1). Since BAB = A−1, we obtain +B = [X : νZ : ν−1Y ] for some ν ∈ C∗. +Through a projective transformation +ψ = diag(1, λν, λ), which is in Norm (⟨A⟩, PGL3(C)), we can further reduce to +ν = 1. Eventually, we conclude: +Lemma 4.1. For each fixed integer n ≥ 3, there is, up to PGL3(C)-conjugation, a +unique dihedral group D2n of order 2n. More precisely, any such group is conjugate +to the group generated by B = [X : Z : Y ] and A = diag(1, ζn, ζ−1 +n ). +Now, we will prove that a dihedral group G = ⟨ A, B⟩ as above has a real field +of moduli, moreover, it descends to R. +Proof. Since σ A = A−1 and σ B = B−1, then σG = G and G has a real field of +moduli. +On the other hand, we have seen in Theorem 2.2 that φ−1 A φ ∈ PGL3(R) +through a projective transformation φ of the shape: +φ = + + +1 +0 +0 +0 +α +β +0 +α +β + + . +It remains to see that φ−1 B φ ∈ PGL3(R) so that φ−1 G φ is a model of G over R. +Indeed, we have + +ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) +7 +φ−1 B φ += + + +2ℑ(α β) i +0 +0 +0 +β +−β +0 +−α +α + + [X : Z : Y ] + + +1 +0 +0 +0 +α +β +0 +α +β + + += + + +2ℑ(α β) +0 +0 +0 +−β +β +0 +α +−α + + + + +1 +0 +0 +0 +α +β +0 +α +β + + += + + +2ℑ(α β) i +0 +0 +0 +−2ℑ(α β) i +−2ℑ(β2) i +0 +2ℑ(α2) i +2ℑ(α β) i + + += + + +2ℑ(α β) +0 +0 +0 +−2ℑ(α β) +−2ℑ(β2) +0 +2ℑ(α2) +2ℑ(α β) + + ∈ PGL3(R). +□ +This completes the proof of Theorem 2.4. +5. The cases when G is a finite primitive subgroup of PGL3(C) +Recall that the finite primitive subgroups PGL3(C) are the Hessian groups Hess∗, +for ∗ = 216, 72, 36, the alternating groups A∗, for ∗ = 5, 6, and the Klein group +PSL(2, 7) of order 168. We study their definability over R in this section. +5.1. The Hessian groups Hess∗. The Hessian group of order 216, denoted by +Hess216, is unique up to conjugation in PGL3(C). See [23, p. 217] or [18, Lemma +2.3.7] for more details. For instance, we fix Hess216 = ⟨S, T, U, V ⟩ where +S = diag(1, ζ3, ζ−1 +3 ), U = diag(1, 1, ζ3), V = + + +1 +1 +1 +1 +ζ3 +ζ−1 +3 +1 +ζ−1 +3 +ζ3 + + , T = [Y : Z : X]. +Also, we consider the Hessian subgroup of order 72, Hess72 = ⟨S, T, V, UV U −1⟩, +and the Hessian subgroup of order 36, Hess36 = ⟨S, T, V ⟩. +Concerning the Hessian groups Hess∗, for ∗ ∈ {36, 72, 216}. We first show +Proposition 5.1. Any of the Hessian groups Hess∗ has a real field of moduli. +Proof. It is easy to see that σS = S−1, σU = U −1, and σT = T . Furthermore +σV = 3V −1 in GL3(C), hence we also have σV = V −1 in PGL3(C). It follows that +σ Hess∗ = Hess∗ if ∗ = 216 or 36. So Hess216 and Hess36 indeed have a real field of +moduli. For Hess72, we get σ Hess72 = ⟨S, T, V, U −1V −1U⟩ ⊂ Hess216; another copy +of Hess72 inside Hess216. The Group structure of Hess216 [10] assures that all copies +of Hess72 are Hess216-conjugates, that is to say, there is a projective transformation +ψ ∈ Hess216 such that ψ−1 Hess72 ψ = σ Hess72. From this we obtain that Hess72 +has a real field of moduli as well. +□ +As a consequence, +Corollary 5.2. The Hessian groups Hess∗ for ∗ = 216, 72 and 36 are all pseudo- +real. +Proof. It is easy to see that ST = T S, so ⟨S, T ⟩ is isomorphic to C3 × C3. By [17, +Lemma 5.2] (see also [25, Section 4]), C3 ×C3 is a subgroup of PGL3(K) if and only +if the field K contains a nontrivial cube root of unity. Since ζ3 /∈ R, we can’t reduce +⟨S, T ⟩ to a subgroup of PGL3(R) as ζ3 /∈ R. In particular, φ−1 Hess∗ φ ⊈ PGL3(R) +for any φ ∈ PGL3(C). Combining with Proposition 5.1, we conclude that Hess∗ is +pseudo-real for ∗ = 216, 72 and 36 as claimed. +□ + +8 +E. BADR AND A. ELGUINDY +5.2. The alternating groups A5 and A6. We first note that PGL3(C) possesses +a single conjugacy class isomorphic to each of A5 and A6, see [23, p. 224, 225] or [18, +Lemma 2.3.7]. Therefore, for i ∈ {5, 6} Ai and σ Ai must be PGL3(C)-conjugates. +In other words, Ai has a real field of moduli. +Since A6 contains C3 × C3 as a subgroup, then we can use the same argument +as in Corollary 5.2 to deduce the following. +Corollary 5.3. The alternating group A6 is pseudo-real. +For the icosahedral group A5, the situation is different. To study it we fix the +copy G := ⟨A, B, C⟩ in PGL3(C), where +A = diag(1, ζ−1 +5 , ζ5), B = [X : Z : Y ], C = + + +2 +2 +2 +1 +cos(4π/5) +cos(2π/5) +1 +cos(2π/5) +cos(4π/5) + + . +According to [18, Lemma 2.3.7 ], G is PGL3(C)-conjugate to A5. Any subgroup of +PGL3(C) isomorphic to A5 is PGL3(C) conjugate to G. +Now, we are going to construct an explicit model for G over R. +Recall, from our study above of the Dihedral group in §4, that ⟨ A, B⟩ descends to +R via a transformation of the shape +φ = + + +1 +0 +0 +0 +α +β +0 +α +β + + ∈ PGL3(C). +Moreover, one can check that φ−1 C φ equals + + +4ℑ(α β) +8ℑ(α β) ℜ(α) +8ℑ(α β) ℜ(β) +2ℑ(β) +2 +� +cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ) +� +−2 cos(2π/5)ℑ(β2) +2ℑ(α) +2 cos(2π/5)ℑ(α2) +2 +� +cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ) +� + + , +in PGL3(R). Thus all generators of G when conjugated by the same φ become in +PGL3(R), hence the same is true for the whole group and the result follows. +5.3. The Klein group PSL(2, 7). Again, there is a single conjugacy class of +PSL(2, 7) in PGL3(C). Thus it has a real field of moduli. Also, we know by [18, +Lemma 2.3.7] that a representative of such a class contains the element diag(1, ζ7, ζ3 +7). +Theorem 2.2 applies to n = 7, a = 1, b = 3 to conclude that PSL(2, 7) is not defin- +able over R. +6. Connection to Arithmetic Geometry +Let C : F(X, Y, Z) = 0 be a non-singular plane curve defined over C with non- +trivial automorphism group Aut(C) in PGL3(C), +Lemma 6.1. We have Aut(σC) = σ Aut(C) +Proof. For any φ ∈ Aut(C), φF(X, Y, Z) = cF(X, Y, Z) for some c ∈ C∗. Applying +σ to both sides yields +σ(c) σF(X, Y, Z) = σ �φF(X, Y, Z) +� += +σφ (σF(X, Y, Z)) . +That is, σφ leaves invariant σC : σF(X, Y, Z) = 0. Equivalently, σφ ∈ Aut(σC), +hence σ Aut(C) ⊆ Aut(σC). +By a similar argument we can show the other inclusion. +□ +Theorem 6.2. Let C : F(X, Y, Z) = 0 be a smooth plane curve over C. If C has +a real field of moduli in the Arithmetic Geometry sense, then Aut(C) has +a real +field of moduli in the Group Theory sense. +The converse need not be true. + +ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) +9 +Proof. Since C : F(X, Y, Z) = 0 has a real field of moduli, then it must be the case +that σC : σF(X, Y, Z) = 0 and C : F(X, Y, Z) = 0 are C-projectively equivalent +(isomorphic over C). Moreover, any isomorphism between complex non-singular +plane curves C and C′ is always given by a projective transformation φ ∈ PGL3(C) +such that their automorphism groups are conjugates via this φ. As a consequence, +we obtain that φ−1 Aut(C) φ = Aut(σC), which equals σ Aut(C) by Lemma 6.1. +Thus Aut(C) has a real field of moduli as claimed. +To complete the argument, Example 6.3 below provides infinitely many counter +examples that Aut(C) can descend R, but C : F(X, Y, Z) = 0 does not even have +a real field of moduli. +□ +Example 6.3. Consider the two-dimensional family Ca,b of smooth plane quintic +curves given by +Ca,b : X5 + Y 5 + Z5 + iaX(Y Z)2 + ibX3(Y Z), +where a, b ∈ R∗ such that a/b ̸= (c5 − 3)c2 +2c5 − 1 ζm +10 for any c ∈ C∗ and m ∈ {±1, ±3, 5}. +• Non-singularity. We first note that no singular points lie over Y = 0. +Indeed, if C has singularity at (α : 0 : β), then α and β must be 0 in +order to satisfy FX = FZ = 0, a contradiction. Second, the resultant of +f1(X, Z) := FY (X, 1, Z) and f2(X, Z) := FZ(X, 1, Z) with respect to X is +given by +ResX(f1, f2) = −125 i b3 (Z5 − 1)3. +Using MATHEMATICA, one can verify that we have singular points over +Z5 = 1 only if a/b = (c5 − 3)c2 +2c5 − 1 ζm +10 for some c ∈ C∗ and m ∈ {±1, ±3, 5}, +which is absurd by assumption. +• Automorphism group. The stratification of smooth plane quintics by +their automorphism groups in [3, 6] assures that the group D10 gener- +ated by ρ1 = diag(1, ζ5, ζ−1 +5 ) and ρ2 = [X : Z : Y ] is a always a sub- +group of automorphisms for Ca,b. Moreover, if Ca,b admits a larger auto- +morphism group, then it should be GAP(150, 5) = (Z/5Z)2 ⋊ S3, where +in this situation Ca,b is K-isomorphic to the Fermat quintic curve F5; +the most symmetric smooth quintic curve. +In particular, there must +be an extra automorphism ρ3 /∈ ⟨ρ1⟩ of order 5 that commutes with +ρ1 as any Z/5Z inside (Z/5Z)2 ⋊ S3 is contained in a (Z/5Z)2. +See +Group Structure of (Z/5Z)2 ⋊ S3 [10]. Straightforward calculations in PGL3(C) +lead to ρ3 = diag(1, α, β) with α5 = β5 = 1. Checking the action of such +an automorphism on the defining equation of Ca,b tells us that a = b = 0 +or ρ3 ∈ ⟨ρ1⟩. Therefore, Aut(Ca,b) = D10 = ⟨ρ1, ρ2⟩. +Now, we conclude by Theorem 2.4 that Aut(Ca,b) descends to R. +• Ca,b does not have a real field of moduli. Suppose that C is a member +of the family Ca,b such that C has a real field of moduli. Hence C and σC +are C-projectively equivalent via some φ ∈ PGL3(C). Since C and σC +belong to the same family Ca,b, we have σ Aut(C) = Aut(C) = ⟨ρ1, ρ2⟩. +In particular, φ should be in the normalizer of ⟨ρ1, ρ2⟩ in PGL3(C). We +reduce to the case φ−1ρ1φ = ρ1 or ρ−1 as {c, cζ5, cζ−1 +5 } ̸= {1, ζ2 +5, ζ−2 +5 } or +{1, ζ3 +5, ζ−3 +5 } for any c ∈ C∗ by Lemma 3.1. Consequently, φ = diag(1, α, β) +or [X : αZ : βY ] for some α, β ∈ C∗. Because φC = σC, we must have +α5 = β5 = 1 and αβ = (αβ)2 = −1. The last condition is inconsistent, +which means that C and σC are never C-isomorphic. + +10 +E. BADR AND A. ELGUINDY +References +[1] M. Artebani and S. Qusipe, Fields of moduli and fields of definition of odd signature curves, +Arch. Math. 99 (2012), 333-343. +[2] M. Artebani, M. Carvacho, R. A. Hidalgo, and S. Quispe, A tower of Riemann surfaces which +cannot be defined over their field of moduli, Glasgow Math. J. 59 (2017), 379-393. +[3] E. Badr and F. Bars, Automorphism groups of nonsingular plane curves of degree 5. Comm. +Algebra 44 (2016), no. 10, 4327-4340. MR 3508302. +[4] E. Badr and F. Bars, On fake ES-irreducibile components of certain strata of smooth plane +sextics. Preprint 2022, https://doi.org/10.48550/arXiv.2208.08904. +[5] E. Badr and F. Bars, The stratification by automorphism groups of smooth plane sextics +curves. Preprint 2022, https://doi.org/10.48550/arXiv.2208.12749. +[6] E. Badr and E. Lorenzo. A note on the stratification of smooth plane curves of genus 6. +Colloq. Math. 192, (2020), 207-222. +[7] E. Badr, R. A. Hidalgo, and S. Quispe, Non-hyperelliptic Riemann surfaces with real field of +moduli but not definable over the reals, Arch. Math. 110 (2018), 219-222. +[8] H. C. Chang, On plane algebraic curves, Chinese J. Math. 6 (1978), no. 2, 185- 189. MR +529972. +[9] C. J. Earle, On the moduli of closed Riemann surfaces with symmetries, In: Advances in +the Theory of Riemann Surfaces, L.V. Ahlfors et al. (Eds.), 119-130, Princeton Univ. Press, +Princeton, 1971. +[10] T. Dokchitser, GroupNames, https://people.maths.bris.ac.uk/ matyd/GroupNames/about.html +[11] R. Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Graduate +Texts in Mathematics, No. 52. MR 0463157. +[12] T. Harui, Automorphism groups of plane curves. Kodai Math. J. 42 (2), (2019), 308-331. +[13] P. Henn, Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht +3. Inagural-dissertation, Heidelberg, 1976. +[14] R. A. Hidalgo, Non-hyperelliptic Riemann surfaces with real field of moduli but not definable +over the reals, Arch. Math. 93 (2009), 219-222. +[15] R. A. Hidalgo and S. Quispe, Fields of moduli of some special curves, J. Pure Appl. Algebra +220 (2022), 55-60. +[16] R. A. Hidalgo and T. Shaska, On the field of moduli of superelliptic curves: In higher genus +curves in mathematical physics and arithmetic geometry. Commun. Contemp. Math. 703 +(2018). +[17] Y. Hu, Jordan constant for PGL3(K), arXiv:2206.02186v1 [math.RT], 5 June 2022. +[18] B. Huggins, Fields of moduli and fields of definition of curves. ProQuest LLC, Ann Arbor, +MI, 2005, PhD Thesis University of California, Berkeley. MR2708514 +[19] A. Hurwitz, ¨Uber algebraische Gebilde mit eindeutigen Transformationen in sich, Math. Ann. +41 (1892), no. 3, 403-442. MR 1510753. +[20] S. Koizumi, +Fields of moduli for polarized abelian varieties and for curves, Nagoya Math. +J. 48 (1972), 37-55. +[21] A. Kontogeorgis, Field of moduli versus field of definition for cyclic covers of the projective +line, J. Th´eor. Nombres Bordeaux 21 (2009), 679-692. +[22] A. Kuribayashi and K. Komiya, On Weierstrass points and automorphisms of curves of genus +three. Algebraic geometry (Proc. Summer Meeting, Univ. Copenhagen, Copenhagen, 1978), +[23] H. Mitchell, Determination of the ordinary and modular ternary linear groups, Trans. Amer. +Math. Soc. 12 (1911), no. 2, 207-242. MR 1500887. +[24] V. Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of +algebraic varieties. Affine Algebraic Geometry: The Russell Festschrift, CRM Proceedings +and Lecture Notes, 54 (2011), 289-311. +[25] E. Yasinsky, The Jordan constant for Cremona group of rank 2, Korean Math. Soc. 54, no. +5 (2017), 1859-1871. +• Eslam Badr +Mathematics Department, Faculty of Science, Cairo University, Giza-Egypt +Email address: eslam@sci.cu.edu.eg +Mathematics and Actuarial Science Department (MACT), American University in +Cairo (AUC), New Cairo-Egypt +Email address: eslammath@aucegypt.edu +• Ahmad El-Guindy + +ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) +11 +Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt +Email address: aelguindy@sci.cu.edu.eg + diff --git a/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/load_file.txt b/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39d72732f68f58ce15d56449f0d8fd037192bb47 --- /dev/null +++ b/-NAyT4oBgHgl3EQfqfhf/content/tmp_files/load_file.txt @@ -0,0 +1,510 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf,len=509 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='00543v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='GR] 2 Jan 2023 ON PSEUDO-REAL FINITE SUBGROUPS OF PGL3(C) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let G be a finite subgroup of PGL3(C), and let σ be the generator of Gal(C/R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We say that G has a real field of moduli if σG and G are PGL3(C)-conjugates, that is, if ∃ φ ∈ PGL3(C) such that φ−1 G φ = σG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we say that R is a field of definition for G or that G is definable over R if G is PGL3(C)-conjugate to some G′ ⊂ PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this situation, we call G′ a model for G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If G has R as a field of definition but is not definable over R, then we call G pseudo-real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this paper, we first show that any finite cyclic subgroup G = Z/nZ in PGL3(C) has a real field of moduli and we provide a necessary and sufficient condition for G = Z/nZ to be definable over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We also prove that any dihedral group D2n with n ≥ 3 in PGL3(C) is definable over R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we study all six classes of finite primitive subgroups of PGL3(C), and show that all of them except the icosahedral group A5 are pseudo-real;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='5, whereas A5 is definable over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Finally, we explore the connection of these notions in group theory with their analogues in arithmetic geometry;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='6 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Introduction The projective general linear group over the complex numbers PGL3(C) is widely studied in several branches of mathematics for many reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Some of these mo- tivations come from algebraic geometry, arithmetic geometry, and also from group theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We give some examples of such motivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (1) In complex algebraic geometry, PGL3(C) can be viewed as the automorphism group Aut(P2(C)) of the complex projective plane P2(C), see [11, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1] for example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, any isomorphism between two smooth complex plane curves C and C′ of a fixed degree d ≥ 4 is induced by an element of PGL3(C), see [8, Theorem 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For such a curve we have the finiteness result | Aut(C)| < +∞ due to Hurwitz [19], hence we can view Aut(C) as a finite subgroup of PGL3(C) acting on a non-singular plane model F(X, Y, Z) = 0 for C inside P2(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is thus natural to classify finite subgroups G in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Based on geometrical methods, Mitchell [23] achieved such classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Recently, Harui [12] made Mitchell’s classification more precise under the assumption that G = Aut(C) for some smooth plane curves C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' However, some of these groups live in a short exact sequence, hence group extension problems arise, which can sometimes be hard to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Another parallel line of research is to obtain the stratification of C-isomorphism classes of smooth plane curves of a fixed degree d by their automorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Henn in his PhD dissertation [13] and Komiya-Kuribayashi [22] accomplished this task for smooth quartic curves (d = 4), Badr-Bars [3, 4, 5] for smooth quinitcs (d = 5) and for smooth sextics (d = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 20G20, 14L35, 14H37, 22F50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Projective linear groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Field of moduli;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Fields of definitions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Pseudo- real;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Smooth plane curves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Automorphism groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 1 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY (2) In complex arithmetic geometry, the problem of studying fields of definition versus fields of moduli for a Riemann surface S has attracted a lot of recent research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For example, we refer to [1, 2, 7, 9, 14, 15, 16, 18, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More precisely, a subfield K of C is called a field of definition for S if there exists a model of S defined by polynomials with coefficients in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The field of moduli for S is the intersection of all fields of definition for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The work of Koizumi [20] guarantee the existence of a model for S over a finite extension of its field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this direction, the surface S is said to be pseudo-real if its field of moduli is a subfield of R, but S does not have R as a field of definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The above aspects from algebraic geometry and arithmetic geometry are the main motivation for us to extend the notions of fields of definition, fields of moduli, pseudo-real, to the study of arithmetic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, there has been other in- stances in which it has been fruitful to translate concepts from arithmetic geometry to group theory, as we illustrate next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (3) In group theory, we can measure to which extent an infinite group Γ is similar to an abelian group by computing its Jordan constant, denoted by J(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is defined to be the smallest positive integer such that any finite subgroup of Γ has an abelian normal subgroup with index not exceeding J(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This definition originated from the theory of abelian varieties, more specifically, [24, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Concerning the Jordan constant J(PGL3(K)), where K is a field of characteristic 0, Hu [17] showed that it assumes only one of the values: 360, 168, 60, 24, 12, 6, depending on whether √ 5 or ζ3 belongs to K or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Here ζ3 denotes a primitive 3rd root of unity in K, a fixed algebraic closure of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, J(PGL3(C)) = 360, see [17, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2] for full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Throughout the paper, we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Norm(G, PGL3(C)) is the normalizer of G inside PGL3(C), ζn = e 2πi n , a fixed primitive nth root of unity in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We shall view C× as a subgroup of GL3(C) by identifying 0 ̸= c ∈ C with diag(c, c, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If A is in GL3(C), we let π(A) denote its image under the canonical projection onto PGL3(C), namely π(A) is the coset (or equiva- lence class) C×A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To ease notation, we occasionally continue to use A in place of π(A) when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If A = (ai,j) ∈ GL3(C), then the projective linear transformation π(A) ∈ PGL3(K) is sometimes written as [a1,1X + a1,2Y + a1,3Z : a2,1X + a2,2Y + a2,3Z : a3,1X + a3,2Y + a3,3Z].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Galois group Gal(C/R)- action on PGL3(C) is a left action, denoted by σφ for any φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For c ∈ C, ℜ(c) and ℑ(c) denote the real and the imaginary parts of c respectively, and |c| denotes the absolute value of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Main results Let G ⊂ PGL3(C) be cyclic of order 1 < n < +∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Up to PGL3(C)-conjugation, such G is generated by a diagonal element A := diag(1, ζa n, ζb n), for some 0 ≤ a < b ≤ n − 1 such that gcd(a, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, we have that (1) G always has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 3 (2) R is a field of definition for G if and only if A and A−1 are conjugates via a transformation of the shape φ σφ−1 for some φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In this situation, φ−1 G φ would give a model for G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' An homology of period n is a projective linear transformation of the plane P2(C), which is PGL3(C)-conjugate to diag(1, 1, ζn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Such a transformation fixes point- wise a projective line L, its axis, and a point P ∈ P2(C) − L, its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In its canonical form, the line is L : Z = 0 and the point is P = (0 : 0 : 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Otherwise, it is a non-homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, we have: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let G = ⟨A⟩ ⊂ PGL3(C) be a cyclic group of order n as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, there exists a model for G over R if and only if n = 2 or n > 2 such that a + b, a − 2b or 2a − b equals 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, any cyclic group generated by a homology of period n ≥ 3 is pseudo-real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we can get a model for G over R generated by φ−1 A φ = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 2ℑ(α β ζa n) 2|β|2 sin(2πa/n) 0 −2|α|2 sin(2πa/n) 2ℑ(α β ζ−a n ) \uf8f6 \uf8f8 for some α, β ∈ C∗ The above results can be reformulated using characteristic polynomials of lifts to B ∈ GL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If we denote the characteristic polynomial of such B by fB(t), then it is straightforward to see that for c ∈ C∗ fcB(t) = c3fB(t/c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1) So while we can not attach a single polynomial as a characteristic polynomial to an element A ∈ PGL3(C), we can attach to such an A an equivalence class of polynomials in C[t] coming from the action given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Such classes are preserved under conjugation in PGL3(C), and we can prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A finite cyclic group G of order n ≥ 3 is definable over R if there exists A ∈ GL3(C) such that π(A) (the image of A in PGL3(C) under the natural projection) generates G in PGL3(C) and the characteristic polynomial fA(t) ∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The converse is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For G = D2n, a dihedral group in PGL3(C), we prove: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any dihedral group D2n of order 2n with n ≥ 3 in PGL3(C) is conjugate to ⟨B, π(A)⟩, where B = [X : Z : Y ] and A = diag(1, ζa n, ζ−a n ) for some integer a such that gcd(n, a) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, we always can descend it to R as ⟨ φ−1 B φ, φ−1 A φ⟩, where φ−1 A φ is as given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 and φ−1 B φ = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 −2ℑ(α β) −2ℑ(β2) 0 2ℑ(α2) 2ℑ(α β) \uf8f6 \uf8f8 for some α, β ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' When G is one of the finite primitive subgroup of PGL3(C), we show the follow- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any of the finite primitive subgroups namely, the Hessian groups Hess∗, for ∗ = 216, 72 and 36, the Klein group PSL(2, 7) of order 168, the icosa- hedral group A5 of order 60 and the alternating group A6 of order 360, has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, none of them descends to R except A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More concretely, 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY we always can descend A5 to R as φ−1 ⟨ A, B, C⟩ φ, such that φ−1 A φ and φ−1 B φ are as given in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4 with n = 5 and a = 4, and φ−1 C φ equals \uf8eb \uf8ed 4ℑ(α β) 8ℑ(α β) ℜ(α) 8ℑ(α β) ℜ(β) 2ℑ(β) 2 � cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ) � −2 cos(2π/5)ℑ(β2) 2ℑ(α) 2 cos(2π/5)ℑ(α2) 2 � cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ) � \uf8f6 \uf8f8 , for some α, β ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A connection with these notions in arithmetic geometry is described by the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let C : F(X, Y, Z) = 0 be a smooth plane curve over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If C has a real field of moduli in the Arithmetic Geometry sense, then its automorphism group Aut(C) has a real field of moduli in the Group Theory sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The converse of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='6 is not necessarily true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Below is a counter example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' There are infinitely many smooth plane quintic curves defined over C by an equation of the form Cα,β : X5 + Y 5 + Z5 + αX(Y Z)2 + βX3(Y Z) = 0, such that the automorphism group Aut(Cα,β) = D10 has a real field of moduli, but Cα,β does not have a real field of moduli as its field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The case when G is cyclic Suppose that G = ⟨diag(1, ζa n, ζb n)⟩ in PGL3(C) such that 0 ≤ a < b ≤ n − 1 and gcd(a, b) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since the complex conjugation automorphism σ : C → C sends ζn �→ ζ−1 n , then σG = ⟨diag(1, ζ−a n , ζ−b n )⟩ = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, G has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This proves Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1-(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1-(2), we assume that G descends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' That is, there exists φ ∈ PGL3(C) satisfying φ−1 A φ ∈ PGL3(R), where A = diag(1, ζa n, ζb n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This holds if and only if φ−1 A φ = σ � φ−1 A φ � = σφ−1 A−1 σφ, which we can read in two different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' First as � φ σφ−1�−1 A � φ σφ−1� = A−1, which shows that A and A−1 are conjugates via φ σφ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Second as φ−1 A φ = σ � φ−1 A φ � , which shows that φ−1 A φ ∈ PGL3(R) as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We need the following lemma to discuss Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Assume A and B are matrices in GL3(C) such that π(A) and π(B) are PGL3(C)-conjugates (where π denotes the natural projection from GL3(C) to PGL3(C)), then there is a constant c ∈ C∗ such that the eigenvalues of B are precisely cν1, cν2, cν3, where ν1, ν2, ν3 are the eigenvalues of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Suppose that there is an ψ ∈ PGL3(C) such that ψ−1 π(A) ψ = π(B) in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, this equation corresponds to ψ−1 A ψ = (1/c)B in GL3(C) for some c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hence, A and (1/c)B are similar matrices in GL3(C), so by elementary linear algebra, we guarantee that their characteristic polynomials have the same roots, say ν1, ν2, ν3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, the eigenvalues of B are cν1, cν2, cν3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ We now present the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (of the necessity direction) First, assume that G is generated by a homology A = diag(1, 1, ζn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since {c, c, c ζn} ̸= {1, 1, ζ−1 n } for any c ∈ C∗ unless n = 2, then A and A−1 are never PGL3(C)-conjugates for n ≥ 3 by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, G does not have a model over R by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Secondly, assume that G is generated by a non-homology A = diag(1, ζa n, ζb n) such that {c, c ζa n, c ζb n} = {1, ζ−a n , ζ−b n } for some c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Then, c is either 1, ζ−a n or ζ−b n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, if c = 1, then ζa n = ζ−a n , ζb n = ζ−b n or ζa n = ζ−b n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' That is, 2a = 2b = 0 mod n or a + b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We discard the case 2a = 2b = 0 mod n as it implies that n or n/2 would divide gcd(a, b) = 1, a contradiction because n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This leaves us with a + b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' if c = ζ−a n , then ζb−a n = ζ−b n , and n | a − 2b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' if c = ζ−b n , then ζa−b n = ζ−a n , and 2a − b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This completes the necessity part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (of the sufficiency direction) If G is cyclic generated by a homology of period 2, then G is PGL3(C)-conjugate to ⟨diag(1, 1, −1)⟩ in PGL3(R), and we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Otherwise, G is generated by a non-homology A = diag(1, ζa n, ζb n) of order n ≥ 3 such that a + b, a − 2b or 2a − b equals 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' First, we show that any of the last two situation can be reduced to the first one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, if A = diag(1, ζ2b n , ζb n), then one can take ψ = [Y : Z : X] so that ψ−1 A ψ = diag(ζb n, 1, ζ2b n ) = diag(1, ζ−b n , ζb n) = diag(1, ζa′ n , ζ−a′ n ) in PGL3(C), where a′ := −b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Similarly, if A = diag(1, ζa n, ζ2a n ), then take ψ = [Z : X : Y ] to get ψ−1 A ψ = diag(ζa n, ζ2a n , 1) = diag(1, ζa n, ζ−a n ) in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now we are going to handle the situation when n divides a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Take φ = \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' One easily verifies that φ σφ−1 = [X : Z : Y ] ∈ Norm(G, PGL3(C)) such that [X : Z : Y ] A [X : Z : Y ] = A−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, we deduce by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1 that φ−1 G φ ≤ PGL3(R) is a model of G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More specifically, φ−1 A φ = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 β −β 0 −α α \uf8f6 \uf8f8 diag(1, ζa n, ζ−a n ) \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 ζa n β −ζ−a n β 0 −ζa n α ζ−a n α \uf8f6 \uf8f8 \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 2ℑ(α β ζa n) i 2|β|2 sin(2πa/n) i 0 −2|α|2 sin(2πa/n) i 2ℑ(α β ζ−a n ) i \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 2ℑ(α β ζa n) 2|β|2 sin(2πa/n) 0 −2|α|2 sin(2πa/n) 2ℑ(α β ζ−a n ) \uf8f6 \uf8f8 ∈ PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Next, assume that G is generated by a non-homology π(A) ∈ PGL3(C) of order n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' As a consequence Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, we can say that fA(t) ∈ R[t] is a sufficient (rather than necessary) condition for G to descend to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (of Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3) By Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1, there exists c ∈ C∗ such that fA(t) = (t − c)(t − cζa n)(t − cζb n) ∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, the roots c, c ζa n, c ζb n of fA(t) are pairwise distinct, since π(A) is a non- homology in PGL3(C) by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, the coefficients c3ζa+b n , c(1+ζa n +ζb n), c2(ζa+b n +ζa n +ζb n) belong to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus there are r, r′ ∈ R such that ζa+b n = r/c3 and ζa n + ζb n = r′/c− 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Consequently, the last condition becomes c2(r/c3+r′/c−1) ∈ R, in other words, c3−r′c2+r′′c−r = 0 for some r, r′, r′′ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' This means that c ∈ C is algebraic over R of degree dividing 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since C/R is a field extension of degree 2, then c must be algebraic over R of degree 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, c ∈ R, which in turns implies that ζa+b n , ζa n + ζb n ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Clearly, ζa+b n ∈ R only if a+b = k( n 2 ) with k = 1, 2 or 3, since 3 ≤ a+b ≤ 2n−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If k = 1 or 3, then ζa+b n = −1 and ζa n + ζb n = ζa n − ζ−a n = 2 sin(2π a/n) i /∈ R, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hence k = 1 and a + b = 0 mod n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 we deduce that G descends to R, which was to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To see that the converse does not hold in general, take A = diag(ζ3 5, ζ4 5, ζ2 5) in GL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Clearly, fA(t) /∈ R[t].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' However, G = ⟨π(A)⟩ is definable over R by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, since π(A) = diag(1, ζ5, ζ−1 5 ) = diag(1, ζa n, ζb n) with n | a + b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The case when G is a Dihedral group Suppose that G = ⟨A, B : An = B2 = 1, BAB = A−1⟩ is a dihedral group D2n in PGL3(C) with n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' There is no loss of generality to take A = diag(1, ζa n, ζb n) up to conjugation and projective equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since A and A−1 are PGL3(C)- conjugates via B, then, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2, A must be a non-homology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, we can always reduce to the case b = −a modulo n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore, we can assume by [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7] that B belongs to PBD(2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since BAB = A−1, we obtain B = [X : νZ : ν−1Y ] for some ν ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Through a projective transformation ψ = diag(1, λν, λ), which is in Norm (⟨A⟩, PGL3(C)), we can further reduce to ν = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Eventually, we conclude: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For each fixed integer n ≥ 3, there is, up to PGL3(C)-conjugation, a unique dihedral group D2n of order 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' More precisely, any such group is conjugate to the group generated by B = [X : Z : Y ] and A = diag(1, ζn, ζ−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, we will prove that a dihedral group G = ⟨ A, B⟩ as above has a real field of moduli, moreover, it descends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since σ A = A−1 and σ B = B−1, then σG = G and G has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' On the other hand, we have seen in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 that φ−1 A φ ∈ PGL3(R) through a projective transformation φ of the shape: φ = \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It remains to see that φ−1 B φ ∈ PGL3(R) so that φ−1 G φ is a model of G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, we have ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 7 φ−1 B φ = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 β −β 0 −α α \uf8f6 \uf8f8 [X : Z : Y ] \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 −β β 0 α −α \uf8f6 \uf8f8 \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) i 0 0 0 −2ℑ(α β) i −2ℑ(β2) i 0 2ℑ(α2) i 2ℑ(α β) i \uf8f6 \uf8f8 = \uf8eb \uf8ed 2ℑ(α β) 0 0 0 −2ℑ(α β) −2ℑ(β2) 0 2ℑ(α2) 2ℑ(α β) \uf8f6 \uf8f8 ∈ PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ This completes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The cases when G is a finite primitive subgroup of PGL3(C) Recall that the finite primitive subgroups PGL3(C) are the Hessian groups Hess∗, for ∗ = 216, 72, 36, the alternating groups A∗, for ∗ = 5, 6, and the Klein group PSL(2, 7) of order 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We study their definability over R in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Hessian groups Hess∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Hessian group of order 216, denoted by Hess216, is unique up to conjugation in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' See [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 217] or [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For instance, we fix Hess216 = ⟨S, T, U, V ⟩ where S = diag(1, ζ3, ζ−1 3 ), U = diag(1, 1, ζ3), V = \uf8eb \uf8ed 1 1 1 1 ζ3 ζ−1 3 1 ζ−1 3 ζ3 \uf8f6 \uf8f8 , T = [Y : Z : X].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Also, we consider the Hessian subgroup of order 72, Hess72 = ⟨S, T, V, UV U −1⟩, and the Hessian subgroup of order 36, Hess36 = ⟨S, T, V ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Concerning the Hessian groups Hess∗, for ∗ ∈ {36, 72, 216}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We first show Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any of the Hessian groups Hess∗ has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is easy to see that σS = S−1, σU = U −1, and σT = T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Furthermore σV = 3V −1 in GL3(C), hence we also have σV = V −1 in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It follows that σ Hess∗ = Hess∗ if ∗ = 216 or 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' So Hess216 and Hess36 indeed have a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For Hess72, we get σ Hess72 = ⟨S, T, V, U −1V −1U⟩ ⊂ Hess216;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' another copy of Hess72 inside Hess216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Group structure of Hess216 [10] assures that all copies of Hess72 are Hess216-conjugates, that is to say, there is a projective transformation ψ ∈ Hess216 such that ψ−1 Hess72 ψ = σ Hess72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' From this we obtain that Hess72 has a real field of moduli as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ As a consequence, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Hessian groups Hess∗ for ∗ = 216, 72 and 36 are all pseudo- real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' It is easy to see that ST = T S, so ⟨S, T ⟩ is isomorphic to C3 × C3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' By [17, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2] (see also [25, Section 4]), C3 ×C3 is a subgroup of PGL3(K) if and only if the field K contains a nontrivial cube root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since ζ3 /∈ R, we can’t reduce ⟨S, T ⟩ to a subgroup of PGL3(R) as ζ3 /∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, φ−1 Hess∗ φ ⊈ PGL3(R) for any φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Combining with Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1, we conclude that Hess∗ is pseudo-real for ∗ = 216, 72 and 36 as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The alternating groups A5 and A6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We first note that PGL3(C) possesses a single conjugacy class isomorphic to each of A5 and A6, see [23, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 224, 225] or [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, for i ∈ {5, 6} Ai and σ Ai must be PGL3(C)-conjugates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In other words, Ai has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since A6 contains C3 × C3 as a subgroup, then we can use the same argument as in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 to deduce the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The alternating group A6 is pseudo-real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For the icosahedral group A5, the situation is different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To study it we fix the copy G := ⟨A, B, C⟩ in PGL3(C), where A = diag(1, ζ−1 5 , ζ5), B = [X : Z : Y ], C = \uf8eb \uf8ed 2 2 2 1 cos(4π/5) cos(2π/5) 1 cos(2π/5) cos(4π/5) \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' According to [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7 ], G is PGL3(C)-conjugate to A5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Any subgroup of PGL3(C) isomorphic to A5 is PGL3(C) conjugate to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, we are going to construct an explicit model for G over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Recall, from our study above of the Dihedral group in §4, that ⟨ A, B⟩ descends to R via a transformation of the shape φ = \uf8eb \uf8ed 1 0 0 0 α β 0 α β \uf8f6 \uf8f8 ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, one can check that φ−1 C φ equals \uf8eb \uf8ed 4ℑ(α β) 8ℑ(α β) ℜ(α) 8ℑ(α β) ℜ(β) 2ℑ(β) 2 � cos(4π/5)ℑ(αβ) − cos(2π/5)ℑ(αβ) � −2 cos(2π/5)ℑ(β2) 2ℑ(α) 2 cos(2π/5)ℑ(α2) 2 � cos(4π/5)ℑ(αβ) + cos(2π/5)ℑ(αβ) � \uf8f6 \uf8f8 , in PGL3(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus all generators of G when conjugated by the same φ become in PGL3(R), hence the same is true for the whole group and the result follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The Klein group PSL(2, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Again, there is a single conjugacy class of PSL(2, 7) in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus it has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Also, we know by [18, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='7] that a representative of such a class contains the element diag(1, ζ7, ζ3 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2 applies to n = 7, a = 1, b = 3 to conclude that PSL(2, 7) is not defin- able over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Connection to Arithmetic Geometry Let C : F(X, Y, Z) = 0 be a non-singular plane curve defined over C with non- trivial automorphism group Aut(C) in PGL3(C), Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We have Aut(σC) = σ Aut(C) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' For any φ ∈ Aut(C), φF(X, Y, Z) = cF(X, Y, Z) for some c ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Applying σ to both sides yields σ(c) σF(X, Y, Z) = σ �φF(X, Y, Z) � = σφ (σF(X, Y, Z)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' That is, σφ leaves invariant σC : σF(X, Y, Z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Equivalently, σφ ∈ Aut(σC), hence σ Aut(C) ⊆ Aut(σC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' By a similar argument we can show the other inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Let C : F(X, Y, Z) = 0 be a smooth plane curve over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' If C has a real field of moduli in the Arithmetic Geometry sense, then Aut(C) has a real field of moduli in the Group Theory sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The converse need not be true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since C : F(X, Y, Z) = 0 has a real field of moduli, then it must be the case that σC : σF(X, Y, Z) = 0 and C : F(X, Y, Z) = 0 are C-projectively equivalent (isomorphic over C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, any isomorphism between complex non-singular plane curves C and C′ is always given by a projective transformation φ ∈ PGL3(C) such that their automorphism groups are conjugates via this φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' As a consequence, we obtain that φ−1 Aut(C) φ = Aut(σC), which equals σ Aut(C) by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Thus Aut(C) has a real field of moduli as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' To complete the argument, Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3 below provides infinitely many counter examples that Aut(C) can descend R, but C : F(X, Y, Z) = 0 does not even have a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' □ Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Consider the two-dimensional family Ca,b of smooth plane quintic curves given by Ca,b : X5 + Y 5 + Z5 + iaX(Y Z)2 + ibX3(Y Z), where a, b ∈ R∗ such that a/b ̸= (c5 − 3)c2 2c5 − 1 ζm 10 for any c ∈ C∗ and m ∈ {±1, ±3, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Non-singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We first note that no singular points lie over Y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Indeed, if C has singularity at (α : 0 : β), then α and β must be 0 in order to satisfy FX = FZ = 0, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Second, the resultant of f1(X, Z) := FY (X, 1, Z) and f2(X, Z) := FZ(X, 1, Z) with respect to X is given by ResX(f1, f2) = −125 i b3 (Z5 − 1)3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Using MATHEMATICA, one can verify that we have singular points over Z5 = 1 only if a/b = (c5 − 3)c2 2c5 − 1 ζm 10 for some c ∈ C∗ and m ∈ {±1, ±3, 5}, which is absurd by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Automorphism group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The stratification of smooth plane quintics by their automorphism groups in [3, 6] assures that the group D10 gener- ated by ρ1 = diag(1, ζ5, ζ−1 5 ) and ρ2 = [X : Z : Y ] is a always a sub- group of automorphisms for Ca,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Moreover, if Ca,b admits a larger auto- morphism group, then it should be GAP(150, 5) = (Z/5Z)2 ⋊ S3, where in this situation Ca,b is K-isomorphic to the Fermat quintic curve F5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' the most symmetric smooth quintic curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, there must be an extra automorphism ρ3 /∈ ⟨ρ1⟩ of order 5 that commutes with ρ1 as any Z/5Z inside (Z/5Z)2 ⋊ S3 is contained in a (Z/5Z)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' See Group Structure of (Z/5Z)2 ⋊ S3 [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Straightforward calculations in PGL3(C) lead to ρ3 = diag(1, α, β) with α5 = β5 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Checking the action of such an automorphism on the defining equation of Ca,b tells us that a = b = 0 or ρ3 ∈ ⟨ρ1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Therefore, Aut(Ca,b) = D10 = ⟨ρ1, ρ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Now, we conclude by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='4 that Aut(Ca,b) descends to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Ca,b does not have a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Suppose that C is a member of the family Ca,b such that C has a real field of moduli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hence C and σC are C-projectively equivalent via some φ ∈ PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Since C and σC belong to the same family Ca,b, we have σ Aut(C) = Aut(C) = ⟨ρ1, ρ2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' In particular, φ should be in the normalizer of ⟨ρ1, ρ2⟩ in PGL3(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' We reduce to the case φ−1ρ1φ = ρ1 or ρ−1 as {c, cζ5, cζ−1 5 } ̸= {1, ζ2 5, ζ−2 5 } or {1, ζ3 5, ζ−3 5 } for any c ∈ C∗ by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Consequently, φ = diag(1, α, β) or [X : αZ : βY ] for some α, β ∈ C∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Because φC = σC, we must have α5 = β5 = 1 and αβ = (αβ)2 = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' The last condition is inconsistent, which means that C and σC are never C-isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' BADR AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ELGUINDY References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Artebani and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Qusipe, Fields of moduli and fields of definition of odd signature curves, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 99 (2012), 333-343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Artebani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Carvacho, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hidalgo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Quispe, A tower of Riemann surfaces which cannot be defined over their field of moduli, Glasgow Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 59 (2017), 379-393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Badr and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Bars, Automorphism groups of nonsingular plane curves of degree 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Algebra 44 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 10, 4327-4340.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' MR 3508302.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Badr and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Bars, On fake ES-irreducibile components of certain strata of smooth plane sextics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Preprint 2022, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='08904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Badr and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Bars, The stratification by automorphism groups of smooth plane sextics curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Preprint 2022, https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='2208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='12749.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Badr and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Lorenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A note on the stratification of smooth plane curves of genus 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Colloq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 192, (2020), 207-222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Badr, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hidalgo, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Quispe, Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 110 (2018), 219-222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [8] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Chang, On plane algebraic curves, Chinese J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 6 (1978), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 2, 185- 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' MR 529972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [9] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Earle, On the moduli of closed Riemann surfaces with symmetries, In: Advances in the Theory of Riemann Surfaces, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Ahlfors et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ), 119-130, Princeton Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Press, Princeton, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Dokchitser, GroupNames, https://people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='maths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='bris.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='uk/ matyd/GroupNames/about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='html [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hartshorne, Algebraic geometry, Springer-Verlag, New York-Heidelberg, 1977, Graduate Texts in Mathematics, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' MR 0463157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Harui, Automorphism groups of plane curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Kodai Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 42 (2), (2019), 308-331.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [13] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Henn, Die Automorphismengruppen dar algebraischen Functionenkorper vom Geschlecht 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Inagural-dissertation, Heidelberg, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hidalgo, Non-hyperelliptic Riemann surfaces with real field of moduli but not definable over the reals, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 93 (2009), 219-222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hidalgo and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Quispe, Fields of moduli of some special curves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Algebra 220 (2022), 55-60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [16] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hidalgo and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Shaska, On the field of moduli of superelliptic curves: In higher genus curves in mathematical physics and arithmetic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 703 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [17] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hu, Jordan constant for PGL3(K), arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='02186v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='RT], 5 June 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [18] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Huggins, Fields of moduli and fields of definition of curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' ProQuest LLC, Ann Arbor, MI, 2005, PhD Thesis University of California, Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' MR2708514 [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Hurwitz, ¨Uber algebraische Gebilde mit eindeutigen Transformationen in sich, Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 41 (1892), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 3, 403-442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' MR 1510753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [20] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Koizumi, Fields of moduli for polarized abelian varieties and for curves, Nagoya Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 48 (1972), 37-55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Kontogeorgis, Field of moduli versus field of definition for cyclic covers of the projective line, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Th´eor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Nombres Bordeaux 21 (2009), 679-692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [22] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Kuribayashi and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Komiya, On Weierstrass points and automorphisms of curves of genus three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Algebraic geometry (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Summer Meeting, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Copenhagen, Copenhagen, 1978), [23] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Mitchell, Determination of the ordinary and modular ternary linear groups, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 12 (1911), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 2, 207-242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' MR 1500887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [24] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Popov, On the Makar-Limanov, Derksen invariants, and finite automorphism groups of algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Affine Algebraic Geometry: The Russell Festschrift, CRM Proceedings and Lecture Notes, 54 (2011), 289-311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' [25] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Yasinsky, The Jordan constant for Cremona group of rank 2, Korean Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' 5 (2017), 1859-1871.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content=' Eslam Badr Mathematics Department, Faculty of Science, Cairo University, Giza-Egypt Email address: eslam@sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='eg Mathematics and Actuarial Science Department (MACT), American University in Cairo (AUC), New Cairo-Egypt Email address: eslammath@aucegypt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='edu Ahmad El-Guindy ON PSEUDO-REAL FINITE SUBGROUPS IN PGL3(C) 11 Mathematics Department, Faculty of Science, Cairo University, Giza, Egypt Email address: aelguindy@sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='cu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} +page_content='eg' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-NAyT4oBgHgl3EQfqfhf/content/2301.00543v1.pdf'} diff --git a/.gitattributes b/.gitattributes index f57b67444813d65e98a7d2d2d79552ddd7136ead..38e5600115cf4c7310a8c42c204c5d0811293252 100644 --- a/.gitattributes +++ b/.gitattributes @@ -1078,3 +1078,59 @@ ndE0T4oBgHgl3EQfqAGQ/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -tex kb_37/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text 89AzT4oBgHgl3EQf-_4J/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text aNE4T4oBgHgl3EQfnw2m/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +v9FJT4oBgHgl3EQffixw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf filter=lfs diff=lfs merge=lfs -text +5dE1T4oBgHgl3EQf6gWk/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +kb_37/content/kb_37.pdf filter=lfs diff=lfs merge=lfs -text +yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf filter=lfs diff=lfs merge=lfs -text +rNFRT4oBgHgl3EQffDdP/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf filter=lfs diff=lfs merge=lfs -text +o9FPT4oBgHgl3EQfLTSh/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +kdE3T4oBgHgl3EQfJgnA/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf filter=lfs diff=lfs merge=lfs -text +yNAzT4oBgHgl3EQftP0T/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +GdAyT4oBgHgl3EQfSvfs/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf filter=lfs diff=lfs merge=lfs -text +j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf filter=lfs diff=lfs merge=lfs -text +4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf filter=lfs diff=lfs merge=lfs -text +hdA0T4oBgHgl3EQfIP8h/content/2301.02071v1.pdf filter=lfs diff=lfs merge=lfs -text +OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf filter=lfs diff=lfs merge=lfs -text +itAzT4oBgHgl3EQfbPyr/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf filter=lfs diff=lfs merge=lfs -text +BtE0T4oBgHgl3EQfQAAH/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf filter=lfs diff=lfs merge=lfs -text +u9E3T4oBgHgl3EQf-QtO/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf filter=lfs diff=lfs merge=lfs -text +OtFOT4oBgHgl3EQf3zQR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf filter=lfs diff=lfs merge=lfs -text +UNAyT4oBgHgl3EQfVvd8/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf filter=lfs diff=lfs merge=lfs -text +l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf filter=lfs diff=lfs merge=lfs -text +QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf filter=lfs diff=lfs merge=lfs -text +XNE0T4oBgHgl3EQf3QLX/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +4tFIT4oBgHgl3EQf7Ctv/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf filter=lfs diff=lfs merge=lfs -text +H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf filter=lfs diff=lfs merge=lfs -text +rdAzT4oBgHgl3EQfrf1b/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +j9E1T4oBgHgl3EQf0QVw/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +mdAyT4oBgHgl3EQfYvcB/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +qtFPT4oBgHgl3EQf9DUp/content/2301.13210v1.pdf filter=lfs diff=lfs merge=lfs -text +mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf filter=lfs diff=lfs merge=lfs -text +H9AzT4oBgHgl3EQfU_x9/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf filter=lfs diff=lfs merge=lfs -text +-9AzT4oBgHgl3EQfhPxU/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ANE4T4oBgHgl3EQf4w6U/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +rdAzT4oBgHgl3EQfrf1b/content/2301.01644v1.pdf filter=lfs diff=lfs merge=lfs -text +ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf filter=lfs diff=lfs merge=lfs -text +qtFPT4oBgHgl3EQf9DUp/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +-9AzT4oBgHgl3EQfhPxU/content/2301.01481v1.pdf filter=lfs diff=lfs merge=lfs -text +jdE4T4oBgHgl3EQfsg0n/content/2301.05216v1.pdf filter=lfs diff=lfs merge=lfs -text +LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf filter=lfs diff=lfs merge=lfs -text +W9E1T4oBgHgl3EQfvgXD/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +T9FIT4oBgHgl3EQffis3/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +ENE0T4oBgHgl3EQfQgDK/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf filter=lfs diff=lfs merge=lfs -text +99E0T4oBgHgl3EQfwwHR/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +1dA0T4oBgHgl3EQfMv9X/vector_store/index.faiss filter=lfs diff=lfs merge=lfs -text +6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf filter=lfs diff=lfs merge=lfs -text +cdAyT4oBgHgl3EQf-foD/content/2301.00891v1.pdf filter=lfs diff=lfs merge=lfs -text diff --git a/0dAzT4oBgHgl3EQfefzh/content/tmp_files/2301.01439v1.pdf.txt b/0dAzT4oBgHgl3EQfefzh/content/tmp_files/2301.01439v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..71b7723e36a9c2f8d53046a3cc5d0c1654c5a518 --- /dev/null +++ b/0dAzT4oBgHgl3EQfefzh/content/tmp_files/2301.01439v1.pdf.txt @@ -0,0 +1,1737 @@ +Convergence of Adaptive Mixed Interior Penalty Dis- +continuous Galerkin Methods for H(cur l)-Elliptic +Problems +K. Liu1, M. Tang2,, X. Q. Xing2 and L. Q. Zhong2 +1 School of Sciece, East China University of Technology, Nanchang, 330013, China +2 School of Mathematical Sciences, South China Normal University, Guangzhou, +510631, China +Abstract. In this paper, we study the convergence of adaptive mixed interior penalty +discontinuous Galerkin method for H(cur l)-elliptic problems. We first get the mixed +model of H(cur l)-elliptic problem by introducing a new intermediate variable. Then +we discuss the continuous variational problem and discrete variational problem, which +based on interior penalty discontinuous Galerkin approximation. Next, we construct the +corresponding posteriori error indicator, and prove the contraction of the summation of +the energy error and the scaled error indicator. At last, we confirm and illustrate the +theoretical result through some numerical experiments. +AMS subject classifications: 65M15,65N12,65N30 +Key words: Adaptive mixed interior penalty discontinuous Galerkin methods, Convergence, H(cur l)- +elliptic problems. +1. Introduction +Let Ω ⊂ �3 be Lipschitz bounded polygonal domain with a single connected boundary +∂ Ω. We consider the following H(cur l)-elliptic problem +∇ × µ∇ × u + κu = f +in +Ω, +(1.1) +u × n = 0 +on +∂ Ω, +(1.2) +where n is the unit normal vector of the boundary ∂ Ω, f ∈ L2(Ω), µ and κ are piecewise +constants is consistent with the initial partition �0 for Ω and satisfy µ1 < µ < µ2 and +κ1 < κ < κ2, here, µi and κi(i = 1,2) are positive constants. By introducing an auxiliary +∗Corresponding author. Email addresses: liukai@ecut.edu.cn (K. Liu), mingtang@m.scnu.edu.cn (M. +Tang),xingxq@scnu.edu.cn(X. Q. Xing), zhong@m.scnu.edu.cn (L. Q. Zhong) +1 +arXiv:2301.01439v1 [math.NA] 4 Jan 2023 + +2 +K Liu et al. +variable p = µ∇ × u, then we get the mixed scheme with the boundary value problem +(1.1)-(1.2) +p = µ∇ × u +in +Ω, +(1.3) +∇ × p + κu = f +in +Ω, +(1.4) +u × n = 0 +on +∂ Ω. +(1.5) +The mixed finite element method is very convenient for processing high-order equations +and equations containing two or more unknown functions, which has attracted widespread +attention. For mixed finite element method, there are only few research results for Maxwell +problem [13] and Maxwell’s eigenvalue problem [12,14,15]. +Adaptive finite element method automatically refines and optimizes meshes accord- +ing to the singularity of solutions. It is a highly reliable and efficient numerical calculation +method. At present, the convergence analysis research of the adaptive mixed finite element +method for the elliptic equation is relatively complete. Chen, Holst and Xu [7] proved the +convergence analysis of the adaptive mixed finite element algorithm for elliptic equations. +Du and Xie [10] proved the convergence analysis of the adaptive mixed finite element +algorithm for the convection diffusion equation. However, there are only few research +results on the posterior error estimator of Maxwell’s equations for the adaptive mixed fi- +nite element method. For example, Carstensen and Ma [5] establishes the convergence of +adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite +elliptic problems. Carstensen, Hoppe, Sharma and Warburton [4] designs and analyzes +the posterior error estimation of the adaptive hybrid conforming finite element method of +H(cur l)-elliptic problem. Recently, Chung, Yuen and Zhong [8] present a-posteriori error +analysis for the staggered discontinuous Galerkin method. As far as we know, there are not +any published literatures for the convergence analysis of the adaptive mixed finite element +method for the boundary value problem(1.3)-(1.5). Our contributions in this paper are to +• construct a new error estimator, which does not include the negative power of the +local mesh size in the jump term for the traditional DG method; +• get the convergence of the Adaptive Mixed Interior Penalty Discontinuous Galerkin +(AMIPDG) method by using the similar technique used in [2]. However, this tech- +nique in [2] can not be used directly for mixed forms. +We present our main result in the following theorem. +Theorem 1.1. Let {�k,Uk,Qk, uk, pk,η(uk, pk;�k)}k≥0 be the sequence of meshes, finite +element space, mixed discrete solution and posterior error estimate indicator produced by the +AMIPDG algorithm. Then there exist constants ρ > 0 and δ ∈ (0,1), which depend on +marking parameter and the shape regularity of the initial mesh �0, such that +∥|u − uk+1|∥2 +k+1 + ρη2(uk+1, pk+1;�k+1) ≤ δ +� +∥|u − uk|∥2 +k + ρη2(uk, pk;�k) +� +. +Therefore, for a given precision, the AMIPDG method will terminate after a finite number of +operations. + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +3 +For convenience, we let C denote a generic positive constant which may be different +at different occurrences and adopt the following notation. The subscripted constant Ci +represents a particularly important constant. a ≲ b means a ≤ C b for some constants C +which are independent of mesh sizes. +The rest of this paper is organized as follows. In Section 2, we first present the contin- +uous variational problem, the discrete variational problem, and the procedure of AMIPDG. +In Section 3, we first show the upper bound estimate of the error, which is key to the con- +vergence analysis, then we prove the indicator reduction and the convergence of AMIPDG +algorithm. In Section 4, we provide some numerical experiments to illustrate the effective- +ness of the AMIPDG. +2. Adaptive Mixed interior penalty discontinuous Galerkin method +In this section, we introduce the continuous variational problem, the discrete variational +problem of mixed internal penalty discontinuous finite element method, and the procedure +of AMIPDG. +2.1. Continuous variational problem +For an open and connected bounded domain D ⊂ �3, we denote by L2(D) (resp. +L2(D) := (L2(D))3) the spaces of square-integrable functions (resp. vector fields) on D +with inner product (·,·)0,D. We define the spaces +H(cur l; D) = {u ∈ L2(D) : ∇ × u ∈ L2(D)}, +H(div; D) = {u ∈ L2(D) : ∇ · u ∈ L2(D)}, +with +(u, v)cur l,D := (u, v)0,D + (∇ × u,∇ × v)0,D, +∀u, v ∈ H(cur l; D), +(u, v)div,D := (u, v)0,D + (∇ · u,∇ · v)0,D, +∀u, v ∈ H(div; D), +and the induced norm as: +∥u∥2 +cur l,D := ∥u∥2 +0,D + ∥∇ × u∥2 +0,D, ∀u ∈ H(cur l, D), +∥u∥2 +div,D := ∥u∥2 +0,D + ∥∇ · u∥2 +0,D, +∀u ∈ H(div, D), +respectively, where ∥ · ∥L2(D) := (·,·)1/2 +D +denotes the norm of the space L2(D) or L2(D). We +also define H0(cur l; D) = {v ∈ H(cur l; D) : v × n = 0 on ∂ D} in the trace sense. +Next, we first define two space U := H0(curl;Ω),Q := L2(Ω). Then, the mixed vari- +ational problem of the mixed boundary value problem (1.3)-(1.5) reads as: find (u, p) ∈ +U × Q such that: +a(p,q) − b(u,q) = ℓ1(q), +∀q ∈ Q, +(2.1) +d(v, p) + c(u, v) = ℓ2(v), +∀v ∈ U. +(2.2) + +4 +K Liu et al. +The bilinear forms a, b, c and the functionals ℓ1(·),ℓ2(·) are given by +a(p,q) := (p,q), +(2.3) +b(u,q) := (µ∇ × u,q), +(2.4) +c(u, v) := (κu, v), +(2.5) +d(v, p) := (∇ × v, p) +(2.6) +ℓ1(q) := 0, +(2.7) +ℓ2(v) := ( f , v). +(2.8) +The operator-theoretic framework involves operator � : (U × Q) → (U × Q)∗ defined +by +(� (u, p))(v,q) := a(p,q) − b(u,q) + d(v, p) + c(u, v),∀u, v ∈ U, p,q ∈ Q, +(2.9) +where (Q × U)∗ is the dual spaces of (Q × U). Then we can rewrite (2.1)-(2.2) as +(� (u, p))(v,q) = ℓ(v,q), +(2.10) +with ℓ(v,q) = ℓ1(q) + ℓ2(v), and ℓi are given by (2.7)-(2.8). +Then, we state the well-posedness of the variational problem (2.1)-(2.2) in the follow- +ing lemma, and it can be found in section 3 of [3]. +Lemma 2.1. Under the assumptions on the problem of (1.1)-(1.2), � is a continuous and +bijective linear operator. Hence, for any ℓ = (ℓ1,ℓ2) ∈ (Q×U)∗, the mixed variational problem +(2.1)-(2.2) has a unique solution (u, p) ∈ (U × Q), which satisfy the following continuously +∥(u, p)∥U×Q := (∥u∥2 +curl,Ω + ∥p∥2 +0)1/2 ≲ ∥ℓ1∥Q∗ + ∥ℓ2∥U∗. +(2.11) +2.2. Discrete variational problem +We suppose that �h is a family of shape regularity, quasi-uniform and conform tetrahe- +dral generation on Ω. Let hτ = |τ|1/3 denote the mesh size with |τ| being the volume of +τ ∈ �h. +Define the discontinuous finite element function space �(�h) as: +�(�h) = {v ∈ L2(Ω) : vτ = v|τ ∈ (Pl(τ))3, +∀τ ∈ �h}, +where Pl(τ) is the set of polynomials defined in the volume τ whose degree does not exceed +l, where l ≥ 1 is an integer. +Let �h, � 0 +h and � ∂ +h denote the set of the all faces of its volumes, and the set of internal +faces, and the set of boundary faces, respectively. Thus, �h = � 0 +h +� +� ∂ +h . Let H1(Ω;�h) be +the space of piecewise Sobolev functions defined by +H1(Ω;�h) = +� +v ∈ L2(Ω) : vτ = v|τ ∈ H1(τ), +∀ τ ∈ �h +� +. + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +5 +and H1(Ω;�h) = (H1(Ω;�h))3. Let L2(�h) be the set of L2 functions defined on �h. More- +over, we define the following inner products +(v, w)� ′ +h += +� +τ∈� ′ +h +� +τ +v · wdx, +∀v, w ∈ L2(Ω), ∀� +′ +h ⊂ �h, +< v, w >� ′ +h += +� +f ∈� ′ +h +� +f +v · wds, +∀v, w ∈ L2(�h), ∀� +′ +h ⊂ �h. +For f ∈ � 0 +h , we have τi ∈ �h(i = 1,2), such that f = ∂ τ1 ∩ ∂ τ2. Then we denote the +jump and average of v as: +[[v]] += +v1 × n1 + v2 × n2, +∀v ∈ H1(Ω;�h), +{{v}} += +v1 + v2 +2 +, +∀v ∈ H1(Ω;�h), +where v i denote the values of v on v|τi(i = 1,2) and ni denote the out unit normal vectors +on f exterior v|τi. +For f ∈ � ∂ +h , we have τ ∈ �h, such that f = ∂ τ ∩ ∂ Ω. Then we denote the jump and +average of v as: +[[v]] = vτ × n∂ Ω, {{v}} = vτ. +(2.12) +Next, we give the corresponding discrete scheme of (2.1)-(2.2). Firstly, we define the +corresponding discrete space as follow +Uh := {vh ∈ �(�h)| +[[vh]]|f = 0,∀f ∈ � ∂ +h }, +Qh := �(�h). +Then, the formulation of the discrete Mixed Interior Penalty Discontinuous Galerkin (MIPDG) +method reads: find (uh, ph) ∈ (Uh,Qh) such that +ah(ph,qh) − bh(uh,qh) = ℓ1,h(qh) + d1,h(uh,qh), +∀qh ∈ Qh, +(2.13) +dh(vh, ph) + ch(uh, vh) = ℓ2,h(vh) + d2,h(uh, vh), +∀vh ∈ Uh, +(2.14) +where +ah(ph,qh) := (ph,qh)�h, +bh(uh,qh) := (µ∇ × uh,qh)�h, +ch(uh, vh) := (κuh, vh)�h, +dh(vh, ph) := (∇ × vh, ph)�h, +ℓ1,h(qh) := 0, +ℓ2,h(vh) := ( f , vh)�h, +d1,h(uh,qh) := − < {{µqh}},[[uh]] >�h, +d2,h(uh, vh) :=< ({{µ∇ × uh}} − αh−1 +f [[uh]]),[[vh]] >�h, + +6 +K Liu et al. +here the constant α > 0 denote the penalty parameter, hf denote the diameter of the +circumcircle of f . Thus hτ ≈ hf . +Remark 2.1. The calculation of ∇ × uh in the bilinear terms are piecewise derivations. +The standard symmetric Interior Penalty Discontinuous Galerkin (IPDG) method of the +boundary value problem (1.1)-(1.2) is to find uh ∈ Uh, such that +aIP(uh, vh) +:= (κuh, vh)�h + (µ∇ × uh,∇ × vh)�h− < {{µ∇ × vh}},[[uh]] >�h +− < {{µ∇ × uh}},[[vh]] >�h +αh−1 +f +< [[uh]],[[vh]] >�h +(2.15) += ( f , vh)�h. +The following lemma shows that the discrete variational problems (2.13)-(2.14) and (2.15) +are equivalent. +Lemma 2.2. [ [3], Theorem 4.1] The formulations (2.13)-(2.14) and (2.15) are formally +equivalent in the following sense. If (uh, ph) ∈ (Uh,Qh) are the solution of discrete variational +problem (2.13)-(2.14), then uh ∈ Uh solves (2.15). Conversely, if uh ∈ Uh solves (2.15), then +there exists some ph ∈ Qh such that (uh, ph) ∈ (Uh,Qh) are the solution of (2.13)-(2.14). +Ayuso de Dios, Hiptmair and Pagliantini proved the well-posedness of (2.15) in section +2 of [1]. Therefore, by combining Lemma 2.2, we obtain the well-posedness of discrete +variational problems (2.13)-(2.14). +2.3. Adaptive Mixed Interior Penalty Discontinuous Galerkin method(AMIPDG) +Our adaptive cycle can be implemented by the following algorithm: +Next, we will discuss each step in AEFEM in detail. +2.3.1. Procedure SOLVE +For f ∈ L2(Ω), and a shape regular mesh �k, Let (uk, pk) be the exact MIPDG solution of +(2.13)-(2.14). Here, we assume that the solutions (uk, pk) can be solved accurately. +2.3.2. Procedure ESTIMATE +A posteriori error indicator is an essential ingredient of adaptivity. They are computable +quantities depending on the computed solution(s) and data that provide information about +the quality of approximation and may consequently be used to make judicious mesh modi- +fications. Here, we design a new posteriori error estimation indicator for equations (2.13)- +(2.14), which is similar to that in [20]. For τ ∈ �h, f ∈ �h and (vh,qh) ∈ Uh × Qh, the +residual a posteriori error estimator for the symmetric AMIPDG method is given by +η2(vh,qh;τ) : += +∥R1(vh,qh)∥2 +L2(τ) + h2 +τ +� +∥R2(vh,qh)∥2 +L2(τ) + ∥R3(vh)∥2 +L2(τ) +� ++ +� +f ∈∂ τ +hf +� +∥J1(qh)∥2 +L2(f ) + ∥J2(vh)∥2 +L2(f ) +� +. +(2.16) + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +7 +Algorithm 2.1 Adaptive Mixed Interior Penalty Discontinuous Galerkin Method (AMIPDG) +cycle +Input initial triangulation �0; data f ; tolerance tol; marking parameter θ ∈ (0,1). +Output a triangulation �J; MIPDG solution (uJ, pJ). +η = 1; k = 0; +while η ≥ tol +SOLVE solve discrete varational problem (2.13)-(2.14) on �k to get the solution (uk, pk); +ESTIMATE compute the posterior error estimator η = η(uk, pk,�k) by using (2.17); +MARK seek a minimum cardinality �k ⊂ �k such that +η2 � +uk, pk,�k +� +≥ θη2 � +uk, pk,�k +� +; +REFINE bisect elements in �k and the neighboring elements to form a conforming �k+1; +k = k + 1; +end +uJ = uk; pJ = pk; �J = �k; +They consist of the element residuals and face jump residuals as +R1(vh,qh)|τ := qh|τ − µ∇ × vh|τ, +R2(vh,qh)|τ := f |τ − (∇ × qh + κvh)|τ, +R3(vh)|τ := ∇ · ( f |τ − κvh|τ), +J1(qh)|f := [[qh]], +J2(vh)|f := [[(f − κvh)]]. +where hf denote the diameter of the circumcircle of f , and hτ ≈ hf . +For any set � ′ +h ⊆ �h, the error indicator is defined as +η2(vh,qh;� ′ +h ) = +� +τ∈� ′ +h +η2(vh,qh;τ). +(2.17) +2.3.3. Procedure MARK +We use the Dörfler mark which was proposed by Dörfler [9]. Set marking parameter θ ∈ +(0,1), the module MARK outputs a subset of marked elements �k ⊂ �k with minimal +cardinality, such that +η2(v k,q k;�k) ≥ θη2(v k,q k;�k). +(2.18) +2.3.4. Procedure REFINE +Our implementation of REFINE uses the longest edge bisection strategy. A detailed intro- +duction about the longest edge bisection strategy was provided in [6]. To avoid confusion, +the relationship between the two tetrahedral meshes �h and �H that are nested into each + +8 +K Liu et al. +other is defined as: �h is the new mesh division of �H after one cycle of the above cycle +process, abbreviated as �H ≤ �h. +3. Convergence of AMIPDG algorithm +In this section, we establish the upper bound estimate of the error. Subsequently, we +demonstrate that the sum of the energy error and the error estimator between two consec- +utive adaptive loops is a contraction. Finally, we proof that the AMIPDG is convergence. +3.1. The upper bound estimate of the error +In this subsection, before establishing the reliability of a posteriori error estimator, we +need to define the corresponding DG norm, for any (v,q) ∈ U × Q and (vh,qh) ∈ Uh × Qh, +∥(v,q) +− +(vh,qh)∥2 +DG := ∥q − qh∥2 +L2(Ω) + ∥κ(v − vh)∥2 +L2(Ω) ++ +� +τ∈�h +∥µ∇ × (v − vh)∥2 +L2(τ) + +� +f ∈�h +αh−1 +f +< [[vh]],[[vh]] >f . +(3.1) +Remark 3.1. For any v ∈ U and vh ∈ Uh, we have +∥[[vh]]∥2 +L2(f ) = ∥[[(v − vh)]]∥2 +L2(f ), +∀f ∈ �h. +In fact, v ∈ U implies that [[v]]|f = 0 (see Chapter 5 of [16]). +We summarize our main result in this subsection as follows. +Theorem 3.1. Let (u, p) ∈ U×Q and (uh, ph) ∈ Uh ×Qh be the solutions of (2.1)-(2.2) and +(2.13)-(2.14), respectively. Let η(uh, ph;�h) be the residual error indicator of (2.17). Then +we have the following estimate +∥(u, p) − (uh, ph)∥2 +DG ≤ C1η2(uh, ph;�h), +(3.2) +where the constant C1 depending on the shape regularity of mesh. +Let (uh, ph) ∈ Uh × Qh be the solution of (2.13)-(2.14), similarly to [4], we introduce +the nonconformity of the MSIPDG method results in some consistency error: +ζ := min +˜vh∈U +� � +τ∈�h +(∥uh − ˜vh∥2 +L2(τ) + ∥∇ × (uh − ˜vh)∥2 +L2(τ)) +�1/2. +(3.3) +We denote that ˜uh ∈ U is the unique minimizer of (3.3), namely +˜ζ = +� � +τ∈�h +(∥uh − ˜uh∥2 +L2(τ) + ∥∇ × (uh − ˜uh)∥2 +L2(τ)) +�1/2. +(3.4) + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +9 +Lemma 3.1. Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.1)-(2.2) and +(2.13)-(2.14), respectively, let ˜uh be the unique minimizer of (3.3), then +∥(u − ˜uh, p − ph)∥U×Q = (∥u − ˜uh∥2 +curl,Ω + ∥p − ph∥2 +0)1/2 ≲ ∥˜ℓ1∥Q∗ + ∥˜ℓ2∥U∗, +where the residuals ˜ℓ1 ∈ Q∗ and ˜ℓ2 ∈ U∗ defined by +˜ℓ1(q) = ℓ1(q) − a(ph,q) + b(˜uh,q), +∀q ∈ Q, +(3.5) +˜ℓ2(v) = ℓ2(v) − d(v, ph) − c(˜uh, v), +∀v ∈ U. +(3.6) +Proof. For any q1,q2,q ∈ Q and any v1, v2, v ∈ U. we have the following property by +(2.9) +(� (v1 + v2,q1 + q2))(v,q) += a(q1 + q2,q) − b(v1 + v2,q) + d(v,q1 + q2) + c(v1 + v2, v) += a(q1,q) − b(v1,q) + d(v,q1) + c(v1, v) ++a(q2,q) − b(v2,q) + d(v,q2) + c(v2, v) += (� (v1,q1))(v,q) + (� (v2,q2))(v,q). +Thus, +(� (u − ˜uh, p − ph))(v,q) += (� (u, p))(v,q) − (� (˜uh, ph))(v,q) += (ℓ1(q) + ℓ2(v)) − (a(ph,q) − b(˜uh,q) + d(v, ph) + c(˜uh, v)) += ˜ℓ1(q) + ˜ℓ2(v). +In fact that (u − ˜uh, p −ph) ∈ U ×Q and combining the Lemma 2.1 can concludes the proof. +Next, we will provide upper bounds for ∥˜ℓ1∥Q∗ and ∥˜ℓ2∥U∗ in Lemmas 3.2 and 3.4, +respectively. +Lemma 3.2. Let (uh, ph) ∈ Uh × Qh be the solutions of (2.13)-(2.14), and ˜uh be the unique +minimizer of (3.3). Then we get the estimate of the linear functional ˜ℓ1 defined in (3.5) as +following +∥˜ℓ1∥Q∗ ≲ +� � +τ∈�h +∥R1(uh, ph)∥2 +L2(τ) +�1/2 + +� � +τ∈�h +∥∇ × (˜uh − uh)∥2 +L2(τ) +�1/2. +(3.7) +Proof. For any q ∈ Q, by the definition of ˜ℓ1, we have +˜ℓ1(q) = +� +τ∈�h +� +τ +� +(µ∇ × uh − ph) + µ∇ × (˜uh − uh) +� +· qdx. + +10 +K Liu et al. +Then applying the Hölder inequality and the Cauchy-Schwarz inequality, +|˜ℓ1(q)| ≤ +� +τ∈�h +∥µ∇ × uh − ph∥L2(τ)∥q∥L2(Ω) + +� +τ∈�h +∥µ∇ × (˜uh − uh)∥L2(τ)∥q∥L2(Ω) +≲ +�� � +τ∈�h +∥R1(uh, ph)∥2 +L2(τ) +�1/2 + +� � +τ∈�h +∥∇ × (˜uh − uh)∥2 +L2(τ) +�1/2� +∥q∥L2(Ω), +conclude the proof. +Before estimating the term ∥˜ℓ2∥U∗, we need to introduce the following interpolation +operator with the corresponding approximations. +Lemma 3.3. [ [19], Theorem 1] Let Nd1 +0(Ω;�h) be the lowest order edge elements of Nédélec +first family. Then there exists an operator Πh : H0(curl;Ω) → Nd1 +0(Ω;�h) with the following +properties: For every v ∈ H0(curl;Ω), there exist ϕ ∈ H1 +0(Ω) and z ∈ H1 +0(Ω), such that +v − Πhv = ∇ϕ + z. +And for any τ ∈ �h and f ∈ �h, we have +h−1 +τ ∥ϕ∥L2(τ) + ∥∇ϕ∥L2(τ) ≲ hτ∥v∥L2(Ωτ), +h−1 +τ ∥z∥L2(τ) + ∥∇z∥L2(τ) ≲ hτ∥∇ × v∥L2(Ωτ), +where Ωτ = +� +f ∈τ +Ωf , Ωf = {τ′ ∈ �h, f ∈ τ′}, and the constants depending on the shape +regularity of the mesh. +Lemma 3.4. Let (uh, ph) ∈ Uh × Qh be the solution of (2.13)-(2.14), and ˜uh be the unique +solution of (3.3). Then the linear functional ˜ℓ2 defined in (3.6) satisfies the following estimate +∥˜ℓ2∥U∗ ≲ +� � +τ∈� +h2 +τ(∥R2(uh, ph)∥2 +L2(τ) + ∥R2(uh)∥2 +L2(τ)) ++ +� +f ∈� +hf (∥J1(ph)∥2 +L2(f ) + ∥J2(uh)∥2 +L2(f )) + +� +τ∈� +∥uh − ˜uh∥2 +L2(τ) +�1/2 +. +(3.8) +Proof. For any v ∈ U and Πh given by Lemma 3.3, we have +v − Πhv = ∇ϕ + z, +(3.9) +where ϕ ∈ H1 +0(Ω) and z ∈ H1 +0(Ω). According to linearity of the operator ˜ℓ2 and (3.9), we +have +˜ℓ2(v) = ˜ℓ2(Πhv) + ˜ℓ2(v − Πhv) = ˜ℓ2(Πhv) + ˜ℓ2(∇ϕ) + ˜ℓ2(z). +(3.10) +We will next estimate the three terms on the right hand side of (3.10). + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +11 +For the first term ˜ℓ2(Πhv) of (3.10), using the definition of ˜ℓ2, we have +˜ℓ2(Πhv) += +ℓ2(Πhv) − d(Πhv, ph) − c(˜uh,Πhv) += +ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) + c(uh − ˜uh,Πhv). +Noting that Πhv ∈ Nd1 +0(Ω;�h) ⊆ Uh has zero jumps, and combining (2.14), we have +ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) = ℓ2,h(Πhv) − dh(Πhv, ph) − ch(uh,Πhv) = 0. +Thus, we have +˜ℓ2(Πhv) += +c(vh − ˜uh,Πhv) += +c(vh − ˜uh, v) + c(vh − ˜uh,Πhv − v) +≤ +∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥Πhv − v∥0,�h). +Then using (3.9), triangle inequality and Lemma 3.3, we get +˜ℓ2(Πhv) +≤ +∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ + z∥0,�h) +≤ +∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ∥0,�h + ∥z∥0,�h) +≤ +∥κ∥0,∞∥vh − ˜uh∥0,�h∥v∥curl,�h. +(3.11) +For the second term ˜ℓ2(∇ϕ) of (3.10), using the definition of ˜ℓ2, (2.8), (2.4), (2.6) and +the fact ∇ × ∇ϕ = 0, which implies +˜ℓ2(∇ϕ) += +ℓ2(∇ϕ) − d(∇ϕ, ph) − c(˜uh,∇ϕ) += +( f ,∇ϕ) − (∇ × ∇ϕ, ph) − (κ˜uh,∇ϕ) += +( f ,∇ϕ) − (κ˜uh,∇ϕ). +(3.12) +By (3.12) and Green’s formula, we have +˜ℓ2(∇ϕ) += +( f ,∇ϕ) − (κuh,∇ϕ) + (κ(uh − ˜uh),∇ϕ) +≤ +� +τ∈�h +(R3(uh),ϕ)0,τ + +� +f ∈�h +< J2(uh),ϕ >0,f +(κ(uh − ˜uh),∇ϕ). +Applying the Cauchy-Schwarz inequality, Lemma 3.3 and trace inequality, we have +˜ℓ2(∇ϕ) ≤ +� � +τ∈�h +h2 +τ∥R3(uh)∥2 +0,τ + +� +f ∈�h +hf ∥J2(uh)∥2 +0,f ++ +� +τ∈�h +∥κ∥0,∞∥uh − ˜uh∥2 +0,τ +�1/2 +∥v∥curl,�h. +(3.13) + +12 +K Liu et al. +Similarly, for the third term ˜ℓ2(z) of (3.10), we have +˜ℓ2(z) += +( f , z) − (∇ × z, ph) − (κ˜uh, z) += +( f , z) − (∇ × z, ph) − (κuh, z) + (κ(uh − ˜uh), z) +≤ +� � +τ∈�h +h2 +τ∥R2(uh, ph)∥2 +0,τ + +� +f ∈�h +hf ∥J1(ph)∥2 +0,f ++ +� +τ∈�h +∥κ∥0,∞∥uh − ˜uh∥2 +0,τ +�1/2 +∥v∥curl,�h. +(3.14) +Substituting (3.11), (3.13) and (3.14) into (3.10), the proof is completed. +Notice that both (3.7) and (3.8) are related to the terms +� +τ∈�h +∥∇ × (˜uh − uh)∥2 +L2(τ) and +� +τ∈� +∥uh − ˜uh∥2 +L2(τ), which are a part of ˜ζ. Therefore, we prove upper bounds for ˜ζ in the +following Lemma. +Lemma 3.5. Let (uh, ph) ∈ Uh × Qh be the solutions of (2.13)-(2.14) and ˜ζ be consistency +error of (3.4), we have +˜ζ2 ≲ η2(uh, ph;�h). +(3.15) +Proof. For any vh ∈ Uh, there exit an interpolation operator �h : H1(Ω;�h) → Uc +h, such +that(see Proposition 4.5 of [11]) +∥vh − �hvh∥2 +L2(Ω) ≲ +� +f ∈�h +hf ∥[[vh]]∥2 +L2(f ), +(3.16) +� +τ∈�h +∥∇ × (vh − �hvh)∥2 +L2(τ) ≲ +� +f ∈�h +h−1 +f ∥[[vh]]∥2 +L2(f ). +(3.17) +Then, combining (3.3), (3.4), (3.16), (3.17), and the fact hf < 1, we get +˜ζ2 += +� +τ∈�h +(∥uh − ˜uh∥2 +L2(τ) + ∥∇ × (uh − ˜uh)∥2 +L2(τ)) +≤ +� +τ∈�h +(∥uh − �huh∥2 +L2(τ) + ∥∇ × (uh − �huh)∥2 +L2(τ)) +≲ +� +f ∈�h +hf ∥[[uh]]∥2 +L2(f ) + +� +f ∈�h +h−1 +f ∥[[uh]]∥2 +L2(f ) +≲ +� +f ∈�h +h−1 +f ∥[[uh]]∥2 +L2(f ). +(3.18) +Noting that (uh, ph) ∈ Uh × Qh is the solution of discrete variational problem (2.13)- +(2.14). Then by using Lemma 2.2, we know that uh is the solution of discrete variational +problem (2.15). Hence, we have ( see Lemma 5 of [20]) +α∥h−1/2 +f +[[uh]]∥L2(�h) ≲ η(uh, ph;�h). +(3.19) + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +13 +At last, combining (3.18) and (3.19), we have +˜ζ2 +≲ +η2(uh, ph;� ). +Combining Lemmas 3.1, 3.2, 3.4 and 3.5, we will prove Theorem 3.1. +Proof. [ Proof of Theorem 3.1:] By using (3.1), the triangle inequality, (3.4), Lemmas +3.1, 3.2, 3.4, 3.5 and (3.19), we get +∥(u, p) − (uh, ph)∥2 +DG +≲ +∥p − ph∥2 +L2(Ω) + ∥κ(u − uh)∥2 +L2(Ω) ++ +� +τ∈�h +∥∇ × µ(u − uh)∥2 +L2(τ) + +� +f ∈�h +αh−1 +f +< [[uh]],[[uh]] >f +≲ +∥p − ph∥2 +L2(Ω) + ∥u − ˜uh∥2 +cur l,Ω + ˜ζ2 + +� +f ∈�h +αh−1 +f +< [[uh]],[[uh]] >f += +∥(u − ˜uh, p − ph)∥U×Q + ˜ζ2 + +� +f ∈�h +αh−1 +f +< [[uh]],[[uh]] >f +≲ +∥˜ℓ1∥2 +Q∗ + ∥˜ℓ2∥2 +U∗ + ˜ζ2 + +� +f ∈�h +αh−1 +f +< [[uh]],[[uh]] >f +≤ +C1η2(uh, ph;�h). +3.2. The error reduces on two successive meshes +For convenience, for any v ∈ U and vh ∈ Uh, we denote +∥|v − vh|∥2 +h += +∥κ(v − vh)∥2 +L2(Ω) + +� +τ∈�h +∥∇ × µ(v − vh)∥2 +L2(τ) ++ +� +f ∈�h +αh−1 +f +< [[vh]],[[vh]] >f . +(3.20) +Let Uc +h be the H(cur l) conforming subspace of Uh given by +Uc +h := Uh ∩ H0(curl;Ω). +Then, there is a subspace U⊥ +h which can orthogonally decompose Uh under L2 inner product +such that Uh := Uc +h ⊕ U⊥ +h . Especially, if (uh, ph) ∈ Uh × Qh is the solution of (2.13)-(2.14), +then we have +∥|u⊥ +h |∥2 +h ≲ α +� +f ∈∂ τ +∥h−1/2 +f +[[uh]]∥2 +L2(f ). +(3.21) +In fact, from the Lemma 2.2, notice that uh satisfies the IPDG scheme of (2.15), and ac- +cording to Lemma 2 in [20], we can obtain (3.21). + +14 +K Liu et al. +In order to easily estimate the jump term of face �h, we need to introduce the lifting +operators and the corresponding stability estimates, more details are referenced to Propo- +sition 12 in [18]. +Let �h : H1(Ω;�h) → Uh be the lifting operators, which satisfies the following equality +� +Ω +�h(v) · wdx =< [[v]],{{w}} >�h, +∀w ∈ Uh, +(3.22) +and +∥�h(v)∥L2(Ω) ≤ C� ∥h−1/2[[v]]∥L2(�h), +(3.23) +where the constant C� depending on the shape regularity of mesh �h and the degree of +polynomial l. +Lemma 3.6. Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.1)-(2.2) and +(2.13)-(2.14), respectively, we have +∥p − ph∥L2(Ω) +≲ +∥∇ × (u − uh)∥L2(Ω) + η(uh, ph;�h), +(3.24) +∥ph − pH∥L2(Ω) +≲ +∥∇ × (uh − uH)∥L2(Ω) ++ +� +η(uh, ph;�h) + η(uH, pH;�H) +� +. +(3.25) +Proof. Noting that Qh ⊆ Q, and using (2.1), the definition of R1(uh, ph) and (2.16), we +have +∥p − ph∥L2(�h) +≤ +sup +∀q∈Q +(p − ph,q)�h +∥q∥L2(�h) += +sup +∀q∈Q +(µ∇ × u,q)�h − +� +R1(uh, ph) + µ∇ × uh,q +� +�h +∥q∥L2(�h) +≤ +sup +∀q∈Q +(µ∇ × (u − uh),q)�h − +� +R1(uh, ph),q +� +�h +∥q∥L2(�h) +≲ +∥∇ × (u − uh)∥L2(�h) + η(uh, ph;�h). +Similarly, using the definition of R1(uh, ph), (2.13), (3.21)-(3.23), and the fact [[uh]] = + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +15 +[[uc +h + u⊥ +h ]] = [[u⊥ +h ]], we have +∥ph − pH∥L2(�h) ≤ +sup +∀qh∈Qh +(ph − pH,qh)�h +∥qh∥L2(�h) +≤ +sup +∀qh∈Qh +(ph,qh)�h − +� +R1(uH, pH) + µ∇ × uH,qh +� +�h +∥qh∥L2(�h) +≤ +sup +∀qh∈Qh +(µ∇ × uh,qh)�h+ < {{qh}},[[µuh]] >�h − +� +R1(uH, pH) + µ∇ × uH,qh +� +�h +∥qh∥L2(�h) += +sup +∀qh∈Qh +(µ∇ × (uh − uH),qh)�h+ < {{qh}},[[µuh]] >�h − +� +R1(uH, pH),qh +� +�h +∥qh∥L2(�h) +≲ +∥∇ × (uh − uH)∥L2(�h) + ∥h−1/2 +τ +[[uh]]∥L2(�h) + η(uH, pH;�H) +≲ +∥∇ × (uh − uH)∥L2(�h) + C� ∥h−1/2 +τ +[[u⊥ +h ]]∥L2(�h) + η(uH, pH;�H) +≲ +∥∇ × (uh − uH)∥L2(τ) + +� +η(uh, ph;�h) + η(uH, pH;�H) +� +. +Remark 3.2. Noting that ∥(u, p)−(uh, ph)∥2 +DG+η2(uh, ph;�h) and ∥|u−uh|∥2 +h+η2(uh, ph;�h) +are equivalent. In fact, by (3.24), we first know that +∥(u, p) − (uh, ph)∥2 +DG + η2(uh, ph;�h) += ∥|u − uh|∥2 +h + ∥p − ph∥2 +L2(�h) + η2(uh, ph;�h) +≲ ∥|u − uh|∥2 +h + η2(uh, ph;�h). +Secondly, it is shown by the definition of ∥ · ∥DG +∥|u − uh|∥2 +h + η2(uh, ph;�h) ≤ ∥(u, p) − (uh, ph)∥2 +DG + η2(uh, ph;�h). +Thus, we next only need to consider the convergence of ∥|u − uh|∥2 +h + η2(uh, ph;�h). +We first show that the error plus some quantity reduces with a fixed factor on two +successive meshes. +Lemma 3.7. Given f ∈ L2(Ω) and two tetrahedral mesh �h and �H, where �H ≤ �h. Let +(u, p) ∈ U × Q be the solution of (2.1)-(2.2), and (uh, ph) ∈ Uh × Qh, (uH, pH) ∈ UH × QH +be the solutions of (2.13)-(2.14), respectively. Then there exit two constants δ1,δ2 ∈ (0,1), +such that +∥|u − uh|∥2 +h +≤ +(1 + δ1)∥|u − uH|∥2 +H − 1 − δ2 +2 +∥|uh − uH|∥2 +h ++ +C3 +δ1δ2α +� +η2(uh, ph;�h) + η2(uH, pH;�H) +� +. +(3.26) +where C3 depending on the C� . + +16 +K Liu et al. +Proof. Choosing that q = ∇ × v, and subtracting (2.1) from (2.2), we obtain +(κu, v) + (µ∇ × u,∇ × v) = ( f , v). +(3.27) +Subtracting (2.15) from (3.27) with v = vh = uc +h − uc +H, and using [[uc +h − uc +H]] = 0, we +have +(κ(u − uh), uc +h − uc +H)0,�h + (µ∇ × (u − uh),∇ × (uc +h − uc +H))0,�h ++ < [[uh]],{{µ∇ × (uc +h − uc +H)}} >�h= 0, +which leads to +(κ(u − uh), uc +h − uc +H)0,�h + (µ∇ × (u − uh),∇ × (uc +h − uc +H))0,�h += − < [[uh]],{{µuc +h − uc +H}} >�h . +(3.28) +Using (3.22) and (3.23), we have +< [[uh]],{{∇ × (uc +h − uc +H)}} >�h += +(�h(uh),∇ × (uc +h − uc +H))0,�h +≤ C� ∥h−1/2[[uh]]∥0,�h∥∇ × (uc +h − uc +H)∥0,�h. +(3.29) +Let uh = uc +h + u⊥ +h and uH = uc +H + u⊥ +H, we have +uh + uc +H − uc +h = uH − u⊥ +H + u⊥ +h , +(3.30) +where uc +H ∈ Uc +H, uc +h ∈ Uc +h, u⊥ +H ∈ U⊥ +H, u⊥ +h ∈ U⊥ +h . By (3.30), (3.28), (3.29) and Young’s +inequality, we get +∥|u − uh|∥2 +h += ∥κ(u − uh)∥2 +L2(Ω) + ∥∇ × µ(u − uh)∥2 +L2(Ω) ++ +� +f ∈�h +αh−1 +f +< [[(u − uh)]],[[u − uh]] >�h += ∥|u − uh − uc +H + uc +h|∥2 +h − ∥|uc +h − uc +H|∥2 +h − 2(κ(u − uh), uc +h − uc +H)0,�h +−2(µ∇ × (u − uh),∇ × (uc +h − uc +H))0,�h +−2 +� +f ∈�h +αh−1 +f +< [[(u − uh)]],[[uc +h − uc +H]] > +≲ ∥|u − uH|∥2 +H + 2∥|u − uH|∥H∥|u⊥ +h − u⊥ +H|∥h + ∥|u⊥ +h − u⊥ +H|∥2 +h − ∥|uc +h − uc +H|∥2 +h ++2∥h−1/2[[uh]]∥0,�h∥∇ × (uc +h − uc +H)∥0,�h +≤ (1 + δ1)∥|u − uH|∥2 +H + (1 + 1 +δ1 +)∥|u⊥ +h − u⊥ +H|∥2 +h − (1 − ˆδ2C� )∥|uc +h − uc +H|∥2 +h ++C� +ˆδ2 +∥h−1/2[[uh]]∥2 +0,�h += (1 + δ1)∥|u − uH|∥2 +H + (1 + 1 +δ1 +)∥|u⊥ +h − u⊥ +H|∥2 +h − (1 − δ2)∥|uc +h − uc +H|∥2 +h ++ +C2 +� +δ2 +∥h−1/2[[uh]]∥2 +0,�h, + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +17 +where δ2 = ˆδ2C� . Using uc +H = uH − u⊥ +H, uc +h = uh − u⊥ +h , triangle inequality and average +inequality, we have +∥|uc +h − uc +H|∥2 +h ≥ 1 +2∥|uh − uH|∥2 +h − ∥|u⊥ +h − u⊥ +H|∥2 +h. +By triangle inequality and (3.21), we obtain +∥|u⊥ +h − u⊥ +H|∥2 +h +≤ +2(∥|u⊥ +h |∥2 +h + ∥|u⊥ +H|∥2 +H) +≤ +2α∥h−1/2[[u⊥ +h ]]∥2 +0,�h + 2α∥h−1/2[[u⊥ +H]]∥2 +0,�h. +Combining [[uH]] = [[u⊥ +H + uc +H]] = [[u⊥ +H]] and (3.19), we have +∥|u − uh|∥2 +h +≤ +(1 + δ1)∥|u − uH|∥2 +H − 1 − δ2 +2 +∥|uh − uH|∥2 +h ++ +C3 +δ1δ2α +� +η2(uh, ph;�h) + η2(uH, pH;�H) +� +. +3.3. Contraction of the error estimator +In this subsection, we prove the reduction of error indicators. Let us first consider the +effect of changing the finite element function used in the estimator. +Lemma 3.8. Given f ∈ L2(Ω) and two tetrahedral mesh �h, �H with �H ≤ �h. Let (vh,qh) ∈ +Uh × Qh and (v H,q H) ∈ UH × QH. For any ε > 0, we have +η2(vh,qh;�h) ≤ (1 + ε)η2(v H,q H;�h) + Cε∥(vh,qh) − (v H,q H)∥2 +DG, +(3.31) +where Cε depending on the ε, and the mesh size h < 1. +Proof. +For any τ∗ ∈ �h, we will discuss each of the five components of the mark +η2(vh,qh;�h). +Firstly, using the definition of R1(vh,qh) and triangle inequality, we have +∥R1(vh,qh)∥L2(τ∗) +(3.32) += ∥qh − µ∇ × vh∥L2(τ∗) += ∥qh − q H + µ∇ × (v H − vh) + q H − µ∇ × v H∥L2(τ∗) +≲ ∥q H − ∇ × v H∥L2(τ∗) + ∥qh − q H∥L2(τ∗) + ∥∇ × (vh − v H)∥L2(τ∗). +Secondly, using the definition of R2(vh,qh), triangle inequality and inverse inequality, +we get +hτ∗∥R2(vh,qh)∥L2(τ∗) +(3.33) += hτ∗(∥ f − ∇ × qh − κvh∥L2(τ∗)) += hτ∗(∥ f − ∇ × (qh − q H) − κ(vh − v H) − ∇ × q H − κv H∥L2(τ∗)) +≤ hτ∗(∥ f − ∇ × q H − κv H∥L2(τ∗) + ∥∇ × (qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗)) +≲ hτ∗(∥R2(v H,q H)∥L2(τ∗) + h−1 +τ∗ ∥(qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗)) +≲ hτ∗∥R2(v H,q H)∥L2(τ∗) + ∥(qh − q H)∥L2(τ∗) + hτ∗∥κ(vh − v H)∥L2(τ∗). + +18 +K Liu et al. +Similarly, using the definition of R3(vh), triangle inequality and inverse inequality, we +get +hτ∗∥R3(vh)∥L2(τ∗) +(3.34) += hτ∗∥∇ · ( f − κvh)∥L2(τ∗) += hτ∗∥∇ · ( f − κv H + κv H − κvh)∥L2(τ∗) +≤ hτ∗(∥∇ · ( f − κv H)∥L2(τ∗) + ∥∇ · κ(v H − vh)∥L2(τ∗)) +≲ hτ∗(∥R3(v H)∥L2(τ∗) + h−1 +τ∗ ∥κ(v H − vh)∥L2(τ∗)) +≲ hτ∗∥R3(v H)∥L2(τ∗) + ∥κ(v H − vh)∥L2(τ∗). +Next, we discuss the jump J1(qh) and J2(vh). For any f ∈ �(�h), we let f = τ1 +∗ +� +τ2 +∗ +with τ1 +∗,τ2 +∗ ∈ �h. Furthermore, using the definition of J1(qh), triangle inequality and trace +inequality, we have +h1/2 +f +∥J1(qh)∥L2(f ) +(3.35) += h1/2 +f +∥[[qh]]∥L2(f ) += h1/2 +f +∥[[q H + qh − q H]]∥L2(f ) +≤ h1/2 +f +(∥[[q H]]∥L2(f ) + ∥[[qh − q H]]∥L2(f )) +≤ h1/2 +f +∥[[q H]]∥L2(f ) + h1/2 +f +∥(qh − q H)|τ1 +∗∥L2(f ) + h1/2 +f +∥(qh − q H)|τ2 +∗∥L2(f ) +≲ h1/2 +f +∥J1(q H)∥L2(f ) + ∥(qh − q H)∥L2(τ1 +∗∪τ2 +∗). +Similarly, using the definition of J2(vh), triangle inequality and trace inequality, we +have +h1/2 +f +∥J2(vh)∥L2(f ) +(3.36) += h1/2 +f +∥[[( f − κvh)]]∥L2(f ) += h1/2 +f +∥[[( f − κv H + κv H − κvh)]]∥L2(f ) +≤ h1/2 +f +(∥[[(f − κv H)]]∥L2(f ) + ∥[[κ(v H − vh)]]∥L2(f )) +≤ h1/2 +f +∥J2(v H)∥L2(f ) + h1/2 +f +(∥κ(v H − vh)|τ1 +∗∥L2(f ) + ∥κ(v H − vh)|τ2 +∗∥L2(f )) +≲ h1/2 +f +∥J2(v H)∥L2(f ) + ∥κv H − κvh∥L2(τ1 +∗∪τ2 +∗). +Finally, the desired result (3.31) is obtained by combining (3.32)-(3.36), Young’s in- +equality and the shape regularity of mesh �h. +We then prove the contraction of the error estimator under the assumptions on the +problem of (2.13)-(2.14). +Lemma 3.9. Given constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h). Let +(uH, pH) ∈ UH × QH be the solution of (2.13)-(2.14), and ��H−→�h = �H \ (�h ∩ �H) be the + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +19 +set of all element refined into �h on �H. Then, there is a constant λ ∈ (0,1) independent of +mesh size, such that +η2(uH, pH;�h) ≤ η2(uH, pH;�H) − λη2(uH, pH;��H→�h). +(3.37) +Proof. Assume that the tetrahedral mesh τ ∈ �H is divided into two new tetrahedral +mesh τ1 +∗ and τ2 +∗ with equal volumes, where τ1 +∗,τ2 +∗ ∈ �h. Thus, h3 +τ1 +∗ = |τ1 +∗| = |τ2 +∗| = h3 +τ2 +∗ = +2−1h3 +τ by the shape regularity of mesh, which implies hτ1 +∗ = hτ2 +∗ = 2−1/3hτ. Then, we have +∥R1(uH, pH)∥2 +L2(τ1 +∗) + ∥R1(uH, pH)∥2 +L2(τ2 +∗) ≤ ∥R1(uH, pH)∥2 +L2(τ), +(3.38) +and +h2 +τ1 +∗(∥R2(uH, pH)∥2 +L2(τ1 +∗) + ∥R3(uH)∥2 +L2(τ1 +∗)) ++ h2 +τ2 +∗(∥R2(uH, pH)∥2 +L2(τ2 +∗) + ∥R3(uH)∥2 +L2(τ2 +∗)) +≤ 2−2/3h2 +τ(∥R2(uH, pH)∥2 +L2(τ) + ∥R3(uH)∥2 +L2(τ)). +(3.39) +For any f ∈ ∂ (τ1 +∗ ∪ τ2 +∗), which can be divided into three parts; +(1) For the first part, there are two of the faces are constant and belong to τ . +(2) For the second part, there are two new faces that overlap and are used to divide the +mesh τ. Since (uH, ph) ∈ UH × QH is a continuous polynomial in the region τ, it follows +that the value of [[ph]] and [[( f − κuH)]] on this surface is equal to zero. +(3) For the third part, there are four faces that are obtained by dividing the two faces +in the τ into two. +Furthermore, we obtain +η2(uH, pH;τ1 +∗) + η2(uH, pH;τ2 +∗) ≤ γη2(uH, pH;τ). +(3.40) +where constant γ ∈ (0,1) independent of mesh τ. +Next, since ��H→�h represents the part of the set in the tetrahedral set �H that will +be used to be refined, it follows that ��H→�h ⊂ �H. Let ��H→�h denote the part of the +cell set that has been refined in the tetrahedral set �H, we have ��h→�H ∈ �h. Obviously, +�H \��H→�h = �h \��H→�h. Then combining the (3.40), and the marking strategy (2.18), +we have +η2(uH, pH;�h) += +η2(uH, pH;�h \ ��H→�h) + η2(uH, pH;��H→�h) +≤ +η2(uH, pH;�H \ ��H→�h) + γη2(uH, pH;��H→�h) +≤ +η2(uH, pH;�H) + (γ − 1)η2(uH, pH;��H→�h) +≤ +η2(uH, pH;�H) − λη2(uH, pH;��H→�h), +where λ = 1 − γ ∈ (0,1) independent of mesh size. +Now, we combine the Lemmas 3.6, 3.8 and 3.9 to prove the reduction of error indicators. + +20 +K Liu et al. +Lemma 3.10. Given a constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h). Let +(uh, ph) ∈ Uh × Qh and (uH, pH) ∈ UH × QH be the solutions of (2.13)-(2.14), respectively. +For any ε > 0 and λ ∈ (0,1), we have +(1 − Cε +α )η2(uh, ph;�h) +≤ +(1 + ε + Cε +α )η2(uH, pH;�H) +− (1 + ε)λη2(uH, pH;��H→�h) + Cε∥|uh − uH|∥2 +h, +(3.41) +where constant Cε depending on the ε and mesh size. +Proof. Using the Lemmas 3.6, 3.8 and 3.9, we have +η2(uh, ph;�h) +≤ +(1 + ε) +� +η2(uH, pH;�H) − λη2(uH, pH;��H→�h) +� ++Cε∥(uh, ph) − (uH, pH)∥2 +DG +≤ +(1 + ε) +� +η2(uH, pH;�H) − λη2(uH, pH;��H→�h) +� ++Cε∥|uh − uH|∥2 +h + ∥ph − pH∥2 +L2(Ω) +≤ +(1 + ε) +� +η2(uH, pH;�H) − λη2(uH, pH;��H→�h) +� ++Cε∥|uh − uH|∥2 +h + Cε +α +� +η2(uh, ph;�h) + η2(uH, pH;�H) +� +, +which completes the proof. +3.4. Convergence result +Now, we proved that the sum of the norm of the error and the scaled error indicator is +attenuated. +Theorem 3.2. For a given θ ∈ (0,1),let {�k,Uk,Qk, uk, pk,η(uk, pk;�k)}k≥0 be the se- +quence of meshes, Mixed discrete solution (defined by (2.13)-(2.14)), and the estimate in- +dicator produced by the AMIPDG algorithm. Then there exist constants ρ > 0, δ ∈ (0,1), +which depend on marking parameter θ and the shape regularity of the initial mesh �0, such +that +∥|u − uk+1|∥2 +k+1 + ρη2(uk+1, pk+1;�k+1) ≤ δ +� +∥|u − uk|∥2 +k + ρη2(uk, pk;�k) +� +. +Proof. Setting �ρ = 1−δ2 +2Cε , then multiply the both sides of the (3.41) inequality by �ρ, we +get +�ρ(1 − Cε +α )η2(uk+1, pk+1;�k+1) +≤ �ρ(1 + ε + Cε +α )η2(uk, pk;�k) − �ρ(1 + ε)λη2(uk, pk;��k→�k+1) ++1 − δ2 +2 +∥|uk+1 − uk|∥2 +h. +(3.42) + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +21 +Next, by the (3.26) and (3.42), we have +∥|u − uk+1|∥2 +k+1 + �ρ(1 − Cε +α )η2(uk+1, pk+1;�k+1) +≤ (1 + δ1)∥|u − uk|∥2 +k + +C3 +δ1δ2α +� +η2(v k+1,q k+1;�k+1) + η2(v k,q k;�k) +� ++�ρ(1 + ε + Cε +α )η2(uk, pk;�k) − �ρ(1 + ε)λη2(uk, pk;��k→�k+1). +(3.43) +First move the term and then according to Dörfler marking strategy (2.18), the Theorem +3.1 and ∥| · |∥h ≤ ∥ · ∥DG, we know −η2(v k,q k;��k→�k+1) ≤ −θη2(v k,q k;�k), then +∥|u − uk+1|∥2 +k+1 ++ +�ρ(1 − Cε +α − +C3 +�ρδ1δ2α)η2(uk+1, pk+1;�k+1) +≤ +(1 + δ1)∥|u − uk|∥2 +k − �ρ(1 + ε)λθ +2 +η2(uk, pk;�k) ++�ρ +� +1 + ε + Cε +α + +C3 +�ρδ1δ2α − (1 + ε)λθ +2 +� +η2(uk, pk;�k) +≤ +(1 + δ1 − +�ρ(1 + ε)λθC−1 +1 +2 +)∥|u − uk|∥2 +k ++�ρ +� +1 + ε + Cε +α + +C3 +�ρδ1δ2α − (1 + ε)λθ +2 +� +η2(uk, pk;�k). +For convenience, denote +β1 += +1 − Cε +α − +C3 +�ρδ1δ2α, +β2 += +1 + δ1 − +�ρ(1 + ε)λθC−1 +1 +2 +, +β3 += +(1 + ε)(1 − λθ +2 ) + Cε +α + +C3 +�ρδ1δ2α. +Thus +∥|u − uk+1|∥2 +k+1 + �ρβ1η2(uk+1, pk+1;�k+1) ≤ β2∥|u − uk|∥2 +k + �ρβ3η2(uk, pk;�k). +Next, we firstly choose δ1 = +�ρ(1+ε)λθC−1 +1 +4 +, then select the appropriate δ2 to make �ρ = +1−δ2 +2Cε smaller to ensure 0 < δ1 < 1, Secondly, we let ε > 0 and (1 + ε)(1 − λθ +2 ) = 1 − λθ +4 ( +λθ ∈ (0,1)), therefore +β2 = 1 − δ1 ∈ (0,1), (1 + ε)(1 − λθ +2 ) < 1. +Furthermore, we choose a sufficiently large penalty parameter α such that +β1 > β3. + +22 +K Liu et al. +Finally, there is a constant δ = max{β2, β1 +β3 }. Then, we let ρ = �ρβ1, and obtain +∥|u − uk+1|∥2 +k+1 + ρη2(uk+1, pk+1;�k+1) ≤ δ +� +∥|u − uk|∥2 +k + ρη2(uk, pk;�k) +� +. +Corollary 3.1. Under the conditions of Theorem 3.2, we have +∥(u, p) − (uk, pk)∥2 +DG + ρη2(uk, pk;�k) ≤ δk �Cδ. +where �Cδ = C +� +∥(u, p) − (u0, p0)∥2 +DG + ρη2(u0, p0;�0) +� +. Therefore, for a given precision, +the AMIPDG method will terminate after a finite number of operations. +Proof. Using the Remark 3.2 and Theorem 3.2, we have +∥(u, p) − (uk, pk)∥2 +DG + ρη2(uk, pk;�k) +≤ +C +� +∥|u − uk|∥2 +k + ρη2(uk, pk;�k) +� +≤ +δk �Cδ. +4. Numerical experiments +In this section, we test some numerical experiments to show the efficiency and the +robustness of AMIPDG. We carry out these numerical experiments by using the MATLAB +software package iFEM [6]. In Experiments 4.1 and 4.2, we take p = ∇ × u. +In Example 4.1, we discuss the influence of the penalty parameter α on the error in +∥ · ∥DG norm, and observe the dependency of the condition number of stiffness matrix on +α. +Example 4.1. Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution +of the model (1.1)-(1.2): +u = +� +� +x(x − 1)y(y − 1)z(z − 1) +sin(πx)sin(πy)sin(πz) +(1 − ex)(1 − ex−1)(1 − e y)(1 − e y−1)(1 − ez)(1 − ez−1) +� +�. +It is easy to see that the solution u satisfies the boundary condition u × n = 0 on ∂ Ω. +In this example, we get a uniform mesh by partitioning the x−, y− and z−axes into +equally distributed M(M ≥ 2) subintervals, and then dividing one cube into six tetrahe- +drons. Let h = 1/M be mesh sizes for different tetrahedrons meshes. We fixed mesh with +h = 1/4 and report the error estimates in ∥ · ∥DG norm and condition number of stiffness +matrices for different penalty parameters α = 1,10,100,500 and 1000 in Table 1. We note +that ∥u − uh∥0 increases at first and then decreases as the penalty parameter α increases. + +Convergence of AMIPDG methods for H(cur l)-elliptic problems +23 +Table 1: The error in ∥ · ∥DG norms and condition number of stiffness matrices with h = 1/4. +α +1 +10 +100 +500 +1000 +∥ +� +p − ph, u − uh +� +∥DG +3.949e+00 +1.133e-00 +8.614e-01 +8.649e-01 +8.659e-01 +Cond +3.235e+04 +7.021e+04 +5.959e+05 +2.995e+06 +6.150e+06 +The condition numbers of stiffness matrices increase with the increase of penalty parame- +ters α. +As a way to balance, in the following numerical tests, we always choose α = 100. +Noting that we only consider uniform meshes in Example 4.1. Next we test adaptive +meshes. +Example 4.2. Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution +of the model (1.1)-(1.2) +u = +� +� +� +x(x−1)y(y−1)z(z−1) +x2+y2+z2+0.001 +x(x−1)y(y−1)z(z−1) +x2+y2+z2+0.001 +− x(x−1)y(y−1)z(z−1) +x2+y2+z2+0.001 +� +� +�. +Note that the solution u satisfies the condition u × n = 0 on ∂ Ω. +The right of Figure 1 shows an adaptively refined mesh with marking parameter- θ = +0.7 after k = 18. The grid is locally refined near the origin. +Figure 1: Left: the initial mesh with 1152 DoFs. Right: the adaptive mesh(θ = 0.7) with 181104 DoFs +after 18 refinements. +The Figure 2 shows the curves of log N−logη +� +uk, pk;�k +� +for parameters θ = 0.3,0.5,0.7. +The curves indicate the convergence and the quasi-optimality of the adaptive algorithm +AMIPDG of η +� +uk, pk;�k +� +. +Acknowledgment +The first author is supported by the East China University of Technology (DHBK2019209) +and Jiangxi Province Education Department (GJJ200755). The second, third and fourth +authors are supported by the National Natural Science Foundation of China (Grant No. +12071160). The third author is also supported by the National Natural Science Foundation +of China (Grant No. 11901212). + +24 +K Liu et al. +Figure 2: Quasi optimality of the AMIPDG of the error η +� +uk, pk;�k +� +with different marking parameters +θ. +References +[1] B. AYUSO DE DIOS, R. HIPTMAIR AND C.L. PAGLIANTINI, Auxiliary space preconditioners +for SIP-DG discretizations of H(curl)-elliptic problems with discontinuous coefficients. IMA J. +Numer. Anal. 37(2017), pp, 646-686. +[2] A. BONITO AND R.H. NOCHETTO, Quasi-optimal convergence rate of an adaptive discontin- +uous Galerkin method. SIAM J. Numer. Anal. 48(2010), pp. 734–771. +[3] C. CARSTENSEN AND R.H. HOPPE, Unified framework for an a posteriori error analysis of +non-standard finite element approximations of H(cur l)-elliptic problems. J. Numer. Math. +17(2009), pp. 27–44. +[4] C. CARSTENSEN, R.H. HOPPR, N. SHARMA AND T. WARBURTON, Adaptive hybridized in- +terior penalty discontinuous galerkin methods for H(cur l)–elliptic problems. Numer. Math. +Theor. Meth. Appl. 4(2011), pp. 13–37. +[5] C. CARSTENSEN AND R. MA, Adaptive mixed finite element methods for non-self-adjoint +indefinite second-order elliptic pdes with optimal rates. SIAM J. Numer. Anal. 59(2021), pp. +955–982. +[6] L. CHEN, iFEM: an innovative finite element method package in MATLAB. Technical report, +University of California at Irvine (2009). +[7] L. CHEN, M. HOLST AND J.C. XU, +Convergence and optimality of adaptive mixed finite +element methods. Math. Comp. 78(2009), pp. 35–53. +[8] E.T. CHUNG, M.C. YUEN AND L.Q. ZHONG, A-posteriori error analysis for a staggered dis- +continuous Galerkin discretization of the time-harmonic Maxwell’s equations. Appl. Math. +Comput. 237(2014), pp. 613–631. +[9] L. DÖRFLER, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. +33(1996), pp. 1106–1124. +[10] S.H. DU AND X.P. XIE, +Convergence of an adaptive mixed finite element method for + +Rate of convergence is CN-0.33 +10 +adaptive refiniment = 0.3 +adaptive refiniment =0.5 +adaptive refiniment =0.7 +CN-0.33 +104 +105 +Number of unknownsConvergence of AMIPDG methods for H(cur l)-elliptic problems +25 +convection-diffusion-reaction equations. Sci. China Math. 58(2015), pp. 1327–1348. +[11] P. HOUSTON, I. PERUGIA, A. SCHNEEBELI ADN D. SCHÖTZAU, Interior penalty method for +the indefinite time-harmonic Maxwell equations. Numer. Math. 100(2005), pp. 485–518. +[12] W. JIANG, N. LIU, Y. TANG AND Q.H. LIU, +Mixed finite element method for 2D vector +Maxwell’s eigenvalue problem in anisotropic media. Progress In Electromagnetics Research +148(2014), pp. 159–170. +[13] C. JOG AND A. NANDY, Mixed finite elements for electromagnetic. Comput. Math. Appl. +68(2014), pp. 887–902. +[14] F. KIKUCHI, +Mixed and penalty formulations for finite element analysis of an eigenvalue +problem in electromagnetism. Comput. Methods Appl. Mech. Engrg. 64(1987), pp. 509–521. +[15] N. LIU, L. TOBÓN, Y. TANG AND Q.H. LIU, Mixed spectral element method for 2D Maxwell’s +eigenvalue problem. Commun. Comput. Phys. 17(2015), pp. 458–486. +[16] P. MONK, Finite Element Methods for Maxwell Equations. Numerical Mathematics and Scientific +Computation. Oxford University Press, Oxford(2003). +[17] J.C. NÉDÉLEC, Mixed finite elements in �3. Numer. Math. 35(1980), pp. 315–341. +[18] I. PERUGIA, D. SCHÖTZAU AND P. MONK, Stabilized interior penalty methods for the time- +harmonic Maxwell equations. Comput. Methods Appl. Mech. Eng. 191(2002), pp. 4675–4697. +[19] J. SCHÖBERL, A posteriori error estimates for Maxwell equations. Math. Comp. 77(2008), +pp. 633–649. +[20] X.Q. XING AND L.Q. ZHONG, A posteriori error estimate of discontinuous Galerkin Method +for H(curl)-elliptic problems (in Chinese). Journal of South China Normal University (Natural +Science Edition). 44(2012), pp. 18–21. + diff --git a/0dAzT4oBgHgl3EQfefzh/content/tmp_files/load_file.txt b/0dAzT4oBgHgl3EQfefzh/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..19afa64a09f3bda61c18ee75c37405a0015f5452 --- /dev/null +++ b/0dAzT4oBgHgl3EQfefzh/content/tmp_files/load_file.txt @@ -0,0 +1,966 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf,len=965 +page_content='Convergence of Adaptive Mixed Interior Penalty Dis- continuous Galerkin Methods for H(cur l)-Elliptic Problems K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Liu1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Tang2,, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Xing2 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Zhong2 1 School of Sciece, East China University of Technology, Nanchang, 330013, China 2 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for H(cur l)-elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We first get the mixed model of H(cur l)-elliptic problem by introducing a new intermediate variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we construct the corresponding posteriori error indicator, and prove the contraction of the summation of the energy error and the scaled error indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' At last, we confirm and illustrate the theoretical result through some numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' AMS subject classifications: 65M15,65N12,65N30 Key words: Adaptive mixed interior penalty discontinuous Galerkin methods, Convergence, H(cur l)- elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Introduction Let Ω ⊂ �3 be Lipschitz bounded polygonal domain with a single connected boundary ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We consider the following H(cur l)-elliptic problem ∇ × µ∇ × u + κu = f in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) u × n = 0 on ∂ Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) where n is the unit normal vector of the boundary ∂ Ω, f ∈ L2(Ω), µ and κ are piecewise constants is consistent with the initial partition �0 for Ω and satisfy µ1 < µ < µ2 and κ1 < κ < κ2, here, µi and κi(i = 1,2) are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' By introducing an auxiliary ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Email addresses: liukai@ecut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Liu), mingtang@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Tang),xingxq@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn(X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Xing), zhong@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Zhong) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='01439v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='NA] 4 Jan 2023 2 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' variable p = µ∇ × u, then we get the mixed scheme with the boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) p = µ∇ × u in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3) ∇ × p + κu = f in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4) u × n = 0 on ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) The mixed finite element method is very convenient for processing high-order equations and equations containing two or more unknown functions, which has attracted widespread attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For mixed finite element method, there are only few research results for Maxwell problem [13] and Maxwell’s eigenvalue problem [12,14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Adaptive finite element method automatically refines and optimizes meshes accord- ing to the singularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' It is a highly reliable and efficient numerical calculation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' At present, the convergence analysis research of the adaptive mixed finite element method for the elliptic equation is relatively complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Chen, Holst and Xu [7] proved the convergence analysis of the adaptive mixed finite element algorithm for elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Du and Xie [10] proved the convergence analysis of the adaptive mixed finite element algorithm for the convection diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' However, there are only few research results on the posterior error estimator of Maxwell’s equations for the adaptive mixed fi- nite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For example, Carstensen and Ma [5] establishes the convergence of adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Carstensen, Hoppe, Sharma and Warburton [4] designs and analyzes the posterior error estimation of the adaptive hybrid conforming finite element method of H(cur l)-elliptic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Recently, Chung, Yuen and Zhong [8] present a-posteriori error analysis for the staggered discontinuous Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' As far as we know, there are not any published literatures for the convergence analysis of the adaptive mixed finite element method for the boundary value problem(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Our contributions in this paper are to construct a new error estimator, which does not include the negative power of the local mesh size in the jump term for the traditional DG method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' get the convergence of the Adaptive Mixed Interior Penalty Discontinuous Galerkin (AMIPDG) method by using the similar technique used in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' However, this tech- nique in [2] can not be used directly for mixed forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We present our main result in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let {�k,Uk,Qk, uk, pk,η(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k)}k≥0 be the sequence of meshes, finite element space, mixed discrete solution and posterior error estimate indicator produced by the AMIPDG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exist constants ρ > 0 and δ ∈ (0,1), which depend on marking parameter and the shape regularity of the initial mesh �0, such that ∥|u − uk+1|∥2 k+1 + ρη2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ δ � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, for a given precision, the AMIPDG method will terminate after a finite number of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 3 For convenience, we let C denote a generic positive constant which may be different at different occurrences and adopt the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The subscripted constant Ci represents a particularly important constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' a ≲ b means a ≤ C b for some constants C which are independent of mesh sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Section 2, we first present the contin- uous variational problem, the discrete variational problem, and the procedure of AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Section 3, we first show the upper bound estimate of the error, which is key to the con- vergence analysis, then we prove the indicator reduction and the convergence of AMIPDG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Section 4, we provide some numerical experiments to illustrate the effective- ness of the AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Adaptive Mixed interior penalty discontinuous Galerkin method In this section, we introduce the continuous variational problem, the discrete variational problem of mixed internal penalty discontinuous finite element method, and the procedure of AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Continuous variational problem For an open and connected bounded domain D ⊂ �3, we denote by L2(D) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' L2(D) := (L2(D))3) the spaces of square-integrable functions (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' vector fields) on D with inner product (·,·)0,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We define the spaces H(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) = {u ∈ L2(D) : ∇ × u ∈ L2(D)}, H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) = {u ∈ L2(D) : ∇ · u ∈ L2(D)}, with (u, v)cur l,D := (u, v)0,D + (∇ × u,∇ × v)0,D, ∀u, v ∈ H(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D), (u, v)div,D := (u, v)0,D + (∇ · u,∇ · v)0,D, ∀u, v ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D), and the induced norm as: ∥u∥2 cur l,D := ∥u∥2 0,D + ∥∇ × u∥2 0,D, ∀u ∈ H(cur l, D), ∥u∥2 div,D := ∥u∥2 0,D + ∥∇ · u∥2 0,D, ∀u ∈ H(div, D), respectively, where ∥ · ∥L2(D) := (·,·)1/2 D denotes the norm of the space L2(D) or L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We also define H0(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) = {v ∈ H(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) : v × n = 0 on ∂ D} in the trace sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we first define two space U := H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω),Q := L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, the mixed vari- ational problem of the mixed boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) reads as: find (u, p) ∈ U × Q such that: a(p,q) − b(u,q) = ℓ1(q), ∀q ∈ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) d(v, p) + c(u, v) = ℓ2(v), ∀v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) 4 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The bilinear forms a, b, c and the functionals ℓ1(·),ℓ2(·) are given by a(p,q) := (p,q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3) b(u,q) := (µ∇ × u,q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4) c(u, v) := (κu, v), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) d(v, p) := (∇ × v, p) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) ℓ1(q) := 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) ℓ2(v) := ( f , v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8) The operator-theoretic framework involves operator � : (U × Q) → (U × Q)∗ defined by (� (u, p))(v,q) := a(p,q) − b(u,q) + d(v, p) + c(u, v),∀u, v ∈ U, p,q ∈ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9) where (Q × U)∗ is the dual spaces of (Q × U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) as (� (u, p))(v,q) = ℓ(v,q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10) with ℓ(v,q) = ℓ1(q) + ℓ2(v), and ℓi are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, we state the well-posedness of the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) in the follow- ing lemma, and it can be found in section 3 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Under the assumptions on the problem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2), � is a continuous and bijective linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Hence, for any ℓ = (ℓ1,ℓ2) ∈ (Q×U)∗, the mixed variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) has a unique solution (u, p) ∈ (U × Q), which satisfy the following continuously ∥(u, p)∥U×Q := (∥u∥2 curl,Ω + ∥p∥2 0)1/2 ≲ ∥ℓ1∥Q∗ + ∥ℓ2∥U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Discrete variational problem We suppose that �h is a family of shape regularity, quasi-uniform and conform tetrahe- dral generation on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let hτ = |τ|1/3 denote the mesh size with |τ| being the volume of τ ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Define the discontinuous finite element function space �(�h) as: �(�h) = {v ∈ L2(Ω) : vτ = v|τ ∈ (Pl(τ))3, ∀τ ∈ �h}, where Pl(τ) is the set of polynomials defined in the volume τ whose degree does not exceed l, where l ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let �h, � 0 h and � ∂ h denote the set of the all faces of its volumes, and the set of internal faces, and the set of boundary faces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, �h = � 0 h � � ∂ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) be the space of piecewise Sobolev functions defined by H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = � v ∈ L2(Ω) : vτ = v|τ ∈ H1(τ), ∀ τ ∈ �h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 5 and H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let L2(�h) be the set of L2 functions defined on �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' More- over, we define the following inner products (v, w)� ′ h = � τ∈� ′ h � τ v · wdx, ∀v, w ∈ L2(Ω), ∀� ′ h ⊂ �h, < v, w >� ′ h = � f ∈� ′ h � f v · wds, ∀v, w ∈ L2(�h), ∀� ′ h ⊂ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For f ∈ � 0 h , we have τi ∈ �h(i = 1,2), such that f = ∂ τ1 ∩ ∂ τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we denote the jump and average of v as: [[v]] = v1 × n1 + v2 × n2, ∀v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), {{v}} = v1 + v2 2 , ∀v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), where v i denote the values of v on v|τi(i = 1,2) and ni denote the out unit normal vectors on f exterior v|τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For f ∈ � ∂ h , we have τ ∈ �h, such that f = ∂ τ ∩ ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we denote the jump and average of v as: [[v]] = vτ × n∂ Ω, {{v}} = vτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='12) Next, we give the corresponding discrete scheme of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Firstly, we define the corresponding discrete space as follow Uh := {vh ∈ �(�h)| [[vh]]|f = 0,∀f ∈ � ∂ h }, Qh := �(�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, the formulation of the discrete Mixed Interior Penalty Discontinuous Galerkin (MIPDG) method reads: find (uh, ph) ∈ (Uh,Qh) such that ah(ph,qh) − bh(uh,qh) = ℓ1,h(qh) + d1,h(uh,qh), ∀qh ∈ Qh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13) dh(vh, ph) + ch(uh, vh) = ℓ2,h(vh) + d2,h(uh, vh), ∀vh ∈ Uh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) where ah(ph,qh) := (ph,qh)�h, bh(uh,qh) := (µ∇ × uh,qh)�h, ch(uh, vh) := (κuh, vh)�h, dh(vh, ph) := (∇ × vh, ph)�h, ℓ1,h(qh) := 0, ℓ2,h(vh) := ( f , vh)�h, d1,h(uh,qh) := − < {{µqh}},[[uh]] >�h, d2,h(uh, vh) :=< ({{µ∇ × uh}} − αh−1 f [[uh]]),[[vh]] >�h, 6 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' here the constant α > 0 denote the penalty parameter, hf denote the diameter of the circumcircle of f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus hτ ≈ hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The calculation of ∇ × uh in the bilinear terms are piecewise derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The standard symmetric Interior Penalty Discontinuous Galerkin (IPDG) method of the boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) is to find uh ∈ Uh, such that aIP(uh, vh) := (κuh, vh)�h + (µ∇ × uh,∇ × vh)�h− < {{µ∇ × vh}},[[uh]] >�h − < {{µ∇ × uh}},[[vh]] >�h +αh−1 f < [[uh]],[[vh]] >�h (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) = ( f , vh)�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The following lemma shows that the discrete variational problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [ [3], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1] The formulations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) are formally equivalent in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' If (uh, ph) ∈ (Uh,Qh) are the solution of discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), then uh ∈ Uh solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Conversely, if uh ∈ Uh solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15), then there exists some ph ∈ Qh such that (uh, ph) ∈ (Uh,Qh) are the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Ayuso de Dios, Hiptmair and Pagliantini proved the well-posedness of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) in section 2 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, by combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we obtain the well-posedness of discrete variational problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Adaptive Mixed Interior Penalty Discontinuous Galerkin method(AMIPDG) Our adaptive cycle can be implemented by the following algorithm: Next, we will discuss each step in AEFEM in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure SOLVE For f ∈ L2(Ω), and a shape regular mesh �k, Let (uk, pk) be the exact MIPDG solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Here, we assume that the solutions (uk, pk) can be solved accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure ESTIMATE A posteriori error indicator is an essential ingredient of adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' They are computable quantities depending on the computed solution(s) and data that provide information about the quality of approximation and may consequently be used to make judicious mesh modi- fications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Here, we design a new posteriori error estimation indicator for equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)- (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), which is similar to that in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For τ ∈ �h, f ∈ �h and (vh,qh) ∈ Uh × Qh, the residual a posteriori error estimator for the symmetric AMIPDG method is given by η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ) : = ∥R1(vh,qh)∥2 L2(τ) + h2 τ � ∥R2(vh,qh)∥2 L2(τ) + ∥R3(vh)∥2 L2(τ) � + � f ∈∂ τ hf � ∥J1(qh)∥2 L2(f ) + ∥J2(vh)∥2 L2(f ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16) Convergence of AMIPDG methods for H(cur l)-elliptic problems 7 Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 Adaptive Mixed Interior Penalty Discontinuous Galerkin Method (AMIPDG) cycle Input initial triangulation �0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' data f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' tolerance tol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' marking parameter θ ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Output a triangulation �J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MIPDG solution (uJ, pJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' η = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' k = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' while η ≥ tol SOLVE solve discrete varational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) on �k to get the solution (uk, pk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' ESTIMATE compute the posterior error estimator η = η(uk, pk,�k) by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MARK seek a minimum cardinality �k ⊂ �k such that η2 � uk, pk,�k � ≥ θη2 � uk, pk,�k � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' REFINE bisect elements in �k and the neighboring elements to form a conforming �k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' k = k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' end uJ = uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pJ = pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' �J = �k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' They consist of the element residuals and face jump residuals as R1(vh,qh)|τ := qh|τ − µ∇ × vh|τ, R2(vh,qh)|τ := f |τ − (∇ × qh + κvh)|τ, R3(vh)|τ := ∇ · ( f |τ − κvh|τ), J1(qh)|f := [[qh]], J2(vh)|f := [[(f − κvh)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' where hf denote the diameter of the circumcircle of f , and hτ ≈ hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any set � ′ h ⊆ �h, the error indicator is defined as η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='� ′ h ) = � τ∈� ′ h η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure MARK We use the Dörfler mark which was proposed by Dörfler [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Set marking parameter θ ∈ (0,1), the module MARK outputs a subset of marked elements �k ⊂ �k with minimal cardinality, such that η2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≥ θη2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure REFINE Our implementation of REFINE uses the longest edge bisection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' A detailed intro- duction about the longest edge bisection strategy was provided in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' To avoid confusion, the relationship between the two tetrahedral meshes �h and �H that are nested into each 8 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' other is defined as: �h is the new mesh division of �H after one cycle of the above cycle process, abbreviated as �H ≤ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG algorithm In this section, we establish the upper bound estimate of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Subsequently, we demonstrate that the sum of the energy error and the error estimator between two consec- utive adaptive loops is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Finally, we proof that the AMIPDG is convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The upper bound estimate of the error In this subsection, before establishing the reliability of a posteriori error estimator, we need to define the corresponding DG norm, for any (v,q) ∈ U × Q and (vh,qh) ∈ Uh × Qh, ∥(v,q) − (vh,qh)∥2 DG := ∥q − qh∥2 L2(Ω) + ∥κ(v − vh)∥2 L2(Ω) + � τ∈�h ∥µ∇ × (v − vh)∥2 L2(τ) + � f ∈�h αh−1 f < [[vh]],[[vh]] >f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any v ∈ U and vh ∈ Uh, we have ∥[[vh]]∥2 L2(f ) = ∥[[(v − vh)]]∥2 L2(f ), ∀f ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In fact, v ∈ U implies that [[v]]|f = 0 (see Chapter 5 of [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We summarize our main result in this subsection as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U×Q and (uh, ph) ∈ Uh ×Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) be the residual error indicator of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we have the following estimate ∥(u, p) − (uh, ph)∥2 DG ≤ C1η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) where the constant C1 depending on the shape regularity of mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), similarly to [4], we introduce the nonconformity of the MSIPDG method results in some consistency error: ζ := min ˜vh∈U � � τ∈�h (∥uh − ˜vh∥2 L2(τ) + ∥∇ × (uh − ˜vh)∥2 L2(τ)) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3) We denote that ˜uh ∈ U is the unique minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3), namely ˜ζ = � � τ∈�h (∥uh − ˜uh∥2 L2(τ) + ∥∇ × (uh − ˜uh)∥2 L2(τ)) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4) Convergence of AMIPDG methods for H(cur l)-elliptic problems 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively, let ˜uh be the unique minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3), then ∥(u − ˜uh, p − ph)∥U×Q = (∥u − ˜uh∥2 curl,Ω + ∥p − ph∥2 0)1/2 ≲ ∥˜ℓ1∥Q∗ + ∥˜ℓ2∥U∗, where the residuals ˜ℓ1 ∈ Q∗ and ˜ℓ2 ∈ U∗ defined by ˜ℓ1(q) = ℓ1(q) − a(ph,q) + b(˜uh,q), ∀q ∈ Q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) ˜ℓ2(v) = ℓ2(v) − d(v, ph) − c(˜uh, v), ∀v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any q1,q2,q ∈ Q and any v1, v2, v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' we have the following property by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9) (� (v1 + v2,q1 + q2))(v,q) = a(q1 + q2,q) − b(v1 + v2,q) + d(v,q1 + q2) + c(v1 + v2, v) = a(q1,q) − b(v1,q) + d(v,q1) + c(v1, v) +a(q2,q) − b(v2,q) + d(v,q2) + c(v2, v) = (� (v1,q1))(v,q) + (� (v2,q2))(v,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, (� (u − ˜uh, p − ph))(v,q) = (� (u, p))(v,q) − (� (˜uh, ph))(v,q) = (ℓ1(q) + ℓ2(v)) − (a(ph,q) − b(˜uh,q) + d(v, ph) + c(˜uh, v)) = ˜ℓ1(q) + ˜ℓ2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In fact that (u − ˜uh, p −ph) ∈ U ×Q and combining the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 can concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we will provide upper bounds for ∥˜ℓ1∥Q∗ and ∥˜ℓ2∥U∗ in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), and ˜uh be the unique minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we get the estimate of the linear functional ˜ℓ1 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) as following ∥˜ℓ1∥Q∗ ≲ � � τ∈�h ∥R1(uh, ph)∥2 L2(τ) �1/2 + � � τ∈�h ∥∇ × (˜uh − uh)∥2 L2(τ) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any q ∈ Q, by the definition of ˜ℓ1, we have ˜ℓ1(q) = � τ∈�h � τ � (µ∇ × uh − ph) + µ∇ × (˜uh − uh) � qdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 10 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then applying the Hölder inequality and the Cauchy-Schwarz inequality, |˜ℓ1(q)| ≤ � τ∈�h ∥µ∇ × uh − ph∥L2(τ)∥q∥L2(Ω) + � τ∈�h ∥µ∇ × (˜uh − uh)∥L2(τ)∥q∥L2(Ω) ≲ �� � τ∈�h ∥R1(uh, ph)∥2 L2(τ) �1/2 + � � τ∈�h ∥∇ × (˜uh − uh)∥2 L2(τ) �1/2� ∥q∥L2(Ω), conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Before estimating the term ∥˜ℓ2∥U∗, we need to introduce the following interpolation operator with the corresponding approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [ [19], Theorem 1] Let Nd1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) be the lowest order edge elements of Nédélec first family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exists an operator Πh : H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω) → Nd1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) with the following properties: For every v ∈ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω), there exist ϕ ∈ H1 0(Ω) and z ∈ H1 0(Ω), such that v − Πhv = ∇ϕ + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' And for any τ ∈ �h and f ∈ �h, we have h−1 τ ∥ϕ∥L2(τ) + ∥∇ϕ∥L2(τ) ≲ hτ∥v∥L2(Ωτ), h−1 τ ∥z∥L2(τ) + ∥∇z∥L2(τ) ≲ hτ∥∇ × v∥L2(Ωτ), where Ωτ = � f ∈τ Ωf , Ωf = {τ′ ∈ �h, f ∈ τ′}, and the constants depending on the shape regularity of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), and ˜uh be the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then the linear functional ˜ℓ2 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) satisfies the following estimate ∥˜ℓ2∥U∗ ≲ � � τ∈� h2 τ(∥R2(uh, ph)∥2 L2(τ) + ∥R2(uh)∥2 L2(τ)) + � f ∈� hf (∥J1(ph)∥2 L2(f ) + ∥J2(uh)∥2 L2(f )) + � τ∈� ∥uh − ˜uh∥2 L2(τ) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any v ∈ U and Πh given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3, we have v − Πhv = ∇ϕ + z, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9) where ϕ ∈ H1 0(Ω) and z ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' According to linearity of the operator ˜ℓ2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9), we have ˜ℓ2(v) = ˜ℓ2(Πhv) + ˜ℓ2(v − Πhv) = ˜ℓ2(Πhv) + ˜ℓ2(∇ϕ) + ˜ℓ2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10) We will next estimate the three terms on the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 11 For the first term ˜ℓ2(Πhv) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), using the definition of ˜ℓ2, we have ˜ℓ2(Πhv) = ℓ2(Πhv) − d(Πhv, ph) − c(˜uh,Πhv) = ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) + c(uh − ˜uh,Πhv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that Πhv ∈ Nd1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ⊆ Uh has zero jumps, and combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), we have ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) = ℓ2,h(Πhv) − dh(Πhv, ph) − ch(uh,Πhv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, we have ˜ℓ2(Πhv) = c(vh − ˜uh,Πhv) = c(vh − ˜uh, v) + c(vh − ˜uh,Πhv − v) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥Πhv − v∥0,�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9), triangle inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3, we get ˜ℓ2(Πhv) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ + z∥0,�h) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ∥0,�h + ∥z∥0,�h) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h∥v∥curl,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='11) For the second term ˜ℓ2(∇ϕ) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), using the definition of ˜ℓ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) and the fact ∇ × ∇ϕ = 0, which implies ˜ℓ2(∇ϕ) = ℓ2(∇ϕ) − d(∇ϕ, ph) − c(˜uh,∇ϕ) = ( f ,∇ϕ) − (∇ × ∇ϕ, ph) − (κ˜uh,∇ϕ) = ( f ,∇ϕ) − (κ˜uh,∇ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='12) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='12) and Green’s formula, we have ˜ℓ2(∇ϕ) = ( f ,∇ϕ) − (κuh,∇ϕ) + (κ(uh − ˜uh),∇ϕ) ≤ � τ∈�h (R3(uh),ϕ)0,τ + � f ∈�h < J2(uh),ϕ >0,f +(κ(uh − ˜uh),∇ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Applying the Cauchy-Schwarz inequality, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3 and trace inequality, we have ˜ℓ2(∇ϕ) ≤ � � τ∈�h h2 τ∥R3(uh)∥2 0,τ + � f ∈�h hf ∥J2(uh)∥2 0,f + � τ∈�h ∥κ∥0,∞∥uh − ˜uh∥2 0,τ �1/2 ∥v∥curl,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13) 12 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, for the third term ˜ℓ2(z) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), we have ˜ℓ2(z) = ( f , z) − (∇ × z, ph) − (κ˜uh, z) = ( f , z) − (∇ × z, ph) − (κuh, z) + (κ(uh − ˜uh), z) ≤ � � τ∈�h h2 τ∥R2(uh, ph)∥2 0,τ + � f ∈�h hf ∥J1(ph)∥2 0,f + � τ∈�h ∥κ∥0,∞∥uh − ˜uh∥2 0,τ �1/2 ∥v∥curl,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Notice that both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8) are related to the terms � τ∈�h ∥∇ × (˜uh − uh)∥2 L2(τ) and � τ∈� ∥uh − ˜uh∥2 L2(τ), which are a part of ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, we prove upper bounds for ˜ζ in the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) and ˜ζ be consistency error of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), we have ˜ζ2 ≲ η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any vh ∈ Uh, there exit an interpolation operator �h : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) → Uc h, such that(see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5 of [11]) ∥vh − �hvh∥2 L2(Ω) ≲ � f ∈�h hf ∥[[vh]]∥2 L2(f ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16) � τ∈�h ∥∇ × (vh − �hvh)∥2 L2(τ) ≲ � f ∈�h h−1 f ∥[[vh]]∥2 L2(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17) Then, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17), and the fact hf < 1, we get ˜ζ2 = � τ∈�h (∥uh − ˜uh∥2 L2(τ) + ∥∇ × (uh − ˜uh)∥2 L2(τ)) ≤ � τ∈�h (∥uh − �huh∥2 L2(τ) + ∥∇ × (uh − �huh)∥2 L2(τ)) ≲ � f ∈�h hf ∥[[uh]]∥2 L2(f ) + � f ∈�h h−1 f ∥[[uh]]∥2 L2(f ) ≲ � f ∈�h h−1 f ∥[[uh]]∥2 L2(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18) Noting that (uh, ph) ∈ Uh × Qh is the solution of discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)- (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we know that uh is the solution of discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Hence, we have ( see Lemma 5 of [20]) α∥h−1/2 f [[uh]]∥L2(�h) ≲ η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19) Convergence of AMIPDG methods for H(cur l)-elliptic problems 13 At last, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19), we have ˜ζ2 ≲ η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Combining Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5, we will prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1:] By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1), the triangle inequality, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19), we get ∥(u, p) − (uh, ph)∥2 DG ≲ ∥p − ph∥2 L2(Ω) + ∥κ(u − uh)∥2 L2(Ω) + � τ∈�h ∥∇ × µ(u − uh)∥2 L2(τ) + � f ∈�h αh−1 f < [[uh]],[[uh]] >f ≲ ∥p − ph∥2 L2(Ω) + ∥u − ˜uh∥2 cur l,Ω + ˜ζ2 + � f ∈�h αh−1 f < [[uh]],[[uh]] >f = ∥(u − ˜uh, p − ph)∥U×Q + ˜ζ2 + � f ∈�h αh−1 f < [[uh]],[[uh]] >f ≲ ∥˜ℓ1∥2 Q∗ + ∥˜ℓ2∥2 U∗ + ˜ζ2 + � f ∈�h αh−1 f < [[uh]],[[uh]] >f ≤ C1η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The error reduces on two successive meshes For convenience, for any v ∈ U and vh ∈ Uh, we denote ∥|v − vh|∥2 h = ∥κ(v − vh)∥2 L2(Ω) + � τ∈�h ∥∇ × µ(v − vh)∥2 L2(τ) + � f ∈�h αh−1 f < [[vh]],[[vh]] >f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='20) Let Uc h be the H(cur l) conforming subspace of Uh given by Uc h := Uh ∩ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, there is a subspace U⊥ h which can orthogonally decompose Uh under L2 inner product such that Uh := Uc h ⊕ U⊥ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Especially, if (uh, ph) ∈ Uh × Qh is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), then we have ∥|u⊥ h |∥2 h ≲ α � f ∈∂ τ ∥h−1/2 f [[uh]]∥2 L2(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21) In fact, from the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, notice that uh satisfies the IPDG scheme of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15), and ac- cording to Lemma 2 in [20], we can obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 14 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In order to easily estimate the jump term of face �h, we need to introduce the lifting operators and the corresponding stability estimates, more details are referenced to Propo- sition 12 in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let �h : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) → Uh be the lifting operators, which satisfies the following equality � Ω �h(v) · wdx =< [[v]],{{w}} >�h, ∀w ∈ Uh, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='22) and ∥�h(v)∥L2(Ω) ≤ C� ∥h−1/2[[v]]∥L2(�h), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='23) where the constant C� depending on the shape regularity of mesh �h and the degree of polynomial l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively, we have ∥p − ph∥L2(Ω) ≲ ∥∇ × (u − uh)∥L2(Ω) + η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='24) ∥ph − pH∥L2(Ω) ≲ ∥∇ × (uh − uH)∥L2(Ω) + � η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that Qh ⊆ Q, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1), the definition of R1(uh, ph) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16), we have ∥p − ph∥L2(�h) ≤ sup ∀q∈Q (p − ph,q)�h ∥q∥L2(�h) = sup ∀q∈Q (µ∇ × u,q)�h − � R1(uh, ph) + µ∇ × uh,q � �h ∥q∥L2(�h) ≤ sup ∀q∈Q (µ∇ × (u − uh),q)�h − � R1(uh, ph),q � �h ∥q∥L2(�h) ≲ ∥∇ × (u − uh)∥L2(�h) + η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, using the definition of R1(uh, ph), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' and the fact [[uh]] = Convergence of AMIPDG methods for H(cur l)-elliptic problems 15 [[uc h + u⊥ h ]] = [[u⊥ h ]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' we have ∥ph − pH∥L2(�h) ≤ sup ∀qh∈Qh (ph − pH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h ∥qh∥L2(�h) ≤ sup ∀qh∈Qh (ph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h − � R1(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH) + µ∇ × uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh � �h ∥qh∥L2(�h) ≤ sup ∀qh∈Qh (µ∇ × uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h+ < {{qh}},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[µuh]] >�h − � R1(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH) + µ∇ × uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh � �h ∥qh∥L2(�h) = sup ∀qh∈Qh (µ∇ × (uh − uH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h+ < {{qh}},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[µuh]] >�h − � R1(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh � �h ∥qh∥L2(�h) ≲ ∥∇ × (uh − uH)∥L2(�h) + ∥h−1/2 τ [[uh]]∥L2(�h) + η(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) ≲ ∥∇ × (uh − uH)∥L2(�h) + C� ∥h−1/2 τ [[u⊥ h ]]∥L2(�h) + η(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) ≲ ∥∇ × (uh − uH)∥L2(τ) + � η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that ∥(u, p)−(uh, ph)∥2 DG+η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) and ∥|u−uh|∥2 h+η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In fact, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='24), we first know that ∥(u, p) − (uh, ph)∥2 DG + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = ∥|u − uh|∥2 h + ∥p − ph∥2 L2(�h) + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≲ ∥|u − uh|∥2 h + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Secondly, it is shown by the definition of ∥ · ∥DG ∥|u − uh|∥2 h + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ ∥(u, p) − (uh, ph)∥2 DG + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, we next only need to consider the convergence of ∥|u − uh|∥2 h + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We first show that the error plus some quantity reduces with a fixed factor on two successive meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given f ∈ L2(Ω) and two tetrahedral mesh �h and �H, where �H ≤ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U × Q be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2), and (uh, ph) ∈ Uh × Qh, (uH, pH) ∈ UH × QH be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exit two constants δ1,δ2 ∈ (0,1), such that ∥|u − uh|∥2 h ≤ (1 + δ1)∥|u − uH|∥2 H − 1 − δ2 2 ∥|uh − uH|∥2 h + C3 δ1δ2α � η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='26) where C3 depending on the C� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 16 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Choosing that q = ∇ × v, and subtracting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2), we obtain (κu, v) + (µ∇ × u,∇ × v) = ( f , v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='27) Subtracting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='27) with v = vh = uc h − uc H, and using [[uc h − uc H]] = 0, we have (κ(u − uh), uc h − uc H)0,�h + (µ∇ × (u − uh),∇ × (uc h − uc H))0,�h + < [[uh]],{{µ∇ × (uc h − uc H)}} >�h= 0, which leads to (κ(u − uh), uc h − uc H)0,�h + (µ∇ × (u − uh),∇ × (uc h − uc H))0,�h = − < [[uh]],{{µuc h − uc H}} >�h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='28) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='23), we have < [[uh]],{{∇ × (uc h − uc H)}} >�h = (�h(uh),∇ × (uc h − uc H))0,�h ≤ C� ∥h−1/2[[uh]]∥0,�h∥∇ × (uc h − uc H)∥0,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='29) Let uh = uc h + u⊥ h and uH = uc H + u⊥ H, we have uh + uc H − uc h = uH − u⊥ H + u⊥ h , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='30) where uc H ∈ Uc H, uc h ∈ Uc h, u⊥ H ∈ U⊥ H, u⊥ h ∈ U⊥ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='30), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='28), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='29) and Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' we get ∥|u − uh|∥2 h = ∥κ(u − uh)∥2 L2(Ω) + ∥∇ × µ(u − uh)∥2 L2(Ω) + � f ∈�h αh−1 f < [[(u − uh)]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[u − uh]] >�h = ∥|u − uh − uc H + uc h|∥2 h − ∥|uc h − uc H|∥2 h − 2(κ(u − uh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' uc h − uc H)0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h −2(µ∇ × (u − uh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='∇ × (uc h − uc H))0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h −2 � f ∈�h αh−1 f < [[(u − uh)]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[uc h − uc H]] > ≲ ∥|u − uH|∥2 H + 2∥|u − uH|∥H∥|u⊥ h − u⊥ H|∥h + ∥|u⊥ h − u⊥ H|∥2 h − ∥|uc h − uc H|∥2 h +2∥h−1/2[[uh]]∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h∥∇ × (uc h − uc H)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h ≤ (1 + δ1)∥|u − uH|∥2 H + (1 + 1 δ1 )∥|u⊥ h − u⊥ H|∥2 h − (1 − ˆδ2C� )∥|uc h − uc H|∥2 h +C� ˆδ2 ∥h−1/2[[uh]]∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h = (1 + δ1)∥|u − uH|∥2 H + (1 + 1 δ1 )∥|u⊥ h − u⊥ H|∥2 h − (1 − δ2)∥|uc h − uc H|∥2 h + C2 � δ2 ∥h−1/2[[uh]]∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 17 where δ2 = ˆδ2C� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Using uc H = uH − u⊥ H, uc h = uh − u⊥ h , triangle inequality and average inequality, we have ∥|uc h − uc H|∥2 h ≥ 1 2∥|uh − uH|∥2 h − ∥|u⊥ h − u⊥ H|∥2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' By triangle inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21), we obtain ∥|u⊥ h − u⊥ H|∥2 h ≤ 2(∥|u⊥ h |∥2 h + ∥|u⊥ H|∥2 H) ≤ 2α∥h−1/2[[u⊥ h ]]∥2 0,�h + 2α∥h−1/2[[u⊥ H]]∥2 0,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Combining [[uH]] = [[u⊥ H + uc H]] = [[u⊥ H]] and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19), we have ∥|u − uh|∥2 h ≤ (1 + δ1)∥|u − uH|∥2 H − 1 − δ2 2 ∥|uh − uH|∥2 h + C3 δ1δ2α � η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Contraction of the error estimator In this subsection, we prove the reduction of error indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let us first consider the effect of changing the finite element function used in the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given f ∈ L2(Ω) and two tetrahedral mesh �h, �H with �H ≤ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (vh,qh) ∈ Uh × Qh and (v H,q H) ∈ UH × QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any ε > 0, we have η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ (1 + ε)η2(v H,q H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + Cε∥(vh,qh) − (v H,q H)∥2 DG, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='31) where Cε depending on the ε, and the mesh size h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any τ∗ ∈ �h, we will discuss each of the five components of the mark η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Firstly, using the definition of R1(vh,qh) and triangle inequality, we have ∥R1(vh,qh)∥L2(τ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='32) = ∥qh − µ∇ × vh∥L2(τ∗) = ∥qh − q H + µ∇ × (v H − vh) + q H − µ∇ × v H∥L2(τ∗) ≲ ∥q H − ∇ × v H∥L2(τ∗) + ∥qh − q H∥L2(τ∗) + ∥∇ × (vh − v H)∥L2(τ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Secondly, using the definition of R2(vh,qh), triangle inequality and inverse inequality, we get hτ∗∥R2(vh,qh)∥L2(τ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='33) = hτ∗(∥ f − ∇ × qh − κvh∥L2(τ∗)) = hτ∗(∥ f − ∇ × (qh − q H) − κ(vh − v H) − ∇ × q H − κv H∥L2(τ∗)) ≤ hτ∗(∥ f − ∇ × q H − κv H∥L2(τ∗) + ∥∇ × (qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗)) ≲ hτ∗(∥R2(v H,q H)∥L2(τ∗) + h−1 τ∗ ∥(qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗)) ≲ hτ∗∥R2(v H,q H)∥L2(τ∗) + ∥(qh − q H)∥L2(τ∗) + hτ∗∥κ(vh − v H)∥L2(τ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 18 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, using the definition of R3(vh), triangle inequality and inverse inequality, we get hτ∗∥R3(vh)∥L2(τ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='34) = hτ∗∥∇ · ( f − κvh)∥L2(τ∗) = hτ∗∥∇ · ( f − κv H + κv H − κvh)∥L2(τ∗) ≤ hτ∗(∥∇ · ( f − κv H)∥L2(τ∗) + ∥∇ · κ(v H − vh)∥L2(τ∗)) ≲ hτ∗(∥R3(v H)∥L2(τ∗) + h−1 τ∗ ∥κ(v H − vh)∥L2(τ∗)) ≲ hτ∗∥R3(v H)∥L2(τ∗) + ∥κ(v H − vh)∥L2(τ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we discuss the jump J1(qh) and J2(vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any f ∈ �(�h), we let f = τ1 ∗ � τ2 ∗ with τ1 ∗,τ2 ∗ ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Furthermore, using the definition of J1(qh), triangle inequality and trace inequality, we have h1/2 f ∥J1(qh)∥L2(f ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='35) = h1/2 f ∥[[qh]]∥L2(f ) = h1/2 f ∥[[q H + qh − q H]]∥L2(f ) ≤ h1/2 f (∥[[q H]]∥L2(f ) + ∥[[qh − q H]]∥L2(f )) ≤ h1/2 f ∥[[q H]]∥L2(f ) + h1/2 f ∥(qh − q H)|τ1 ∗∥L2(f ) + h1/2 f ∥(qh − q H)|τ2 ∗∥L2(f ) ≲ h1/2 f ∥J1(q H)∥L2(f ) + ∥(qh − q H)∥L2(τ1 ∗∪τ2 ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, using the definition of J2(vh), triangle inequality and trace inequality, we have h1/2 f ∥J2(vh)∥L2(f ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='36) = h1/2 f ∥[[( f − κvh)]]∥L2(f ) = h1/2 f ∥[[( f − κv H + κv H − κvh)]]∥L2(f ) ≤ h1/2 f (∥[[(f − κv H)]]∥L2(f ) + ∥[[κ(v H − vh)]]∥L2(f )) ≤ h1/2 f ∥J2(v H)∥L2(f ) + h1/2 f (∥κ(v H − vh)|τ1 ∗∥L2(f ) + ∥κ(v H − vh)|τ2 ∗∥L2(f )) ≲ h1/2 f ∥J2(v H)∥L2(f ) + ∥κv H − κvh∥L2(τ1 ∗∪τ2 ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Finally, the desired result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='31) is obtained by combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='32)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='36), Young’s in- equality and the shape regularity of mesh �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We then prove the contraction of the error estimator under the assumptions on the problem of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uH, pH) ∈ UH × QH be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), and ��H−→�h = �H \\ (�h ∩ �H) be the Convergence of AMIPDG methods for H(cur l)-elliptic problems 19 set of all element refined into �h on �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, there is a constant λ ∈ (0,1) independent of mesh size, such that η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Assume that the tetrahedral mesh τ ∈ �H is divided into two new tetrahedral mesh τ1 ∗ and τ2 ∗ with equal volumes, where τ1 ∗,τ2 ∗ ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, h3 τ1 ∗ = |τ1 ∗| = |τ2 ∗| = h3 τ2 ∗ = 2−1h3 τ by the shape regularity of mesh, which implies hτ1 ∗ = hτ2 ∗ = 2−1/3hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, we have ∥R1(uH, pH)∥2 L2(τ1 ∗) + ∥R1(uH, pH)∥2 L2(τ2 ∗) ≤ ∥R1(uH, pH)∥2 L2(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='38) and h2 τ1 ∗(∥R2(uH, pH)∥2 L2(τ1 ∗) + ∥R3(uH)∥2 L2(τ1 ∗)) + h2 τ2 ∗(∥R2(uH, pH)∥2 L2(τ2 ∗) + ∥R3(uH)∥2 L2(τ2 ∗)) ≤ 2−2/3h2 τ(∥R2(uH, pH)∥2 L2(τ) + ∥R3(uH)∥2 L2(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='39) For any f ∈ ∂ (τ1 ∗ ∪ τ2 ∗), which can be divided into three parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (1) For the first part, there are two of the faces are constant and belong to τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2) For the second part, there are two new faces that overlap and are used to divide the mesh τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Since (uH, ph) ∈ UH × QH is a continuous polynomial in the region τ, it follows that the value of [[ph]] and [[( f − κuH)]] on this surface is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3) For the third part, there are four faces that are obtained by dividing the two faces in the τ into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Furthermore, we obtain η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ1 ∗) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ2 ∗) ≤ γη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='40) where constant γ ∈ (0,1) independent of mesh τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, since ��H→�h represents the part of the set in the tetrahedral set �H that will be used to be refined, it follows that ��H→�h ⊂ �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let ��H→�h denote the part of the cell set that has been refined in the tetrahedral set �H, we have ��h→�H ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Obviously, �H \\��H→�h = �h \\��H→�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then combining the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='40), and the marking strategy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18), we have η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h \\ ��H→�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H \\ ��H→�h) + γη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) + (γ − 1)η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h), where λ = 1 − γ ∈ (0,1) independent of mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Now, we combine the Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9 to prove the reduction of error indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 20 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given a constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh and (uH, pH) ∈ UH × QH be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any ε > 0 and λ ∈ (0,1), we have (1 − Cε α )η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ (1 + ε + Cε α )η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − (1 + ε)λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) + Cε∥|uh − uH|∥2 h, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='41) where constant Cε depending on the ε and mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Using the Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9, we have η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ (1 + ε) � η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) � +Cε∥(uh, ph) − (uH, pH)∥2 DG ≤ (1 + ε) � η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) � +Cε∥|uh − uH|∥2 h + ∥ph − pH∥2 L2(Ω) ≤ (1 + ε) � η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) � +Cε∥|uh − uH|∥2 h + Cε α � η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � , which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence result Now, we proved that the sum of the norm of the error and the scaled error indicator is attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For a given θ ∈ (0,1),let {�k,Uk,Qk, uk, pk,η(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k)}k≥0 be the se- quence of meshes, Mixed discrete solution (defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14)), and the estimate in- dicator produced by the AMIPDG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exist constants ρ > 0, δ ∈ (0,1), which depend on marking parameter θ and the shape regularity of the initial mesh �0, such that ∥|u − uk+1|∥2 k+1 + ρη2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ δ � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Setting �ρ = 1−δ2 2Cε , then multiply the both sides of the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='41) inequality by �ρ, we get �ρ(1 − Cε α )η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ �ρ(1 + ε + Cε α )η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) − �ρ(1 + ε)λη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��k→�k+1) +1 − δ2 2 ∥|uk+1 − uk|∥2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='42) Convergence of AMIPDG methods for H(cur l)-elliptic problems 21 Next, by the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='42), we have ∥|u − uk+1|∥2 k+1 + �ρ(1 − Cε α )η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ (1 + δ1)∥|u − uk|∥2 k + C3 δ1δ2α � η2(v k+1,q k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) + η2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � +�ρ(1 + ε + Cε α )η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) − �ρ(1 + ε)λη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��k→�k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='43) First move the term and then according to Dörfler marking strategy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18), the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 and ∥| · |∥h ≤ ∥ · ∥DG, we know −η2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��k→�k+1) ≤ −θη2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k), then ∥|u − uk+1|∥2 k+1 + �ρ(1 − Cε α − C3 �ρδ1δ2α)η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ (1 + δ1)∥|u − uk|∥2 k − �ρ(1 + ε)λθ 2 η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) +�ρ � 1 + ε + Cε α + C3 �ρδ1δ2α − (1 + ε)λθ 2 � η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≤ (1 + δ1 − �ρ(1 + ε)λθC−1 1 2 )∥|u − uk|∥2 k +�ρ � 1 + ε + Cε α + C3 �ρδ1δ2α − (1 + ε)λθ 2 � η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For convenience, denote β1 = 1 − Cε α − C3 �ρδ1δ2α, β2 = 1 + δ1 − �ρ(1 + ε)λθC−1 1 2 , β3 = (1 + ε)(1 − λθ 2 ) + Cε α + C3 �ρδ1δ2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus ∥|u − uk+1|∥2 k+1 + �ρβ1η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ β2∥|u − uk|∥2 k + �ρβ3η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we firstly choose δ1 = �ρ(1+ε)λθC−1 1 4 , then select the appropriate δ2 to make �ρ = 1−δ2 2Cε smaller to ensure 0 < δ1 < 1, Secondly, we let ε > 0 and (1 + ε)(1 − λθ 2 ) = 1 − λθ 4 ( λθ ∈ (0,1)), therefore β2 = 1 − δ1 ∈ (0,1), (1 + ε)(1 − λθ 2 ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Furthermore, we choose a sufficiently large penalty parameter α such that β1 > β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 22 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Finally, there is a constant δ = max{β2, β1 β3 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, we let ρ = �ρβ1, and obtain ∥|u − uk+1|∥2 k+1 + ρη2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ δ � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we have ∥(u, p) − (uk, pk)∥2 DG + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≤ δk �Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' where �Cδ = C � ∥(u, p) − (u0, p0)∥2 DG + ρη2(u0, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, for a given precision, the AMIPDG method will terminate after a finite number of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Using the Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we have ∥(u, p) − (uk, pk)∥2 DG + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≤ C � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � ≤ δk �Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numerical experiments In this section, we test some numerical experiments to show the efficiency and the robustness of AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We carry out these numerical experiments by using the MATLAB software package iFEM [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we take p = ∇ × u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1, we discuss the influence of the penalty parameter α on the error in ∥ · ∥DG norm, and observe the dependency of the condition number of stiffness matrix on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2): u = � � x(x − 1)y(y − 1)z(z − 1) sin(πx)sin(πy)sin(πz) (1 − ex)(1 − ex−1)(1 − e y)(1 − e y−1)(1 − ez)(1 − ez−1) � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' It is easy to see that the solution u satisfies the boundary condition u × n = 0 on ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In this example, we get a uniform mesh by partitioning the x−, y− and z−axes into equally distributed M(M ≥ 2) subintervals, and then dividing one cube into six tetrahe- drons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let h = 1/M be mesh sizes for different tetrahedrons meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We fixed mesh with h = 1/4 and report the error estimates in ∥ · ∥DG norm and condition number of stiffness matrices for different penalty parameters α = 1,10,100,500 and 1000 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We note that ∥u − uh∥0 increases at first and then decreases as the penalty parameter α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 23 Table 1: The error in ∥ · ∥DG norms and condition number of stiffness matrices with h = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' α 1 10 100 500 1000 ∥ � p − ph, u − uh � ∥DG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='949e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='133e-00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='614e-01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='649e-01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='659e-01 Cond 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='235e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='021e+04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='959e+05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='995e+06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='150e+06 The condition numbers of stiffness matrices increase with the increase of penalty parame- ters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' As a way to balance, in the following numerical tests, we always choose α = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that we only consider uniform meshes in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next we test adaptive meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) u = � � � x(x−1)y(y−1)z(z−1) x2+y2+z2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='001 x(x−1)y(y−1)z(z−1) x2+y2+z2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='001 − x(x−1)y(y−1)z(z−1) x2+y2+z2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='001 � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Note that the solution u satisfies the condition u × n = 0 on ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The right of Figure 1 shows an adaptively refined mesh with marking parameter- θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7 after k = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The grid is locally refined near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Figure 1: Left: the initial mesh with 1152 DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Right: the adaptive mesh(θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) with 181104 DoFs after 18 refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The Figure 2 shows the curves of log N−logη � uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k � for parameters θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The curves indicate the convergence and the quasi-optimality of the adaptive algorithm AMIPDG of η � uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Acknowledgment The first author is supported by the East China University of Technology (DHBK2019209) and Jiangxi Province Education Department (GJJ200755).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The second, third and fourth authors are supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 12071160).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The third author is also supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 11901212).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 24 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Figure 2: Quasi optimality of the AMIPDG of the error η � uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k � with different marking parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' AYUSO DE DIOS, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' HIPTMAIR AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' PAGLIANTINI, Auxiliary space preconditioners for SIP-DG discretizations of H(curl)-elliptic problems with discontinuous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 37(2017), pp, 646-686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' BONITO AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' NOCHETTO, Quasi-optimal convergence rate of an adaptive discontin- uous Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 48(2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 734–771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [3] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' CARSTENSEN AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' HOPPE, Unified framework for an a posteriori error analysis of non-standard finite element approximations of H(cur l)-elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 17(2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 27–44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [4] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' CARSTENSEN, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' HOPPR, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SHARMA AND T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' WARBURTON, Adaptive hybridized in- terior penalty discontinuous galerkin methods for H(cur l)–elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 4(2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 13–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' CARSTENSEN AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MA, Adaptive mixed finite element methods for non-self-adjoint indefinite second-order elliptic pdes with optimal rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 59(2021), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 955–982.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [6] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' CHEN, iFEM: an innovative finite element method package in MATLAB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Technical report, University of California at Irvine (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [7] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' CHEN, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' HOLST AND J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' XU, Convergence and optimality of adaptive mixed finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 78(2009), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 35–53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [8] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' CHUNG, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' YUEN AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' ZHONG, A-posteriori error analysis for a staggered dis- continuous Galerkin discretization of the time-harmonic Maxwell’s equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 237(2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 613–631.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' DÖRFLER, A convergent adaptive algorithm for Poisson’s equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 33(1996), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 1106–1124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' DU AND X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' XIE, Convergence of an adaptive mixed finite element method for Rate of convergence is CN-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='33 10 adaptive refiniment = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3 adaptive refiniment =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5 adaptive refiniment =0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7 CN-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='33 104 105 Number of unknownsConvergence of AMIPDG methods for H(cur l)-elliptic problems 25 convection-diffusion-reaction equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' China Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 58(2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 1327–1348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [11] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' HOUSTON, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' PERUGIA, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SCHNEEBELI ADN D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SCHÖTZAU, Interior penalty method for the indefinite time-harmonic Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 100(2005), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 485–518.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [12] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' JIANG, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' LIU, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' TANG AND Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' LIU, Mixed finite element method for 2D vector Maxwell’s eigenvalue problem in anisotropic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Progress In Electromagnetics Research 148(2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 159–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' JOG AND A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' NANDY, Mixed finite elements for electromagnetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 68(2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 887–902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [14] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' KIKUCHI, Mixed and penalty formulations for finite element analysis of an eigenvalue problem in electromagnetism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Engrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 64(1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 509–521.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [15] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' LIU, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' TOBÓN, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' TANG AND Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' LIU, Mixed spectral element method for 2D Maxwell’s eigenvalue problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 17(2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 458–486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [16] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MONK, Finite Element Methods for Maxwell Equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numerical Mathematics and Scientific Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Oxford University Press, Oxford(2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' NÉDÉLEC, Mixed finite elements in �3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 35(1980), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 315–341.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [18] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' PERUGIA, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SCHÖTZAU AND P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MONK, Stabilized interior penalty methods for the time- harmonic Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Methods Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 191(2002), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 4675–4697.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' SCHÖBERL, A posteriori error estimates for Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 77(2008), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 633–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [20] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' XING AND L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' ZHONG, A posteriori error estimate of discontinuous Galerkin Method for H(curl)-elliptic problems (in Chinese).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Journal of South China Normal University (Natural Science Edition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 44(2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 18–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} diff --git a/0tAyT4oBgHgl3EQf1PlJ/content/tmp_files/2301.00731v1.pdf.txt b/0tAyT4oBgHgl3EQf1PlJ/content/tmp_files/2301.00731v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b833eb02b420a5b2a17018ee930811d7066ea088 --- /dev/null +++ b/0tAyT4oBgHgl3EQf1PlJ/content/tmp_files/2301.00731v1.pdf.txt @@ -0,0 +1,742 @@ +arXiv:2301.00731v1 [math.DS] 2 Jan 2023 +Feuerbach’s and Poncelet’s theorems meet in space +(On the occasion of their bicentennial) +E. A. Avksentyev +December 29, 2022 +Abstract +Three-dimensional analogues of the Feuerbach theorem are proposed in this paper. One of them +concerns some tetrahedron analogue of the Euler circle. Another one is pretty interesting «up-in-ex- +touch» construction. And the third one, it turns out, is closely related to Poncelet’s theorem. This is +very beautiful Grace’s theorem. It seems that this theorem is not widely known, and that no elementary +proof has been given. Such an elementary proof of the Grace theorem is obtained in this paper by using +properties of imaginary generators on a sphere and of isotropic tangents to a conic. An applying of the +Grace theorem leads to several corollaries. One of them is Laguerre’s theorem, which generalizes the +Euler-Chapple formulas. Further, we consider a spatial analog of Poncelet’s theorem. We prove that +the Grace spheres touch some fixed sphere under the Poncelet rotation of bicentric tetrahedron. Finaly, +going out from a plane into the third dimension, we obtain a new proof of Feuerbach’s theorem and +perhaps the shortest proof of Euler-Chapple formulas. +Введение +Данная работа посвящена двум знаменитым геометрическим теоремам, кажется никак не связанным +между собой, разве что они были опубликованны в один год двести лет назад [5, 14]. Приведем их +формулировки +Теорема (Feuerbach, 1822). Окружность девяти точек произвольного треугольника касается его +вписанной и трех вневписанных окружностей. +Теорема (Poncelet, 1822). Пусть для двух данных коник существует вписано-описанный в них +многоугольник. Тогда этот многоугольник может динамически «вращаться» около данных коник, +оставаясь вписано-описанным в них. +У обеих теорем есть масса обобщений, но пространственные аналоги, насколько нам известно, +имеются только у теоремы Понселе. Их довольно много (см., например, [6,8,9,15]) и среди них есть +множество замечательных, но малоизвестных результатов. +Задача трехмерного обобщения теоремы Фейербаха поставлена еще более ста лет назад в моно- +графии Кулиджа [2]: +«The geometry of the tetrahedron lags far behind that of the triangle... Is there an analogue +to Feuerbach’s theorem? Above all what corresponds to the Hart systems? ...These difficult +but important and interesting questions offer ample scope for serious work» (p. 247). +Теорема Фейербаха содержит в себе два удивительных геометрических факта. Первый состоит в +том, что четыре замечательные окружности треугольника – вписанная и три описанные – имеют об- +щую касательную окружность. Второй же заключается в том, что эта общая касательная окружность +является еще и окружностью девяти точек, которая и без того сама по себе замечательна. +Первая попытка найти аналог теоремы Фейербаха в пространстве приводит к вопросу: существу- +ет ли сфера, которая касалась бы вписанной и вневписанных сфер? +Но здесь нас ожидает первый «сюрприз»: у произвольного тетраэдра кроме обычных четырех +вневписанных сфер, аналогичных трем вневписанным сферам треугольника, существует еще три +дважды-вневписанные сферы или чердачные (от англ. «roof»), как они названы в [20] (см. также [21]). +Т.е., всего существует целых восемь сфер (см. рис. 1), касающихся граней тетраэдра! Назовем +их касательными сферами. Было бы слишком оптимистично ожидать, что все восемь касательных +1 + +Рис. 1: Восемь касательных сфер тетраэдра +сфер могли бы касаться одной сферы. И действительно, ответ на поставленный вопрос оказывается +отрицательным: в общем случае произвольного тетраэдра такой сферы не существует. +Проверить это очень легко: для этого достаточно рассмотреть лишь один пример подходящего +тетраэдра. И нет сомнений, что такой знаток геометрии как Кулидж хорошо знал, что такой сферы +в общем случае нет. Однако, он все-таки поставил вопрос поиска трехмерных аналогов теоремы +Фейербаха, находя его важным, интересным и открывающим «широкие возможности для серьезной +работы». +В каком же направлении искать тогда аналоги теоремы Фейербаха в пространстве? Кажется, +что осталась лишь задача описания частных случаев тетраэдров, у которых существует сфера, ка- +сающаяся внутренним или внешним образом пяти, шести, семи или всех восьми касательных сфер. +В работе [11] есть некоторое продвижение в этой задаче и для существования такой сферы получе- +ны аналитические условия в специальных связанных с тетраэдром пентасферических координатах. +К сожалению, эти условия весьма громоздкие и из них совершенно не ясно, существуют ли такие +тетраэдры и как они устроены. Таким образом, задача в такой постановке остается незакрытой. +Возникает еще идея поискать пространственный аналог теоремы Фейербаха в таком направле- +нии: существует ли окружность, действительная или мнимая, которая касалась бы всех восьми +касательных сфер? Кажется маловероятным, что ответ мог бы быть положительным, но задача пред- +ставляется интересной. +Оставив пока эти вопросы, мы приведем далее целых три трехмерных аналога теоремы Фейербаха. +Первый аналог, которую мы хотим предложить в § 1 в качестве трехмерного обобщения теоремы +Фейербаха, является довольно интересным фактом. У него очень простое доказательство, которое, +2 + +тем не менее, раскрывает связь этой конструкции с неевклидовой геометрией и приводит к трехмер- +ному обобщению окружности Эйлера. Поэтому из трех аналогов этот наиболее аутентичен. +Второй является очень красивой теоремой геометрии тетраэдра, открытой сто двадцать пять лет +назад, но, кажется, до сих пор малоизвестной. Ее единственное оригинальное доказательство столь +сложно, что есть целая статья с его реконструкцией. В § 2 мы получим элементарное доказательство +этой теоремы, в котором обнаружится ее связь с теоремой Понселе. Второй аналог выглядит наименее +аутентичным, но на наш взгляд, он ближе и роднее к теореме Фейербаха, чем другие два. +Третий аналог представляет из себя довольно интересную конструкцию касающихся сфер, кото- +рую мы назвали «up-in-ex-touch»-конструкция. Мы приведем ее в конце § 3, в котором мы также +получим, возможно, самое короткое доказательство формул Эйлера-Чаппла. +С помощью теоремы Грейса мы в §4 получим короткое и простое доказательство теоремы Лагерра, +обобщающей формулы Эйлера-Чаппла. §5 посвящен трехмерному аналогу формул Эйлера-Чаппла. +Далее в §6 мы рассмотрим пространственные аналоги теоремы Понселе. Мы покажем, что при +вращении Понселе вписано-вневписанного тетраэдра его сферы Грейса касаются некоторой фикси- +рованной сферы. +В конце, совершая «выход в пространство», мы дадим новое доказательство теоремы Фейербаха. +1 Первый аналог теоремы Фейербаха для тетраэдра +Итак, рассмотрим произвольный тетраэдр общего вида, у которого имеется восемь касательных сфер. +В качестве первого аналога теоремы Фейербаха для тетраэдра предлагаем следующую теорему. +Теорема 1.1. Существует четыре круговых конуса, каждый из которых касается всех восьми его +касательных сфер. +Доказательство. Рассмотрим сферу ζD с центром в вершине D тетраэдра ABCD и спроектируем +из центра D на сферу ζD все восемь касательных сфер. Их проекциями будут четыре окружности +на сфере ζD, поскольку каждая пара гомотетичных относительно D сфер спроектируются в одну и +ту же окружность. Эти четыре окружности касаются сторон сферического треугольника, стороны +которого являются проекциями плоскостей трехгранного угла при вершине D. По теореме Фейербаха +для сферического треугольника существует окружность, касающаяся этих четырех окружностей. +Конус с вершиной D, содержащий эту окружность, очевидно удовлетворяет утверждению теоремы. +Такой конус есть у каждой вершины. +✷ +Теорема Фейербаха в сферической геометрии, в той облегченной форме, которую мы использо- +вали в доказательстве, равносильна теореме Харта (см. [2]). Таким образом, в какой-то степени мы +ответили на оба вопроса Кулиджа, которые мы цитировали во введении. На самом деле, можно про- +двинуться еще дальше в этом направлении, если применить результат Акопяна [19], в котором он +нашел такие свойства окружности Харта, которые во многом аналогичны свойствам окружности +девяти точек. Хотя в [19] все утверждения формулируются для плоскости Лобачевского, но мы их +естественным образом адаптируем применительно к трехгранным углам нашего тетраэдра. +Избытком трехгранного угла называется величина, равная разнице между суммой его двух- +гранных углов и 180◦. Медиатором трехгранного угла назовем плоскость, содержащую его ребро +и делящую его на два трехгранных угла с равными избытками. При рассмотренной выше проекции +трехгранного угла на сферу медиатор переходит в сферическую чевиану, делящую пополам пло- +щадь соответственного треугольника (в [19] эта чевиана называется биссектором или биссекторным +отрезком). Три медиатора пересекаются по прямой, которую можно назвать псевдоцентроидалью, +поскольку ей соответствует псевдоцентроид сферического треугольника. +Четыре прямые из одного пучка назовем вписанной четверкой, если все они являются образую- +щими одного кругового конуса. Следующее утверждение является аналогом Леммы 5 из [19]. +3 + +Предложение 1.2. Пусть a, b, c – ребра трехгранного угла с вершиной D. Тогда существует един- +ственная тройка прямых ha, hb, hc, лежащих в плоскостях ⟨ab⟩, ⟨ac⟩, ⟨bc⟩ соответственно, таких +что четверки {a, b, ha, hb}; {a, c, ha, hc}; {b, c, hb, hc} являются вписанными. +Плоскости aha, bhb, chc являются аналогами так называемых псевдовысот, которым в [19] дается +еще и другое определение через углы. Эти три плоскости пересекаются по общей прямой, назовем ее +псевдоортоцентралью по аналогии с псевдоортоцентрами гиперболических треугольников. +Круговой конус, содержащий все три ребра трехгранного угла в качестве своих образующих, +назовем описанным. +В [19, §§ 4-6] показано, что основания трех псевдовысот и трех биссекторных чевиан лежат на +одной окружности. Центр этой окружности лежит на одной прямой с центром описанной, псевдоцен- +троидом и всевдоортоцентром. Сформулируем аналогичные утверждение для тетраэдра. +Теорема 1.3 (Конус Эйлера трехгранного угла). У любого трехгранного угла основания трех его +медиаторов и трех его псевдовысот лежат на одном круговом конусе. +Теорема 1.4 (Плоскость Эйлера трехгранного угла). У произвольного трехгранного угла четыре +прямых – псевдоцентроидаль, псевдоортоцентраль, ось описанного конуса и ось конуса Эйлера – +лежат в одной плоскости. +Главным же результатом работы [19] является гиперболический аналог теоремы Фейербаха, со- +гласно которому окружность Эйлера гиперболического треугольника касается его вписанной и трех +вневписанных окружностей. Применительно к тетраэдру мы получаем следующее усиление Теоре- +мы 1.1 +Теорема 1.5 (Аналог теоремы Фейербаха для тетраэдра). Четыре конуса Эйлера трехгранных углов +тетраэдра касаются всех восьми его касательных сфер. +Отметим несколько вопросов, которые возникают в связи с рассмотренными конструкциями. +Вопрос 1.6. Инцидентны ли какие либо из следующих четверок замечательных прямых тетраэд- +ра: псевдоцентроидали, псевдоортоцентрали, оси четырех описанных конусов, оси четырех конусов +Эйлера? +Вопрос 1.7. Существуют ли еще какие-либо квадрики, касающиеся всех касательных сфер, отлич- +ные от четырех конусов Эйлера и четырех плоскостей граней? +Вопрос 1.8. Любые три конуса общего положения пересекаются в восьми точках. Не окажется +ли так, что четыре конуса Эйлера тетраэдра имеют восемь общих точек? Есть ли какие-то +примечательные свойства биквадратических кривых, по которым пересекаются конусы Эйлера? +2 Теорема Грейса как трехмерный аналог теоремы Фейербаха +Более ста лет назад, британский математик Джон Хилтон Грейс в своей работе [7] открыл и доказал +следующее замечательное свойство касательных сфер тетраэдра. +Теорема 2.1 (Grace, 1897). Касательные сферы тетраэдра ABCD могут быть разбиты на че- +тыре пары так, что парные сферы гомотетичны с центром D, и для каждой пары существует +касающаяся их сфера, проходящая через вершины A, B, C. +Замечание 2.2. Все касательные сферы можно разбить на две группы по четыре сферы. В одну +входят вписанная и три дважды-вневписанные сферы, а в другую – четыре вневписанные. Любые +две сферы из разных групп гомотетичны относительно одной из вершин тетраэдра. Для каждой +4 + +такой пары сфер существует единственная касающаяся их сфера Грейса, которая проходит через +вершины грани, противоположной к той вершине, относительно которой данная пара касательных +сфер гомотетична. Таким образом, всего получается шестнадцать сфер Грейса: для каждой из +четырех граней тетраэдра через ее вершины проходит четыре различные сферы Грейса. +Теорема Грейса связывает касательные сферы тетраэдра с замечательными точками, его верши- +нами, с помощью общих касающихся их сфер. Это ее сближает с теоремой Фейербаха, с которой она, +на наш взгляд, сравнима по красоте и имеет некоторое сходство. В этом смысле, можно было бы +считать теорему Грейса неким трехмерным аналогом теоремы Фейербаха. +Рис. 2: Сфера Грейса GD, касающаяся вписанной сферы σ, вневписанной сферы σD и проходящая через A, B, C. +В недавней статье [13] Maehara и Martini замечают, что «по-видимому, эта теорема малоизвестна +и до сих пор не имеет элементарного доказательства». В качестве результата они приводят такое +доказательство, но лишь для частного случая триортогонального тетраэдра, пользуясь при этом +аналитической техникой. +Оригинальное же доказательство Грейса очень красивое и геометрическое, но довольно трудное. +Поскольку Грейс дал лишь его набросок, Maehara и Tokushige в работе [12] подробно реконструиро- +вали это доказательство. +Мы получим элементарное и вполне короткое геометрическое доказательство теоремы Грейса, +но сначала напомним некоторые определения и факты проективной геометрии. Пусть E3 – веще- +ственное трехмерное евклидово пространство. Мы будет рассматривать его проективное пополнение +«бесконечно удаленной» плоскостью. Эта модель проективного пространства получается переходом +от декартовых координат (x, y, z) в E3 к однородным координатам (x : y : z : w), в которых бесконеч- +но удаленной плоскости соответствуют точки с координатами (x : y : z : 0). Кроме того рассмотрим +комплексификацию пространства, позволяя координатам принимать комплексные значения. Добав- +ленные точки будем называть мнимыми. +Записывая в однородных координатах (x : y : z : w) общее уравнение сферы +x2 + y2 + z2 + 2axw + 2byw + 2czw + dw2 = 0, +легко видеть, что она пересекает бесконечно-удаленную плоскость w = 0 по кривой +x2 + y2 + z2 = 0, w = 0, +5 + +D +GD +A +C +B +ODкоторая является общей для всех сфер. Она называется абсолютной окружностью. +Всякая плоскость пересекает абсолютную окружность в двух сточках – круговых точках этой +плоскости. В однородных координатах (x : y : z) на плоскости ее круговыми точками являются точки +I = (1 : i : 0) и J = (1 : −i : 0). Все окружности плоскости проходят через ее круговые точки и каждая +коника плоскости, проходящая через ее круговые точки, является окружностью (см. [16, § 4·8]). +Прямая, пересекающая абсолютную окружность, называется изотропной. Каждая такая прямая +является, естественно, мнимой. +Предложение 2.3 ( [22, Гл. 12, § 2]). Касательные к невырожденной конике, проведенные из любого +ее фокуса, являются изотропными. +Таким образом, каждая прямая, проходящая через фокус коники и круговую точку ее плоскости, +является изотропной. Для окружности это означает, что касательные из ее центра проходят через +круговые точки. +Образующей квадрики называется прямая, которая целиком принадлежит поверхности этой квад- +рики. В комплексном проективном пространстве все невырожденные квадрики эквивалентны. +Предложение 2.4 ( [9, § 2]). +(i) Через каждую точку невырожденной квадрики проходят ровно две образующие, действительны +или мнимые. Касательная плоскость пересекает квадрику по двум образующим, проходящим +через точку касания. +(ii) Все образующие квадрики распадаются на два семейства таким образом, что любые две обра- +зующие из одного семейства не пересекаются, а любые две образующие из разных семейств +пересекаются. Через любую точку образующей одного семейства проходит единственная об- +разующая другого семейства. +(iii) Любая плоскость, проходящая через образующую квадрики касается этой квадрики в некото- +рой точке этой образующей. +Пусть даны две сферы γ и η. Рассмотрим множество M(γ, η) сфер, которые касаются обеих сфер +γ и η. Заметим что множество M(γ, η) распадается на два класса эквивалентности по типу касаний. +Если сфера α касается γ и η одинаковым образом (обеих внутренним, или обеих внешним), то α +принадлежит одному классу. Если же α касается γ и η различным образом (одной сферы внутренним, +а другой внешним, или наоборот), то α принадлежит другому классу. Прямые, проходящие через +точки касания γ и η со сферами одного класса, проходят через общую точку. Для сфер одного класса +эта точка – один из двух центров инверсии, переводящей γ и η друг в друга, а для сфер другого +класса – второй такой центр (эти точки – центры подобия сфер γ и η). +Замечание 2.5. Все это имеет место быть и в случае, если, скажем, сфера η вырождается в +плоскость π (сферу бесконечно большого радиуса). Тогда рассмотренные выше инверсные центры γ +и π – это точки сферы γ, касательные плоскости в которых параллельны π. +Следующая теорема является главным результатом этого параграфа. Она описывает семейство +коник σ, которые вместе с данной окружностью Σ образуют 3-пару Понселе (Σ, σ), т.е. для них суще- +ствует треугольник, вписанный в Σ и описанный около σ. Из этой теоремы практически мгновенно +следует теорема Грейса, что мы сразу покажем после ее формулировки. +Теорема 2.6 (О 3-парах Понселе). Пусть даны плоскость π и окружность Σ на ней. Фиксируем +сферу γ, содержащую окружность Σ, и рассмотрим множество M(γ, π) сфер, касающихся сферы +γ и плоскости π. Тогда если сферы α и β пробегают разные классы множества M(γ, π), то описан- +ный около них конус K высекает на плоскости π семейство коник σ, образующих 3-пару Понселе с +окружностью Σ. +6 + +Доказательство Теоремы Грейса. Пусть α и β – две касательные сферы тетраэдра ABCD, гомо- +тетичные относительно вершины D. Рассмотрим сферу γ, касающуюся сфер α и β и проходящую +через вершины A и B. Таких сфер, вообще говоря, целых четыре. Но две из них в данном случае +вырождены в плоскости ⟨DAB⟩ и ⟨ABC⟩, которые принадлежат разным классам множества M(α, β). +Тогда оставшиеся две сферы тоже принадлежат разным классам и в качестве γ выберем ту, которая +принадлежит другому, нежели плоскость ⟨ABC⟩, классу. Пусть она пересекает плоскость ⟨ABC⟩ по +окружности Σ. Описанный около α и β конус с вершиной D пересекает плоскость ⟨ABC⟩ по конике +σ, касающейся сторон треугольника ABC. По Теореме о 3-парах Понселе вершина C также должна +лежать на окружности Σ. +✷ +Доказательство Теоремы 2.6 о 3-парах Понселе. +Пусть Fα и Fβ – тоски касания сфер α и β с плоскостью π, которые по теореме Данделена (1822, [3]) +являются фокусами коники σ. Далее будем считать, что точки Fα и Fβ не совпадают друг с другом +и с центром окружности Σ. Эти частные случаи сводятся к общему малым шевелением сфер α и β и +утверждение теоремы для них получается предельным переходом. Если I – одна из круговых точек +плоскости π, то I ∈ Σ. Обозначим через Pα и Pβ точки вторичного пересечения прямых IFα и IFβ с +коникой Σ. Тогда треугольник IPαPβ вписан в окружность Σ, прямые IPα и IPβ касаются коники σ, +и нам достаточно доказать, в силу теоремы Понселе, что прямая PαPβ тоже касается коники σ. +Рис. 3: 3-пары Понселе (Σ, σ). Мнимые касательные представлены дугообразными розовыми отрезками. +Пусть A и B – точки касания сферы γ со сферами α и β. Заметим, что прямая IFα является +образующей сферы α. Обозначим через lA одну из двух образующих сферы α в точке A, которая +пересекает образующую IFα (т.е. lA и IFα принадлежат разным семействам образующих сферы γ). +Поскольку lA является также образующей и сферы γ, точка пересечения lA ∩ IFα – это одна из двух +точек пересечения прямой IFα со сферой γ, т.е. это либо точка I, либо точка Pα. +Заметим, что первый случай не возможен в силу нашей договоренности считать, что точка Fα +отлична от центра окружности Σ. В самом деле, I лежала бы тогда в пересечении касательных плос- +костей сферы α в точках A и Fα, т.е. полярно-сопряженная к AFα относительно α прямая содержала +бы круговую точку I. А так как она вещественная и потому не может быть изотропной, она являлась +7 + +人 +T +A +P +a +P +Fp +D +Fa +Bбы бесконечно-удаленной, т.е. касательные плоскости сферы α в точках A и Fα были бы параллельны, +а точка Fα совпадала бы с центром окружности Σ. +Таким образом, прямая APα является общей образующей lA сфер α и γ в точке A, и аналогично, +прямая BPβ совпадает с lB – одной из двух общих образующих сфер β и γ в точке B. Покажем, что +lA и lB компланарны. +Для этого рассмотрим гомотетию с центром A, переводящую α в γ. Пусть gA – образующая +сферы γ, в которую переходит образующая IFα сферы α. Заметим, что +1) I ∈ gA, поскольку gA ∥ IFα, +2) прямая gA инцидентна с прямой lA, т. к. прямая lA инвариантна при рассмотренной гомоте- +тии и инцидентна с прямой IFα. Т. е. gA и lA – две образующие сферы γ, принадлежащие разным +семействам. +Аналогично, если gB – образующая сферы γ, в которую переходит образующая IFβ сферы β при +гомотетии с центром B, переводящей β в γ, то +3) I ∈ gB, +4) gB и lB – тоже две образующие сферы γ, принадлежащие разным семействам. +Из замечания 2.5 следует, что прямые gA и gB проходят через различные инверсные центры +сферы γ и плоскости π, а потому различны. Тогда из 1) и 3) следует, что образующие gA и gB сферы +α имеют общую точку и, значит, принадлежат разным семействам, откуда в силу 2) и 4) следует, что +образующие lA и lB тоже из разных семейств, а потому компланарны. +Теперь рассмотрим плоскость ⟨lA; lB⟩, которая в силу утверждения [iii] Предложения 2.4 касается +обеих сфер α и β. Заметим, что вершина конуса K содержит прямую AB. Действительно, поскольку +конус K пересекает π по невырожденной конике, его вершина не лежит на π. Так как α и β из +разных классов множества M(γ, π), то γ и π из разных классов множества M(α, β). Значит, прямая +AB проходит через инверсный центр сфер α и β, который не лежит на плоскости π. +Т.о., ⟨lA; lB⟩ – касательная плоскость конуса K, а потому пересекает плоскость π по прямой, +касающейся коники σ. Осталось заметить, что ⟨lA; lB⟩ пересекает π по прямой PαPβ, и таким образом, +треугольник IPαPβ является вписано-описанным. +✷ +3 Формулы Эйлера-Чаппла и up-in-ex-touch-аналог теоремы Фейер- +баха +Теорема 3.1 (Euler, Chapple). Пусть R, r и ra – радиусы описанной, вписанной и вневписанной +окружностей произвольного треугольника, d и da – расстояния от центра описанной окружности +до центров вписанной и вневписанной. Тогда выполняются следующие соотношения +d2 = R2 − 2Rr +(1) +d2 +a = R2 + 2Rra +(2) +Мы приведем два, наверное, самых коротких доказательства этой теоремы. Для этого рассмотрим +сферу ∆, построенную диаметрально на описанной окружности, наовем ее описанной сферой тре- +угольника, сферу δ радиуса r, касающуюся плоскости треугольника в центре его вписанной окруж- +ности, наовем ее вписано-поднятой, и сферу δa радиуса ra, касающуюся плоскости треугольника в +центре соответствующей вневписанной окружности, наовем ее вневписано-поднятой. +Заметим, что соотношения (1), (2) можно переписать в виде равенств +d2 + r2 = (R − r)2, +d2 + r2 +a = (R + ra)2, +которые равносильны касанию сфер ∆ и δ, ∆ и δa. +8 + +Рис. 4: Сферы ∆ и δ касаются друг друга +Доказательство 1. Касания ∆ и δ, ∆ и δa сразу следует +из Теоремы Грейса. Действительно, рассмотрим тетраэдр с +основанием ABC и вершиной D на бесконечности в перпен- +дикулярном к плоскости (ABC) направлении. Тогда сфера +δ является его вписанной сферой, симметричная ей относи- +тельно плоскости (ABC) – его вневписанной сферой, а сле- +довательно, сфера ∆ – его сферой Грейса. Для пары ∆ и δa +рассуждение аналогично. +✷ +Это доказательство примечательно своей лаконичностью +и красотой, но использование сложной Теоремы Грейса мо- +жет выглядеть как «стрельба из пушки по воробьям». Поэто- +му приводим другое +Доказательство 2. Сделаем инверсию относительно сфе- +ры, построенной диаметрально на вписанной окружности. +Заметим, что сфера ∆ переходит в сферу ∆′, построенную +диаметрально на окружности, проходящей через середины сторон треугольника Жергона (верши- +нами которого являются точки касания вписанной окружности △ABC со сторонами). А сфера δ +переходит в плоскость δ′, удаленную от плоскости (ABC) параллельно на расстояние r +2 . Поскольку, +радиус сферы ∆′, очевидно, тоже равен r +2, сферы ∆′ и δ′, а следовательно, и сферы ∆ и δ касаются +друг друга. +✷ +Заметим, что доказанное свойство касания сферы ∆ с четырьмя сферами δ, δa, δb, δc является +своего рода тоже неким аналогом теоремы Фейербаха в пространстве. +Теорема 3.2 (Up-in-ex-touch). Описанная сфера треугольника касается его вписано-поднятой и че- +тырех вневписано-поднятых сфер. +Рис. 5: Up-in-ex-touch-аналог теоремы Фейербаха. +9 + +Заметим также, что сфера ∆ касается не только сфер δ, δa, δb, δc, но и еще четырех симметричных +им относительно плоскости треугольника, т.е. целых восьми сфер. +4 Теорема Лагерра и ее применение к тетраэдру +Теорема 4.1 (Laguerre [10], 1879). Окружность Σ радиуса R с центром в точке O и коника σ с +фокусами Fα, Fβ и малой полуосью b образуют 3-пару Понселе тогда и только тогда, когда выпол- +няется соотношение +(R2 − d2 +α)(R2 − d2 +β) = 4R2b2, +(3) +где dα = |OFα|, dβ = |OFβ|. +Замечание 4.2. Малая полуось b может быть как действительной (у эллипсов), так и мнимой (у +гипербол). В первом случае из формулы Лагерра видно, что фокусы эллипса должны лежать либо +оба внутри окружности, либо оба вне. Во втором случае, у гиперболы, один фокус должен лежать +внутри окружности, другой – снаружи. +Замечание 4.3. Если коника σ является параболой, то условие существования вписано-описанных +треугольников для пары (Σ, σ) становится совсем простым: d = R, где d = |OF|, т.е. фокус F +параболы должен лежать на окружности. Это следует из известной теоремы Ламбера. +Доказательство Теоремы Лагерра (⇒) Пусть γ – произвольная сфера, содержащая окружность Σ, +а cфера α касается в точке Fα плоскости π, содержащей окружность Σ, а также касается сферы γ. +Рассмотрим произвольный вписано-описанный треугольник ABC и проведем через его стороны ка- +сательные плоскости к сфере α. Они пересекаются в некоторой точке D, образуя тетраэдр ABCD, +у которого сфера α является одной из касательных сфер, а γ – сферой Грейса, которая касает- +ся также другой касатеьной сферы β тетраэдра ABCD, гомотетичной α относительно вершины D. +Как известно, сферы α и β касаются плоскости π в точках, изогонально сопряженных относительно +△ABC. Кроме того, поскольку Fα и Fβ – фокусы вписанной в △ABC коники σ, они также изого- +нально сопряжены. Отсюда заключаем, что сфера β касается плоскости π в точке Fβ. +Нам понадобится одна очень простая лемма +Лемма 4.4 (Thebault [17], 1922). Для малой полуоси b коники, высекаемой описанным около сфер α +и β конусом на их общей касательной плоскости, выполняется соотношение +|b2| = rαrβ +(4) +Пусть Sα и Sβ – две диаметрально противоположные точки на γ в перпендикулярном к плоскости +π направлении, которые являются инверсными центрами сферы γ и плоскости π (см. замечание 2.5). +Учитывая, что сферы α и β принадлежат разным классам множества M(γ, π) (см. доказательство +теоремы Грейса), легко выразить радиусы сфер α и β: +rα = +���� +Σ(Fα) +2π(Sα) +���� , +rβ = +���� +Σ(Fβ) +2π(Sβ) +���� , +(5) +где Σ(Fα) = d2 +α − R2 и Σ(Fβ) = d2 +β − R2 – степени точек Fα и Fβ относительно окружности Σ, +а π(Sα), π(Sβ) – расстояния от точек Sα и Sβ до плоскости π. +Перемножим равенства (5) и учтем, что π(Sα)π(Sβ) = R2. Получим, что в равенстве (3) левая и +правая части равны по модулю. Правая часть отрицательна только в случае, если коника σ является +гиперболой. Такое происходит только тогда, когда сферы α и β касаются описанного около них +конуса с вершиной D по разные стороны от D, а плоскости π – по одну сторону. Тогда сферы γ они +должны касаться по разные стороны, а следовательно, точки Fα и Fβ их касания с π относительно +10 + +окружности Σ лежат тоже по разные стороны и левая часть (3) в этом случае также отрицательна. +Таким образом, модули можно снять и равенство (3) считать доказанным. +(⇐) Пусть выполняется (3). Если коники (Σ, σ) не образуют 3-пару Понселе, то можно изменить +малую полуось b коники σ так, чтобы они образовали 3-пару Понселе. Тогда по уже доказанному +тоже должно выполняться равенство (3), следовательно величина b не изменилась, т.е. (Σ, σ) как раз +и образуют 3-пару Понселе +✷ +Теорема Лаггера, примененная к тетраэдру, позволяет получить следующее интересное метриче- +ское соотношение для касательных сфер тетраэдра. +Теорема 4.5. Пусть ∆D – сфера, описанная около грани ABC тетраэдра ABCD, α и β – две +касательные сферы, гомотетичные относительно D. Тогда произведение косинусов углов, которые +сфера ∆D образует с α и β (среди них один угол мнимый), равно 1, если α и β касаются ⟨ABC⟩ с +одной стороны, или −1, если с разных. +cos(� +∆D, α) cos(� +∆D, β) = sign k, +(6) +где k – коэффициент упомянутой гомотетии с центром D. +Доказательство Пусть OD и R – центр и радиус сферы ∆D; rα, rβ – радиусы сфер α и β; Dα, Dβ – +расстояния между центрами ∆D и α, ∆D и α; dα, dβ – расстояния от OD до точек Fα и Fβ касания +плоскости ⟨ABC⟩ со сферами α и β. Пусть Σ = ⊙(ABC), а конус K с вершиной D, описанный около +α и β, пересекает плоскость ⟨ABC⟩ по конике σ. +Воспользуемся леммой 4.4 и заметим, что в нашей конструкции с тетраэдром равенство (4) можно +уточнить +b2 = rαrβ sign k, +(7) +поскольку b2 может быть отрицательным, только если коника σ является гиперболой, что возможно +лишь в том случае, если вершина конуса K является центром отрицательной гомотетии сфер α и β, +т.е. они вписаны в K по разные стороны от его вершины. +По теореме Лагерра для пары (Σ, σ) имеем +(R2 − d2 +α)(R2 − d2 +β) = 4R2b2, +(8) +По теореме Пифагора +d2 +α = D2 +α − r2 +α, +d2 +β = D2 +β − r2 +β +Подставляя эти равенства и (7) в соотношение (8), получаем требуемое соотношение +R2 + r2 +α − D2 +α +2R rα +· +R2 + r2 +β − D2 +β +2R rβ += sign k +✷ +5 Трехмерный аналог формулы Эйлера-Чаппла +В связи с теоремой Эйлера-Чаппла возникает естественный вопрос о возможности ее трехмерного +обобщения на случай тетраэдра. Этот вопрос был поставлен впервые Ж. Д. Жергонном в 1816 году +в издаваемом им журнале1 в виде краткой сноски, относящейся к тетраэдру с радиусами описанной +сферы R, вписанной – r и расстоянием d между их центрами: +1Annales de math´ematiques pures et appliqu´ees, 6 (1815-1816), p. 228. +11 + +«Il +serait +sur-tout +int´eressant +de +savoir +si +d +peut +ˆetre +exprim´e +uniquement +en fonction de R et r. +J. D. G.» +Спустя восемь лет в том же журнале было опубликовано положительное решение этой задачи в +работе Дюрранда [4], где он доказал следующее соотношение: +d2 = (R + r)(R − 3r). +(9) +Этот результат получил широкое признание и в течение многих лет на него ссылались в литерату- +ре, например, в таких почтенных изданиях как Математическая энциклопедия Клейна «Encyklop¨adie +der mathematischen Wissenschaften» [18] (первая в мире математическая энциклопедия) и «Enciclopedia +delle matematiche elementari» [1] (крупнейшая энциклопедия по математике, изданная в Италии). Од- +нако, формула Дюрранда (9) оказалась неверной, а ответ на вопрос Жергонна – отрицательным: не +существует общей для всех тетраэдров функциональной зависимости между R, r и d. Доказатель- +ство Дюрранда было практически безупречным, но незаметная ошибка заключалась в его убежден- +ности, что описанная и вписанная сфера непременно должны иметь некоторую зависимость. Вопрос +Жергонна можно было бы сформулировать так: каковы условия существования вписано-описанного +тетраэдра для двух данных сфер? +Оказывается никаких необходимых условий для этого не требуется. +Теорема 5.1. Для любых двух невырожденных квадрик общего положения существует бесконечное +семейство вписано-описанных тетраэдров. Любая касательная плоскость ко вписанной квадрике +может содержать грань такого тетраэдра, а его вершиной может быть произвольная точка +описанной квадрики. +В работе Фонтене [6] 1899 года эта теорема считается уже известной (см. также [8]). +Итак, в отличие от плоского случая в пространстве для любых двух произвольных сфер всегда +существует вписано-описанный в них тетраэдр, причем он может динамически вращаться около этих +сфер, все время оставаясь вписано-описанным. При этом, любая точка описанной сферы может быть +вершиной такого тетраэдра. +Но оказывается, что не для любых двух вещественных сфер такой тетраэдр может быть веще- +ственным. Критерием существования вещественного вписано-описанного тетраэдра является следу- +ющее условие Грейса, исправляющее соотношение Дюрранда (9): +Теорема 5.2 (Grace [8], 1917). Для данных двух сфер S и T необходимым и достаточным условием +существования вписано-описанного вещественного тетраэдра, у которого вершины лежат на S, а +плоскости граней касаются T, является следующее условие в зависимости от взаимного располо- +жения S и T: +(a) T вложена в S и +d2 ⩽ (R + r)(R − 3r); +(b) T и S расположены одна вне другой; +(c) T и S пересекаются по действительной окружности и +d2 ⩽ (R − r)(R + 3r). +6 Вращение Понселе вписано-описанного тетраэдра +Теорема 5.1 позволяет рассмотреть динамику «вращения» вписано-описанного тетраэдра. Эта дина- +мика не столь однозначна, как в плоской теореме Понселе. Это показывает следующая теорема. +12 + +Теорема 6.1 ( [8]). Пусть вершины тетраэдра лежат на квадрике S, а грани касаются квадрики +T. Тогда при фиксации плоскости π одной из его граней противоположная вершина P может при +этом варьироваться, пробегая плоское сечение π′ квадрики S. +Таким образом, тетраэдр вращается с намного большей свободой, чем вписано-описанный мно- +гоугольник. Когда выбрана плоскость π, существует целая коника для выбора произвольной точки +на ней в качестве вершины P, а для каждой такой пары P и π существует однопараметрическое +семейство вписано-описанных треугольников, каждый из которых может быть противоположной к +вершине P гранью вписано-описанного тетраэдра. Таким образом, в общем случае существует 4- +параметрическое семейство тетраэдров. +У плоской теоремы Понселе есть такой «эффект замыкания»: если начиная с некоторой начальной +точки A1 строится последовательно вписано-описанная ломаная A1A2 . . . An и оказывается, что звено +A1An тоже касается вписанной коники, замыкая ее, то такое замыкание будет происходить всегда. +Если же, по аналогии, строить вписано-описанный тетраэдр для двух данных квадрик S и T, +последовательно выбирая касательные плоскости его граней, то возникает следующий вопрос. Когда +мы провели уже три плоскости, которые образовали вписано-описанный трехгранный угол, всегда +ли можно его замкнуть четвертой плоскостью, чтобы образовался вписано-описанный тетраэдр? +Ответ дает следующая теорема Фонтене. +Теорема 6.2 (Fonten´e [6]). Последовательный процесс построения вписано-описанного тетраэдра +всегда замыкается тогда и только тогда, когда квадрики S и T имеют четыре общих образующих. +В этом случае, плоскость π и вершина P могут быть выбраны совсем произвольно и, таким +образом, существует 5-параметрическое семейство вписано-описанных тетраэдров. +Теорема 6.3. Пусть фиксированы описанная сфера S тетраэдра и одна из восьми его касательных +сфер T, а тетраэдр динамически «вращается» около них, оставаясь вписано-описанным. Тогда все +четыре касающиеся T сферы Грейса все время касаются некоторой фиксированной сферы, концен- +тричной с описанной сферой S. +Доказательство. Пусть сфера Грейса G проходит через вершины грани a и пусть вписанная +сфера S касается сферы G в точке P, а плоскости ⟨a⟩ – в точке Q. Обозначим центры сфер S +и T через OS и OT . Прямая PQ при вращении тетраэдра проходит через фиксированную точку – +предельную точку K пучка сфер ⟨S, T⟩. Кроме того, на прямой PQ лежит инверсный центр E сферы +G и плоскости ⟨a⟩, касательная в котором к G параллельна плоскости ⟨a⟩. Следовательно, OSE∥OT Q +и △OSEK ∼ △OT QK, откуда получаем такое выражение +OSE = OSK +OT K · rT , +правая часть которого является величиной постоянной при вращении тетраэдра. Тогда, сфера с ради- +усом, равным этой величине, и центром в точке OS касается сферы Грейса в любой момент вращения. +✷ +7 Доказательство теоремы Фейербаха через выход в пространство +Пусть δ – вписанная окружность треугольника ABC с центром в точке I и радиусом r, H – ор- +тоцентр треугольника ABC, точки A1, B1, C1, I1 – середины отрезков AH, BH, CH, IH (I1 – инцентр +△A1B1C1). Описанная около △A1B1C1 окружность ϑ – это окружность девяти точек △ABC. Пусть +также ⊙a, ⊙b, ⊙c – окружности с диаметрами BC, CA, AB, ∆ и Θ – сферы, построенные диаметраль- +но на окружностях δ и θ. +13 + +Доказательство Теоремы Фейербаха. +Заметим, что касание окружностей δ и θ равносильно касанию сфер ∆ и Θ. По Теореме 3.2 для +△A1B1C1 его описанная сфера Θ касается его вписано-поднятой сферы Υ. Поэтому касание Θ и ∆ +равносильно тому, что сфера Θ инвариантна при инверсии, переводящей сферы ∆ и Υ друг в друга. +Заметим, что центр S этой инверсии расположен над точкой H на высоте r (т.е. SH⊥(ABC), |SH| = +r), а коэффициент инверсии (квадрат радиуса сферы инверсии) равен |IH| · |I1H| = |IH|2 +2 +. Таким +образом, достаточно доказать равенство Θ(S) = |IH|2 +2 +, которое в силу того, что Θ(S) = θ(H) + r2, +равносильно соотношению +|IH|2 − 2r2 = 2θ(H) +(10) +Заметим, что левая часть равенства (10) равна степени точки H относительно окружности ξ +радиуса r +√ +2 с центром I (ξ высекает на сторонах △ABC равные отрезки длины 2r). Осталось вос- +пользоваться следующим замечательным свойством окружности ξ. +Теорема 7.1. Окружности ξ, ⊙a, ⊙b, ⊙c имеют общий радикальный центр в точке H. +Тогда заметим, что степень точки H относительно окружности θ в два раза меньше ее степени +относительно окружностей ⊙a, ⊙b, ⊙c и равенство (10) равносильно утверждению ξ(H) = ⊙a(H) = +⊙b(H) = ⊙c(H) Теоремы 7.1. +✷ +Для доказательства Теоремы 7.1 рассмотрим окружность χa, диаметром которой является жер- +гониана вершины A (т.е. отрезок, соединяющий A с точкой касания вписанной окружности δ со +стороной BC) и воспользуемся следующим свойством окружности χa, возможно, имеющим и само- +стоятельный интерес. +Лемма 7.2 (χa-лемма). Окружности χa, ξ, ⊙a принадлежат одному пучку. +Доказательство Теоремы 7.1. Достаточно проверить, что H ∈ rad(ξ, ⊙a). +Заметим, что rad(χa, ⊙b) – это высота AH, rad(⊙a, ⊙b) – это высота CH, следовательно, +H = rad(χa, ⊙a, ⊙b) ∈ rad(χa, ⊙a) = rad(ξ, ⊙a), +где последнее равенство верно в силу χa-леммы. +✷ +Доказательство χa-леммы. +Воспользуемся следующим известным метрическим соотношением для пучков окружностей. +Лемма 7.3 (О пучке). Если окружности α, β, γ лежат в одном пучке, то для любой точки P ∈ γ +отношение ее степеней относительно α и β постоянно, причем +α(P) +β(P) = dαγ +dβγ +, +(11) +где dαγ и dβγ – расстояния между центрами α, γ и β, γ. +Верно и обратное утверждение. +Лемма 7.4 (Обратная лемма о пучке). Пусть центры окружностей α, β, γ коллинеарны, и на +окружности γ имеется такая точка P, для которой выполняется соотношение (11). Тогда окруж- +ности α, β, γ принадлежат одному пучку. +14 + +Рис. 6: Окружности ξ, χa, ⊙a принадлежат одному пучку +В качестве окружностей α, β, γ из Обратной леммы о пучке возьмем окружности ⊙a, ξ, χa, центры +M, I, L которых лежат на средней линии ML треугольника APQ. При этом, +LM +LI = AQ +AN = ra +r . +Для точки P ∈ χa имеем +α(P) = −(p − b)(p − c), +β(P) = −r2. +Тогда (11) запишется в виде соотношения +(p − b)(p − c) +r2 += ra +r , +которое равносильно легко проверяемому равенству +(p − b)(p − c) = r ra. +✷ +Доказательство χa-леммы выходом в пространство. Заметим, что окружности χa, δ и окруж- +ность ⊙P Q с диаметром на отрезке PQ лежат в одном пучке. Поднимем их центры перпендикулярно +плоскости ⟨ABC⟩, сохраняя коллинеарность: L → L, I → I, M → M, и пусть LL = r +2, II = r. Тогда +легко найти, что MM = r + ra +2 +. При этом сферы S(L), S(J), S(M) с центрами L, I, M, содержащие +окружности χa, δ, ⊙P Q соответственно, также принадлежат одному пучку. Рассмотрим плоскость +π∥⟨ABC⟩, проходящую через I, и ортогональную проекцию △ABC → △A′B′C′ на плоскость π. Оста- +лось заметить, что сечениями сфер S(L), S(J), S(M) плоскостью π являются окружности ξ′, χ′ +a, ⊙′ +a. +Действительно, для сечений S(L), S(I) это очевидно, а для S(M) это легко проверить, поскольку +квадрат радиуса окружности ее сечения плоскостью π равен +|MP|2 + +�ra + r +2 +�2 +− +�ra − r +2 +�2 += |MP|2 +rar = |MP|2 +(p−b)(p−c) = |MP|2 +|BP|·|CP| = +� a +2 +�2 +. +Так как при пересечении сфер пучка плоскостью получается пучок окружностей, то ξ′, χ′ +a, ⊙′ +a +принадлежат одному пучку. +✷ +15 + +3 +N +L +H +B +M +P +CСписок литературы +[1] +Biggiogero G., +La geometria del tetraedro. In: Enciclopedia delle Matematiche Elementari. A cura di +L. Berzotari, G. Vivanti e D. Gigli. Volume II, parte I. Milano 1937. Ristampa anastatica, Maggio 1943, p. 237. +[2] Coolidge J. L., A treatise on the circle and the sphere, Oxford: Clarendon Press, 1916. +[3] Dandelin G., M´emoire sur quelques propri´et´es remarquables de la focale parabolique, Nouveaux m´emoires de +l’Acad´emie rouale des sciences et belles-lettres de Bruxelles, T. 2 (1822) 171-200. +[4] Durrande J. B., D´emonstrations ´el´ementaires des principales propriet´es des hexagones inscrits et circonscrits +au cercle, suivies de la solution de divers probl`emes de la g´eometrie. Dissertation de la g´eometrie pure. Annales +de math´ematiques pures et appliqu´ees, 14 (1823- 1824) 29-63. +[5] Feuerbach K. W., Eigenschaften einiger merkw¨urdigen Punkte des geradlinigen Dreiecks, N¨urnberg, 1822. +[6] +Fonten´e G., +Sur des poly`edres mobiles comparables aux polygones de Poncelet, +Nouvelles annales de +math´ematiques 3e s´erie, tome 18 (1899) 67-74. +[7] Grace J. H., Circles, spheres, and linear complexes, Trans. Cambridge Philosophical Soc. 14 (1898) 153–190. +[8] Grace J. H., Tetrahedra in relation to spheres and quadrics, Proc. London Math. Soc. 17 (1918) 259-271. +[9] Griffiths Ph., Harris J., A Poncelet theorem in space, Comm Math. Helv., 52 (1977) 145-160. +[10] Laguerre E. N., Sur la relation qui existe entre un cercle circonscrit `а un triangle et les ´el´ements d’une conique +inscrite dans ce triangle, Nouvelles annales de math´ematiques 2e s´erie, tome 18 (1879) 241-246. +[11] Lewis T. C., Is there an analogue in solid geometry to Feuerbach’s theorem, Messenger of Mathematics, volume +49 (1919) 187-192. +[12] +Hiroshi Maehara, Norihide Tokushige, +Schl¨afli’s double six, Lie’s line-sphere transformation, and Grace’s +theorem, European Journal of Combinatorics, 30 (2009) 1337–1351. +[13] +Hiroshi Maehara, Horst Martini, Tangent Spheres of Tetrahedra and a Theorem of Grace, The American +Mathematical Monthly, 127:10 (2020) 897-910 +[14] Poncelet J. - V., Trait´e des propri´et´es projectives des figures, Gauthier-Villars, Paris, 1822. +[15] Protasov V. Yu., Generalized closing theorems, Elem. Math., 66 (2011) 98-117. +[16] Sommerville, D. M. Y., Analytical geometry of three dimensions, Cambridge University Press, 1943. +[17] Thebault V., Sur un theoreme classique de Dandelin, Nouvelles annales de mathematiques 5e serie, t. 1 (1922) +200-205. +[18] Zacharias M. Elementargeometrie und elementare nicht-euklidische Geometrie in synthetischer Behandlung. In: +Encyklop¨adie der Mathematischen Wissenschaften mit Einschluß ihrer Anwendungen. Drifter Band. Geometric. +Redigiert yon W. Fr. Meyer und H. Mohrmann. B. G. ˙Teubner, Leipzig, 1914-1931, S. 1059. +[19] +Акопян А. В., +О некоторых классических конструкциях в геометрии Лобачевского, +Матем. просв., +выпуск 13 (2009) 155–170. +[20] Берже М., Геометрия, М. Мир, 1984. +[21] Заславский А. А., Сравнительная геометрия треугольника и тетраэдра, Матем. просв., вып. 8 (2004) +78–92 +[22] Фиников С. П., Аналитическая геометрия, Москва, 1952. +16 + diff --git a/0tAyT4oBgHgl3EQf1PlJ/content/tmp_files/load_file.txt b/0tAyT4oBgHgl3EQf1PlJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..74fec8ce7ff0db7eafe416e7d3f5d6ba20ef5513 --- /dev/null +++ b/0tAyT4oBgHgl3EQf1PlJ/content/tmp_files/load_file.txt @@ -0,0 +1,554 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf,len=553 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='00731v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='DS] 2 Jan 2023 Feuerbach’s and Poncelet’s theorems meet in space (On the occasion of their bicentennial) E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Avksentyev December 29, 2022 Abstract Three-dimensional analogues of the Feuerbach theorem are proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' One of them concerns some tetrahedron analogue of the Euler circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Another one is pretty interesting «up-in-ex- touch» construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' And the third one, it turns out, is closely related to Poncelet’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' This is very beautiful Grace’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' It seems that this theorem is not widely known, and that no elementary proof has been given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Such an elementary proof of the Grace theorem is obtained in this paper by using properties of imaginary generators on a sphere and of isotropic tangents to a conic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' An applying of the Grace theorem leads to several corollaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' One of them is Laguerre’s theorem, which generalizes the Euler-Chapple formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Further, we consider a spatial analog of Poncelet’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' We prove that the Grace spheres touch some fixed sphere under the Poncelet rotation of bicentric tetrahedron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Finaly, going out from a plane into the third dimension, we obtain a new proof of Feuerbach’s theorem and perhaps the shortest proof of Euler-Chapple formulas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Введение Данная работа посвящена двум знаменитым геометрическим теоремам, кажется никак не связанным между собой, разве что они были опубликованны в один год двести лет назад [5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Приведем их формулировки Теорема (Feuerbach, 1822).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Окружность девяти точек произвольного треугольника касается его вписанной и трех вневписанных окружностей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема (Poncelet, 1822).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть для двух данных коник существует вписано-описанный в них многоугольник.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда этот многоугольник может динамически «вращаться» около данных коник, оставаясь вписано-описанным в них.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' У обеих теорем есть масса обобщений, но пространственные аналоги, насколько нам известно, имеются только у теоремы Понселе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Их довольно много (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', например, [6,8,9,15]) и среди них есть множество замечательных, но малоизвестных результатов.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Задача трехмерного обобщения теоремы Фейербаха поставлена еще более ста лет назад в моно- графии Кулиджа [2]: «The geometry of the tetrahedron lags far behind that of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Is there an analogue to Feuerbach’s theorem?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Above all what corresponds to the Hart systems?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='These difficult but important and interesting questions offer ample scope for serious work» (p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 247).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема Фейербаха содержит в себе два удивительных геометрических факта.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Первый состоит в том, что четыре замечательные окружности треугольника – вписанная и три описанные – имеют об- щую касательную окружность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Второй же заключается в том, что эта общая касательная окружность является еще и окружностью девяти точек, которая и без того сама по себе замечательна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Первая попытка найти аналог теоремы Фейербаха в пространстве приводит к вопросу: существу- ет ли сфера, которая касалась бы вписанной и вневписанных сфер?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Но здесь нас ожидает первый «сюрприз»: у произвольного тетраэдра кроме обычных четырех вневписанных сфер, аналогичных трем вневписанным сферам треугольника, существует еще три дважды-вневписанные сферы или чердачные (от англ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' «roof»), как они названы в [20] (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' также [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', всего существует целых восемь сфер (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 1), касающихся граней тетраэдра!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Назовем их касательными сферами.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Было бы слишком оптимистично ожидать, что все восемь касательных 1 Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 1: Восемь касательных сфер тетраэдра сфер могли бы касаться одной сферы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' И действительно, ответ на поставленный вопрос оказывается отрицательным: в общем случае произвольного тетраэдра такой сферы не существует.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Проверить это очень легко: для этого достаточно рассмотреть лишь один пример подходящего тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' И нет сомнений, что такой знаток геометрии как Кулидж хорошо знал, что такой сферы в общем случае нет.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Однако, он все-таки поставил вопрос поиска трехмерных аналогов теоремы Фейербаха, находя его важным, интересным и открывающим «широкие возможности для серьезной работы».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В каком же направлении искать тогда аналоги теоремы Фейербаха в пространстве?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Кажется, что осталась лишь задача описания частных случаев тетраэдров, у которых существует сфера, ка- сающаяся внутренним или внешним образом пяти, шести, семи или всех восьми касательных сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В работе [11] есть некоторое продвижение в этой задаче и для существования такой сферы получе- ны аналитические условия в специальных связанных с тетраэдром пентасферических координатах.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' К сожалению, эти условия весьма громоздкие и из них совершенно не ясно, существуют ли такие тетраэдры и как они устроены.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, задача в такой постановке остается незакрытой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Возникает еще идея поискать пространственный аналог теоремы Фейербаха в таком направле- нии: существует ли окружность, действительная или мнимая, которая касалась бы всех восьми касательных сфер?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Кажется маловероятным, что ответ мог бы быть положительным, но задача пред- ставляется интересной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Оставив пока эти вопросы, мы приведем далее целых три трехмерных аналога теоремы Фейербаха.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Первый аналог, которую мы хотим предложить в § 1 в качестве трехмерного обобщения теоремы Фейербаха, является довольно интересным фактом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' У него очень простое доказательство, которое, 2 тем не менее, раскрывает связь этой конструкции с неевклидовой геометрией и приводит к трехмер- ному обобщению окружности Эйлера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поэтому из трех аналогов этот наиболее аутентичен.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Второй является очень красивой теоремой геометрии тетраэдра, открытой сто двадцать пять лет назад, но, кажется, до сих пор малоизвестной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Ее единственное оригинальное доказательство столь сложно, что есть целая статья с его реконструкцией.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В § 2 мы получим элементарное доказательство этой теоремы, в котором обнаружится ее связь с теоремой Понселе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Второй аналог выглядит наименее аутентичным, но на наш взгляд, он ближе и роднее к теореме Фейербаха, чем другие два.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Третий аналог представляет из себя довольно интересную конструкцию касающихся сфер, кото- рую мы назвали «up-in-ex-touch»-конструкция.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Мы приведем ее в конце § 3, в котором мы также получим, возможно, самое короткое доказательство формул Эйлера-Чаппла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' С помощью теоремы Грейса мы в §4 получим короткое и простое доказательство теоремы Лагерра, обобщающей формулы Эйлера-Чаппла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' §5 посвящен трехмерному аналогу формул Эйлера-Чаппла.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Далее в §6 мы рассмотрим пространственные аналоги теоремы Понселе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Мы покажем, что при вращении Понселе вписано-вневписанного тетраэдра его сферы Грейса касаются некоторой фикси- рованной сферы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В конце, совершая «выход в пространство», мы дадим новое доказательство теоремы Фейербаха.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 1 Первый аналог теоремы Фейербаха для тетраэдра Итак, рассмотрим произвольный тетраэдр общего вида, у которого имеется восемь касательных сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В качестве первого аналога теоремы Фейербаха для тетраэдра предлагаем следующую теорему.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Существует четыре круговых конуса, каждый из которых касается всех восьми его касательных сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Доказательство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рассмотрим сферу ζD с центром в вершине D тетраэдра ABCD и спроектируем из центра D на сферу ζD все восемь касательных сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Их проекциями будут четыре окружности на сфере ζD, поскольку каждая пара гомотетичных относительно D сфер спроектируются в одну и ту же окружность.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Эти четыре окружности касаются сторон сферического треугольника, стороны которого являются проекциями плоскостей трехгранного угла при вершине D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' По теореме Фейербаха для сферического треугольника существует окружность, касающаяся этих четырех окружностей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Конус с вершиной D, содержащий эту окружность, очевидно удовлетворяет утверждению теоремы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Такой конус есть у каждой вершины.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Теорема Фейербаха в сферической геометрии, в той облегченной форме, которую мы использо- вали в доказательстве, равносильна теореме Харта (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, в какой-то степени мы ответили на оба вопроса Кулиджа, которые мы цитировали во введении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' На самом деле, можно про- двинуться еще дальше в этом направлении, если применить результат Акопяна [19], в котором он нашел такие свойства окружности Харта, которые во многом аналогичны свойствам окружности девяти точек.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Хотя в [19] все утверждения формулируются для плоскости Лобачевского, но мы их естественным образом адаптируем применительно к трехгранным углам нашего тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Избытком трехгранного угла называется величина, равная разнице между суммой его двух- гранных углов и 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Медиатором трехгранного угла назовем плоскость, содержащую его ребро и делящую его на два трехгранных угла с равными избытками.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' При рассмотренной выше проекции трехгранного угла на сферу медиатор переходит в сферическую чевиану, делящую пополам пло- щадь соответственного треугольника (в [19] эта чевиана называется биссектором или биссекторным отрезком).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Три медиатора пересекаются по прямой, которую можно назвать псевдоцентроидалью, поскольку ей соответствует псевдоцентроид сферического треугольника.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Четыре прямые из одного пучка назовем вписанной четверкой, если все они являются образую- щими одного кругового конуса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Следующее утверждение является аналогом Леммы 5 из [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 3 Предложение 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть a, b, c – ребра трехгранного угла с вершиной D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда существует един- ственная тройка прямых ha, hb, hc, лежащих в плоскостях ⟨ab⟩, ⟨ac⟩, ⟨bc⟩ соответственно, таких что четверки {a, b, ha, hb};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' {a, c, ha, hc};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' {b, c, hb, hc} являются вписанными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Плоскости aha, bhb, chc являются аналогами так называемых псевдовысот, которым в [19] дается еще и другое определение через углы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Эти три плоскости пересекаются по общей прямой, назовем ее псевдоортоцентралью по аналогии с псевдоортоцентрами гиперболических треугольников.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Круговой конус, содержащий все три ребра трехгранного угла в качестве своих образующих, назовем описанным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В [19, §§ 4-6] показано, что основания трех псевдовысот и трех биссекторных чевиан лежат на одной окружности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Центр этой окружности лежит на одной прямой с центром описанной, псевдоцен- троидом и всевдоортоцентром.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Сформулируем аналогичные утверждение для тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='3 (Конус Эйлера трехгранного угла).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' У любого трехгранного угла основания трех его медиаторов и трех его псевдовысот лежат на одном круговом конусе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='4 (Плоскость Эйлера трехгранного угла).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' У произвольного трехгранного угла четыре прямых – псевдоцентроидаль, псевдоортоцентраль, ось описанного конуса и ось конуса Эйлера – лежат в одной плоскости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Главным же результатом работы [19] является гиперболический аналог теоремы Фейербаха, со- гласно которому окружность Эйлера гиперболического треугольника касается его вписанной и трех вневписанных окружностей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Применительно к тетраэдру мы получаем следующее усиление Теоре- мы 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 Теорема 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='5 (Аналог теоремы Фейербаха для тетраэдра).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Четыре конуса Эйлера трехгранных углов тетраэдра касаются всех восьми его касательных сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Отметим несколько вопросов, которые возникают в связи с рассмотренными конструкциями.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Вопрос 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Инцидентны ли какие либо из следующих четверок замечательных прямых тетраэд- ра: псевдоцентроидали, псевдоортоцентрали, оси четырех описанных конусов, оси четырех конусов Эйлера?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Вопрос 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Существуют ли еще какие-либо квадрики, касающиеся всех касательных сфер, отлич- ные от четырех конусов Эйлера и четырех плоскостей граней?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Вопрос 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Любые три конуса общего положения пересекаются в восьми точках.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Не окажется ли так, что четыре конуса Эйлера тетраэдра имеют восемь общих точек?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Есть ли какие-то примечательные свойства биквадратических кривых, по которым пересекаются конусы Эйлера?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 2 Теорема Грейса как трехмерный аналог теоремы Фейербаха Более ста лет назад, британский математик Джон Хилтон Грейс в своей работе [7] открыл и доказал следующее замечательное свойство касательных сфер тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 (Grace, 1897).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Касательные сферы тетраэдра ABCD могут быть разбиты на че- тыре пары так, что парные сферы гомотетичны с центром D, и для каждой пары существует касающаяся их сфера, проходящая через вершины A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Замечание 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Все касательные сферы можно разбить на две группы по четыре сферы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В одну входят вписанная и три дважды-вневписанные сферы, а в другую – четыре вневписанные.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Любые две сферы из разных групп гомотетичны относительно одной из вершин тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для каждой 4 такой пары сфер существует единственная касающаяся их сфера Грейса, которая проходит через вершины грани, противоположной к той вершине, относительно которой данная пара касательных сфер гомотетична.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, всего получается шестнадцать сфер Грейса: для каждой из четырех граней тетраэдра через ее вершины проходит четыре различные сферы Грейса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема Грейса связывает касательные сферы тетраэдра с замечательными точками, его верши- нами, с помощью общих касающихся их сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Это ее сближает с теоремой Фейербаха, с которой она, на наш взгляд, сравнима по красоте и имеет некоторое сходство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В этом смысле, можно было бы считать теорему Грейса неким трехмерным аналогом теоремы Фейербаха.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 2: Сфера Грейса GD, касающаяся вписанной сферы σ, вневписанной сферы σD и проходящая через A, B, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В недавней статье [13] Maehara и Martini замечают, что «по-видимому, эта теорема малоизвестна и до сих пор не имеет элементарного доказательства».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В качестве результата они приводят такое доказательство, но лишь для частного случая триортогонального тетраэдра, пользуясь при этом аналитической техникой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Оригинальное же доказательство Грейса очень красивое и геометрическое, но довольно трудное.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поскольку Грейс дал лишь его набросок, Maehara и Tokushige в работе [12] подробно реконструиро- вали это доказательство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Мы получим элементарное и вполне короткое геометрическое доказательство теоремы Грейса, но сначала напомним некоторые определения и факты проективной геометрии.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть E3 – веще- ственное трехмерное евклидово пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Мы будет рассматривать его проективное пополнение «бесконечно удаленной» плоскостью.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Эта модель проективного пространства получается переходом от декартовых координат (x, y, z) в E3 к однородным координатам (x : y : z : w), в которых бесконеч- но удаленной плоскости соответствуют точки с координатами (x : y : z : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Кроме того рассмотрим комплексификацию пространства, позволяя координатам принимать комплексные значения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Добав- ленные точки будем называть мнимыми.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Записывая в однородных координатах (x : y : z : w) общее уравнение сферы x2 + y2 + z2 + 2axw + 2byw + 2czw + dw2 = 0, легко видеть, что она пересекает бесконечно-удаленную плоскость w = 0 по кривой x2 + y2 + z2 = 0, w = 0, 5 D GD A C B ODкоторая является общей для всех сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Она называется абсолютной окружностью.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Всякая плоскость пересекает абсолютную окружность в двух сточках – круговых точках этой плоскости.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В однородных координатах (x : y : z) на плоскости ее круговыми точками являются точки I = (1 : i : 0) и J = (1 : −i : 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Все окружности плоскости проходят через ее круговые точки и каждая коника плоскости, проходящая через ее круговые точки, является окружностью (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [16, § 4·8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Прямая, пересекающая абсолютную окружность, называется изотропной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Каждая такая прямая является, естественно, мнимой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Предложение 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='3 ( [22, Гл.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 12, § 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Касательные к невырожденной конике, проведенные из любого ее фокуса, являются изотропными.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, каждая прямая, проходящая через фокус коники и круговую точку ее плоскости, является изотропной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для окружности это означает, что касательные из ее центра проходят через круговые точки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Образующей квадрики называется прямая, которая целиком принадлежит поверхности этой квад- рики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В комплексном проективном пространстве все невырожденные квадрики эквивалентны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Предложение 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='4 ( [9, § 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (i) Через каждую точку невырожденной квадрики проходят ровно две образующие, действительны или мнимые.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Касательная плоскость пересекает квадрику по двум образующим, проходящим через точку касания.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (ii) Все образующие квадрики распадаются на два семейства таким образом, что любые две обра- зующие из одного семейства не пересекаются, а любые две образующие из разных семейств пересекаются.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Через любую точку образующей одного семейства проходит единственная об- разующая другого семейства.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (iii) Любая плоскость, проходящая через образующую квадрики касается этой квадрики в некото- рой точке этой образующей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть даны две сферы γ и η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рассмотрим множество M(γ, η) сфер, которые касаются обеих сфер γ и η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим что множество M(γ, η) распадается на два класса эквивалентности по типу касаний.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если сфера α касается γ и η одинаковым образом (обеих внутренним, или обеих внешним), то α принадлежит одному классу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если же α касается γ и η различным образом (одной сферы внутренним, а другой внешним, или наоборот), то α принадлежит другому классу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Прямые, проходящие через точки касания γ и η со сферами одного класса, проходят через общую точку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для сфер одного класса эта точка – один из двух центров инверсии, переводящей γ и η друг в друга, а для сфер другого класса – второй такой центр (эти точки – центры подобия сфер γ и η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Замечание 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Все это имеет место быть и в случае, если, скажем, сфера η вырождается в плоскость π (сферу бесконечно большого радиуса).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда рассмотренные выше инверсные центры γ и π – это точки сферы γ, касательные плоскости в которых параллельны π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Следующая теорема является главным результатом этого параграфа.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Она описывает семейство коник σ, которые вместе с данной окружностью Σ образуют 3-пару Понселе (Σ, σ), т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' для них суще- ствует треугольник, вписанный в Σ и описанный около σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Из этой теоремы практически мгновенно следует теорема Грейса, что мы сразу покажем после ее формулировки.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='6 (О 3-парах Понселе).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть даны плоскость π и окружность Σ на ней.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Фиксируем сферу γ, содержащую окружность Σ, и рассмотрим множество M(γ, π) сфер, касающихся сферы γ и плоскости π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда если сферы α и β пробегают разные классы множества M(γ, π), то описан- ный около них конус K высекает на плоскости π семейство коник σ, образующих 3-пару Понселе с окружностью Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 6 Доказательство Теоремы Грейса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть α и β – две касательные сферы тетраэдра ABCD, гомо- тетичные относительно вершины D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рассмотрим сферу γ, касающуюся сфер α и β и проходящую через вершины A и B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таких сфер, вообще говоря, целых четыре.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Но две из них в данном случае вырождены в плоскости ⟨DAB⟩ и ⟨ABC⟩, которые принадлежат разным классам множества M(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда оставшиеся две сферы тоже принадлежат разным классам и в качестве γ выберем ту, которая принадлежит другому, нежели плоскость ⟨ABC⟩, классу.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть она пересекает плоскость ⟨ABC⟩ по окружности Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Описанный около α и β конус с вершиной D пересекает плоскость ⟨ABC⟩ по конике σ, касающейся сторон треугольника ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' По Теореме о 3-парах Понселе вершина C также должна лежать на окружности Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Доказательство Теоремы 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='6 о 3-парах Понселе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть Fα и Fβ – тоски касания сфер α и β с плоскостью π, которые по теореме Данделена (1822, [3]) являются фокусами коники σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Далее будем считать, что точки Fα и Fβ не совпадают друг с другом и с центром окружности Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Эти частные случаи сводятся к общему малым шевелением сфер α и β и утверждение теоремы для них получается предельным переходом.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если I – одна из круговых точек плоскости π, то I ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Обозначим через Pα и Pβ точки вторичного пересечения прямых IFα и IFβ с коникой Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда треугольник IPαPβ вписан в окружность Σ, прямые IPα и IPβ касаются коники σ, и нам достаточно доказать, в силу теоремы Понселе, что прямая PαPβ тоже касается коники σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 3: 3-пары Понселе (Σ, σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Мнимые касательные представлены дугообразными розовыми отрезками.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть A и B – точки касания сферы γ со сферами α и β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что прямая IFα является образующей сферы α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Обозначим через lA одну из двух образующих сферы α в точке A, которая пересекает образующую IFα (т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' lA и IFα принадлежат разным семействам образующих сферы γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поскольку lA является также образующей и сферы γ, точка пересечения lA ∩ IFα – это одна из двух точек пересечения прямой IFα со сферой γ, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' это либо точка I, либо точка Pα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что первый случай не возможен в силу нашей договоренности считать, что точка Fα отлична от центра окружности Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В самом деле, I лежала бы тогда в пересечении касательных плос- костей сферы α в точках A и Fα, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' полярно-сопряженная к AFα относительно α прямая содержала бы круговую точку I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' А так как она вещественная и потому не может быть изотропной, она являлась 7 人 T A P a P Fp D Fa Bбы бесконечно-удаленной, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' касательные плоскости сферы α в точках A и Fα были бы параллельны, а точка Fα совпадала бы с центром окружности Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, прямая APα является общей образующей lA сфер α и γ в точке A, и аналогично, прямая BPβ совпадает с lB – одной из двух общих образующих сфер β и γ в точке B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Покажем, что lA и lB компланарны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для этого рассмотрим гомотетию с центром A, переводящую α в γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть gA – образующая сферы γ, в которую переходит образующая IFα сферы α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что 1) I ∈ gA, поскольку gA ∥ IFα, 2) прямая gA инцидентна с прямой lA, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' к.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' прямая lA инвариантна при рассмотренной гомоте- тии и инцидентна с прямой IFα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' gA и lA – две образующие сферы γ, принадлежащие разным семействам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Аналогично, если gB – образующая сферы γ, в которую переходит образующая IFβ сферы β при гомотетии с центром B, переводящей β в γ, то 3) I ∈ gB, 4) gB и lB – тоже две образующие сферы γ, принадлежащие разным семействам.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Из замечания 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='5 следует, что прямые gA и gB проходят через различные инверсные центры сферы γ и плоскости π, а потому различны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда из 1) и 3) следует, что образующие gA и gB сферы α имеют общую точку и, значит, принадлежат разным семействам, откуда в силу 2) и 4) следует, что образующие lA и lB тоже из разных семейств, а потому компланарны.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теперь рассмотрим плоскость ⟨lA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' lB⟩, которая в силу утверждения [iii] Предложения 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='4 касается обеих сфер α и β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что вершина конуса K содержит прямую AB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Действительно, поскольку конус K пересекает π по невырожденной конике, его вершина не лежит на π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Так как α и β из разных классов множества M(γ, π), то γ и π из разных классов множества M(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Значит, прямая AB проходит через инверсный центр сфер α и β, который не лежит на плоскости π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='о.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', ⟨lA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' lB⟩ – касательная плоскость конуса K, а потому пересекает плоскость π по прямой, касающейся коники σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Осталось заметить, что ⟨lA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' lB⟩ пересекает π по прямой PαPβ, и таким образом, треугольник IPαPβ является вписано-описанным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ 3 Формулы Эйлера-Чаппла и up-in-ex-touch-аналог теоремы Фейер- баха Теорема 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 (Euler, Chapple).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть R, r и ra – радиусы описанной, вписанной и вневписанной окружностей произвольного треугольника, d и da – расстояния от центра описанной окружности до центров вписанной и вневписанной.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда выполняются следующие соотношения d2 = R2 − 2Rr (1) d2 a = R2 + 2Rra (2) Мы приведем два, наверное, самых коротких доказательства этой теоремы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для этого рассмотрим сферу ∆, построенную диаметрально на описанной окружности, наовем ее описанной сферой тре- угольника, сферу δ радиуса r, касающуюся плоскости треугольника в центре его вписанной окруж- ности, наовем ее вписано-поднятой, и сферу δa радиуса ra, касающуюся плоскости треугольника в центре соответствующей вневписанной окружности, наовем ее вневписано-поднятой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что соотношения (1), (2) можно переписать в виде равенств d2 + r2 = (R − r)2, d2 + r2 a = (R + ra)2, которые равносильны касанию сфер ∆ и δ, ∆ и δa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 8 Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 4: Сферы ∆ и δ касаются друг друга Доказательство 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Касания ∆ и δ, ∆ и δa сразу следует из Теоремы Грейса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Действительно, рассмотрим тетраэдр с основанием ABC и вершиной D на бесконечности в перпен- дикулярном к плоскости (ABC) направлении.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда сфера δ является его вписанной сферой, симметричная ей относи- тельно плоскости (ABC) – его вневписанной сферой, а сле- довательно, сфера ∆ – его сферой Грейса.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для пары ∆ и δa рассуждение аналогично.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Это доказательство примечательно своей лаконичностью и красотой, но использование сложной Теоремы Грейса мо- жет выглядеть как «стрельба из пушки по воробьям».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поэто- му приводим другое Доказательство 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Сделаем инверсию относительно сфе- ры, построенной диаметрально на вписанной окружности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что сфера ∆ переходит в сферу ∆′, построенную диаметрально на окружности, проходящей через середины сторон треугольника Жергона (верши- нами которого являются точки касания вписанной окружности △ABC со сторонами).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' А сфера δ переходит в плоскость δ′, удаленную от плоскости (ABC) параллельно на расстояние r 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поскольку, радиус сферы ∆′, очевидно, тоже равен r 2, сферы ∆′ и δ′, а следовательно, и сферы ∆ и δ касаются друг друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Заметим, что доказанное свойство касания сферы ∆ с четырьмя сферами δ, δa, δb, δc является своего рода тоже неким аналогом теоремы Фейербаха в пространстве.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2 (Up-in-ex-touch).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Описанная сфера треугольника касается его вписано-поднятой и че- тырех вневписано-поднятых сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 5: Up-in-ex-touch-аналог теоремы Фейербаха.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 9 Заметим также, что сфера ∆ касается не только сфер δ, δa, δb, δc, но и еще четырех симметричных им относительно плоскости треугольника, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' целых восьми сфер.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 4 Теорема Лагерра и ее применение к тетраэдру Теорема 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 (Laguerre [10], 1879).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Окружность Σ радиуса R с центром в точке O и коника σ с фокусами Fα, Fβ и малой полуосью b образуют 3-пару Понселе тогда и только тогда, когда выпол- няется соотношение (R2 − d2 α)(R2 − d2 β) = 4R2b2, (3) где dα = |OFα|, dβ = |OFβ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Замечание 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Малая полуось b может быть как действительной (у эллипсов), так и мнимой (у гипербол).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В первом случае из формулы Лагерра видно, что фокусы эллипса должны лежать либо оба внутри окружности, либо оба вне.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Во втором случае, у гиперболы, один фокус должен лежать внутри окружности, другой – снаружи.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Замечание 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если коника σ является параболой, то условие существования вписано-описанных треугольников для пары (Σ, σ) становится совсем простым: d = R, где d = |OF|, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' фокус F параболы должен лежать на окружности.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Это следует из известной теоремы Ламбера.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Доказательство Теоремы Лагерра (⇒) Пусть γ – произвольная сфера, содержащая окружность Σ, а cфера α касается в точке Fα плоскости π, содержащей окружность Σ, а также касается сферы γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рассмотрим произвольный вписано-описанный треугольник ABC и проведем через его стороны ка- сательные плоскости к сфере α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Они пересекаются в некоторой точке D, образуя тетраэдр ABCD, у которого сфера α является одной из касательных сфер, а γ – сферой Грейса, которая касает- ся также другой касатеьной сферы β тетраэдра ABCD, гомотетичной α относительно вершины D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Как известно, сферы α и β касаются плоскости π в точках, изогонально сопряженных относительно △ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Кроме того, поскольку Fα и Fβ – фокусы вписанной в △ABC коники σ, они также изого- нально сопряжены.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Отсюда заключаем, что сфера β касается плоскости π в точке Fβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Нам понадобится одна очень простая лемма Лемма 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='4 (Thebault [17], 1922).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для малой полуоси b коники, высекаемой описанным около сфер α и β конусом на их общей касательной плоскости, выполняется соотношение |b2| = rαrβ (4) Пусть Sα и Sβ – две диаметрально противоположные точки на γ в перпендикулярном к плоскости π направлении, которые являются инверсными центрами сферы γ и плоскости π (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' замечание 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Учитывая, что сферы α и β принадлежат разным классам множества M(γ, π) (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' доказательство теоремы Грейса), легко выразить радиусы сфер α и β: rα = ���� Σ(Fα) 2π(Sα) ���� , rβ = ���� Σ(Fβ) 2π(Sβ) ���� , (5) где Σ(Fα) = d2 α − R2 и Σ(Fβ) = d2 β − R2 – степени точек Fα и Fβ относительно окружности Σ, а π(Sα), π(Sβ) – расстояния от точек Sα и Sβ до плоскости π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Перемножим равенства (5) и учтем, что π(Sα)π(Sβ) = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Получим, что в равенстве (3) левая и правая части равны по модулю.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Правая часть отрицательна только в случае, если коника σ является гиперболой.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Такое происходит только тогда, когда сферы α и β касаются описанного около них конуса с вершиной D по разные стороны от D, а плоскости π – по одну сторону.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда сферы γ они должны касаться по разные стороны, а следовательно, точки Fα и Fβ их касания с π относительно 10 окружности Σ лежат тоже по разные стороны и левая часть (3) в этом случае также отрицательна.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, модули можно снять и равенство (3) считать доказанным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (⇐) Пусть выполняется (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если коники (Σ, σ) не образуют 3-пару Понселе, то можно изменить малую полуось b коники σ так, чтобы они образовали 3-пару Понселе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда по уже доказанному тоже должно выполняться равенство (3), следовательно величина b не изменилась, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (Σ, σ) как раз и образуют 3-пару Понселе ✷ Теорема Лаггера, примененная к тетраэдру, позволяет получить следующее интересное метриче- ское соотношение для касательных сфер тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть ∆D – сфера, описанная около грани ABC тетраэдра ABCD, α и β – две касательные сферы, гомотетичные относительно D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда произведение косинусов углов, которые сфера ∆D образует с α и β (среди них один угол мнимый), равно 1, если α и β касаются ⟨ABC⟩ с одной стороны, или −1, если с разных.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' cos(� ∆D, α) cos(� ∆D, β) = sign k, (6) где k – коэффициент упомянутой гомотетии с центром D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Доказательство Пусть OD и R – центр и радиус сферы ∆D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' rα, rβ – радиусы сфер α и β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Dα, Dβ – расстояния между центрами ∆D и α, ∆D и α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' dα, dβ – расстояния от OD до точек Fα и Fβ касания плоскости ⟨ABC⟩ со сферами α и β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть Σ = ⊙(ABC), а конус K с вершиной D, описанный около α и β, пересекает плоскость ⟨ABC⟩ по конике σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Воспользуемся леммой 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='4 и заметим, что в нашей конструкции с тетраэдром равенство (4) можно уточнить b2 = rαrβ sign k, (7) поскольку b2 может быть отрицательным, только если коника σ является гиперболой, что возможно лишь в том случае, если вершина конуса K является центром отрицательной гомотетии сфер α и β, т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' они вписаны в K по разные стороны от его вершины.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' По теореме Лагерра для пары (Σ, σ) имеем (R2 − d2 α)(R2 − d2 β) = 4R2b2, (8) По теореме Пифагора d2 α = D2 α − r2 α, d2 β = D2 β − r2 β Подставляя эти равенства и (7) в соотношение (8), получаем требуемое соотношение R2 + r2 α − D2 α 2R rα R2 + r2 β − D2 β 2R rβ = sign k ✷ 5 Трехмерный аналог формулы Эйлера-Чаппла В связи с теоремой Эйлера-Чаппла возникает естественный вопрос о возможности ее трехмерного обобщения на случай тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Этот вопрос был поставлен впервые Ж.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Д.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Жергонном в 1816 году в издаваемом им журнале1 в виде краткой сноски, относящейся к тетраэдру с радиусами описанной сферы R, вписанной – r и расстоянием d между их центрами: 1Annales de math´ematiques pures et appliqu´ees, 6 (1815-1816), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 11 «Il serait sur-tout int´eressant de savoir si d peut ˆetre exprim´e uniquement en fonction de R et r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='» Спустя восемь лет в том же журнале было опубликовано положительное решение этой задачи в работе Дюрранда [4], где он доказал следующее соотношение: d2 = (R + r)(R − 3r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (9) Этот результат получил широкое признание и в течение многих лет на него ссылались в литерату- ре, например, в таких почтенных изданиях как Математическая энциклопедия Клейна «Encyklop¨adie der mathematischen Wissenschaften» [18] (первая в мире математическая энциклопедия) и «Enciclopedia delle matematiche elementari» [1] (крупнейшая энциклопедия по математике, изданная в Италии).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Од- нако, формула Дюрранда (9) оказалась неверной, а ответ на вопрос Жергонна – отрицательным: не существует общей для всех тетраэдров функциональной зависимости между R, r и d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Доказатель- ство Дюрранда было практически безупречным, но незаметная ошибка заключалась в его убежден- ности, что описанная и вписанная сфера непременно должны иметь некоторую зависимость.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Вопрос Жергонна можно было бы сформулировать так: каковы условия существования вписано-описанного тетраэдра для двух данных сфер?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Оказывается никаких необходимых условий для этого не требуется.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для любых двух невырожденных квадрик общего положения существует бесконечное семейство вписано-описанных тетраэдров.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Любая касательная плоскость ко вписанной квадрике может содержать грань такого тетраэдра, а его вершиной может быть произвольная точка описанной квадрики.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В работе Фонтене [6] 1899 года эта теорема считается уже известной (см.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' также [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Итак, в отличие от плоского случая в пространстве для любых двух произвольных сфер всегда существует вписано-описанный в них тетраэдр, причем он может динамически вращаться около этих сфер, все время оставаясь вписано-описанным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' При этом, любая точка описанной сферы может быть вершиной такого тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Но оказывается, что не для любых двух вещественных сфер такой тетраэдр может быть веще- ственным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Критерием существования вещественного вписано-описанного тетраэдра является следу- ющее условие Грейса, исправляющее соотношение Дюрранда (9): Теорема 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2 (Grace [8], 1917).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для данных двух сфер S и T необходимым и достаточным условием существования вписано-описанного вещественного тетраэдра, у которого вершины лежат на S, а плоскости граней касаются T, является следующее условие в зависимости от взаимного располо- жения S и T: (a) T вложена в S и d2 ⩽ (R + r)(R − 3r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (b) T и S расположены одна вне другой;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' (c) T и S пересекаются по действительной окружности и d2 ⩽ (R − r)(R + 3r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 6 Вращение Понселе вписано-описанного тетраэдра Теорема 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 позволяет рассмотреть динамику «вращения» вписано-описанного тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Эта дина- мика не столь однозначна, как в плоской теореме Понселе.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Это показывает следующая теорема.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 12 Теорема 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 ( [8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть вершины тетраэдра лежат на квадрике S, а грани касаются квадрики T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда при фиксации плоскости π одной из его граней противоположная вершина P может при этом варьироваться, пробегая плоское сечение π′ квадрики S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, тетраэдр вращается с намного большей свободой, чем вписано-описанный мно- гоугольник.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Когда выбрана плоскость π, существует целая коника для выбора произвольной точки на ней в качестве вершины P, а для каждой такой пары P и π существует однопараметрическое семейство вписано-описанных треугольников, каждый из которых может быть противоположной к вершине P гранью вписано-описанного тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, в общем случае существует 4- параметрическое семейство тетраэдров.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' У плоской теоремы Понселе есть такой «эффект замыкания»: если начиная с некоторой начальной точки A1 строится последовательно вписано-описанная ломаная A1A2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' An и оказывается, что звено A1An тоже касается вписанной коники, замыкая ее, то такое замыкание будет происходить всегда.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если же, по аналогии, строить вписано-описанный тетраэдр для двух данных квадрик S и T, последовательно выбирая касательные плоскости его граней, то возникает следующий вопрос.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Когда мы провели уже три плоскости, которые образовали вписано-описанный трехгранный угол, всегда ли можно его замкнуть четвертой плоскостью, чтобы образовался вписано-описанный тетраэдр?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Ответ дает следующая теорема Фонтене.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2 (Fonten´e [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Последовательный процесс построения вписано-описанного тетраэдра всегда замыкается тогда и только тогда, когда квадрики S и T имеют четыре общих образующих.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В этом случае, плоскость π и вершина P могут быть выбраны совсем произвольно и, таким образом, существует 5-параметрическое семейство вписано-описанных тетраэдров.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть фиксированы описанная сфера S тетраэдра и одна из восьми его касательных сфер T, а тетраэдр динамически «вращается» около них, оставаясь вписано-описанным.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда все четыре касающиеся T сферы Грейса все время касаются некоторой фиксированной сферы, концен- тричной с описанной сферой S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Доказательство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть сфера Грейса G проходит через вершины грани a и пусть вписанная сфера S касается сферы G в точке P, а плоскости ⟨a⟩ – в точке Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Обозначим центры сфер S и T через OS и OT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Прямая PQ при вращении тетраэдра проходит через фиксированную точку – предельную точку K пучка сфер ⟨S, T⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Кроме того, на прямой PQ лежит инверсный центр E сферы G и плоскости ⟨a⟩, касательная в котором к G параллельна плоскости ⟨a⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Следовательно, OSE∥OT Q и △OSEK ∼ △OT QK, откуда получаем такое выражение OSE = OSK OT K · rT , правая часть которого является величиной постоянной при вращении тетраэдра.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда, сфера с ради- усом, равным этой величине, и центром в точке OS касается сферы Грейса в любой момент вращения.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ 7 Доказательство теоремы Фейербаха через выход в пространство Пусть δ – вписанная окружность треугольника ABC с центром в точке I и радиусом r, H – ор- тоцентр треугольника ABC, точки A1, B1, C1, I1 – середины отрезков AH, BH, CH, IH (I1 – инцентр △A1B1C1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Описанная около △A1B1C1 окружность ϑ – это окружность девяти точек △ABC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть также ⊙a, ⊙b, ⊙c – окружности с диаметрами BC, CA, AB, ∆ и Θ – сферы, построенные диаметраль- но на окружностях δ и θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 13 Доказательство Теоремы Фейербаха.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что касание окружностей δ и θ равносильно касанию сфер ∆ и Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' По Теореме 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2 для △A1B1C1 его описанная сфера Θ касается его вписано-поднятой сферы Υ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поэтому касание Θ и ∆ равносильно тому, что сфера Θ инвариантна при инверсии, переводящей сферы ∆ и Υ друг в друга.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что центр S этой инверсии расположен над точкой H на высоте r (т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' SH⊥(ABC), |SH| = r), а коэффициент инверсии (квадрат радиуса сферы инверсии) равен |IH| · |I1H| = |IH|2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Таким образом, достаточно доказать равенство Θ(S) = |IH|2 2 , которое в силу того, что Θ(S) = θ(H) + r2, равносильно соотношению |IH|2 − 2r2 = 2θ(H) (10) Заметим, что левая часть равенства (10) равна степени точки H относительно окружности ξ радиуса r √ 2 с центром I (ξ высекает на сторонах △ABC равные отрезки длины 2r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Осталось вос- пользоваться следующим замечательным свойством окружности ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Теорема 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Окружности ξ, ⊙a, ⊙b, ⊙c имеют общий радикальный центр в точке H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда заметим, что степень точки H относительно окружности θ в два раза меньше ее степени относительно окружностей ⊙a, ⊙b, ⊙c и равенство (10) равносильно утверждению ξ(H) = ⊙a(H) = ⊙b(H) = ⊙c(H) Теоремы 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Для доказательства Теоремы 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1 рассмотрим окружность χa, диаметром которой является жер- гониана вершины A (т.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='е.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' отрезок, соединяющий A с точкой касания вписанной окружности δ со стороной BC) и воспользуемся следующим свойством окружности χa, возможно, имеющим и само- стоятельный интерес.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Лемма 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='2 (χa-лемма).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Окружности χa, ξ, ⊙a принадлежат одному пучку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Доказательство Теоремы 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Достаточно проверить, что H ∈ rad(ξ, ⊙a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что rad(χa, ⊙b) – это высота AH, rad(⊙a, ⊙b) – это высота CH, следовательно, H = rad(χa, ⊙a, ⊙b) ∈ rad(χa, ⊙a) = rad(ξ, ⊙a), где последнее равенство верно в силу χa-леммы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Доказательство χa-леммы.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Воспользуемся следующим известным метрическим соотношением для пучков окружностей.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Лемма 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='3 (О пучке).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Если окружности α, β, γ лежат в одном пучке, то для любой точки P ∈ γ отношение ее степеней относительно α и β постоянно, причем α(P) β(P) = dαγ dβγ , (11) где dαγ и dβγ – расстояния между центрами α, γ и β, γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Верно и обратное утверждение.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Лемма 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content='4 (Обратная лемма о пучке).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Пусть центры окружностей α, β, γ коллинеарны, и на окружности γ имеется такая точка P, для которой выполняется соотношение (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда окруж- ности α, β, γ принадлежат одному пучку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 14 Рис.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 6: Окружности ξ, χa, ⊙a принадлежат одному пучку В качестве окружностей α, β, γ из Обратной леммы о пучке возьмем окружности ⊙a, ξ, χa, центры M, I, L которых лежат на средней линии ML треугольника APQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' При этом, LM LI = AQ AN = ra r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Для точки P ∈ χa имеем α(P) = −(p − b)(p − c), β(P) = −r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда (11) запишется в виде соотношения (p − b)(p − c) r2 = ra r , которое равносильно легко проверяемому равенству (p − b)(p − c) = r ra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ Доказательство χa-леммы выходом в пространство.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Заметим, что окружности χa, δ и окруж- ность ⊙P Q с диаметром на отрезке PQ лежат в одном пучке.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Поднимем их центры перпендикулярно плоскости ⟨ABC⟩, сохраняя коллинеарность: L → L, I → I, M → M, и пусть LL = r 2, II = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Тогда легко найти, что MM = r + ra 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' При этом сферы S(L), S(J), S(M) с центрами L, I, M, содержащие окружности χa, δ, ⊙P Q соответственно, также принадлежат одному пучку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Рассмотрим плоскость π∥⟨ABC⟩, проходящую через I, и ортогональную проекцию △ABC → △A′B′C′ на плоскость π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Оста- лось заметить, что сечениями сфер S(L), S(J), S(M) плоскостью π являются окружности ξ′, χ′ a, ⊙′ a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Действительно, для сечений S(L), S(I) это очевидно, а для S(M) это легко проверить, поскольку квадрат радиуса окружности ее сечения плоскостью π равен |MP|2 + �ra + r 2 �2 − �ra − r 2 �2 = |MP|2 +rar = |MP|2 +(p−b)(p−c) = |MP|2 +|BP|·|CP| = � a 2 �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Так как при пересечении сфер пучка плоскостью получается пучок окружностей, то ξ′, χ′ a, ⊙′ a принадлежат одному пучку.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ✷ 15 3 N L H B M P CСписок литературы [1] Biggiogero G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', La geometria del tetraedro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' In: Enciclopedia delle Matematiche Elementari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' A cura di L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Berzotari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Vivanti e D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Gigli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Volume II, parte I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Milano 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Ristampa anastatica, Maggio 1943, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [2] Coolidge J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', A treatise on the circle and the sphere, Oxford: Clarendon Press, 1916.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [3] Dandelin G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', M´emoire sur quelques propri´et´es remarquables de la focale parabolique, Nouveaux m´emoires de l’Acad´emie rouale des sciences et belles-lettres de Bruxelles, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 2 (1822) 171-200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [4] Durrande J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', D´emonstrations ´el´ementaires des principales propriet´es des hexagones inscrits et circonscrits au cercle, suivies de la solution de divers probl`emes de la g´eometrie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Dissertation de la g´eometrie pure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Annales de math´ematiques pures et appliqu´ees, 14 (1823- 1824) 29-63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [5] Feuerbach K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Eigenschaften einiger merkw¨urdigen Punkte des geradlinigen Dreiecks, N¨urnberg, 1822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [6] Fonten´e G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Sur des poly`edres mobiles comparables aux polygones de Poncelet, Nouvelles annales de math´ematiques 3e s´erie, tome 18 (1899) 67-74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [7] Grace J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Circles, spheres, and linear complexes, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Cambridge Philosophical Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 14 (1898) 153–190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [8] Grace J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Tetrahedra in relation to spheres and quadrics, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 17 (1918) 259-271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [9] Griffiths Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Harris J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', A Poncelet theorem in space, Comm Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Helv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', 52 (1977) 145-160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [10] Laguerre E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Sur la relation qui existe entre un cercle circonscrit `а un triangle et les ´el´ements d’une conique inscrite dans ce triangle, Nouvelles annales de math´ematiques 2e s´erie, tome 18 (1879) 241-246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [11] Lewis T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Is there an analogue in solid geometry to Feuerbach’s theorem, Messenger of Mathematics, volume 49 (1919) 187-192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [12] Hiroshi Maehara, Norihide Tokushige, Schl¨afli’s double six, Lie’s line-sphere transformation, and Grace’s theorem, European Journal of Combinatorics, 30 (2009) 1337–1351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [13] Hiroshi Maehara, Horst Martini, Tangent Spheres of Tetrahedra and a Theorem of Grace, The American Mathematical Monthly, 127:10 (2020) 897-910 [14] Poncelet J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' - V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Trait´e des propri´et´es projectives des figures, Gauthier-Villars, Paris, 1822.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [15] Protasov V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Generalized closing theorems, Elem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', 66 (2011) 98-117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [16] Sommerville, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Analytical geometry of three dimensions, Cambridge University Press, 1943.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [17] Thebault V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Sur un theoreme classique de Dandelin, Nouvelles annales de mathematiques 5e serie, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 1 (1922) 200-205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [18] Zacharias M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Elementargeometrie und elementare nicht-euklidische Geometrie in synthetischer Behandlung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' In: Encyklop¨adie der Mathematischen Wissenschaften mit Einschluß ihrer Anwendungen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Drifter Band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Redigiert yon W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Fr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Meyer und H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Mohrmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' ˙Teubner, Leipzig, 1914-1931, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 1059.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [19] Акопян А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' В.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', О некоторых классических конструкциях в геометрии Лобачевского, Матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' просв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', выпуск 13 (2009) 155–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [20] Берже М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Геометрия, М.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' Мир, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' [21] Заславский А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Сравнительная геометрия треугольника и тетраэдра, Матем.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' просв.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', вып.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 8 (2004) 78–92 [22] Фиников С.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' П.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=', Аналитическая геометрия, Москва, 1952.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} +page_content=' 16' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tAyT4oBgHgl3EQf1PlJ/content/2301.00731v1.pdf'} diff --git a/1dA0T4oBgHgl3EQfMv9X/vector_store/index.faiss b/1dA0T4oBgHgl3EQfMv9X/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..632b04d9236a81553580e77e41e0c1a78dcad97a --- /dev/null +++ b/1dA0T4oBgHgl3EQfMv9X/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e029f237af3297266b95dfbf08e89b275cb74db1efba6f1448b06efacfba3c93 +size 4784173 diff --git a/1dA0T4oBgHgl3EQfMv9X/vector_store/index.pkl b/1dA0T4oBgHgl3EQfMv9X/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3a1656c7438223895b2864088ba55844c0602ee1 --- /dev/null +++ b/1dA0T4oBgHgl3EQfMv9X/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e9c5e99ee97fadf397dab70a0c2f87bf125968574238a05ec6f08689b346a326 +size 167994 diff --git a/1dAzT4oBgHgl3EQfevzu/content/tmp_files/2301.01443v1.pdf.txt b/1dAzT4oBgHgl3EQfevzu/content/tmp_files/2301.01443v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d57d77d84da44644b777f2471baabe413db5a1e9 --- /dev/null +++ b/1dAzT4oBgHgl3EQfevzu/content/tmp_files/2301.01443v1.pdf.txt @@ -0,0 +1,551 @@ +A QUANTUM APPROACH FOR STOCHASTIC CONSTRAINED BINARY OPTIMIZATION +Sarthak Gupta and Vassilis Kekatos +Bradley Dept. of ECE, Virginia Tech, Blacksburg, VA 24061, USA; {gsarthak,kekatos}@vt.edu +ABSTRACT +Analytical and practical evidence indicates the advantage +of quantum computing solutions over classical alternatives. +Quantum-based heuristics relying on the variational quantum +eigensolver (VQE) and the quantum approximate optimiza- +tion algorithm (QAOA) have been shown numerically to +generate high-quality solutions to hard combinatorial prob- +lems, yet incorporating constraints to such problems has +been elusive. To this end, this work puts forth a quantum +heuristic to cope with stochastic binary quadratically con- +strained quadratic programs (QCQP). Identifying the strength +of quantum circuits to efficiently generate samples from prob- +ability distributions that are otherwise hard to sample from, +the variational quantum circuit is trained to generate binary- +valued vectors to approximately solve the aforesaid stochastic +program. The method builds upon dual decomposition and +entails solving a sequence of judiciously modified standard +VQE tasks. Tests on several synthetic problem instances us- +ing a quantum simulator corroborate the near-optimality and +feasibility of the method, and its potential to generate feasible +solutions for the deterministic QCQP too. +Index Terms— QAOA, VQE, dual decomposition, quan- +tum unconstrained binary optimization (QUBO). +1. INTRODUCTION +Quantum computers exhibit an innate ability to handle ex- +ponentially large computations in a parallel fashion yet with +a strong probabilistic flavor. +Quantum algorithms such as +Shor’s integer factorization, Grover’s search, and the linear +system solver of Harrow-Hassidim-Lloyd (HHL) can attain +polynomial or even exponential speedups over the best known +algorithms on classical computers [1]. Nonetheless, some of +these quantum algorithms presume a large number of qubits +on fault-tolerant quantum computers. In the near-term inter- +mediate scale (NISQ) era, quantum computers are noisy and +thus oftentimes limited in terms of number of gates and/or +qubits. With such limitations in mind, variational quantum +algorithms have been suggested as promising tools to practi- +cally showcase quantum advantage in the NISQ setup [2]. +This work was supported by a seed funding grant from the Virginia Com- +monwealth Cybersecurity Initiative (CCI) – Southwest Virginia node. +Variational quantum computers involve a sequence of pa- +rameterized gates. Their parameters are updated externally +by a classical computer in a closed-loop fashion to steer the +quantum state towards a desirable direction. The variational +quantum eigensolver (VQE) used to provide high-quality +solutions to combinatorial problems is a representative ex- +ample. The Quantum Approximate Optimization Algorithm +(QAOA) [3] is a special instance of VQE. In QAOA, not +only the parameters but also the architecture of the quan- +tum circuit become problem-dependent. The quantum circuit +trained by QAOA operates as a sampler to efficiently gener- +ate near-optimal solutions of binary quadratic problems (e.g., +MAXCUT); see [4] for a summary of claims on QAOA. +While most VQE/QAOA schemes target unconstrained +problems, dealing with constraints is crucial to several appli- +cations in machine learning, wireless communications, and +financial (stock trading) optimization. +Adding constraints +to QAOA or adiabetic quantum computing [5] (the QAOA +counterpart for non-gate-based quantum computers) has been +pursued in two ways. +One approach has been to convert +the constrained problem into an unconstrained minimization +of a Lagrangian-like function [6, 7]. However, the weights +for constraint penalties can be safely selected only if con- +straints are expressed as Boolean functions or linear equal- +ities. An alternative approach modifies the architecture of +the quantum circuit (via the mixer Hamiltonian of QAOA) +to confine quantum states on the subspace spanned by con- +straints [8, 9, 4, 10]. Nonetheless, constructing such ‘driver’ +mixer Hamiltonians is again highly problem-dependent and +often limited to equality constraints. Reference [11] devel- +ops a quantum adiabetic approach to tackle binary linearly- +constrained quadratic programs. It targets the dual problem +and interfaces the quantum computer with a branch-and- +bound scheme ran classically. Reference [12] treats mixed- +binary quadratic-plus-convex problems using the alternating +direction method of multipliers (ADMM) to split binary +and continuous variables into separate minimizations, solved +by QAOA and classical convex optimizers respectively per +ADMM iteration. +Relation to prior work. +Addressing binary QCQPs by +quantum heuristics has been largely unexplored to the au- +thors’ knowledge. We put forth a quantum-based heuristic +to solve a stochastic binary QCQP. Harnessing the power of +quantum circuits to sample from probability mass functions +arXiv:2301.01443v1 [quant-ph] 4 Jan 2023 + +(PMF) that are hard to sample classically, we devise a dual +decomposition technique that solves a sequence of standard +VQE tasks to systematically adjust Lagrangian multipliers. +Numerical tests using quantum computer simulators pro- +vided by IBM evaluate this technique on randomly generated +stochastic and deterministic binary QCQPs. +2. QUANTUM COMPUTING PRELIMINARIES +A quantum system consisting of n quantum bits (qubits) is de- +scribed by an exponentially large state vector |x⟩ ∈ CN with +N = 2n assuming the system is in a pure state. The Dirac no- +tation |x⟩ named ket emphasizes that vector x is unit-norm or +�N−1 +k=0 |xk|2 = 1. If ek is the k-th canonical vector of length +N, we can write |x⟩ = �N−1 +k=0 xk |ek⟩. The vector ek is of- +tentimes alternatively expressed as |ek⟩ = |k⟩, where k is the +binary representation of index k. For example, a system with +n = 2 qubits has a state in C4, which is spanned by canonical +vectors {ek}3 +k=0 and e0 = [1 0 0 0]⊤ = |00⟩. Vector |x⟩ +provides a statistical characterization for the quantum state: +If we measure the quantum system output, its qubits will be +in configuration |k⟩ with probability |xk|2 for all k. Symbol +⟨x| termed bra denotes the conjugate transpose of |x⟩, while +the braket ⟨x|y⟩ denotes the inner product between states. +The fundamental operations we can perform on a quan- +tum system is evolution and measurement. The former can +be described by the application of a unitary U on state |x⟩ +to produce state |y⟩ = U |x⟩. Although U is exponentially +large, it is usually implemented efficiently using quantum +gates. Among various types of measurements, we focus on +projective measurements. A projective measurement is asso- +ciated with a Hermitian matrix (named observable) and its +eigenvalue decomposition H = �M +m=1 λmvmvH +m. If such +measurement is performed on |x⟩, outcome m is observed +with probability pm := | ⟨x|vm⟩ |2. Define a random variable +taking value λm when outcome m is observed. The expected +value of this variable is ⟨x|H|x⟩ = �M +m=1 pmλm. If H is di- +agonal, the measurement is on the computational basis. This +is practically important because now vm = em, outcome m +relates to |m⟩, and each qubit can be measured individually. +If quantum system i has been prepared in state |xi⟩ for +i = 1, 2, their joint state would be |x1⟩ ⊗ |x2⟩, where ⊗ +is the Kronecker product. This is oftentimes represented as +|x1⟩ |x2⟩ or |x1, x2⟩. The Kronecker product rule generalizes +to the composition of n systems. For example, |1⟩ |1⟩ |0⟩ = +e1 ⊗ e1 ⊗ e0 = e6 = |110⟩, where the canonical vectors +shown in the middle are in R2 and those at the end are in R8. +3. VARIATIONAL QUANTUM EIGENSOLVER (VQE) +VQE is a heuristic approach to find near-optimal solutions for +combinatorial problems of the general form +min +b∈{0,1}n f(b). +(1) +A particular example of interest is the quadratic unconstrained +binary optimization (QUBO) problem with +f(b) = b⊤Ab + b⊤c + d +(2) +which is known to be NP-hard. For later developments, it is +convenient to reformulate QUBO in terms of the spin {±1} +variables through the transformation +si = 1 − 2bi = (−1)bi for i = 0, . . . , n − 1. +(3) +Collecting the spin variables in vector s = 1 − 2b, the +quadratic objective can be equivalently expressed as +f(b) = ¯f(s) = s⊤ ¯As + s⊤¯c + ¯d +(4) +where ¯A := 1 +4A; ¯c := − 1 +2(A1 + c); and ¯d := 1 +41⊤A1 + +1 +21⊤c + d. We next explain how VQE samples high-quality +solutions of (1) by solving an eigenvalue minimization task. +The VQE method falls under the family of variational +quantum algorithms. The term variational pertains to the fact +that the quantum circuit is not fixed, but parameterized by +relatively few parameters collected in vector θ ∈ RP . These +parameters are iteratively adjusted by classical computer in +a closed-loop fashion so that the quantum system eventually +reaches a desirable state. The process resembles the training +of a neural network whose weights are updated by an opti- +mization algorithm. Similarly to neural networks where the +learner has to select an architecture (e.g., network depth/width +and type of activations), the parameterized form (also termed +ansatz) of the variational quantum circuit is specified a pri- +ori. We will be using a 2-local ansatz where single-qubit RY +gates are applied to all qubits, followed by a full entanglement +circuit, all repeated for 3 layers (iterations) [2]. +Given θ and driven by input |0⟩n, the quantum circuit pro- +duces at its output the quantum state |x(θ)⟩ = U(θ) |0⟩n for +a unitary N × N matrix U(θ). To simplify notation, we will +oftentimes write |x⟩ in lieu of |x(θ)⟩. Albeit |x⟩ ∈ CN is +exponentially long, it can be easily generated by the quan- +tum circuit though it cannot be read out of the circuit as a +vector in a computationally efficient manner. Instead, it is rel- +atively easy to sample from it. Every time we run the quan- +tum circuit driven by |0⟩n, we will be observing one of the +binary outputs |k⟩ = |ek⟩ with probability pk := |xk|2 for +k = 0, . . . , N − 1. The quantum circuit thus serves as an ef- +ficient sampler from the exponentially large probability mass +function (PMF) {pk}N−1 +k=0 . +To exploit this sampling property, we next relate the cost +f(b) with a so-termed Hamiltonian matrix H so that +H |ek⟩ = f(|k⟩) |ek⟩ +for all k. +(5) +Matrix H is apparently diagonal and carries all N function +evaluations f(ek) on its diagonal. Moreover, the canonical +vectors ek constitute the eigenvectors of H, each with cor- +responding eigenvalue f(|k⟩). Therefore, the minimization + +in (1) can be reformulated as the problem of finding the eigen- +vector corresponding to the minimum eigenvalue of H +min +|x⟩ ⟨x| H |x⟩ . +(6) +As long as |x⟩ is allowed to take any of the values {ek}N−1 +k=0 , +the minimizer of (6) corresponds to the minimizer of (1). For +example, if a quantum system has n = 3 qubits, its state +would be |x⟩ ∈ C8. Here ek’s are the columns of the identity +matrix I8. If the minimizer of (6) is |e5⟩ = |b1b2b3⟩ = |101⟩, +then the minimizer of (1) is b = [1 0 1]⊤; and vice versa. +Although H is exponentially large, it can be implemented +using only O(n2) quantum gates since it can be expressed as +H = +n−1 +� +i=0 +n−1 +� +j=0 +¯AijZiZj + +n−1 +� +i=0 +¯ciZi + ¯dIN +(7) +where the N × N Hermitian matrix Zi is defined as +Zi = I2 ⊗ · · · ⊗ Z ⊗ · · · ⊗ I2 with Z = +� 1 +0 +0 +−1 +� +. +This is a Kronecker product involving (n − 1) identity matri- +ces I2 and one Pauli-Z operator Z applied to the i-th qubit. +Matrix H as defined in (7) is obviously diagonal. To estab- +lish (5), note first that Z |0⟩ = |0⟩ and Z |1⟩ = − |1⟩, or +more compactly, Z |b⟩ = (−1)b |b⟩. Consequently, when Zi +is applied to a state |b⟩ = |b1b2 · · · bn⟩, the effect is Zi |b⟩ = +(−1)bi |b⟩ = si |b⟩ from (3). Similarly, it also holds that +ZiZj |b⟩ = sisj |b⟩. Property (5) now follows immediately +by postmultiplying (7) by any |ek⟩ and using f(b) = ¯f(s). +If |x⟩ in (6) is restricted to set E := {ek}N−1 +k=0 , problem +(6) is as hard as (1). VQE relaxes (6) to the set of all quantum +states |x(θ)⟩ that can be parameterized by the chosen ansatz +and via θ. Problem (6) is then solved over θ rather than |x⟩ +min +θ +F(θ) := ⟨x(θ)|H|x(θ)⟩ . +(8) +From the eigenvalue property (5), it follows ⟨en| H |ek⟩ = +f(|k⟩) for all k. How about ⟨x| H |x⟩ for a general state |x⟩? +Because |x⟩ = �N−1 +k=0 xk |ek⟩, it is easy to show that +⟨x|H|x⟩ = +N−1 +� +k=0 +|xk|2f(|k⟩) = +N−1 +� +k=0 +pkf(|k⟩). +(9) +In other words, function F(θ) is the average of f under the +PMF defined by |x⟩. For instance, the random outcome |k⟩ = +|101⟩ occurring with probability |x5|2 is assigned to the ran- +dom variable f taking the value f([1 0 1]⊤). Hence, func- +tion F(θ) is really an expectation (an observable in the quan- +tum computation parlance) of function f(b) when b is drawn +from the PMF {|xk(θ)|2}N−1 +k=0 . Ideally, the global minimizer +θ of (8) defines a PMF via |x(θ)⟩ that samples with non-zero +probability only the canonical vectors |ek⟩ associated with the +smallest eigenvalue of H. +Problem (8) is solved in a hybrid fashion: The quantum +computer samples from |x(θ)⟩ and estimates F(θ) and pos- +sibly its gradient ∇θF. A classical computer uses the pre- +vious information and iteratively updates θ based on a zero- +or first-order optimization algorithm, such as gradient descent +or Bayesian optimization. As with training neural networks, +F(θ) is nonconvex due to the form of the ansatz. Moreover, +the ensemble statistic F(θ) cannot be computed exactly, but +estimated as the sample average ˆF(θ) := �R +r=1 f(br)/R +over R runs, where br is the quantum output after run r. +4. CONSTRAINED VQE +As discussed earlier, VQE provides a successful heuristic for +solving QUBO through the variational formulation of (8). +Can VQE be generalized to deal with a binary QCQP of the +ensuing form? +min +b∈{0,1}n f0(b) +(10) +s.to fm(b) ≤ 0, +m = 1 : M. +Here fm(b) := b⊤Amb + b⊤cm + dm for m = 0, . . . , M. +Solving such problems is also known to be NP-hard. Provid- +ing a quantum heuristic to directly deal with (10) seems to +be challenging. To this end, we relax expectations and aim +at designing a quantum state |x⟩ from which we can draw +binary-valued b that solve the stochastic binary QCQP: +min +|x⟩ +Ex[f0(b)] +(11) +s.to Ex[fm(b)] ≤ 0, +m = 1 : M. +As in the unconstrained setup, rather than minimizing over +|x⟩, we propose optimizing over a PMF parameterized by θ +and captured by quantum state |x(θ)⟩. Specifically, we sug- +gest solving the constrained minimization +min +θ +F0(θ) +(12) +s.to Fm(θ) ≤ 0 : +λm, +m = 1 : M +where each observable Fm(θ) := ⟨x(θ)|Hm|x(θ)⟩ depends +on the Hamiltonian Hm defined similar to H in (7) for all +m. Heed that problem (12) can be reformulated and solved +as a linear program (LP) over the PMF of b. Nonetheless, +that requires evaluating {fm(b)}M +m=0 for all 2n values of b. +Moreover, the optimization variable of this LP is the vector +of PMF values that is exponentially large too. That is also the +case with standard VQE/QAOA. +Contrary to (10), problem (12) is over the continuous vari- +able θ, and thus, we can associate a dual variable λm for each +constraint and define its Lagrangian function +L(θ; λ) := F0(θ) + +M +� +m=1 +λmFm(θ) +(13) + +where λ ∈ RM collects all dual variables. Problem (12) could +be solved via dual decomposition, according to which λ is +updated iteratively via a subgradient ascent step on L as +λt+1 +m +:= max +� +λt +m + µtFm(θt), 0 +� +, m = 1 : M +(14) +for a positive step size µt = µ0/(t + α) with α > 0, and θt +is a minimizer of the Lagrangian L(θ; λt) evaluated at λt: +θt ∈ arg min +θ ⟨x(θ)|H0 + +M +� +m=1 +λt +mHm|x(θ)⟩ . +(15) +Problem (15) takes the QUBO form of (8), and is therefore +amenable to standard VQE or even the celebrated QAOA ap- +proach. Under the latter, the ansatz takes a particular form that +depends on the problem Hamiltonian H0 + �M +m=1 λt +mHm. +Here, we used a problem-independent ansatz under the gen- +eral VQE framework and leave QAOA for future work. +5. NUMERICAL TESTS +The novel solver for (12) was implemented in Python us- +ing the Qiskit library [13]. +The VQE class in Qiskit was +used to solve the minimization for the primal update (15). +In addition to providing the ansatz described in Section 3, +the VQE class was configured with the ‘SLSQP’ optimizer. +The maximum number of iterations was set to 1, 000, and we +used the aer simulator statevector quantum simu- +lation backend. For the dual update in (14), constraint vi- +olations were measured over the observables Hm using the +minimum eigenstate returned by VQE. The stopping criteria +∥λt −λt−1∥2 ≤ 1·10−5 was utilized to ascertain the conver- +gence of the dual updates (14). +To illustrate the application of the proposed strategy to +solving the stochastic binary QCQP in (11), several 2-bit +problem instances were sampled randomly by drawing the +entries of {A0, c0, d0} and {A1, c1, d1} from the standard +normal distribution, while ensuring the resulting problem was +feasible. The VQE approach was compared against a linear +program that finds a PMF solving (12); this was possible due +to the small value of 2n. For the two approaches, the obtained +PMFs along with the associated dual variables are reported in +Table 1 for 4 randomly sampled problem instances. +To study the scalability of the approach and to verify the +compatibility of the solutions with the deterministic QCQP +in (10), we also sampled 30 feasible 5-bit problem instances +with three constraints each. The quadratic cost and constraint +functions were generated as in the previous test. To avoid +instances with non-binding constraints, the constants dm in +the constraint functions were manually adjusted so that at +least one of the constraints was active and yielded a non-zero +dual variable. From the sampled problems, it was found that +the dual decomposition involving VQE was able to produce +the optimal solutions for 28 out of the 30 problem instances +Table 1. Comparing the exact solution of (12) obtained via a +linear program and the proposed quantum-based approach. +# +Found PMF +Dual +Quantum +LP +Quantum +LP +1 +[0.44, 0, 0.56, 0] +[0.44, 0, 0.56, 0] +0.854 +0.851 +2 +[0.71, 0, 0.29, 0] +[0.70, 0, 0.30, 0] +0.337 +0.337 +3 +[0, 0.80, 0, 0.20] +[0, 0.80, 0, 0.20] +0.459 +0.459 +4 +[0, 0, 0.61, 0.39] +[0, 0, 0.60, 0.40] +0.566 +0.566 +Fig. 1. Convergence of dual variables under dual updates (14) +for a stochastic binary QCQP with M = 3 constraints. +tested, whereas infeasible binary candidates were obtained for +the remaining 2 instances. Figure 1 illustrates the conver- +gence of the dual variables for one of the problem instances, +where all three constraints were found to be active. +6. CONCLUSIONS +A novel generalization of VQE to address the need for dealing +with stochastic binary QCQPs has been developed. Lever- +aging dual decomposition, the approach entails solving a +sequence of judiciously modified VQE tasks. Numerical tests +demonstrate that upon convergence of the constrained VQE +algorithm, the variational quantum circuit is able to sample +from a stochastic policy to generate binary-valued vectors +that minimize the binary QCQP and satisfy its constraints +in expectation. Some of these samples seem to be feasible +for the deterministic binary QCQP too. This novel heuristic +sets the foundation for further developments towards con- +strained discrete optimization. +We are currently exploring +several exciting directions: i) Coupling this approach with +QAOA rather than VQE; ii) skipping the nested optimization +in (15) through a primal-dual decomposition alternative as +in [14, 15]; and iii) dealing with mixed-binary setups. + +Convergence of dual variables +入1 +1.2 +入2 +入3 +1.0 +0.8 +0.6 +0.4 +0.2 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +Iterations7. REFERENCES +[1] Michael A. Nielsen and Isaac L. Chuang, +Quantum +Computation and Quantum Information, +Cambridge +University Press, 2000. +[2] Osvaldo Simeone, +“An introduction to quantum ma- +chine learning for engineers,” Foundations and Trends +in Signal Processing, vol. 16, no. 1–2, pp. 1–223, 2022. +[3] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, “A +quantum approximate optimization algorithm applied +to a bounded occurrence constraint problem,” +arXiv: +Quantum Physics, 2014. +[4] S. Hadfield, Z. Wang, B. O’Gorman, E. G. Rieffel, +D. Venturelli, and R. Biswas, “From the quantum ap- +proximate optimization algorithm to a quantum alternat- +ing operator ansatz,” Algorithms, vol. 12, no. 2, pp. 34, +2019. +[5] C. C. McGeoch, +Adiabatic quantum computation +and quantum annealing: Theory and practice, vol. 5, +Springer, Switzerland, 2014. +[6] A. Lucas, “Ising formulations of many NP problems,” +Frontiers in Physics, vol. 2, no. 5, pp. 1–15, 2014. +[7] M. Ohzeki, “Breaking limitation of quantum annealer in +solving optimization problems under constraints,” Sci- +entific reports, vol. 10, no. 1, pp. 1–12, 2020. +[8] I. Hen and M. S. Sarandy, +“Driver Hamiltonians for +constrained optimization in quantum annealing,” Phys. +Rev. A, vol. 93, no. 6, pp. 062312, 2016. +[9] I. Hen and F. M. Spedalieri, “Quantum annealing for +constrained optimization,” Phys. Rev. Appl., vol. 5, no. +63, pp. 034007, 2016. +[10] S. Hadfield, Z.Wang, E. G. Rieffel, B. O’Gorman, +D. Venturelli, and R. Biswas, “Quantum approximate +optimization with hard and soft constraints,” in ACM +Intl. Workshop on Post Moore’s Era Supercomputing, +New York, NY, 2017, pp. 15–21. +[11] Pooya Ronagh, Brad Woods, and Ehsan Iranmanesh, +“Solving constrained quadratic binary problems via +quantum adiabatic evolution,” Quantum Info. Comput., +vol. 16, no. 11–12, pp. 1029–1047, Sept. 2016. +[12] Claudio Gambella and Andrea Simonetto, +“Multi- +block ADMM heuristics for mixed-binary optimization +on classical and quantum computers,” IEEE Trans. on +Quantum Engineering, vol. 1, pp. 1–22, 10 2020. +[13] “Qiskit: An open-source framework for quantum com- +puting,” 2021. +[14] S. Gupta, S. Misra, D. Deka, and V. Kekatos, “DNN- +based policies for stochastic AC-OPF,” in Proc. Power +Syst. Comput. Conf., Porto, Portugal, June 2021, +(to +appear also in the Elsevier Electric Power Systems Re- +search). +[15] S. Gupta, V. Kekatos, and M. Jin, “Controlling smart +inverters using proxies: A chance-constrained DNN- +based approach,” IEEE Trans. Smart Grid, vol. 13, no. +2, pp. 1310–1321, Mar. 2022. + diff --git a/1dAzT4oBgHgl3EQfevzu/content/tmp_files/load_file.txt b/1dAzT4oBgHgl3EQfevzu/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eef1469b6098b081ac53bbf4d5d2f48b00ec02a2 --- /dev/null +++ b/1dAzT4oBgHgl3EQfevzu/content/tmp_files/load_file.txt @@ -0,0 +1,303 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf,len=302 +page_content='A QUANTUM APPROACH FOR STOCHASTIC CONSTRAINED BINARY OPTIMIZATION Sarthak Gupta and Vassilis Kekatos Bradley Dept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' of ECE, Virginia Tech, Blacksburg, VA 24061, USA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' {gsarthak,kekatos}@vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='edu ABSTRACT Analytical and practical evidence indicates the advantage of quantum computing solutions over classical alternatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Quantum-based heuristics relying on the variational quantum eigensolver (VQE) and the quantum approximate optimiza- tion algorithm (QAOA) have been shown numerically to generate high-quality solutions to hard combinatorial prob- lems, yet incorporating constraints to such problems has been elusive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To this end, this work puts forth a quantum heuristic to cope with stochastic binary quadratically con- strained quadratic programs (QCQP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Identifying the strength of quantum circuits to efficiently generate samples from prob- ability distributions that are otherwise hard to sample from, the variational quantum circuit is trained to generate binary- valued vectors to approximately solve the aforesaid stochastic program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The method builds upon dual decomposition and entails solving a sequence of judiciously modified standard VQE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Tests on several synthetic problem instances us- ing a quantum simulator corroborate the near-optimality and feasibility of the method, and its potential to generate feasible solutions for the deterministic QCQP too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Index Terms— QAOA, VQE, dual decomposition, quan- tum unconstrained binary optimization (QUBO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' INTRODUCTION Quantum computers exhibit an innate ability to handle ex- ponentially large computations in a parallel fashion yet with a strong probabilistic flavor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Quantum algorithms such as Shor’s integer factorization, Grover’s search, and the linear system solver of Harrow-Hassidim-Lloyd (HHL) can attain polynomial or even exponential speedups over the best known algorithms on classical computers [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Nonetheless, some of these quantum algorithms presume a large number of qubits on fault-tolerant quantum computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' In the near-term inter- mediate scale (NISQ) era, quantum computers are noisy and thus oftentimes limited in terms of number of gates and/or qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' With such limitations in mind, variational quantum algorithms have been suggested as promising tools to practi- cally showcase quantum advantage in the NISQ setup [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' This work was supported by a seed funding grant from the Virginia Com- monwealth Cybersecurity Initiative (CCI) – Southwest Virginia node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Variational quantum computers involve a sequence of pa- rameterized gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Their parameters are updated externally by a classical computer in a closed-loop fashion to steer the quantum state towards a desirable direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The variational quantum eigensolver (VQE) used to provide high-quality solutions to combinatorial problems is a representative ex- ample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The Quantum Approximate Optimization Algorithm (QAOA) [3] is a special instance of VQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' In QAOA, not only the parameters but also the architecture of the quan- tum circuit become problem-dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The quantum circuit trained by QAOA operates as a sampler to efficiently gener- ate near-optimal solutions of binary quadratic problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=', MAXCUT);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' see [4] for a summary of claims on QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' While most VQE/QAOA schemes target unconstrained problems, dealing with constraints is crucial to several appli- cations in machine learning, wireless communications, and financial (stock trading) optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Adding constraints to QAOA or adiabetic quantum computing [5] (the QAOA counterpart for non-gate-based quantum computers) has been pursued in two ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' One approach has been to convert the constrained problem into an unconstrained minimization of a Lagrangian-like function [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' However, the weights for constraint penalties can be safely selected only if con- straints are expressed as Boolean functions or linear equal- ities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' An alternative approach modifies the architecture of the quantum circuit (via the mixer Hamiltonian of QAOA) to confine quantum states on the subspace spanned by con- straints [8, 9, 4, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Nonetheless, constructing such ‘driver’ mixer Hamiltonians is again highly problem-dependent and often limited to equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Reference [11] devel- ops a quantum adiabetic approach to tackle binary linearly- constrained quadratic programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' It targets the dual problem and interfaces the quantum computer with a branch-and- bound scheme ran classically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Reference [12] treats mixed- binary quadratic-plus-convex problems using the alternating direction method of multipliers (ADMM) to split binary and continuous variables into separate minimizations, solved by QAOA and classical convex optimizers respectively per ADMM iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Relation to prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Addressing binary QCQPs by quantum heuristics has been largely unexplored to the au- thors’ knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' We put forth a quantum-based heuristic to solve a stochastic binary QCQP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Harnessing the power of quantum circuits to sample from probability mass functions arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='01443v1 [quant-ph] 4 Jan 2023 (PMF) that are hard to sample classically, we devise a dual decomposition technique that solves a sequence of standard VQE tasks to systematically adjust Lagrangian multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Numerical tests using quantum computer simulators pro- vided by IBM evaluate this technique on randomly generated stochastic and deterministic binary QCQPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' QUANTUM COMPUTING PRELIMINARIES A quantum system consisting of n quantum bits (qubits) is de- scribed by an exponentially large state vector |x⟩ ∈ CN with N = 2n assuming the system is in a pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The Dirac no- tation |x⟩ named ket emphasizes that vector x is unit-norm or �N−1 k=0 |xk|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' If ek is the k-th canonical vector of length N, we can write |x⟩ = �N−1 k=0 xk |ek⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The vector ek is of- tentimes alternatively expressed as |ek⟩ = |k⟩, where k is the binary representation of index k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For example, a system with n = 2 qubits has a state in C4, which is spanned by canonical vectors {ek}3 k=0 and e0 = [1 0 0 0]⊤ = |00⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Vector |x⟩ provides a statistical characterization for the quantum state: If we measure the quantum system output, its qubits will be in configuration |k⟩ with probability |xk|2 for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Symbol ⟨x| termed bra denotes the conjugate transpose of |x⟩, while the braket ⟨x|y⟩ denotes the inner product between states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The fundamental operations we can perform on a quan- tum system is evolution and measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The former can be described by the application of a unitary U on state |x⟩ to produce state |y⟩ = U |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Although U is exponentially large, it is usually implemented efficiently using quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Among various types of measurements, we focus on projective measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' A projective measurement is asso- ciated with a Hermitian matrix (named observable) and its eigenvalue decomposition H = �M m=1 λmvmvH m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' If such measurement is performed on |x⟩, outcome m is observed with probability pm := | ⟨x|vm⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Define a random variable taking value λm when outcome m is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The expected value of this variable is ⟨x|H|x⟩ = �M m=1 pmλm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' If H is di- agonal, the measurement is on the computational basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' This is practically important because now vm = em, outcome m relates to |m⟩, and each qubit can be measured individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' If quantum system i has been prepared in state |xi⟩ for i = 1, 2, their joint state would be |x1⟩ ⊗ |x2⟩, where ⊗ is the Kronecker product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' This is oftentimes represented as |x1⟩ |x2⟩ or |x1, x2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The Kronecker product rule generalizes to the composition of n systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For example, |1⟩ |1⟩ |0⟩ = e1 ⊗ e1 ⊗ e0 = e6 = |110⟩, where the canonical vectors shown in the middle are in R2 and those at the end are in R8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' VARIATIONAL QUANTUM EIGENSOLVER (VQE) VQE is a heuristic approach to find near-optimal solutions for combinatorial problems of the general form min b∈{0,1}n f(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (1) A particular example of interest is the quadratic unconstrained binary optimization (QUBO) problem with f(b) = b⊤Ab + b⊤c + d (2) which is known to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For later developments, it is convenient to reformulate QUBO in terms of the spin {±1} variables through the transformation si = 1 − 2bi = (−1)bi for i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' , n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (3) Collecting the spin variables in vector s = 1 − 2b, the quadratic objective can be equivalently expressed as f(b) = ¯f(s) = s⊤ ¯As + s⊤¯c + ¯d (4) where ¯A := 1 4A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' ¯c := − 1 2(A1 + c);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' and ¯d := 1 41⊤A1 + 1 21⊤c + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' We next explain how VQE samples high-quality solutions of (1) by solving an eigenvalue minimization task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The VQE method falls under the family of variational quantum algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The term variational pertains to the fact that the quantum circuit is not fixed, but parameterized by relatively few parameters collected in vector θ ∈ RP .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' These parameters are iteratively adjusted by classical computer in a closed-loop fashion so that the quantum system eventually reaches a desirable state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The process resembles the training of a neural network whose weights are updated by an opti- mization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Similarly to neural networks where the learner has to select an architecture (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=', network depth/width and type of activations), the parameterized form (also termed ansatz) of the variational quantum circuit is specified a pri- ori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' We will be using a 2-local ansatz where single-qubit RY gates are applied to all qubits, followed by a full entanglement circuit, all repeated for 3 layers (iterations) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Given θ and driven by input |0⟩n, the quantum circuit pro- duces at its output the quantum state |x(θ)⟩ = U(θ) |0⟩n for a unitary N × N matrix U(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To simplify notation, we will oftentimes write |x⟩ in lieu of |x(θ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Albeit |x⟩ ∈ CN is exponentially long, it can be easily generated by the quan- tum circuit though it cannot be read out of the circuit as a vector in a computationally efficient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Instead, it is rel- atively easy to sample from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Every time we run the quan- tum circuit driven by |0⟩n, we will be observing one of the binary outputs |k⟩ = |ek⟩ with probability pk := |xk|2 for k = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' , N − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The quantum circuit thus serves as an ef- ficient sampler from the exponentially large probability mass function (PMF) {pk}N−1 k=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To exploit this sampling property, we next relate the cost f(b) with a so-termed Hamiltonian matrix H so that H |ek⟩ = f(|k⟩) |ek⟩ for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (5) Matrix H is apparently diagonal and carries all N function evaluations f(ek) on its diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Moreover, the canonical vectors ek constitute the eigenvectors of H, each with cor- responding eigenvalue f(|k⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Therefore, the minimization in (1) can be reformulated as the problem of finding the eigen- vector corresponding to the minimum eigenvalue of H min |x⟩ ⟨x| H |x⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (6) As long as |x⟩ is allowed to take any of the values {ek}N−1 k=0 , the minimizer of (6) corresponds to the minimizer of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For example, if a quantum system has n = 3 qubits, its state would be |x⟩ ∈ C8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Here ek’s are the columns of the identity matrix I8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' If the minimizer of (6) is |e5⟩ = |b1b2b3⟩ = |101⟩, then the minimizer of (1) is b = [1 0 1]⊤;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Although H is exponentially large, it can be implemented using only O(n2) quantum gates since it can be expressed as H = n−1 � i=0 n−1 � j=0 ¯AijZiZj + n−1 � i=0 ¯ciZi + ¯dIN (7) where the N × N Hermitian matrix Zi is defined as Zi = I2 ⊗ · · · ⊗ Z ⊗ · · · ⊗ I2 with Z = � 1 0 0 −1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' This is a Kronecker product involving (n − 1) identity matri- ces I2 and one Pauli-Z operator Z applied to the i-th qubit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Matrix H as defined in (7) is obviously diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To estab- lish (5), note first that Z |0⟩ = |0⟩ and Z |1⟩ = − |1⟩, or more compactly, Z |b⟩ = (−1)b |b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Consequently, when Zi is applied to a state |b⟩ = |b1b2 · · · bn⟩, the effect is Zi |b⟩ = (−1)bi |b⟩ = si |b⟩ from (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Similarly, it also holds that ZiZj |b⟩ = sisj |b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Property (5) now follows immediately by postmultiplying (7) by any |ek⟩ and using f(b) = ¯f(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' If |x⟩ in (6) is restricted to set E := {ek}N−1 k=0 , problem (6) is as hard as (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' VQE relaxes (6) to the set of all quantum states |x(θ)⟩ that can be parameterized by the chosen ansatz and via θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Problem (6) is then solved over θ rather than |x⟩ min θ F(θ) := ⟨x(θ)|H|x(θ)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (8) From the eigenvalue property (5), it follows ⟨en| H |ek⟩ = f(|k⟩) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' How about ⟨x| H |x⟩ for a general state |x⟩?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Because |x⟩ = �N−1 k=0 xk |ek⟩, it is easy to show that ⟨x|H|x⟩ = N−1 � k=0 |xk|2f(|k⟩) = N−1 � k=0 pkf(|k⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (9) In other words, function F(θ) is the average of f under the PMF defined by |x⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For instance, the random outcome |k⟩ = |101⟩ occurring with probability |x5|2 is assigned to the ran- dom variable f taking the value f([1 0 1]⊤).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Hence, func- tion F(θ) is really an expectation (an observable in the quan- tum computation parlance) of function f(b) when b is drawn from the PMF {|xk(θ)|2}N−1 k=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Ideally, the global minimizer θ of (8) defines a PMF via |x(θ)⟩ that samples with non-zero probability only the canonical vectors |ek⟩ associated with the smallest eigenvalue of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Problem (8) is solved in a hybrid fashion: The quantum computer samples from |x(θ)⟩ and estimates F(θ) and pos- sibly its gradient ∇θF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' A classical computer uses the pre- vious information and iteratively updates θ based on a zero- or first-order optimization algorithm, such as gradient descent or Bayesian optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' As with training neural networks, F(θ) is nonconvex due to the form of the ansatz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Moreover, the ensemble statistic F(θ) cannot be computed exactly, but estimated as the sample average ˆF(θ) := �R r=1 f(br)/R over R runs, where br is the quantum output after run r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' CONSTRAINED VQE As discussed earlier, VQE provides a successful heuristic for solving QUBO through the variational formulation of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Can VQE be generalized to deal with a binary QCQP of the ensuing form?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' min b∈{0,1}n f0(b) (10) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='to fm(b) ≤ 0, m = 1 : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Here fm(b) := b⊤Amb + b⊤cm + dm for m = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Solving such problems is also known to be NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Provid- ing a quantum heuristic to directly deal with (10) seems to be challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To this end, we relax expectations and aim at designing a quantum state |x⟩ from which we can draw binary-valued b that solve the stochastic binary QCQP: min |x⟩ Ex[f0(b)] (11) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='to Ex[fm(b)] ≤ 0, m = 1 : M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' As in the unconstrained setup, rather than minimizing over |x⟩, we propose optimizing over a PMF parameterized by θ and captured by quantum state |x(θ)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Specifically, we sug- gest solving the constrained minimization min θ F0(θ) (12) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='to Fm(θ) ≤ 0 : λm, m = 1 : M where each observable Fm(θ) := ⟨x(θ)|Hm|x(θ)⟩ depends on the Hamiltonian Hm defined similar to H in (7) for all m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Heed that problem (12) can be reformulated and solved as a linear program (LP) over the PMF of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Nonetheless, that requires evaluating {fm(b)}M m=0 for all 2n values of b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Moreover, the optimization variable of this LP is the vector of PMF values that is exponentially large too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' That is also the case with standard VQE/QAOA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Contrary to (10), problem (12) is over the continuous vari- able θ, and thus, we can associate a dual variable λm for each constraint and define its Lagrangian function L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' λ) := F0(θ) + M � m=1 λmFm(θ) (13) where λ ∈ RM collects all dual variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Problem (12) could be solved via dual decomposition, according to which λ is updated iteratively via a subgradient ascent step on L as λt+1 m := max � λt m + µtFm(θt), 0 � , m = 1 : M (14) for a positive step size µt = µ0/(t + α) with α > 0, and θt is a minimizer of the Lagrangian L(θ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' λt) evaluated at λt: θt ∈ arg min θ ⟨x(θ)|H0 + M � m=1 λt mHm|x(θ)⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' (15) Problem (15) takes the QUBO form of (8), and is therefore amenable to standard VQE or even the celebrated QAOA ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Under the latter, the ansatz takes a particular form that depends on the problem Hamiltonian H0 + �M m=1 λt mHm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Here, we used a problem-independent ansatz under the gen- eral VQE framework and leave QAOA for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' NUMERICAL TESTS The novel solver for (12) was implemented in Python us- ing the Qiskit library [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The VQE class in Qiskit was used to solve the minimization for the primal update (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' In addition to providing the ansatz described in Section 3, the VQE class was configured with the ‘SLSQP’ optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The maximum number of iterations was set to 1, 000, and we used the aer simulator statevector quantum simu- lation backend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For the dual update in (14), constraint vi- olations were measured over the observables Hm using the minimum eigenstate returned by VQE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The stopping criteria ∥λt −λt−1∥2 ≤ 1·10−5 was utilized to ascertain the conver- gence of the dual updates (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To illustrate the application of the proposed strategy to solving the stochastic binary QCQP in (11), several 2-bit problem instances were sampled randomly by drawing the entries of {A0, c0, d0} and {A1, c1, d1} from the standard normal distribution, while ensuring the resulting problem was feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The VQE approach was compared against a linear program that finds a PMF solving (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' this was possible due to the small value of 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' For the two approaches, the obtained PMFs along with the associated dual variables are reported in Table 1 for 4 randomly sampled problem instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To study the scalability of the approach and to verify the compatibility of the solutions with the deterministic QCQP in (10), we also sampled 30 feasible 5-bit problem instances with three constraints each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' The quadratic cost and constraint functions were generated as in the previous test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' To avoid instances with non-binding constraints, the constants dm in the constraint functions were manually adjusted so that at least one of the constraints was active and yielded a non-zero dual variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' From the sampled problems, it was found that the dual decomposition involving VQE was able to produce the optimal solutions for 28 out of the 30 problem instances Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Comparing the exact solution of (12) obtained via a linear program and the proposed quantum-based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' # Found PMF Dual Quantum LP Quantum LP 1 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='44, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='56, 0] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='44, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='56, 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='851 2 [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='71, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='29, 0] [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='70, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='30, 0] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='337 3 [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='80, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='20] [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='80, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='20] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='459 4 [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='61, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='39] [0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='60, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='40] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='566 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Convergence of dual variables under dual updates (14) for a stochastic binary QCQP with M = 3 constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' tested, whereas infeasible binary candidates were obtained for the remaining 2 instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Figure 1 illustrates the conver- gence of the dual variables for one of the problem instances, where all three constraints were found to be active.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' CONCLUSIONS A novel generalization of VQE to address the need for dealing with stochastic binary QCQPs has been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Lever- aging dual decomposition, the approach entails solving a sequence of judiciously modified VQE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Numerical tests demonstrate that upon convergence of the constrained VQE algorithm, the variational quantum circuit is able to sample from a stochastic policy to generate binary-valued vectors that minimize the binary QCQP and satisfy its constraints in expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Some of these samples seem to be feasible for the deterministic binary QCQP too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' This novel heuristic sets the foundation for further developments towards con- strained discrete optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' We are currently exploring several exciting directions: i) Coupling this approach with QAOA rather than VQE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' ii) skipping the nested optimization in (15) through a primal-dual decomposition alternative as in [14, 15];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' and iii) dealing with mixed-binary setups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Convergence of dual variables 入1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='2 入2 入3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='0 0 20 40 60 80 100 120 140 Iterations7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' REFERENCES [1] Michael A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Nielsen and Isaac L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [2] Osvaldo Simeone, “An introduction to quantum ma- chine learning for engineers,” Foundations and Trends in Signal Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1–2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1–223, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [3] Edward Farhi, Jeffrey Goldstone, and Sam Gutmann, “A quantum approximate optimization algorithm applied to a bounded occurrence constraint problem,” arXiv: Quantum Physics, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [4] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Hadfield, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' O’Gorman, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Rieffel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Venturelli, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Biswas, “From the quantum ap- proximate optimization algorithm to a quantum alternat- ing operator ansatz,” Algorithms, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 34, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' McGeoch, Adiabatic quantum computation and quantum annealing: Theory and practice, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 5, Springer, Switzerland, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Lucas, “Ising formulations of many NP problems,” Frontiers in Physics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 2, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1–15, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Ohzeki, “Breaking limitation of quantum annealer in solving optimization problems under constraints,” Sci- entific reports, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 10, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1–12, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [8] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Hen and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Sarandy, “Driver Hamiltonians for constrained optimization in quantum annealing,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' A, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 93, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 062312, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [9] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Hen and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Spedalieri, “Quantum annealing for constrained optimization,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 5, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 63, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 034007, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Hadfield, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content='Wang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Rieffel, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' O’Gorman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Venturelli, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Biswas, “Quantum approximate optimization with hard and soft constraints,” in ACM Intl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Workshop on Post Moore’s Era Supercomputing, New York, NY, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 15–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [11] Pooya Ronagh, Brad Woods, and Ehsan Iranmanesh, “Solving constrained quadratic binary problems via quantum adiabatic evolution,” Quantum Info.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 16, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 11–12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1029–1047, Sept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [12] Claudio Gambella and Andrea Simonetto, “Multi- block ADMM heuristics for mixed-binary optimization on classical and quantum computers,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' on Quantum Engineering, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1–22, 10 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [13] “Qiskit: An open-source framework for quantum com- puting,” 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Gupta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Misra, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Deka, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Kekatos, “DNN- based policies for stochastic AC-OPF,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Power Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=', Porto, Portugal, June 2021, (to appear also in the Elsevier Electric Power Systems Re- search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Gupta, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Kekatos, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Jin, “Controlling smart inverters using proxies: A chance-constrained DNN- based approach,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' Smart Grid, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 13, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 1310–1321, Mar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dAzT4oBgHgl3EQfevzu/content/2301.01443v1.pdf'} diff --git a/4NE3T4oBgHgl3EQfQAnJ/content/tmp_files/2301.04409v1.pdf.txt b/4NE3T4oBgHgl3EQfQAnJ/content/tmp_files/2301.04409v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5186e858c7695869cb3d2211a2d4a68254181e8d --- /dev/null +++ b/4NE3T4oBgHgl3EQfQAnJ/content/tmp_files/2301.04409v1.pdf.txt @@ -0,0 +1,2068 @@ +Mon. Not. R. Astron. Soc. 000, 1–13 (2021) +Printed 12 January 2023 +(MN LATEX style file v2.2) +Comprehensive spectroscopic and photometric study of +pulsating eclipsing binary star AI Hya +F. Kahraman Ali¸cavu¸s1,2⋆, T. Pawar3†, K. G. He�lminiak3, G. Handler4, A. Moharana3, +F. Ali¸cavu¸s1,2, P. De Cat5, F. Leone6,7, G. Catanzaro7, M. Giarrusso7,8, N. Ukita9,10, +E. Kambe11 +1C¸anakkale Onsekiz Mart University, Faculty of Science, Physics Department, 17100, Canakkale, Turkey +2C¸anakkale Onsekiz Mart University, Astrophysics Research Center and Ulupınar Observatory, TR-17100, anakkale, Turkey +3Nicolaus Copernicus Astronomical Center, Department of Astrophysics, ul. Rabia´nska 8, PL-87-100 Toru´n, Poland +4Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warsaw, Poland +5Royal Observatory of Belgium, Ringlaan 3, B-1180 Brussel, Belgium +6Dipartimento di Fisica e Astronomia, Sezione Astrofisica, Universit?a di Catania, Via S. Sofia 78, I-95123 Catania, Italy +7INAF, Osservatorio Astrofisico di Catania, Via S. Sofia 78, I-95123 Catania, Italy +8University of Florence, Department of Physics and Astronomy, Via Giovanni Sansone 1, I-50019 Sesto Fiorentino, Italy +9Okayama Astrophysical Observatory, National Astronomical Observatory of Japan, 3037-5 Honjo, Kamogata, Asakuchi, Okayama 719-0232, Japan +10The Graduate University for Advanced Studies, 2-21-1 Osawa, Mitaka, Tokyo 181-8588, Japan +11Subaru Telescope, National Astronomical Observatory of Japan, 650 North Aohoku Place, Hilo, HI 96720, USA +Accepted ... Received ...; in original form ... +ABSTRACT +The pulsating eclipsing binaries are remarkable systems that provide an opportu- +nity to probe the stellar interior and to determine the fundamental stellar parameters +precisely. Especially the detached eclipsing binary systems with (a) pulsating compo- +nent(s) are significant objects to understand the nature of the oscillations since the +binary effects in these systems are negligible. Recent studies based on space data have +shown that the pulsation mechanisms of some oscillating stars are not completely +understood. Hence, comprehensive studies of a number of pulsating stars within de- +tached eclipsing binaries are important. In this study, we present a detailed analysis +of the pulsating detached eclipsing binary system AI Hya which was studied by two +independent groups with different methods. We carried out a spectroscopic survey to +estimate the orbital parameters via radial velocity measurements and the atmospheric +parameters of each binary component using the composite and/or disentangled spec- +tra. We found that the more luminous component of the system is a massive, cool +and chemically normal star while the hotter binary component is a slightly metal-rich +object. The fundamental parameters of AI Hya were determined by the analysis of +binary variations and subsequently used in the evolutionary modelling. Consequently, +we obtained the age of the system as 850 ± 20 Myr and found that both binary com- +ponents are situated in the δ Scuti instability strip. The frequency analysis revealed +pulsation frequencies between the 5.5 – 13.0 d−1 and we tried to estimate which binary +component is the pulsating one. However, it turned out that those frequencies could +originate from both binary components. +Key words: +stars: binaries: eclipsing – stars: atmospheres – stars: fundamental +parameters – stars: variables: δ Scuti – stars: individual: AI Hya +⋆ E-mail: filizkahraman01@gmail.com +† E-mail: pawar@ncac.torun.pl +1 +INTRODUCTION +To understand the universe, it is necessary to comprehend +stars which are its building blocks. For a deep investigation +of stars, we should know their basic stellar parameters such +© 2021 RAS +arXiv:2301.04409v1 [astro-ph.SR] 11 Jan 2023 + +2 +F. Kahraman Ali¸cavu¸s et. al. +as mass (M), radius (R) and chemical composition. Binary +stars, in particular the eclipsing ones, are the most suitable +objects to derive these parameters as M and R can be de- +rived with an accuracy better than 1% (Torres, Andersen, & +Gim´enez 2010; Southworth 2013). Therefore, these systems +are substantial for a better understanding of the universe, +our Galaxy, and, most directly, stellar evolution. However, +eclipsing binary systems as such do not provide information +about the stellar interior. This is where the pulsating stars +come in. The oscillation frequencies of pulsating stars can be +used to probe the stellar interior by applying asteroseismic +methods, making eclipsing binary systems with (a) pulsat- +ing component(s) one of the most valuable tools to improve +our knowledge of stellar evolution. +Various types of pulsating stars in different evolution- +ary states exist. Some of them, such as β Cephei, δ Scuti, and +γ Doradus stars (Lampens 2021; Southworth 2021), are also +found in eclipsing binary systems. The δ Scuti variables are +the most common pulsating stars found in eclipsing bina- +ries because of their relatively short pulsation periods. The +δ Scuti stars are A to F-type dwarf or giant stars generally +exhibiting pressure mode oscillations with periods between +18 min and 8 h and amplitudes below 0m.1 in the V-band +(Aerts, Christensen-Dalsgaard, & Kurtz 2010). Their the- +oretical instability strip (e.g. Dupret et al. 2005) indicates +the location of objects in the Hertzsprung-Russell (H-R) di- +agram that are expected to show δ Scuti-type oscillations. +Thanks to space missions such as Kepler (Borucki et al. +2010) and the Transiting Exoplanet Survey Satellite (TESS, +Ricker et al. 2014), we learned that δ Scuti stars are also ob- +served beyond the borders of the theoretical instability strip, +showing the necessity to revise them (Uytterhoeven et al. +2011; Antoci et al. 2014; Bowman & Kurtz 2018). Accord- +ing to the latest catalog of δ Scuti stars in eclipsing binaries, +there are around 90 such objects (Kahraman Ali¸cavu¸s et al. +2017). This number is now increasing especially by the dis- +coveries of new systems from the investigation of the space +data (e.g. Kahraman Ali¸cavu¸s et al. 2022; Gaulme & Guzik +2019). The pulsations of the δ Scuti stars in eclipsing bina- +ries are affected by the other binary component (Kahraman +Ali¸cavu¸s et al. 2017; Liakos & Niarchos 2017). Indeed, their +pulsation period (Ppuls) decreases when the orbital period +(Porb) becomes shorter and, hence, the other component ap- +proaches the pulsating component. It was also thought that +the tidal forces between the binary components can alter the +pulsation axis (Kurtz et al. 2020). The first observational +proof of this was presented by Handler et al. (2020) thanks +to the high-quality data of TESS. These authors showed that +in some binary systems the pulsation axis can align with the +orbital axis because of the tidal forces. This type of object +is now known as tidally tilted pulsators and they are a clear +proof of binary effects on pulsations. +For a deep understanding of the effects of binarity on +pulsations in eclipsing binary systems and on stellar evolu- +tion and structure, comprehensive investigations of such sys- +tems are necessary. AI Hya (V = 9m.35) is an eclipsing binary +system with a δ Scuti component consisting of a F2m and +F0V star (Stancliffe et al. 2015). It has an eccentric orbit and +an orbital period of 8.289649(2) days (Kreiner 2004). Spec- +troscopic observations revealed that AI Hya is a double-lined +binary system (Popper 1988). In a recent study, an updated +photometric analysis based on the TESS data of AI Hya was +given which shows that the secondary component exhibits +multiperiodic oscillations (Lee, Hong, & Kristiansen 2020). +However, no detailed spectral analysis with high-resolution +spectra has been carried out for the system so far. There- +fore, we provide a detailed photometric and spectral analysis +of AI Hya in this study to reveal the true character of this +interesting object. +Two teams were working on this system independently. +One group was led by TP (group-P with KH, AM, NU, and +EK) and the second group by FKA (group-K with GH, FA, +PDC, FL, GC, and MG). We used the same photometric but +different spectroscopic data. We compared our partial re- +sults as the work progressed. However, the overall approach +used by each group was different. In the end, we combined +our results to obtain the final parameters of the system. +The paper is organized as follows. In Sect. 2 the observa- +tional data are introduced. The radial velocity and spectral +analyses are given in Sect. 3 and Sect. 4, respectively. The +binary modelling and the pulsation frequency analysis are +presented in Sect. 5 and Sect. 6. In Sect. 7, discussions and +conclusions are given. +2 +OBSERVATIONAL DATA +In the photometric analysis of AI Hya, TESS data was used +by both groups. TESS was launched in April 2018 mainly to +detect new exoplanets (Ricker et al. 2014). TESS has moni- +tored almost the entire sky which has been subdivided into +sectors that are observed for about 27 days each. The TESS +observations were taken in 2-min. short (SC) and 30-min +long (LC) cadence in the nominal phase of the mission (first +two years). For the extended mission, the LC was reduced +to 10-min. The data are available in the Barbara A. Mikul- +ski Archive for Telescopes (MAST)1 where they are released +in different versions: simple aperture photometry (SAP) and +pre-search data conditioning SAP fluxes (PDCSAP). AI Hya +was observed in one sector only (sector 7). The 2-min SAP +fluxes were used in our analysis since SAP fluxes have lower +flux uncertainty and 2-min data are more suitable for the +analysis of AI Hya (see Sect. 6). They were converted into +magnitude by using the same method as Kahraman Ali¸cavu¸s +et al. (2022). +Photometric data from ground-based surveys also exist, +e.g. from ASAS 3 (Pojma´nski 2002) and ASAS-SN (Jayas- +inghe et al. 2018), but they are of inferior quality and do not +allow for proper analysis of pulsations. The TESS sector 7 +data are the best ones available so far, although AI Hya will +again be visible in the satellite’s field of view in sector 61. +The spectroscopic data of the system were taken from +four different instruments. The list of the instruments and +the basic information about them are given in Table 1. One +spectrum was taken with Catania Astrophysical Observatory +Spectropolarimeter (CAOS, Leone, et al. 2016). The CAOS +is a high-resolution, fibre-fed, cross-dispersed ´echelle spec- +trograph installed to the 91-cm telescope at the Catania +Astrophysical Observatory (Mt. Etna, Italy). Three spectra +of AI Hya were collected from the CORALIE ´echelle spec- +trograph which is mounted on the 1.2-m Leonhard Euler +1 https://mast.stsci.edu +© 2021 RAS, MNRAS 000, 1–13 + +Comprehensive study of AI Hya +3 +Table 1. Information about the spectroscopic observations. N, +R and SNR represent the number of the spectra, resolving power +and the signal-to-noise ratio, respectively. +Spectrometer +N +Observations +R +SNR +Spectral +years +range [˚A] +CAOS +1 +2021 +38000 +50 +415 − 670 +CORALIE +3 +2015 +60000 +20 − 34 +390 − 680 +HERMES +15 +2020 +85000 +50 − 70 +377 − 900 +HIDES +13 +2014 − 2017 +50000 +40 − 88 +408 − 752 +telescope at La Silla Observatory (Chile) (Pepe et al. 2018). +The High Efficiency and Resolution Mercator ´echelle spec- +trograph (HERMES) was also used to obtain high-resolution +spectra of AI Hya. HERMES is mounted on the 1.2-m Mer- +cator telescope at the Roque de Los Muchchos observa- +tory on the Canary Island La Palma in Spain (Raskin et +al. 2011). The last instrument used in this study is the +HIgh-Dispersion ´Echelle spectrograph (HIDES). HIDES is +attached to the 1.88-m telescope of Okayama astrophysical +observatory in Japan (Kambe et al. 2013). The spectra of +CORALIE and HIDES were taken by group-P, while the +spectra of CAOS and HERMES were gathered by group-K. +In total 32 spectra of AI Hya were gathered and these spec- +tra are well distributed in orbital phases of AI Hya. Each +group used the obtained spectra to measure the radial ve- +locity (vr ) changes. Additionally, these data were taken into +account to derive the atmospheric parameters (e.g. effective +temperature Teff, surface gravity log g, metallicity) and the +projected rotational velocity (v sin i) of the components of +AI Hya. +3 +RADIAL VELOCITY ANALYSIS +The vr values of the AI Hya system were measured with +different approaches by both group-P and group-K using +different spectra taken from the distinct instruments. +3.1 +vr measurements +Group-P +calculated +the +vr +values +from +HIDES +and +CORALIE +spectra, +using +the +two-dimensional +cross- +correlation todcor program (Zucker & Mazeh 1994). In the +analysis, a synthetic spectrum was used as a template and +this spectrum was generated using an ATLAS9 model at- +mosphere (Kurucz 1993) having Teff, metallicity [M/H] and +v sin i parameters of 6800 K, 0.0 and 30 km s−1, respectively. +When a template with 60 km s−1(the v sin i value found in +further analysis) was used, the results did not improve in +terms of rms of the orbital fit, nor did the uncertainties of +orbital elements. Moreover, some points, with the smallest +difference in vr measurements, seemed to suffer from sys- +tematic effects, and had to be rejected. We therefore believe +the use of 30 km s−1templates was justified. The calculated +vr values for each binary component are given in Table A1. +Group-K used the RaVeSpAn code (Pilecki et al. 2017) +to determine the vr values of the binary components using +the broadening function formalism. In the analysis, local +thermodynamic equilibrium (LTE) synthetic spectra with +atmospheric parameters similar to that of group-P were used +as templates (Coelho et al. 2005). The spectra of CAOS and +HERMES were used in the vr measurements. The resulting +vr measurements are given in Table A1. +3.2 +vr curve modelling +For the spectroscopic orbital fitting, group-P used all the +available vr measurements, including those made by group- +K and from Popper (1988). Group-P used the v2fit code +(Konacki et al. 2010) which adjusts a double-Keplerian with +a Levenberg-Marquardt algorithm. In this analysis, the am- +plitude of vr curves (K), Porb, the time of phase zero (T0), +mass centre’s velocity (γ), eccentricity (e) and argument of +the periastron (ω) were set as free parameters. Thanks to +the long time span of the data (>51 years), it was possi- +ble to detect the apsidal motion ( ˙ω) of the binary’s orbit: +0.186(56) deg/yr. This is in reasonable agreement (1.75σ) +with the value given by Lee, Hong, & Kristiansen (2020): +0.075(31) deg/yr. The results of the analysis are given in +Table 2 and the theoretical vr curve fits to the measured vr +data are illustrated in Fig.1. +Group-K used the rvfit code2 for the radial velocity +analysis. The rvfit program can analyse single and double- +lined binary systems by using the adaptive simulated an- +nealing method (Iglesias-Marzoa et al. 2015). In the analy- +sis, the Porb taken from Kreiner (2004) was considered as a +fixed parameter. Other orbital parameters such as T0, K, γ, +ω and e were taken as free parameters during the analysis. +Both groups vr measurements were used in the analysis and +as a result, the orbital parameters of the system were ob- +tained. The resulting parameters of the current vr analysis +are given in Table 2. The consistency between the theoretical +vr curve and measurements is shown in Fig. 2. +Both groups found the resulting mass ratio (q += +M2/M1 += +K1/K2)3 larger than 1 (1.075 ± 0.011 and +1.080 ± 0.007 for groups -K and -P, respectively). Accord- +ing to this q value, the vr curve and the results, the star +(generally called secondary) covered by the hotter binary +component at orbital phase 0.5 is more massive than the +hotter binary component (primary). To test these findings, +binary modelling is necessary. Therefore, these results will +be tested in the binary modelling sections. +4 +SPECTRAL ANALYSIS +4.1 +Group-K +4.1.1 +Spectral disentangling +To obtain the atmospheric parameters (Teff, log g), v sin i +and the chemical composition of each binary component of +AI Hya, a detailed spectral analysis is necessary. As AI Hya +is a double-lined binary system, its spectrum consists of the +spectral lines of both binary components. Therefore, group- +K carried out a spectral disentangling analysis to extract the +individual spectra of each binary component from the com- +posite spectra of AI Hya. In the analysis, the code fdbinary +2 http://www.cefca.es/people/riglesias/rvfit html +3 The subscripts 1 and 2 refer to hotter primary and cooler sec- +ondary components, respectively. +© 2021 RAS, MNRAS 000, 1–13 + +4 +F. Kahraman Ali¸cavu¸s et. al. +Table 2. The results of the radial velocity analysis. The sub- +scripts 1 and 2 refer to hotter primary and cooler secondary com- +ponents, respectively. a shows the fixed parameters. +Parameter +Group-P +Group-K +T0 (HJD) +2458491.570 ± 0.028 +2452506.383 ± 0.032 +Porb(d) +8.289761 ± 0.000027 +8.2896490a +γ (km/s) +45.90 ± 0.24 +45.70 ± 0.35 +K1 (km/s) +90.42 ± 0.37 +89.52 ± 0.65 +K2 (km/s) +83.71 ± 0.46 +83.29 ± 0.63 +e +0.2419 ± 0.0036 +0.2432 ± 0.0050 +ω (deg) +254.03 ± 1.30 +250.92 ± 1.63 +˙ω (deg/yr) +0.186 ± 0.056 +a1 sin i (R⊙) +14.380 ± 0.061 +14.222 ± 0.105 +a2 sin i (R⊙) +13.312 ± 0.072 +13.233 ± 0.101 +a sin i (R⊙) +27.692 ± 0.094 +27.454 ± 0.145 +M1 sin3 i (M⊙) +1.992 ± 0.023 +1.950 ± 0.033 +M2 sin3 i (M⊙) +2.151 ± 0.022 +2.095 ± 0.035 +q = M2/M1 +1.080 ± 0.007 +1.075 ± 0.011 +50 +25 +0 +25 +50 +75 +100 +125 +RV (Km/s) +=245.283 +=253.526 +=254.424 +Popper_rv1 +Popper_rv2 +Group-P_rv1 +Group-P_rv2 +Group-K_rv1 +Group-K_rv2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Phase +10 +0 +10 +O-C +Figure 1. Upper panel: The model vr fit to the combined vr mea- +surements from Popper (1988), Group-P (HIDES+CORALIE) +and Group-K (HERMES+CAOS). Lower panel: residuals. Model +made by Group-P. +was used (Ilijic et al. 2004). fdbinary is capable of disen- +tangling a composite spectrum, which includes flux contri- +butions from two or three components, in Fourier space. +Before the analysis with fdbinary, one should know the +light contributions of the binary components at the orbital +phases corresponding to the times the spectra were taken. +These values should be fixed during the analysis. Hence, +to determine the light contributions of both binary compo- +nents at the different orbital phases, we carried out a pre- +liminary binary modelling of AI Hya by taking Teff of the +TESS Input Catalog (TIC; Stassun et al. 2019) as the Teff +of the hotter component. The analysis was performed utiliz- +ing the Wilson-Devinney code (Wilson & Devinney 1971). +As a result of this preliminary analysis, it was found that +the hotter and cooler binary components contribute around +38% and 62% to the total, respectively. However, one should +keep in mind that these light contributions change accord- +ing to the orbital phases. For example, the primary eclipse +Figure 2. Upper panel: The model vr fit to the vr measure- +ments of Groups-K and -P. Lower panel: residuals. Model made +by Group-K. +is a total eclipse where the light contribution of the hotter +components is negligible. +In the analysis, we used the HERMES spectra as they +are well distributed over the orbital phases and have a higher +resolving power. Taking into account the observation time +of each HERMES spectrum, the light contributions at these +times were first determined using the fluxes measured from +the photometric solution and subsequently fixed during the +analysis. In addition to this, we also fixed all results de- +rived in the vr analysis during the spectral disentangling. +For the disentangling progress, we used the spectral inter- +val of ∼4200 − 6400 ˚A by ignoring the parts polluted by tel- +luric lines. For the analysis, this spectral window was di- +vided into 15 spectral parts with steps of ∼ 100 − 150 ˚A. +Each small spectral part was then analysed separately. As +a result, we obtained the individual spectra of each binary +component. The separated spectra derived with fdbinary +were re-normalised by taking into account the light ratio of +the binary components, as described by Ilijic et al. (2004). +4.1.2 +Determination of the atmospheric parameters and +chemical compositions +After the individual spectra of the components of AI Hya +were obtained, we were able to determine the atmospheric +parameters, v sin i, and the chemical composition. To de- +rive these parameters, we used the plane-parallel and line- +blanketed local thermodynamic equilibrium (LTE) ATLAS9 +model atmospheres (Kurucz 1993) and the synthe code +(Kurucz & Avrett 1981) to generate theoretical spectra. +First, the hydrogen lines of the binary components were used +to obtain initial Teff values. +In this analysis, the Hβ lines of the components were +compared with many theoretical Hβ lines which were de- +rived for a wide range of Teff (5000 − 9000 K) with a step +size of 100 K, where log g and metallicity were fixed to 4.0 +and solar, respectively. During the analysis, we took into +account the minimization method described by Catanzaro, +Leone, & Dall (2004) and successfully applied in a series +© 2021 RAS, MNRAS 000, 1–13 + +150 +100 +米 +米 +xnl↓ +50 +Normalized +采 +0 +米 +米 +米 +米 +CAOS +50 +△ CORALIE + HERMES + HIDES +-100 +15 +10 +A +1 +s +5 +米 +中 +uy) +米 +中 +采米 +0 +米 +米 +5 +米 +O-C. +15 +15 +10 +5 +uy) +中谷 +米 +日米日 +米 +-5E +-10 +- +0 +15 +0.0 +0.2 +0.6 +0.8 +1.0 +0.4 +PhaseComprehensive study of AI Hya +5 +Figure 3. Theoretical hydrogen line fits (red dashed lines) to +the Hβ lines (solid black line) of the hotter and cooler binary +components (Group-K). +of papers (i.e., Catanzaro et al. 2022, 2019). Consequently, +the Teff of the hotter and cooler components were found to +be 7500 ± 200 K and 7000 ± 150 K, respectively. We did not +attempt to optimize log g because the hydrogen lines are +not sensitive to this parameter for stars cooler than 8000 K +(Smalley et al. 2002). The best theoretical Hβ line fits to the +separated spectra of the components are shown in Fig. 3. +We also determined values for log g, the microturbulent +velocity ξ, and v sin i by improving the initially determined +Teff value using the excitation potential−abundance rela- +tionship. For the correct atmospheric parameters, different +excitation potentials of the same element should give the +same abundances. Therefore, by using this relation for iron +(Fe), we determined the atmospheric parameters. Detailed +information about this analysis method is given by Kahra- +man Ali¸cavu¸s et al. (2016). The results of this analysis are +listed in Table 3. To determine the errors on the atmospheric +parameters, we checked how their values change for differ- +ences in the excitation potential−abundance correlation of +about 5%. +In the next step, the chemical composition of the binary +components was derived after fixing the atmospheric param- +eters to their final values. For the chemical abundance deter- +mination, we first identified the lines based on the Kurucz +line list4. The spectral synthesizing method and the identi- +fied lines were used in this examination. Consequently, the +chemical compositions of both binary components were ob- +tained and the results are listed in Table 4. The consistency +between the synthetic and observed spectra of both binary +4 http://kurucz.harvard.edu/linelists.html +Figure 4. Consistency between the synthetic (dashed-lines) and +disentangled spectra of the components of AI Hya (Group-K). +Figure 5. Abundance distribution of the components of AI Hya +relative to solar values (Asplund et al. 2009) (Group-K). +components is illustrated in Fig. 4. The abundance distribu- +tions relative to solar abundance (Asplund et al. 2009) are +shown in Fig. 5, indicating that the hotter binary component +has an overabundance compared to the Sun for some ele- +ments. The errors of the chemical compositions were deter- +mined including the uncertainties in the derived atmospheric +parameters and the effects of the resolving power and the +SNR of the spectra, as described by Kahraman Ali¸cavu¸s et +al. (2016). +4.2 +Group-P +For the spectral decomposition and analysis, group-P used +the HIDES data only. Spectral analysis was performed on +both the observed composite spectra and the disentangled +spectra of the individual components. For the spectral disen- +tangling, we used a python wrapper5 made for using version +3 of fdbinary (FD3; Ilijic et al. 2004). A particular por- +tion of the total spectra was taken to ensure good quality +in terms of SNR and spectral features. The light fractions +used for the disentangling procedure were obtained from the +light curve analysis as 38% and 62% for the primary and sec- +ondary respectively. +4.2.1 +gssp +On the other hand, we also modelled the composite spec- +trum using the gssp composite module of the Grid Search +5 https://github.com/ayushmoharana/fd3 initiator +© 2021 RAS, MNRAS 000, 1–13 + +1.0 +0.8 +0.6 +0.4 +xn +T +0.2 +otter +Normalized +0.0 +1.0 +0.8 +0.6 +0.4 +0.2 +Al +cooler +0.0 +4800 +4820 +4840 +4860 +4880 +4900 +4920 +Wavelength (A)Al HyaHot +xnl +1.0 +Normalized +0.9 +Fel +Fel +0.8 +Fel +Fel +0.7 +5439 +5448 +5427 +5430 +5433 +5436 +5444 +Wavelength (A) +xn +00 +.0 +Normalized +Fel +Fel +0.9 +Fel +Fel +Fel +0.8 +5382 +5385 +5400 +5403 +5379 +5391 +5394 +5397 +5388 +Wavelength (A)3 +Hotter star +Cooler star +loge(El)- +O +Mg +Si +Ca +Sc +Cr +Fe +Ti +Mn +Ni +Element6 +F. Kahraman Ali¸cavu¸s et. al. +Table 3. The final atmospheric parameters and v sin i value of the hot (primary) and cool binary components of AI Hya. log ϵ (Fe) +represent the relative abundance with respect to hydrogen (H=12.0) +. +Group-K +Teff (K) +log g (cgs) +ξ (km s−1) +v sin i (km s−1) +log ϵ (Fe) +Primary +7700 ± 100 +3.8 ± 0.1 +3.4 ± 0.3 +57 ± 6 +8.25 ± 0.54 +Secondary +7200 ± 100 +3.6 ± 0.2 +1.9 ± 0.3 +64 ± 4 +7.64 ± 0.20 +Group-P (gssp) +Teff (K) +log g (cgs) +ξ (km s−1) +v sin i (km s−1) +[M/H] +Primary +7350 ± 300 +3.8 (fixed) +4.83 ± 1.15 +50 (fixed) +0.14 ± 0.14 +Secondary +7150 ± 250 +3.6 (fixed) +3.07 ± 0.52 +62 (fixed) +0.06 ± 0.10 +Group-P (iSpec) +Teff (K) +log g (cgs) +ξ (km s−1) +v sin i (km s−1) +[M/H] +Primary +7300 ± 170 +3.83 (fixed) +5.33 ± 0.86 +50 (fixed) +0.15 (fixed) +Secondary +7260 ± 175 +3.58 (fixed) +3.98 ± 0.70 +62 (fixed) +0.01 (fixed) +Table 4. Abundances of individual elements of the binary com- +ponents and Sun (Asplund et al. 2009). +Group-K +Elements +Hotter +Cooler +Solar +component +component +abundance +12Mg +7.96 ± 0.16 +8.01 ± 0.63 +7.60 ± 0.04 +14Si +8.03 ± 0.36 +7.12 ± 0.51 +7.51 ± 0.03 +20Ca +6.93 ± 0.27 +6.69 ± 0.27 +6.34 ± 0.04 +21Sc +3.11 ± 0.32 +3.15 ± 0.04 +22Ti +5.71 ± 0.49 +5.17 ± 0.30 +4.95 ± 0.05 +24Cr +6.63 ± 0.42 +5.80 ± 0.30 +5.64 ± 0.04 +25Mn +6.84 ± 0.82 +6.06 ± 0.45 +5.43 ± 0.05 +26Fe +8.25 ± 0.23 +7.64 ± 0.24 +7.50 ± 0.04 +28Ni +7.44 ± 0.38 +6.73 ± 0.33 +6.22 ± 0.04 +Group-P (iSpec) +Elements +Hotter +Cooler +Solar +component +component +abundance +24Cr +5.95 ± 0.19 +5.63 ± 0.23 +5.64 ± 0.04 +26Fe +7.83 ± 0.16 +7.48 ± 0.17 +7.50 ± 0.04 +28Ni +6.76 ± 0.18 +6.53 ± 0.22 +6.22 ± 0.04 +in Stellar Parameter (gssp) software package (Tkachenko +2015). As its name implies, gssp is based on a grid search +in the fundamental atmospheric parameters. It uses the +method of atmosphere models and spectrum synthesis, +which performs a comparison of the observations with the- +oretical spectra from the grid. These synthetic spectra are +calculated using the synthV LTE-based radiative transfer +code (Tsymbal 1996) and a grid of atmospheric models pre- +computed using llmodels (Shulyak et al. 2004). Specifi- +cally, in the composite module, the user can set the radial +velocity of the components as a free parameter so that all +the possible combinations of the synthetic spectra of primary +and secondary from the computed grid are used to build the +composite theoretical spectra of the binary. This synthetic +spectrum is then compared against the a-priori normalized +observed spectrum and a χ2 merit function is used to judge +the goodness of the fit. +The broadening function (BF) is a representation of +spectral profiles in velocity space. The BF contains signa- +tures of the vr shifts of different lines and also intrinsic stel- +lar effects like rotational broadening, spots, pulsations, etc. +(Rucinski 1999). We calculated the BF for one of the com- +posite spectra of AI Hya to estimate v sin i values for the +primary and secondary components, respectively. This pro- +cess serves to remove the degeneracy between v sin i and +other atmospheric parameters like T eff and [M/H]. A mod- +ified version of the treatment described in Rucinski (1999) +was adopted and a multi Gaussian fit was implemented. The +BF was calculated in a wavelength range of 4080-5000 ˚A. A +synthetic solar-type spectrum with zero projected rotational +velocity v sin i was used as our template. To deal with the +noise in the data, a Gaussian smoother of 3 km s−1 rolling +window was applied to the BF. Two clear peaks were visible +in the velocity space, as shown in Figure 6, corresponding +to the primary and secondary components. The peaks were +fitted with the rotational profile, +G(v) = A +� +�c1 +� +1 − +� +v +vmax +�2 ++ c2 +� +1 − +� +v +vmax +�2�� +�+lv+k +(1) +where A is the area under the profile, vmax is the maximum +velocity shift which occurs at the equator (Gray 2005), c1 +and c2 are constants which are a function of limb darkening +themselves, while l and k are correction factors to the BF +continuum. The BF fit was calculated for the spectra with +the highest SNR and good separation between the compo- +nents in velocity space. The best BF fit to the line profile of +the primary and secondary binary components are shown in +Fig.6. Fixing the obtained values of v sin i from this analysis +and log g from the light curve solution, the gssp composite +fitting routine was applied to obtain stellar temperatures +Teff (1,2), microturbulent velocities ξ and global metallicities +[M/H]. +The step size of the grid gives us a rough idea of the +errors involved. However, to obtain more robust error esti- +mates we plotted the χ2 data for each parameter and fitted +a parabola to obtain the minimum; its distance to the in- +tercepts on the abscissa are taken as the errors. These pa- +rameters are obtained for a total of four spectra and then +averaged out. The remaining spectra were not suitable for +© 2021 RAS, MNRAS 000, 1–13 + +Comprehensive study of AI Hya +7 +100 +50 +0 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +Primary +50 +100 +150 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Secondary +Relative Flux +2457109.96513 BJD +Radial Velocities (km/s) +Figure 6. Broadening functions for the primary and secondary +components of AI Hydrae calculated using HIDES spectra (epoch: +2457109.96513 HJD), which provided a good SNR and velocity +separation between the two components. The blue, dashed line +represents best-fit rotational function (Group-P). +5320 +5330 +5340 +5350 +5360 +5370 +5380 +5390 +5400 +Wavelength (A) +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +Normalized Flux +Data +Model +Figure 7. A snippet of the best-fit model generated by gssp for +the given set of parameters (Group-P). +the analysis in gssp due to lower SNR. The results of the +analysis are compiled in Table 3 and a sample of the fit to +one of the spectra is shown in Figure 7. +4.2.2 +iSpec +A complimentary spectroscopic analysis was performed on +the disentangled spectra of the primary and secondary stars +using iSpec (Blanco-Cuaresma et al. 2014). Before the anal- +ysis, the spectra are treated for vr offset and continuum cor- +rection. Estimates of flux errors were introduced as a sum of +errors calculated from SNR, and flux-scaled residuals from +the disentangled routine. For the spectroscopic analysis we +fixed the log g parameter with values obtained from the light +curve solution and limb darkening parameters with values +adopted from Claret & Bloemen (2011). +We fit the model using the spectral synthesis approach. +This is done by implementing the use of the spectrum code +(Gray & Corbally 1994), a marcs (Gustafsson et al. 2008) +grid of model atmospheres, and solar abundances taken from +Asplund et al. (2009). We adopt a two-step process. The ini- +tial run is aimed at estimating the global metallicity ([M/H]) +by keeping it as a free parameter. The macroturbulent ve- +locity (vmac) and alpha enhancement parameters were set +to zero as vmac has a negligible contribution for stars in the +concerned temperature range and alpha enhancement, when +set as a free parameter, produced implausible values. v sin i +was set to the values obtained by the BF analysis. We com- +pared the obtained value for [M/H] with results from the +gssp analysis and found it to be consistent with the errors. +The average value of [M/H] was calculated and fixed for the +next step where we fit for temperature Teff, microturbulent +velocity ξ, and abundances of Iron (Fe), Nickel (Ni) and +Chromium (Cr), as these were the prominent lines in the +chosen spectral range. +The output parameters obtained from iSpec are given +in Table 3 and Table 4. It is to be noted that Fe, Ni, and Cr +are more abundant in the primary compared to solar values +and those of the secondary star. This trend in the abun- +dances is in agreement with the values obtained by group-K. +The output parameters for the secondary star agree fairly +well with those from the gssp analysis and from the group- +K. The best fit solution for the primary component, as in +the case of gssp analysis, also hinted towards a lower Teff +compared to the group-K solution. +5 +BINARY MODELLING +5.1 +Group-K +To update the fundamental stellar parameters (M, R) of +AI Hya, we performed binary modelling with the help of the +determined atmospheric parameters and the results of the +vr investigation. +In binary modelling, the TESS data were used. How- +ever, the shapes of the eclipses of AI Hya are distorted due +to the pulsations. Thus we first cleaned the pulsations and +only then carried out the binary modelling. Therefore, the +Period04 program (Lenz & Breger 2005) was used to detect +the variations caused by oscillations. The derived pulsation +frequencies6 were cleaned from the light curve and the resid- +uals were used in the binary modelling. +In this analysis, we used the Wilson-Devinney code +(Wilson & Devinney 1971) combined with Monte-Carlo sim- +ulations (Zola et al. 2004, 2010). The pulsation removed data +were binned to around 4000 points to be used in the binary +modelling code. AI Hya is classified as a detached binary +system in the literature (Lee, Hong, & Kristiansen 2020). +According to their results (e.g., for Ω, q, a), both compo- +nents do not seem to fill their Roche lobe, hence the sys- +tem is defined as a detached binary. Also, the morphology +of the light curve, i.e. very small ellipsoidal variations and +eclipses spanning a small fraction of the orbital period, con- +firm this classification. Therefore, a detached binary config- +uration was considered our analysis. In the modelling, we +took some parameters fixed, such as the Teff of the hotter +component, Porb, q taken from our results and bolometric +albedos (Ruci´nski 1969), bolometric gravity-darkening coef- +ficient (von Zeipel 1924), and the logarithmic limb darken- +ing coefficient (van Hamme 1993) taken the same as given +Kahraman Ali¸cavu¸s & Ali¸cavu¸s (2019). The orbital inclina- +tion (i), Teff of the cooler component, phase shift (φ), e, a, +ω, and dimensionless potential (Ω) of the components were +set free. +6 The frequencies given in Sect. 6. +© 2021 RAS, MNRAS 000, 1–13 + +8 +F. Kahraman Ali¸cavu¸s et. al. +Table 5. Results of the light curve analysis and the fundamental +stellar parameters. The Subscripts 1, 2 and 3 represent the hotter, +the cooler, and third binary components, respectively. a Shows +the Fixed Parameters. +Parameter +Value +Value +Group-K +Group-P +i (o) +89.866 ± 0.015 +89.837 ± 0.136 +T 1a (K) +7700 ± 100 +7330 ± 170 +T 2 (K) +7180 ± 230 +7210 ± 150 +Ω1 +11.412 ± 0.046 +- +Ω2 +8.961 ± 0.035 +- +Phase shift +-0.0310 ± 0.0001 +- +q +1.074a +1.075 +r1∗ (mean) +0.1001 ± 0.0036 +0.1015 ± 0.0005 +r2∗ (mean) +0.1412 ± 0.0026 +0.1412 ± 0.0006 +l1 / (l1+l2) +0.381 ± 0.016 +0.374 ±0.02 +l2 / (l1+l2) +0.619 ± 0.016 +0.616 ± 0.02 +l3 +0.0 +0.0 +Derived Quantities +M1 (M⊙) +1.950 ± 0.033 +1.950 ± 0.033 +M2 (M⊙) +2.096 ± 0.035 +2.096 ± 0.035 +R1 (R⊙) +2.754 ± 0.015 +2.787 ± 0.020 +R2 (R⊙) +3.863 ± 0.021 +3.877 ± 0.026 +log (L1/L⊙) +1.381 ± 0.034 +1.311 ± 0.081 +log (L2/L⊙) +1.554 ± 0.035 +1.549 ± 0.097 +log g1 (cgs) +3.848 ± 0.003 +3.838 ± 0.005 +log g2 (cgs) +3.586 ± 0.003 +3.582 ± 0.005 +Mbol1 (mag) +1.30 ± 0.08 +1.474 ± 0.202 +Mbol2 (mag) +0.87 ± 0.08 +0.877 ± 0.243 +MV 1 (mag) +1.25 ± 0.08 +1.424 ± 0.208 +MV 2 (mag) +0.79 ± 0.08 +0.822 ± 0.258 +Distance (pc) +659 ± 30 +642 ± 36 +As a result of this analysis, the fundamental parameters +of both components of AI Hya were calculated. Additionally, +the bolometric (Mbol) and absolute (MV ) magnitudes were +estimated. The jktabsdim code (Southworth, Maxted, & +Smalley 2004b) and the bolometric correction (Eker et al. +2020) are used in the calculations of these parameters. The +outcome of the binary modelling is given in Table 5 and the +consistency of the theoretical light curve with the observa- +tion is shown in Fig. 8. +When the results of this analysis were examined, one +can notice that the more luminous star is the more massive +and also the cooler component. This result is consistent with +the results found in the vr analysis by group-K. +5.2 +Group-P +Aiming to determine precise physical and orbital parame- +ters of AI Hya, we performed its modelling in version 40 of +the jktebop (Southworth, Maxted, & Smalley 2004b). This +program is written by J. Southworth and aimed at modelling +light curves of detached eclipsing binaries and is based on +the ebop program (Popper & Etzel 1981). The code treats +stars as spheres to calculate the eclipse shapes, and biaxial +ellipsoids to calculate proximity effects. The light curves are +calculated by numerical integration of concentric circles over +each stellar surface. It can deal with stellar oblateness of up +Figure 8. Theoretical binary modelling fit without spot assump- +tion (solid-line) (Group-K). +to 4% making it a good choice for AI Hya. The photometric +data remain the same as used by Group-K. +The parameters set as free are Porb, time of minima +of the primary eclipse To, inclination i, eccentricity e, ar- +gument of periastron ω, surface brightness ratio J (sec- +ondary/primary), ratio of radii ( rA +rB ), and the sum of radii +(rA+rB). These radii are relative to the semi-major axis. For +the limb darkening coefficients, we use a logarithmic law and +set their initial values according to Claret (2017). The coef- +ficients were fixed for the initial fit and were perturbed at +the error estimation step. +The code gives an option to include multiple sine and +polynomial functions during the light curve modelling to ac- +count for periodic and long-term trends. We use this func- +tionality to our advantage to pseudo-model the observed +pulsations so that their effect on the binary model is mini- +mal, giving us an improved precision. We analyse the out-of- +eclipse portions of the light curve using pyriod7, and use the +frequencies to initialise the sinusoids in the jktebop input +files. This is done in an iterative way where we add one sine +with a constant period and fit for its epoch and amplitude. +The frequency is kept if the model is improved significantly; +otherwise the next most prominent frequency is taken. In +this analysis, we used a total of 9 sines, which is the limit +for jktebop. The number of independent frequencies of AI +Hya is higher than this maximum limit, hence we are left +with some residual pulsation signals as seen in Figure 9 and +Figure 10. +Once the sines are fixed to the best fit values of epoch, +period and amplitudes, we make the Monte Carlo runs for +error estimation. The results of this analysis are mentioned +in Table 5, in comparison to the values obtained by group-K. +Similarly to the other group, we used the results of vr, and +jktebop solutions to calculate a set of absolute parameters, +including masses, radii, luminosities, and distance. The ef- +fective temperatures mentioned in the table are an average +over the sum of Teff obtained from gssp and iSpec analysis. +7 https://github.com/keatonb/Pyriod +© 2021 RAS, MNRAS 000, 1–13 + +1.0 +0.9 +xnl +Normalised f +0.8 +0.7 +0.6 +Data +Model +0.03 +Res. +0.00 +0.03 +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +1.1 +1.2 +PhaseComprehensive study of AI Hya +9 +Figure 9. jktebop model with 9 sines used to model the pulsa- +tions (Group-P). +Figure 10. Zoomed-in view of the model over an orbit (Group- +P). +6 +FREQUENCY ANALYSIS OF THE +PULSATIONS +AI Hya was observed by TESS during observation sector 7 +in January/February 2019. We used the Simple Aperture +Photometry data from the 2-min cadence light curves avail- +able at the Mikulski Archive for Space Telescopes8 (MAST). +This time series spans 24.45 d and contains 16362 measure- +ments. To determine the pulsation frequencies, we used only +the data that were taken out of eclipse, which reduced the +data set to 14019 measurements (time span 24.07 d). +This time series was analysed using the Period04 soft- +ware (Lenz & Breger 2005) by group-K. This package applies +single-frequency power spectrum analysis and simultaneous +multi-frequency sine-wave fitting. These sine-wave fits are +subtracted from the data and the residuals examined for +the presence of further periodicities. The application of this +procedure to AI Hya is illustrated in Fig. 11. +During such a process, it is important to decide where +to stop. Often this is facilitated via the application of SNR +criteria. In this work, we have adopted the strategy proposed +by Breger et al. (1993) which is to compute the ratio of the +signal amplitude relative to the local noise level to deter- +mine whether the frequency under consideration represents +a significant detection. Whereas Breger et al. (1993) propose +SNR > 4 for a detection, recent findings for space-based data +8 https://mast.stsci.edu/portal/Mashup/Clients/Mast/Portal.html +Figure 11. The Fourier Transform of the out-of-eclipse TESS +light curve of AI Hya (top) and subsequent prewhitening steps. +The blue arrows denote the signals detected. Outside of the fre- +quency range shown no significant signal is present. +Table 6. A least squares fit of the pulsation frequencies of AI +Hya. Formal error estimates for the independent frequencies and +phases (Montgomery & O’Donoghue 1999) are given in braces in +units of the last digits after the comma. +Frequency +Amplitude +SNR +d−1 +mmag +±0.02 +ν1 +6.2412(1) +4.75 +54.2 +ν2 +9.2654(4) +1.18 +9.7 +ν3 +9.9065(4) +1.20 +9.4 +ν4 +12.715(1) +0.48 +4.5 +ν5 +12.928(1) +0.54 +5.4 +ν6 +9.3689(4) +1.42 +11.5 +3νorb +0.3619 +1.76 +7.5 +4νorb +0.4825 +1.32 +5.9 +ν7 +5.5599(7) +0.78 +8.3 +ν8 +5.7804(1) +0.69 +7.5 +2νorb +0.2413 +1.75 +7.3 +ν9 +5.6375(7) +0.73 +7.7 +ν10 +7.136(1) +0.37 +6.0 +ν11 +7.751(1) +0.39 +5.2 +ν12 +9.3051(6) +0.82 +6.7 +ν13 +9.8432(8) +0.69 +5.3 +ν3 + ν7 +15.464(1) +0.43 +5.6 +(e.g., Baran & Koen 2021) suggest that a more conservative +limit must be chosen. Given the restricted frequency range +in which we search for periodicities, our requirement was +SNR > 4.5. Furthermore, in unresolved frequency spectra, +the periodic content present in the time series can easily +be overinterpreted (Balona 2014) which suggests caution re- +garding the present data set. Consequently, we stopped the +frequency search after the detection of 17 signals (lowest +panel of Fig. 11). More periodicities are certainly present, +but these need to await a longer data set for reliable detec- +tion. We list the frequency solution so derived in Table 6. +© 2021 RAS, MNRAS 000, 1–13 + +8.5 +8.6 +8.7 +08.8 +a +M +8.9 +9.0 +Data +9.1 +Model +Residuals +0.01 +Resi. +0.00 +0.01 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +Phase8.48 +8.49 +0 +8.50 +8.51 +Data +Model +1494 +1496 +1495 +1498 +1497 +1499 +1500 +1501 +1502 +Time (ID-2457000) days10 +F. Kahraman Ali¸cavu¸s et. al. +This table also contains three harmonics of the orbital +period. These are not pulsation frequencies, but a conse- +quence of residual binary-induced variability (see Section on +binary modeling for a discussion). The pulsation frequencies +themselves were found in an interval between 5.5 – 13.0 d−1, +with one possible combination frequency. It is however not +clear whether this is a real combination or just a numeri- +cal coincidence keeping in mind the short data set, hence +poor frequency resolution. Our frequency solution is similar +to that reported by Lee, Hong, & Kristiansen (2020) apart +from their identification of possible combination frequencies +that are partly implausible. +To use the pulsations to learn more about the indi- +vidual components by applying asteroseismic methods, it +is essential to know from which star the pulsations orig- +inate. A quick look at the TESS light curve reveals that +pulsations are clearly visible during the total part of the +primary eclipse, meaning that the secondary is the source +of the highest amplitude oscillations. However, both com- +ponents of AI Hya are located within the pulsational in- +stability strip of the δ Scuti stars (Murphy et al. 2019, see +Fig. 12), thus the primary may pulsate as well. δ Scuti stars +generally pulsate in pressure and mixed modes of low ra- +dial order (e.g., Breger 2000). Using the stellar parameters +from Table 5, we can compute the expected frequency of the +radial fundamental mode of both pulsators from the pulsa- +tion constant Q = P +� +ρ/ρ⊙ = PM 1/2R−3/2, assuming Q to +be 0.033 d for this mode (Fitch 1981). We thus expect the +radial fundamental mode frequency of the primary compo- +nent to be around 9.3 d−1, and around 5.8 d−1 for the sec- +ondary component, respectively. In Table 6 oscillation fre- +quencies around both these values are seen, which allows +no more than the educated guess that the pulsations below +∼ 8 d−1 would arise from the secondary component, whereas +the higher frequency modes could originate from either star. +A determination of the origin of the pulsations from the +orbital light time effect is unfortunately out of reach. The +expected light time effect would be about 30 s (cf. Table 2). +An attempt to measure the effect for the strongest pulsa- +tion frequency yielded 35 ± 111 s, a null result. To conclude, +because it is impossible to say with confidence which pulsa- +tion frequencies arise from which component of AI Hya, an +asteroseismic analysis cannot be carried out. +7 +EVOLUTIONARY MODELS +The evolutionary status of the binary components was ex- +amined by utilizing the Modules for Experiments in Stel- +lar Astrophysics (mesa) evolution code (Paxton et al. 2011, +2013) which includes a binary module (Paxton et al. 2015) to +examine the binary orbital evolution and to determine the +initial parameters of binary systems. In this examination, +various evolutionary models were generated considering dif- +ferent metallicity (Z). In the models, MESA equation-of- +state (EOS) were used. The EOS tables are based on the +OPAL EOS tables (Rogers & Nayfonov 2002). The OPAL +opacity tables and the default solar mixtures were adopted +as Z initial fraction from Asplund et al. (2009). Helium +mass fraction were taken Y=0.28, for Z=0.02. Convective +core overshoot was described by the exponentially decaying +prescription of Herwig (2000) and overshooting parameter +adopted 0.20 for both components (Claret & Torres (2016) +find 0.208 for both components). A mixing length αMLT +value of 1.8 was used as the theoretical δ Scuti instability +strip (Dupret et al. 2004, 2005) was obtained with this αMLT +value. +Taking into account the calculated parameters in the +binary modelling for both groups, the evolutionary status +of the binary components was investigated. As a result, we +found that the secondary (more luminous) binary compo- +nent can be represented with the same evolutionary tracks +according to both groups’ results. However, the less lumi- +nous primary component’s position was determined with +different Z parameters as the parameters of this star were +found to be slightly different in the study of the two groups. +According to the evolutionary models, the Z parameters of +both binary components were found similar to solar (As- +plund et al. 2009) within the errors which differs from the +results of the groups as we determined that the less luminous +component’s atmosphere is somewhat enhanced in metals. +The results of this analysis are given in Table 7 and a H-R +diagram is shown in Fig. 12. The observational borders of +the δ Scuti instability strip were taken from Murphy et al. +(2019). As can be seen from the H-R diagram, both binary +components are placed inside the δ Scuti instability strip. +8 +DISCUSSION AND CONCLUSIONS +In this analysis, we present the results of the detailed anal- +ysis of AI Hya carried out by two independent groups. The +system was observed with different high-resolution spectro- +graphs (R≳38000). The radial velocity variations of AI Hya +were modelled using the vr measurements of both groups +and the orbital parameters such as T0, Porb, e and q were +updated. The resulting parameters of the analysis of both +groups are consistent with each other within the errors and +they slightly differ from the results of Popper (1988). Espe- +cially the e value shows a discrepancy. Popper (1988) found +e to be 0.2301 ± 0.0015 while in our study it was determined +as 0.2419 ± 0.0036 and 0.2432 ± 0.0050 by group-P and -K, +respectively. +Since our high-resolution spectra are spread over all or- +bital phases, we were able to derive the atmospheric param- +eters of both binary components by modelling either the +composite spectra or the spectra of the individual compo- +nents after applying spectral disentangling. To derive the +atmospheric parameters, v sin i and the chemical composi- +tion of the binary components, group-K analysed disentan- +gled spectra of the components, while group-P performed +their analysis using both the composite and disentangled +spectra. As a result, group-K found that the more lumi- +nous star is cooler than the less luminous component. They +found the Teff +values from the Hβ line fit and Fe lines +to be 7500 ± 200 K and 7700 ± 100 K for the primary and +7000 ± 150 K and 7200 ± 100 K for the secondary compo- +nent, respectively. Group-P used two different codes in their +analysis. With the gssp code analysis they found a similar +result with group-K even though the resulting Teff values +differ from each other, they determined that the more lu- +minous star is cooler (7150 ± 250 K) and less luminous one +is hotter (7350 ± 300 K). In the iSpec analysis of group-P, +Teff values of both components were found similar to the +© 2021 RAS, MNRAS 000, 1–13 + +Comprehensive study of AI Hya +11 +Table 7. Results obtained from the best-fit evolutionary models. +Parameter +Group-K +Group-P +P initial (days) +8.34 (1) +8.34 (1) +einitial +0.242 (2) +0.243 (2) +Z1 +0.013 (2) +0.016 (2) +Z2 +0.018 (2) +0.018 (2) +Age (Myr) +850 (20) +860 (20) +results of the gssp analysis within error bars. The primary’s +temperature is the most significant discrepancy between the +values derived by the two groups. The exact reason for this +temperature inconsistency is not fully understood, although +it is still only at a level of ∼1.1σ. +In the chemical abundance analysis, both groups found +the less luminous but hotter binary component to show +overabundance while the other component has chemical +abundance similar to solar. Both groups determined the +abundances of some individual elements such as iron (Fe). +They derived Fe abundances as 8.25 ± 0.23 (group-K) and +7.83 ± 0.16 (group-P). These values are consistent with each +other within their 1σ errors, and both demonstrate that the +hotter component has a slightly metal-rich chemical abun- +dance compared to solar values (see Table 4). This comes +somewhat to a surprise, as this binary system should have +been formed in the same interstellar environment and hence +its components should have the same chemical composition. +The difference could be due to the consequences of the evolu- +tion of the system. If AI Hya had a very eccentric orbit when +the system was formed, there could be some material flows +from one component to another that could have changed the +diffusion in one component. Another explanation was given +by Yushchenko et al. (2015) and they pointed out that pos- +sible gas and dust accretion from the circumstellar envelope +could alter the atmospheric composition of one component. +After the determination of the atmospheric parameters, +they were used as input in the binary modelling. Overall, +even though both working groups used different approaches +to estimate the parameters of the binary component of +AI Hya, the values determined by both groups are found to +be consistent with each other within the error bars. The two +groups obtained very similar M and R values with a ⩽1.7% +and ∼0.5% accuracy, respectively. When we compare these +values with the ones found by Lee, Hong, & Kristiansen +(2020), we notice that there are slight differences, especially +in the R parameters, and there is significant diversity in the +calculated distance. These differences could be caused by +the different assumptions of the atmospheric parameters. +The evolutionary status of the system was examined +and it was found that both binary components are inside the +δ Scuti instability strip. The age of the system is determined +as well. According to the determined ages, we could say that +AI Hya is in an important evolutionary phase in terms of +binary evolution. The rapidly evolving massive component +will begin the mass transfer process to the less massive one +approximately 20 Myr from now. This situation could cause +significant variations in the oscillation properties. Increas- +ing the number of such bodies is important in terms of ex- +amining the pulsating structures before the mass transfer +processes. +The pulsation properties of AI Hya were examined us- +Figure 12. The positions of the binary components in the H-R +diagram according the results of both group-K (g-K) and group-P +(g-P). The instability strip (IS) borders of the δ Scuti stars were +taken from Murphy et al. (2019). +ing the TESS data. However, the system has only one sector +of SC data, which offers us a poor frequency resolution. In +the analysis, pulsation frequencies were found between 5.5 +and 13 d−1. As both binary components are placed in the +δ Scuti instability strip, we were unable to say whether one +or both pulsate. Apart from that, we could not find pulsa- +tions related to the orbital frequency. +As a result of this study, we thoroughly examined +a detached binary system showing oscillations. This kind +of objects is particularly important to examine the insta- +bility strip of δ Scuti stars since they allow us to deter- +mine fundamental astrophysical, atmospheric parameters +and the chemical abundances of individual binary compo- +nents. Hence an increasing number of analyses of such sys- +tems is expected to be essential to deeply understand the +nature of pulsations. +ACKNOWLEDGMENTS +The authors would like to thank the reviewer for useful +comments and suggestions that helped to improve the +publication. This study has been supported by the Sci- +entific and Technological Research Council (TUBITAK) +project 120F330. GH thanks the Polish National Center +for Science (NCN) for supporting the study through +grants 2015/18/A/ST9/00578 and 2021/43/B/ST9/02972. +TP’s research is supported through NCN OPUS project +number 2017/27/B/ST9/02727. AM’s acknowledges the +support provided by the Polish National Science Centre +(NCN) OPUS project number 2017/27/B/ST9/02727 and +2021/41/N/ST9/02746. Based on observations made with +the Mercator Telescope, operated on the island of La Palma +by the Flemish Community, at the Spanish Observatorio +del Roque de los Muchachos of the Instituto de Astrof`ısica +de Canarias. The TESS data presented in this paper were +obtained from the Mikulski Archive for Space Telescopes +(MAST). Funding for the TESS mission is provided by +© 2021 RAS, MNRAS 000, 1–13 + +2.0 +tAl Hya +1.8 +1.6 +(L/ Lo) +1.4 +1.2 +Primary (g-K), ☆ Primary (g-P) +Secondary (g-K), ☆ Secondary (g-P) +1.950 MO track (Z=0.013) +1.950 MO track (Z=0.016) +1.0 + 2.096 MO track (Z=0.018) +IS +0.8 +4.00 +3.95 +3.90 +3.85 +3.80 +3.75 +3.70 +3.65 +3.60 +log T (K)12 +F. Kahraman Ali¸cavu¸s et. al. +the NASA Explorer Program. This work has made use +of data from the European Space Agency (ESA) mission +Gaia (http://www.cosmos.esa.int/gaia), processed by the +Gaia Data Processing and Analysis Consortium (DPAC, +http://www.cosmos.esa.int/web/gaia/dpac/consortium). +Funding for the DPAC has been provided by national +institutions, in particular the institutions participating +in the Gaia Multilateral Agreement. This research has +made use of the SIMBAD data base, operated at CDS, +Strasbourq, France. +DATA AVAILABILITY +The data underlying this work will be shared at reasonable +request to the corresponding author. +REFERENCES +Aerts C., Christensen-Dalsgaard J., Kurtz D. W., 2010, +aste.book +Antoci +V., +Cunha +M., +Houdek +G., +Kjeldsen +H., +Trampedach R., Handler G., L¨uftinger T., et al., 2014, +ApJ, 796, 118. doi:10.1088/0004-637X/796/2/118 +Asplund M., Grevesse N., Sauval A. J., Scott P., 2009, +ARA&A, 47, 481 +Balona L. A., 2014, MNRAS, 439, 3453 +Baran, A. S. & Koen, C., 2021, AcA, 71, 113 +Blanco-Cuaresma S., Soubiran C., Heiter U., Jofr´e P., 2014, +A&A, 569, A111. doi:10.1051/0004-6361/201423945 +Borucki W. J., Koch D., Basri G., Batalha N., Brown +T., Caldwell D., Caldwell J., et al., 2010, Sci, 327, 977. +doi:10.1126/science.1185402 +Bowman D. M., Kurtz D. W., 2018, MNRAS, 476, 3169. +doi:10.1093/mnras/sty449 +Breger, M., 2000, in Delta Scuti and Related Stars, ed. M. +Breger & M. H. Montgomery, ASP Conf. Ser. 210, 3 +Breger, M., et al., 1993, A&A, 271, 482 +Catanzaro G., Leone F., Dall T. H., 2004, A&A, 425, 641 +Catanzaro G., Colombo C., Ferrara C., Giarrusso M., 2022, +MNRAS, 515, 4350. doi:10.1093/mnras/stac1723 +Catanzaro +G., +Bus`a +I., +Gangi +M., +Giarrusso +M., +Leone +F., +Munari +M., +2019, +MNRAS, +484, +2530. +doi:10.1093/mnras/stz080 +Claret +A., +2017, +A&A, +600, +A30. +doi:10.1051/0004- +6361/201629705 +Claret +A., +Torres +G., +2016, +A&A, +592, +A15. +doi:10.1051/0004-6361/201628779 +Coelho P., Barbuy B., Mel´endez J., Schiavon R. P., +Castilho B. V., 2005, A&A, 443, 735. doi:10.1051/0004- +6361:20053511 +Dupret M.-A., Grigahc`ene A., Garrido R., Gabriel M., Scu- +flaire R., 2004, A&A, 414, L17 +Dupret M.-A., Grigahc`ene A., Garrido R., Gabriel M., +Scuflaire R., 2005, A&A, 435, 927. doi:10.1051/0004- +6361:20041817 +Eker Z., Soydugan F., Bilir S., Bakı¸s V., Ali¸cavu¸s F., +¨Ozer S., Aslan G., et al., 2020, MNRAS, 496, 3887. +doi:10.1093/mnras/staa1659 +Fitch, W. S., 1981, ApJ 249, 218 +Gaia Collaboration, 2022, Gaia DR3: data release content +and main properties. A&A, in prep. +Gaulme +P., +Guzik +J. +A., +2019, +A&A, +630, +A106. +doi:10.1051/0004-6361/201935821 +Gray +R. +O., +Corbally +C. +J., +1994, +AJ, +107, +742. +doi:10.1086/116893 +Gustafsson B., Edvardsson B., Eriksson K., Jørgensen +U. G., Nordlund ˚A., Plez B., 2008, A&A, 486, 951. +doi:10.1051/0004-6361:200809724 +Grevesse N., Asplund M., Sauval A. J., 2007, SSRv, 130, +105. doi:10.1007/s11214-007-9173-7 +Iglesias-Marzoa, R., L´opez-Morales, M., & Jes´us Ar´evalo +Morales, M. 2015, pasp, 127,567 +Handler G., Kurtz D. W., Rappaport S. A., Saio H., +Fuller J., Jones D., Guo Z., et al., 2020, NatAs, 4, 684. +doi:10.1038/s41550-020-1035-1 +Herwig F., 2000, A&A, 360, 952 +Ilijic S., Hensberge H., Pavlovski K., Freyhammer L. M., +2004, ASPC, 318, 111 +Jayasinghe T., et al., 2018, MNRAS, 477, 3145 +Joergensen H. E., Gronbech B., 1978, A&A, 66, 377 +Kahraman Ali¸cavu¸s F., et al., 2016, MNRAS, 458, 2307 +Kahraman Ali¸cavu¸s F., Soydugan E., Smalley B., Kub´at +J., 2017, MNRAS, 470, 915. doi:10.1093/mnras/stx1241 +Kahraman Ali¸cavu¸s F., Ali¸cavu¸s F., 2019, MNRAS, 488, +5279. doi:10.1093/mnras/stz2003 +Kahraman Ali¸cavu¸s F., G¨um¨u¸s D., Kırmızıta¸s ¨O., Ekinci +¨O., C¸avu¸s S., Kaya Y. T., Ali¸cavu¸s F., 2022, RAA, 22, +085003. doi:10.1088/1674-4527/ac71a4 +Kambe E., et al., 2013, PASJ, 65, 15 ¸s +Konacki +M., +Muterspaugh +M. +W., +Kulkarni +S. +R., +He�lminiak K. G., 2010, ApJ, 719, 1293. doi:10.1088/0004- +637X/719/2/1293 +Kreiner J. M., 2004, AcA, 54, 207 +Kurtz D. W., Handler G., Rappaport S. A., Saio H., Fuller +J., Jacobs T., Schmitt A., et al., 2020, MNRAS, 494, 5118. +doi:10.1093/mnras/staa989 +Kurucz R. L., Avrett E. H., 1981, SAOSR, 391, +Kurucz R., 1993, KurCD, 13, +Lampens +P., +2021, +Galax, +9, +28. +doi:10.3390/galaxies9020028 +Lee J. W., Hong K., Kristiansen M. H., 2020, PASJ, 72, +37. doi:10.1093/pasj/psaa020 +Lenz P., Breger M., 2005, CoAst, 146, 53 +Leone F., et al., 2016, AJ, 151, 116 +Liakos +A., +Niarchos +P., +2017, +MNRAS, +465, +1181. +doi:10.1093/mnras/stw2756 +Montgomery M. H., O’Donoghue D., 1999, Delta Scuti Star +Newsletter 13, 28 +Murphy S., Hey D., Van Reeth, T., Bedding T., 2019, MN- +RAS 485, 2380 +Paxton B., Bildsten L., Dotter A., Herwig F., Lesaffre P., +Timmes F., 2011, ApJS, 192, 3 +Paxton B., et al., 2013, ApJS, 208, 4 +Paxton B., et al., 2015, ApJS, 220, 15 +Pepe F., Bouchy F., Mayor M., Udry S., 2018, haex.book, +190. doi:10.1007/978-3-319-55333-7 190 +Pilecki B., Gieren W., Smolec R., Pietrzy´nski G., Thomp- +son I. B., Anderson R. I., Bono G., et al., 2017, ApJ, 842, +110. doi:10.3847/1538-4357/aa6ff7 +Pojma´nski G., 2002, AcA, 52, 397 +Popper +D. +M., +Etzel +P. +B., +1981, +AJ, +86, +102. +© 2021 RAS, MNRAS 000, 1–13 + +Comprehensive study of AI Hya +13 +doi:10.1086/112862 +Popper D. M., 1988, AJ, 95, 190. doi:10.1086/114627 +Raskin G., et al., 2011, A&A, 526, A69 +Ricker G. R., Winn J. N., Vanderspek R., Latham D. W., +Bakos G. ´A., Bean J. L., Berta-Thompson Z. K., et al., +2014, SPIE, 9143, 914320. doi:10.1117/12.2063489 +Rogers +F. +J., +Nayfonov +A., +2002, +ApJ, +576, +1064. +doi:10.1086/341894 +Ruci´nski S. M., 1969, AcA, 19, 245 +Smalley B., Gardiner R. B., Kupka F., Bessell M. S., 2002, +A&A, 395, 601 +Southworth +J., +Zucker +S., +Maxted +P. +F. +L., +Smal- +ley B., 2004, MNRAS, 355, 986. doi:10.1111/j.1365- +2966.2004.08389.x +Southworth J., Maxted P. F. L., Smalley B., 2004, MNRAS, +351, 1277 +Southworth J., 2013, A&A, 557, A119. doi:10.1051/0004- +6361/201322195 +Southworth +J., +2021, +Univ, +7, +369. +doi:10.3390/universe7100369 +Stancliffe R. J., Fossati L., Passy J.-C., Schneider F. R. N., +2015, A&A, 575, A117. doi:10.1051/0004-6361/201425126 +Stassun K. G., Oelkers R. J., Paegert M., Torres G., Pep- +per J., De Lee N., Collins K., et al., 2019, AJ, 158, 138. +doi:10.3847/1538-3881/ab3467 +Shulyak D., Tsymbal V., Ryabchikova T., St¨utz C., +Weiss W. W., 2004, A&A, 428, 993. doi:10.1051/0004- +6361:20034169 +Torres G., Andersen J., Gim´enez A., 2010, A&ARv, 18, 67. +doi:10.1007/s00159-009-0025-1 +Tsymbal V., 1996, ASPC, 108, 198 +Uytterhoeven K., Moya A., Grigahc`ene A., Guzik J. A., +Guti´errez-Soto J., Smalley B., Handler G., et al., 2011, +A&A, 534, A125. doi:10.1051/0004-6361/201117368 +van Hamme W., 1993, AJ, 106, 2096 +von Zeipel H., 1924, MNRAS, 84, 665 +Wilson R. E., Devinney E. J., 1971, ApJ, 166, 605 +Yushchenko A. V., Gopka V. F., Kang Y.-W., Kim C., Lee +B.-C., Yushchenko V. A., Dorokhova T. N., et al., 2015, +AJ, 149, 59. doi:10.1088/0004-6256/149/2/59 +Zucker +S., +Mazeh +T., +1994, +ApJ, +420, +806. +doi:10.1086/173605 +Zola S., et al., 2004, AcA, 54, 299 +Zola S., Gazeas K., Kreiner J. M., Ogloza W., Siwak M., +Koziel-Wierzbowska D., Winiarski M., 2010, MNRAS, +408, 464 +Rucinski S., 1999, ASPC, 185, 82 +Gray D. F., 2005, oasp.book +Claret +A., +Bloemen +S., +2011, +A&A, +529, +A75. +doi:10.1051/0004-6361/201116451 +Rucinski S., 1999, TJPh, 23, 271 +Tkachenko A., 2015, A&A, 581, A129. doi:10.1051/0004- +6361/201526513 +Table A1. The vr measurements. The subscripts “1” and “2” +represent the more and the less luminous components, respec- +tively. +HJD +vr,1 +vr,2 +Instrument ++2450000 +(km s−1) +(km s−1) +9263.45270 +-12.6 ± 2.8 +109.3 ± 2.7 +CAOS +9161.65803 +132.8 ± 1.8 +-48.3 ± 1.7 +HERMES +9162.64306 +109.8 ± 2.0 +-21.7 ± 1.5 +HERMES +9230.65226 +121.5 ± 1.6 +-24.7 ± 1.8 +HERMES +9231.66393 +124.1 ± 1.7 +-24.9 ± 1.7 +HERMES +9233.62648 +59.8 ± 5.7 +36.1 ± 3.4 +HERMES +9234.55784 +20.6 ± 2.1 +72.1 ± 2.5 +HERMES +9237.61273 +15.0 ± 1.6 +77.3 ± 1.5 +HERMES +9235.43315 +-46.3 ± 1.5 +130.3 ± 1.8 +HERMES +9257.49195 +98.8 ± 1.8 +-2.9 ± 2.0 +HERMES +9260.61123 +-33.9 ± 1.7 +117.9 ± 1.9 +HERMES +9276.55613 +96.6 ± 2.0 +-8.4 ± 1.2 +HERMES +9296.42427 +88.7 ± 1.7 +-3.8 ± 2.0 +HERMES +9297.44747 +126.7 ± 1.8 +-37.4 ± 2.2 +HERMES +9298.45846 +113.5 ± 1.7 +-16.0 ± 1.8 +HERMES +9299.46357 +78.1 ± 2.1 +12.5 ± 2.3 +HERMES +7075.62231 +-39.7 ± 1.5 +129.9 ± 0.5 +CORALIE +7076.63954 +-20.4 ± 1.2 +120.0 ± 1.3 +CORALIE +7109.63123 +-17.9 ± 2.4 +126.0 ± 1.3 +CORALIE +7022.31643 +118.5 ± 1.9 +-31.4 ± 0.5 +HIDES +7060.09414 +-13.6 ± 0.7 +118.2 ± 0.6 +HIDES +7109.96513 +-18.6 ± 1.1 +114.6 ± 0.6 +HIDES +7114.92732 +120.1 ± 1.2 +-34.0 ± 0.7 +HIDES +7146.98403 +118.5 ± 1.3 +-42.9 ± 0.8 +HIDES +7147.96084 +131.3 ± 1.2 +-42.3 ± 0.6 +HIDES +7363.28986 +135.7 ± 0.6 +-48.4 ± 0.7 +HIDES +7755.22744 +-34.0 ± 0.7 +126.5 ± 0.7 +HIDES +7813.13416 +-25.2 ± 0.8 +124.3 ± 0.9 +HIDES +7814.08321 +-28.3 ± 0.9 +126.6 ± 0.9 +HIDES +7846.01822 +-10.8 ± 1.7 +113.4 ± 1.0 +HIDES +8035.34461 +101.6 ± 0.7 +-13.9 ± 0.8 +HIDES +8066.24908 +85.2 ± 0.8 +-4.1 ± 1.3 +HIDES +This paper has been typeset from a TEX/ LATEX file prepared +by the author. +© 2021 RAS, MNRAS 000, 1–13 + diff --git a/4NE3T4oBgHgl3EQfQAnJ/content/tmp_files/load_file.txt b/4NE3T4oBgHgl3EQfQAnJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b35f1ee7292280ca8b2983f78586892fb94b5980 --- /dev/null +++ b/4NE3T4oBgHgl3EQfQAnJ/content/tmp_files/load_file.txt @@ -0,0 +1,1662 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf,len=1661 +page_content='Mon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Astron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 000, 1–13 (2021) Printed 12 January 2023 (MN LATEX style file v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2) Comprehensive spectroscopic and photometric study of pulsating eclipsing binary star AI Hya F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s1,2⋆, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Pawar3†, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' He�lminiak3, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Handler4, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Moharana3, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Ali¸cavu¸s1,2, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' De Cat5, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Leone6,7, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Catanzaro7, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Giarrusso7,8, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Ukita9,10, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kambe11 1C¸anakkale Onsekiz Mart University, Faculty of Science, Physics Department, 17100, Canakkale, Turkey 2C¸anakkale Onsekiz Mart University, Astrophysics Research Center and Ulupınar Observatory, TR-17100, anakkale, Turkey 3Nicolaus Copernicus Astronomical Center, Department of Astrophysics, ul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Rabia´nska 8, PL-87-100 Toru´n, Poland 4Nicolaus Copernicus Astronomical Center, Polish Academy of Sciences, Bartycka 18, PL-00-716 Warsaw, Poland 5Royal Observatory of Belgium, Ringlaan 3, B-1180 Brussel, Belgium 6Dipartimento di Fisica e Astronomia, Sezione Astrofisica, Universit?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='a di Catania, Via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Sofia 78, I-95123 Catania, Italy 7INAF, Osservatorio Astrofisico di Catania, Via S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Sofia 78,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' I-95123 Catania,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Italy 8University of Florence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Via Giovanni Sansone 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' I-50019 Sesto Fiorentino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Italy 9Okayama Astrophysical Observatory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 3037-5 Honjo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kamogata,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Asakuchi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Okayama 719-0232,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Japan 10The Graduate University for Advanced Studies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2-21-1 Osawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Mitaka,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Tokyo 181-8588,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Japan 11Subaru Telescope,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' National Astronomical Observatory of Japan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 650 North Aohoku Place,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Hilo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' HI 96720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' USA Accepted .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Received .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' in original form .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' ABSTRACT The pulsating eclipsing binaries are remarkable systems that provide an opportu- nity to probe the stellar interior and to determine the fundamental stellar parameters precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Especially the detached eclipsing binary systems with (a) pulsating compo- nent(s) are significant objects to understand the nature of the oscillations since the binary effects in these systems are negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Recent studies based on space data have shown that the pulsation mechanisms of some oscillating stars are not completely understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Hence, comprehensive studies of a number of pulsating stars within de- tached eclipsing binaries are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this study, we present a detailed analysis of the pulsating detached eclipsing binary system AI Hya which was studied by two independent groups with different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We carried out a spectroscopic survey to estimate the orbital parameters via radial velocity measurements and the atmospheric parameters of each binary component using the composite and/or disentangled spec- tra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We found that the more luminous component of the system is a massive, cool and chemically normal star while the hotter binary component is a slightly metal-rich object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The fundamental parameters of AI Hya were determined by the analysis of binary variations and subsequently used in the evolutionary modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Consequently, we obtained the age of the system as 850 ± 20 Myr and found that both binary com- ponents are situated in the δ Scuti instability strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The frequency analysis revealed pulsation frequencies between the 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 d−1 and we tried to estimate which binary component is the pulsating one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, it turned out that those frequencies could originate from both binary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Key words: stars: binaries: eclipsing – stars: atmospheres – stars: fundamental parameters – stars: variables: δ Scuti – stars: individual: AI Hya ⋆ E-mail: filizkahraman01@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='com † E-mail: pawar@ncac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='torun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='pl 1 INTRODUCTION To understand the universe, it is necessary to comprehend stars which are its building blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For a deep investigation of stars, we should know their basic stellar parameters such © 2021 RAS arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04409v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='SR] 11 Jan 2023 2 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' as mass (M), radius (R) and chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Binary stars, in particular the eclipsing ones, are the most suitable objects to derive these parameters as M and R can be de- rived with an accuracy better than 1% (Torres, Andersen, & Gim´enez 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Southworth 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Therefore, these systems are substantial for a better understanding of the universe, our Galaxy, and, most directly, stellar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, eclipsing binary systems as such do not provide information about the stellar interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This is where the pulsating stars come in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The oscillation frequencies of pulsating stars can be used to probe the stellar interior by applying asteroseismic methods, making eclipsing binary systems with (a) pulsat- ing component(s) one of the most valuable tools to improve our knowledge of stellar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Various types of pulsating stars in different evolution- ary states exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Some of them, such as β Cephei, δ Scuti, and γ Doradus stars (Lampens 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Southworth 2021), are also found in eclipsing binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The δ Scuti variables are the most common pulsating stars found in eclipsing bina- ries because of their relatively short pulsation periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The δ Scuti stars are A to F-type dwarf or giant stars generally exhibiting pressure mode oscillations with periods between 18 min and 8 h and amplitudes below 0m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 in the V-band (Aerts, Christensen-Dalsgaard, & Kurtz 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Their the- oretical instability strip (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Dupret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2005) indicates the location of objects in the Hertzsprung-Russell (H-R) di- agram that are expected to show δ Scuti-type oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Thanks to space missions such as Kepler (Borucki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2010) and the Transiting Exoplanet Survey Satellite (TESS, Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2014), we learned that δ Scuti stars are also ob- served beyond the borders of the theoretical instability strip, showing the necessity to revise them (Uytterhoeven et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Antoci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Bowman & Kurtz 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Accord- ing to the latest catalog of δ Scuti stars in eclipsing binaries, there are around 90 such objects (Kahraman Ali¸cavu¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This number is now increasing especially by the dis- coveries of new systems from the investigation of the space data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Gaulme & Guzik 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The pulsations of the δ Scuti stars in eclipsing bina- ries are affected by the other binary component (Kahraman Ali¸cavu¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Liakos & Niarchos 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Indeed, their pulsation period (Ppuls) decreases when the orbital period (Porb) becomes shorter and, hence, the other component ap- proaches the pulsating component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' It was also thought that the tidal forces between the binary components can alter the pulsation axis (Kurtz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The first observational proof of this was presented by Handler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2020) thanks to the high-quality data of TESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These authors showed that in some binary systems the pulsation axis can align with the orbital axis because of the tidal forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This type of object is now known as tidally tilted pulsators and they are a clear proof of binary effects on pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For a deep understanding of the effects of binarity on pulsations in eclipsing binary systems and on stellar evolu- tion and structure, comprehensive investigations of such sys- tems are necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' AI Hya (V = 9m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='35) is an eclipsing binary system with a δ Scuti component consisting of a F2m and F0V star (Stancliffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' It has an eccentric orbit and an orbital period of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='289649(2) days (Kreiner 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Spec- troscopic observations revealed that AI Hya is a double-lined binary system (Popper 1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In a recent study, an updated photometric analysis based on the TESS data of AI Hya was given which shows that the secondary component exhibits multiperiodic oscillations (Lee, Hong, & Kristiansen 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, no detailed spectral analysis with high-resolution spectra has been carried out for the system so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' There- fore, we provide a detailed photometric and spectral analysis of AI Hya in this study to reveal the true character of this interesting object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Two teams were working on this system independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' One group was led by TP (group-P with KH, AM, NU, and EK) and the second group by FKA (group-K with GH, FA, PDC, FL, GC, and MG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We used the same photometric but different spectroscopic data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We compared our partial re- sults as the work progressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, the overall approach used by each group was different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the end, we combined our results to obtain the final parameters of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2 the observa- tional data are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The radial velocity and spectral analyses are given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 3 and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The binary modelling and the pulsation frequency analysis are presented in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 5 and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 7, discussions and conclusions are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2 OBSERVATIONAL DATA In the photometric analysis of AI Hya, TESS data was used by both groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' TESS was launched in April 2018 mainly to detect new exoplanets (Ricker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' TESS has moni- tored almost the entire sky which has been subdivided into sectors that are observed for about 27 days each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The TESS observations were taken in 2-min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' short (SC) and 30-min long (LC) cadence in the nominal phase of the mission (first two years).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the extended mission, the LC was reduced to 10-min.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The data are available in the Barbara A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Mikul- ski Archive for Telescopes (MAST)1 where they are released in different versions: simple aperture photometry (SAP) and pre-search data conditioning SAP fluxes (PDCSAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' AI Hya was observed in one sector only (sector 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The 2-min SAP fluxes were used in our analysis since SAP fluxes have lower flux uncertainty and 2-min data are more suitable for the analysis of AI Hya (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' They were converted into magnitude by using the same method as Kahraman Ali¸cavu¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Photometric data from ground-based surveys also exist, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' from ASAS 3 (Pojma´nski 2002) and ASAS-SN (Jayas- inghe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2018), but they are of inferior quality and do not allow for proper analysis of pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The TESS sector 7 data are the best ones available so far, although AI Hya will again be visible in the satellite’s field of view in sector 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The spectroscopic data of the system were taken from four different instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The list of the instruments and the basic information about them are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' One spectrum was taken with Catania Astrophysical Observatory Spectropolarimeter (CAOS, Leone, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The CAOS is a high-resolution, fibre-fed, cross-dispersed ´echelle spec- trograph installed to the 91-cm telescope at the Catania Astrophysical Observatory (Mt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Etna, Italy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Three spectra of AI Hya were collected from the CORALIE ´echelle spec- trograph which is mounted on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2-m Leonhard Euler 1 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='edu © 2021 RAS, MNRAS 000, 1–13 Comprehensive study of AI Hya 3 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Information about the spectroscopic observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' N, R and SNR represent the number of the spectra, resolving power and the signal-to-noise ratio, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Spectrometer N Observations R SNR Spectral years range [˚A] CAOS 1 2021 38000 50 415 − 670 CORALIE 3 2015 60000 20 − 34 390 − 680 HERMES 15 2020 85000 50 − 70 377 − 900 HIDES 13 2014 − 2017 50000 40 − 88 408 − 752 telescope at La Silla Observatory (Chile) (Pepe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The High Efficiency and Resolution Mercator ´echelle spec- trograph (HERMES) was also used to obtain high-resolution spectra of AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' HERMES is mounted on the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2-m Mer- cator telescope at the Roque de Los Muchchos observa- tory on the Canary Island La Palma in Spain (Raskin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The last instrument used in this study is the HIgh-Dispersion ´Echelle spectrograph (HIDES).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' HIDES is attached to the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='88-m telescope of Okayama astrophysical observatory in Japan (Kambe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The spectra of CORALIE and HIDES were taken by group-P, while the spectra of CAOS and HERMES were gathered by group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In total 32 spectra of AI Hya were gathered and these spec- tra are well distributed in orbital phases of AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Each group used the obtained spectra to measure the radial ve- locity (vr ) changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Additionally, these data were taken into account to derive the atmospheric parameters (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' effective temperature Teff, surface gravity log g, metallicity) and the projected rotational velocity (v sin i) of the components of AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 3 RADIAL VELOCITY ANALYSIS The vr values of the AI Hya system were measured with different approaches by both group-P and group-K using different spectra taken from the distinct instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 vr measurements Group-P calculated the vr values from HIDES and CORALIE spectra, using the two-dimensional cross- correlation todcor program (Zucker & Mazeh 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the analysis, a synthetic spectrum was used as a template and this spectrum was generated using an ATLAS9 model at- mosphere (Kurucz 1993) having Teff, metallicity [M/H] and v sin i parameters of 6800 K, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 and 30 km s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' When a template with 60 km s−1(the v sin i value found in further analysis) was used, the results did not improve in terms of rms of the orbital fit, nor did the uncertainties of orbital elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Moreover, some points, with the smallest difference in vr measurements, seemed to suffer from sys- tematic effects, and had to be rejected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We therefore believe the use of 30 km s−1templates was justified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The calculated vr values for each binary component are given in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Group-K used the RaVeSpAn code (Pilecki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2017) to determine the vr values of the binary components using the broadening function formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the analysis, local thermodynamic equilibrium (LTE) synthetic spectra with atmospheric parameters similar to that of group-P were used as templates (Coelho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The spectra of CAOS and HERMES were used in the vr measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The resulting vr measurements are given in Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 vr curve modelling For the spectroscopic orbital fitting, group-P used all the available vr measurements, including those made by group- K and from Popper (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Group-P used the v2fit code (Konacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2010) which adjusts a double-Keplerian with a Levenberg-Marquardt algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this analysis, the am- plitude of vr curves (K), Porb, the time of phase zero (T0), mass centre’s velocity (γ), eccentricity (e) and argument of the periastron (ω) were set as free parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Thanks to the long time span of the data (>51 years), it was possi- ble to detect the apsidal motion ( ˙ω) of the binary’s orbit: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='186(56) deg/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This is in reasonable agreement (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='75σ) with the value given by Lee, Hong, & Kristiansen (2020): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='075(31) deg/yr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The results of the analysis are given in Table 2 and the theoretical vr curve fits to the measured vr data are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Group-K used the rvfit code2 for the radial velocity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The rvfit program can analyse single and double- lined binary systems by using the adaptive simulated an- nealing method (Iglesias-Marzoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the analy- sis, the Porb taken from Kreiner (2004) was considered as a fixed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Other orbital parameters such as T0, K, γ, ω and e were taken as free parameters during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Both groups vr measurements were used in the analysis and as a result, the orbital parameters of the system were ob- tained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The resulting parameters of the current vr analysis are given in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The consistency between the theoretical vr curve and measurements is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Both groups found the resulting mass ratio (q = M2/M1 = K1/K2)3 larger than 1 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='011 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='080 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='007 for groups -K and -P, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Accord- ing to this q value, the vr curve and the results, the star (generally called secondary) covered by the hotter binary component at orbital phase 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 is more massive than the hotter binary component (primary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To test these findings, binary modelling is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Therefore, these results will be tested in the binary modelling sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4 SPECTRAL ANALYSIS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 Group-K 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 Spectral disentangling To obtain the atmospheric parameters (Teff, log g), v sin i and the chemical composition of each binary component of AI Hya, a detailed spectral analysis is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As AI Hya is a double-lined binary system, its spectrum consists of the spectral lines of both binary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Therefore, group- K carried out a spectral disentangling analysis to extract the individual spectra of each binary component from the com- posite spectra of AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the analysis, the code fdbinary 2 http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='cefca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='es/people/riglesias/rvfit html 3 The subscripts 1 and 2 refer to hotter primary and cooler sec- ondary components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' © 2021 RAS, MNRAS 000, 1–13 4 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The results of the radial velocity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The sub- scripts 1 and 2 refer to hotter primary and cooler secondary com- ponents, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' a shows the fixed parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Parameter Group-P Group-K T0 (HJD) 2458491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='570 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='028 2452506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='383 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='032 Porb(d) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='289761 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='000027 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2896490a γ (km/s) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='24 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='35 K1 (km/s) 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='37 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='65 K2 (km/s) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='46 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='29 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63 e 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2419 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2432 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0050 ω (deg) 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='03 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='30 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='92 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63 ˙ω (deg/yr) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='186 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='056 a1 sin i (R⊙) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='380 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='061 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='222 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='105 a2 sin i (R⊙) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='312 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='072 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='233 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='101 a sin i (R⊙) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='692 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='094 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='454 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='145 M1 sin3 i (M⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='992 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='023 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='033 M2 sin3 i (M⊙) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='151 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='022 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='095 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='035 q = M2/M1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='080 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='075 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='011 50 25 0 25 50 75 100 125 RV (Km/s) =245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='283 =253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='526 =254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='424 Popper_rv1 Popper_rv2 Group-P_rv1 Group-P_rv2 Group-K_rv1 Group-K_rv2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 Phase 10 0 10 O-C Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Upper panel: The model vr fit to the combined vr mea- surements from Popper (1988), Group-P (HIDES+CORALIE) and Group-K (HERMES+CAOS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Lower panel: residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Model made by Group-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' was used (Ilijic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' fdbinary is capable of disen- tangling a composite spectrum, which includes flux contri- butions from two or three components, in Fourier space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Before the analysis with fdbinary, one should know the light contributions of the binary components at the orbital phases corresponding to the times the spectra were taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These values should be fixed during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Hence, to determine the light contributions of both binary compo- nents at the different orbital phases, we carried out a pre- liminary binary modelling of AI Hya by taking Teff of the TESS Input Catalog (TIC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Stassun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2019) as the Teff of the hotter component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The analysis was performed utiliz- ing the Wilson-Devinney code (Wilson & Devinney 1971).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As a result of this preliminary analysis, it was found that the hotter and cooler binary components contribute around 38% and 62% to the total, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, one should keep in mind that these light contributions change accord- ing to the orbital phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For example, the primary eclipse Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Upper panel: The model vr fit to the vr measure- ments of Groups-K and -P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Lower panel: residuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Model made by Group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' is a total eclipse where the light contribution of the hotter components is negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the analysis, we used the HERMES spectra as they are well distributed over the orbital phases and have a higher resolving power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Taking into account the observation time of each HERMES spectrum, the light contributions at these times were first determined using the fluxes measured from the photometric solution and subsequently fixed during the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In addition to this, we also fixed all results de- rived in the vr analysis during the spectral disentangling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the disentangling progress, we used the spectral inter- val of ∼4200 − 6400 ˚A by ignoring the parts polluted by tel- luric lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the analysis, this spectral window was di- vided into 15 spectral parts with steps of ∼ 100 − 150 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Each small spectral part was then analysed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As a result, we obtained the individual spectra of each binary component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The separated spectra derived with fdbinary were re-normalised by taking into account the light ratio of the binary components, as described by Ilijic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 Determination of the atmospheric parameters and chemical compositions After the individual spectra of the components of AI Hya were obtained, we were able to determine the atmospheric parameters, v sin i, and the chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To de- rive these parameters, we used the plane-parallel and line- blanketed local thermodynamic equilibrium (LTE) ATLAS9 model atmospheres (Kurucz 1993) and the synthe code (Kurucz & Avrett 1981) to generate theoretical spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' First, the hydrogen lines of the binary components were used to obtain initial Teff values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this analysis, the Hβ lines of the components were compared with many theoretical Hβ lines which were de- rived for a wide range of Teff (5000 − 9000 K) with a step size of 100 K, where log g and metallicity were fixed to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 and solar, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' During the analysis, we took into account the minimization method described by Catanzaro, Leone, & Dall (2004) and successfully applied in a series © 2021 RAS, MNRAS 000, 1–13 150 100 米 米 xnl↓ 50 Normalized 采 0 米 米 米 米 CAOS 50 △ CORALIE HERMES HIDES 100 15 10 A 1 s 5 米 中 uy) 米 中 采米 0 米 米 5 米 O-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 15 15 10 5 uy) 中谷 米 日米日 米 5E 10 0 15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 PhaseComprehensive study of AI Hya 5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Theoretical hydrogen line fits (red dashed lines) to the Hβ lines (solid black line) of the hotter and cooler binary components (Group-K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' of papers (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Catanzaro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2022, 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Consequently, the Teff of the hotter and cooler components were found to be 7500 ± 200 K and 7000 ± 150 K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We did not attempt to optimize log g because the hydrogen lines are not sensitive to this parameter for stars cooler than 8000 K (Smalley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The best theoretical Hβ line fits to the separated spectra of the components are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We also determined values for log g, the microturbulent velocity ξ, and v sin i by improving the initially determined Teff value using the excitation potential−abundance rela- tionship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the correct atmospheric parameters, different excitation potentials of the same element should give the same abundances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Therefore, by using this relation for iron (Fe), we determined the atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Detailed information about this analysis method is given by Kahra- man Ali¸cavu¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The results of this analysis are listed in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To determine the errors on the atmospheric parameters, we checked how their values change for differ- ences in the excitation potential−abundance correlation of about 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the next step, the chemical composition of the binary components was derived after fixing the atmospheric param- eters to their final values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the chemical abundance deter- mination, we first identified the lines based on the Kurucz line list4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The spectral synthesizing method and the identi- fied lines were used in this examination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Consequently, the chemical compositions of both binary components were ob- tained and the results are listed in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The consistency between the synthetic and observed spectra of both binary 4 http://kurucz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='harvard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='edu/linelists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='html Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Consistency between the synthetic (dashed-lines) and disentangled spectra of the components of AI Hya (Group-K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Abundance distribution of the components of AI Hya relative to solar values (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2009) (Group-K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' components is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The abundance distribu- tions relative to solar abundance (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2009) are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 5, indicating that the hotter binary component has an overabundance compared to the Sun for some ele- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The errors of the chemical compositions were deter- mined including the uncertainties in the derived atmospheric parameters and the effects of the resolving power and the SNR of the spectra, as described by Kahraman Ali¸cavu¸s et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 Group-P For the spectral decomposition and analysis, group-P used the HIDES data only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Spectral analysis was performed on both the observed composite spectra and the disentangled spectra of the individual components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the spectral disen- tangling, we used a python wrapper5 made for using version 3 of fdbinary (FD3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Ilijic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A particular por- tion of the total spectra was taken to ensure good quality in terms of SNR and spectral features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The light fractions used for the disentangling procedure were obtained from the light curve analysis as 38% and 62% for the primary and sec- ondary respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 gssp On the other hand, we also modelled the composite spec- trum using the gssp composite module of the Grid Search 5 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='com/ayushmoharana/fd3 initiator © 2021 RAS, MNRAS 000, 1–13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 xn T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 otter Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 Al cooler 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 4800 4820 4840 4860 4880 4900 4920 Wavelength (A)Al HyaHot xnl 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 Normalized 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 Fel Fel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 Fel Fel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 5439 5448 5427 5430 5433 5436 5444 Wavelength (A) xn 00 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 Normalized Fel Fel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 Fel Fel Fel 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 5382 5385 5400 5403 5379 5391 5394 5397 5388 Wavelength (A)3 Hotter star Cooler star loge(El)- O Mg Si Ca Sc Cr Fe Ti Mn Ni Element6 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The final atmospheric parameters and v sin i value of the hot (primary) and cool binary components of AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' log ϵ (Fe) represent the relative abundance with respect to hydrogen (H=12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Group-K Teff (K) log g (cgs) ξ (km s−1) v sin i (km s−1) log ϵ (Fe) Primary 7700 ± 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 57 ± 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='54 Secondary 7200 ± 100 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 64 ± 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='20 Group-P (gssp) Teff (K) log g (cgs) ξ (km s−1) v sin i (km s−1) [M/H] Primary 7350 ± 300 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 (fixed) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='83 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='15 50 (fixed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='14 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='14 Secondary 7150 ± 250 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 (fixed) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='52 62 (fixed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='10 Group-P (iSpec) Teff (K) log g (cgs) ξ (km s−1) v sin i (km s−1) [M/H] Primary 7300 ± 170 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='83 (fixed) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='86 50 (fixed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='15 (fixed) Secondary 7260 ± 175 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='58 (fixed) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='98 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='70 62 (fixed) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='01 (fixed) Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Abundances of individual elements of the binary com- ponents and Sun (Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Group-K Elements Hotter Cooler Solar component component abundance 12Mg 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='16 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='01 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 14Si 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='12 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='51 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='51 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='03 20Ca 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='27 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 21Sc 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 22Ti 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='49 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='17 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='05 24Cr 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='42 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='30 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 25Mn 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='82 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='45 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='05 26Fe 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='23 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='24 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 28Ni 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='44 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='38 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='73 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 Group-P (iSpec) Elements Hotter Cooler Solar component component abundance 24Cr 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='95 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='19 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='23 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='64 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 26Fe 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='16 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='17 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 28Ni 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='53 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='22 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='04 in Stellar Parameter (gssp) software package (Tkachenko 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As its name implies, gssp is based on a grid search in the fundamental atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' It uses the method of atmosphere models and spectrum synthesis, which performs a comparison of the observations with the- oretical spectra from the grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These synthetic spectra are calculated using the synthV LTE-based radiative transfer code (Tsymbal 1996) and a grid of atmospheric models pre- computed using llmodels (Shulyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Specifi- cally, in the composite module, the user can set the radial velocity of the components as a free parameter so that all the possible combinations of the synthetic spectra of primary and secondary from the computed grid are used to build the composite theoretical spectra of the binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This synthetic spectrum is then compared against the a-priori normalized observed spectrum and a χ2 merit function is used to judge the goodness of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The broadening function (BF) is a representation of spectral profiles in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The BF contains signa- tures of the vr shifts of different lines and also intrinsic stel- lar effects like rotational broadening, spots, pulsations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (Rucinski 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We calculated the BF for one of the com- posite spectra of AI Hya to estimate v sin i values for the primary and secondary components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This pro- cess serves to remove the degeneracy between v sin i and other atmospheric parameters like T eff and [M/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A mod- ified version of the treatment described in Rucinski (1999) was adopted and a multi Gaussian fit was implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The BF was calculated in a wavelength range of 4080-5000 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A synthetic solar-type spectrum with zero projected rotational velocity v sin i was used as our template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To deal with the noise in the data, a Gaussian smoother of 3 km s−1 rolling window was applied to the BF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Two clear peaks were visible in the velocity space, as shown in Figure 6, corresponding to the primary and secondary components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The peaks were fitted with the rotational profile, G(v) = A � �c1 � 1 − � v vmax �2 + c2 � 1 − � v vmax �2�� �+lv+k (1) where A is the area under the profile, vmax is the maximum velocity shift which occurs at the equator (Gray 2005), c1 and c2 are constants which are a function of limb darkening themselves, while l and k are correction factors to the BF continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The BF fit was calculated for the spectra with the highest SNR and good separation between the compo- nents in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The best BF fit to the line profile of the primary and secondary binary components are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Fixing the obtained values of v sin i from this analysis and log g from the light curve solution, the gssp composite fitting routine was applied to obtain stellar temperatures Teff (1,2), microturbulent velocities ξ and global metallicities [M/H].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The step size of the grid gives us a rough idea of the errors involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, to obtain more robust error esti- mates we plotted the χ2 data for each parameter and fitted a parabola to obtain the minimum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' its distance to the in- tercepts on the abscissa are taken as the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These pa- rameters are obtained for a total of four spectra and then averaged out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The remaining spectra were not suitable for © 2021 RAS, MNRAS 000, 1–13 Comprehensive study of AI Hya 7 100 50 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 Primary 50 100 150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 Secondary Relative Flux 2457109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='96513 BJD Radial Velocities (km/s) Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Broadening functions for the primary and secondary components of AI Hydrae calculated using HIDES spectra (epoch: 2457109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='96513 HJD), which provided a good SNR and velocity separation between the two components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The blue, dashed line represents best-fit rotational function (Group-P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 5320 5330 5340 5350 5360 5370 5380 5390 5400 Wavelength (A) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='05 Normalized Flux Data Model Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A snippet of the best-fit model generated by gssp for the given set of parameters (Group-P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' the analysis in gssp due to lower SNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The results of the analysis are compiled in Table 3 and a sample of the fit to one of the spectra is shown in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 iSpec A complimentary spectroscopic analysis was performed on the disentangled spectra of the primary and secondary stars using iSpec (Blanco-Cuaresma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Before the anal- ysis, the spectra are treated for vr offset and continuum cor- rection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Estimates of flux errors were introduced as a sum of errors calculated from SNR, and flux-scaled residuals from the disentangled routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the spectroscopic analysis we fixed the log g parameter with values obtained from the light curve solution and limb darkening parameters with values adopted from Claret & Bloemen (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We fit the model using the spectral synthesis approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This is done by implementing the use of the spectrum code (Gray & Corbally 1994), a marcs (Gustafsson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2008) grid of model atmospheres, and solar abundances taken from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We adopt a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The ini- tial run is aimed at estimating the global metallicity ([M/H]) by keeping it as a free parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The macroturbulent ve- locity (vmac) and alpha enhancement parameters were set to zero as vmac has a negligible contribution for stars in the concerned temperature range and alpha enhancement, when set as a free parameter, produced implausible values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' v sin i was set to the values obtained by the BF analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We com- pared the obtained value for [M/H] with results from the gssp analysis and found it to be consistent with the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The average value of [M/H] was calculated and fixed for the next step where we fit for temperature Teff, microturbulent velocity ξ, and abundances of Iron (Fe), Nickel (Ni) and Chromium (Cr), as these were the prominent lines in the chosen spectral range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The output parameters obtained from iSpec are given in Table 3 and Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' It is to be noted that Fe, Ni, and Cr are more abundant in the primary compared to solar values and those of the secondary star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This trend in the abun- dances is in agreement with the values obtained by group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The output parameters for the secondary star agree fairly well with those from the gssp analysis and from the group- K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The best fit solution for the primary component, as in the case of gssp analysis, also hinted towards a lower Teff compared to the group-K solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 5 BINARY MODELLING 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 Group-K To update the fundamental stellar parameters (M, R) of AI Hya, we performed binary modelling with the help of the determined atmospheric parameters and the results of the vr investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In binary modelling, the TESS data were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' How- ever, the shapes of the eclipses of AI Hya are distorted due to the pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Thus we first cleaned the pulsations and only then carried out the binary modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Therefore, the Period04 program (Lenz & Breger 2005) was used to detect the variations caused by oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The derived pulsation frequencies6 were cleaned from the light curve and the resid- uals were used in the binary modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this analysis, we used the Wilson-Devinney code (Wilson & Devinney 1971) combined with Monte-Carlo sim- ulations (Zola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2004, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The pulsation removed data were binned to around 4000 points to be used in the binary modelling code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' AI Hya is classified as a detached binary system in the literature (Lee, Hong, & Kristiansen 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' According to their results (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', for Ω, q, a), both compo- nents do not seem to fill their Roche lobe, hence the sys- tem is defined as a detached binary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Also, the morphology of the light curve, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' very small ellipsoidal variations and eclipses spanning a small fraction of the orbital period, con- firm this classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Therefore, a detached binary config- uration was considered our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the modelling, we took some parameters fixed, such as the Teff of the hotter component, Porb, q taken from our results and bolometric albedos (Ruci´nski 1969), bolometric gravity-darkening coef- ficient (von Zeipel 1924), and the logarithmic limb darken- ing coefficient (van Hamme 1993) taken the same as given Kahraman Ali¸cavu¸s & Ali¸cavu¸s (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The orbital inclina- tion (i), Teff of the cooler component, phase shift (φ), e, a, ω, and dimensionless potential (Ω) of the components were set free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 6 The frequencies given in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' © 2021 RAS, MNRAS 000, 1–13 8 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Results of the light curve analysis and the fundamental stellar parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The Subscripts 1, 2 and 3 represent the hotter, the cooler, and third binary components, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' a Shows the Fixed Parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Parameter Value Value Group-K Group-P i (o) 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='015 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='837 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='136 T 1a (K) 7700 ± 100 7330 ± 170 T 2 (K) 7180 ± 230 7210 ± 150 Ω1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='046 Ω2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='961 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='035 Phase shift 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0310 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0001 q 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='074a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='075 r1∗ (mean) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1001 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1015 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0005 r2∗ (mean) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1412 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0006 l1 / (l1+l2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='381 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='374 ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='02 l2 / (l1+l2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='619 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='616 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='02 l3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 Derived Quantities M1 (M⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='033 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='950 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='033 M2 (M⊙) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='096 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='035 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='096 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='035 R1 (R⊙) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='754 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='015 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='787 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='020 R2 (R⊙) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='863 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='021 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='877 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='026 log (L1/L⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='381 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='034 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='311 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='081 log (L2/L⊙) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='554 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='035 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='549 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='097 log g1 (cgs) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='848 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='003 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='838 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='005 log g2 (cgs) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='586 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='003 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='582 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='005 Mbol1 (mag) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='30 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='474 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='202 Mbol2 (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='877 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='243 MV 1 (mag) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='424 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='208 MV 2 (mag) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='822 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='258 Distance (pc) 659 ± 30 642 ± 36 As a result of this analysis, the fundamental parameters of both components of AI Hya were calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Additionally, the bolometric (Mbol) and absolute (MV ) magnitudes were estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The jktabsdim code (Southworth, Maxted, & Smalley 2004b) and the bolometric correction (Eker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2020) are used in the calculations of these parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The outcome of the binary modelling is given in Table 5 and the consistency of the theoretical light curve with the observa- tion is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' When the results of this analysis were examined, one can notice that the more luminous star is the more massive and also the cooler component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This result is consistent with the results found in the vr analysis by group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 Group-P Aiming to determine precise physical and orbital parame- ters of AI Hya, we performed its modelling in version 40 of the jktebop (Southworth, Maxted, & Smalley 2004b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This program is written by J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Southworth and aimed at modelling light curves of detached eclipsing binaries and is based on the ebop program (Popper & Etzel 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The code treats stars as spheres to calculate the eclipse shapes, and biaxial ellipsoids to calculate proximity effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The light curves are calculated by numerical integration of concentric circles over each stellar surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' It can deal with stellar oblateness of up Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Theoretical binary modelling fit without spot assump- tion (solid-line) (Group-K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' to 4% making it a good choice for AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The photometric data remain the same as used by Group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The parameters set as free are Porb, time of minima of the primary eclipse To, inclination i, eccentricity e, ar- gument of periastron ω, surface brightness ratio J (sec- ondary/primary), ratio of radii ( rA rB ), and the sum of radii (rA+rB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These radii are relative to the semi-major axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' For the limb darkening coefficients, we use a logarithmic law and set their initial values according to Claret (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The coef- ficients were fixed for the initial fit and were perturbed at the error estimation step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The code gives an option to include multiple sine and polynomial functions during the light curve modelling to ac- count for periodic and long-term trends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We use this func- tionality to our advantage to pseudo-model the observed pulsations so that their effect on the binary model is mini- mal, giving us an improved precision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We analyse the out-of- eclipse portions of the light curve using pyriod7, and use the frequencies to initialise the sinusoids in the jktebop input files.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This is done in an iterative way where we add one sine with a constant period and fit for its epoch and amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The frequency is kept if the model is improved significantly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' otherwise the next most prominent frequency is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this analysis, we used a total of 9 sines, which is the limit for jktebop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The number of independent frequencies of AI Hya is higher than this maximum limit, hence we are left with some residual pulsation signals as seen in Figure 9 and Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Once the sines are fixed to the best fit values of epoch, period and amplitudes, we make the Monte Carlo runs for error estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The results of this analysis are mentioned in Table 5, in comparison to the values obtained by group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Similarly to the other group, we used the results of vr, and jktebop solutions to calculate a set of absolute parameters, including masses, radii, luminosities, and distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The ef- fective temperatures mentioned in the table are an average over the sum of Teff obtained from gssp and iSpec analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 7 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='com/keatonb/Pyriod © 2021 RAS, MNRAS 000, 1–13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 xnl Normalised f 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 Data Model 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='03 Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 PhaseComprehensive study of AI Hya 9 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' jktebop model with 9 sines used to model the pulsa- tions (Group-P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Zoomed-in view of the model over an orbit (Group- P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 6 FREQUENCY ANALYSIS OF THE PULSATIONS AI Hya was observed by TESS during observation sector 7 in January/February 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We used the Simple Aperture Photometry data from the 2-min cadence light curves avail- able at the Mikulski Archive for Space Telescopes8 (MAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This time series spans 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='45 d and contains 16362 measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To determine the pulsation frequencies, we used only the data that were taken out of eclipse, which reduced the data set to 14019 measurements (time span 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='07 d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This time series was analysed using the Period04 soft- ware (Lenz & Breger 2005) by group-K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This package applies single-frequency power spectrum analysis and simultaneous multi-frequency sine-wave fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These sine-wave fits are subtracted from the data and the residuals examined for the presence of further periodicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The application of this procedure to AI Hya is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' During such a process, it is important to decide where to stop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Often this is facilitated via the application of SNR criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this work, we have adopted the strategy proposed by Breger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (1993) which is to compute the ratio of the signal amplitude relative to the local noise level to deter- mine whether the frequency under consideration represents a significant detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Whereas Breger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (1993) propose SNR > 4 for a detection, recent findings for space-based data 8 https://mast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='stsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='edu/portal/Mashup/Clients/Mast/Portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='html Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The Fourier Transform of the out-of-eclipse TESS light curve of AI Hya (top) and subsequent prewhitening steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The blue arrows denote the signals detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Outside of the fre- quency range shown no significant signal is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A least squares fit of the pulsation frequencies of AI Hya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Formal error estimates for the independent frequencies and phases (Montgomery & O’Donoghue 1999) are given in braces in units of the last digits after the comma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Frequency Amplitude SNR d−1 mmag ±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='02 ν1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2412(1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='75 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 ν2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2654(4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='18 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ν3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9065(4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='20 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ν4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='715(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ν5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='928(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='54 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ν6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3689(4) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='42 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 3νorb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3619 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='76 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 4νorb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4825 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ν7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5599(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='78 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ν8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7804(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='69 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 2νorb 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2413 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='75 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ν9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6375(7) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='73 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ν10 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='136(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='37 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ν11 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='751(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 ν12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3051(6) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='82 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ν13 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8432(8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='69 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ν3 + ν7 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='464(1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='43 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Baran & Koen 2021) suggest that a more conservative limit must be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Given the restricted frequency range in which we search for periodicities, our requirement was SNR > 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Furthermore, in unresolved frequency spectra, the periodic content present in the time series can easily be overinterpreted (Balona 2014) which suggests caution re- garding the present data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Consequently, we stopped the frequency search after the detection of 17 signals (lowest panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' More periodicities are certainly present, but these need to await a longer data set for reliable detec- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We list the frequency solution so derived in Table 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' © 2021 RAS, MNRAS 000, 1–13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 a M 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 Data 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 Model Residuals 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='01 Resi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 Phase8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='48 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='49 0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='51 Data Model 1494 1496 1495 1498 1497 1499 1500 1501 1502 Time (ID-2457000) days10 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This table also contains three harmonics of the orbital period.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These are not pulsation frequencies, but a conse- quence of residual binary-induced variability (see Section on binary modeling for a discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The pulsation frequencies themselves were found in an interval between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 – 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 d−1, with one possible combination frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' It is however not clear whether this is a real combination or just a numeri- cal coincidence keeping in mind the short data set, hence poor frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Our frequency solution is similar to that reported by Lee, Hong, & Kristiansen (2020) apart from their identification of possible combination frequencies that are partly implausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To use the pulsations to learn more about the indi- vidual components by applying asteroseismic methods, it is essential to know from which star the pulsations orig- inate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A quick look at the TESS light curve reveals that pulsations are clearly visible during the total part of the primary eclipse, meaning that the secondary is the source of the highest amplitude oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, both com- ponents of AI Hya are located within the pulsational in- stability strip of the δ Scuti stars (Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2019, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 12), thus the primary may pulsate as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' δ Scuti stars generally pulsate in pressure and mixed modes of low ra- dial order (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Breger 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Using the stellar parameters from Table 5, we can compute the expected frequency of the radial fundamental mode of both pulsators from the pulsa- tion constant Q = P � ρ/ρ⊙ = PM 1/2R−3/2, assuming Q to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='033 d for this mode (Fitch 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' We thus expect the radial fundamental mode frequency of the primary compo- nent to be around 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 d−1, and around 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 d−1 for the sec- ondary component, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In Table 6 oscillation fre- quencies around both these values are seen, which allows no more than the educated guess that the pulsations below ∼ 8 d−1 would arise from the secondary component, whereas the higher frequency modes could originate from either star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A determination of the origin of the pulsations from the orbital light time effect is unfortunately out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The expected light time effect would be about 30 s (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' An attempt to measure the effect for the strongest pulsa- tion frequency yielded 35 ± 111 s, a null result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To conclude, because it is impossible to say with confidence which pulsa- tion frequencies arise from which component of AI Hya, an asteroseismic analysis cannot be carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 7 EVOLUTIONARY MODELS The evolutionary status of the binary components was ex- amined by utilizing the Modules for Experiments in Stel- lar Astrophysics (mesa) evolution code (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2011, 2013) which includes a binary module (Paxton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2015) to examine the binary orbital evolution and to determine the initial parameters of binary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In this examination, various evolutionary models were generated considering dif- ferent metallicity (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the models, MESA equation-of- state (EOS) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The EOS tables are based on the OPAL EOS tables (Rogers & Nayfonov 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The OPAL opacity tables and the default solar mixtures were adopted as Z initial fraction from Asplund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Helium mass fraction were taken Y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='28, for Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Convective core overshoot was described by the exponentially decaying prescription of Herwig (2000) and overshooting parameter adopted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='20 for both components (Claret & Torres (2016) find 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='208 for both components).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A mixing length αMLT value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 was used as the theoretical δ Scuti instability strip (Dupret et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2004, 2005) was obtained with this αMLT value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Taking into account the calculated parameters in the binary modelling for both groups, the evolutionary status of the binary components was investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As a result, we found that the secondary (more luminous) binary compo- nent can be represented with the same evolutionary tracks according to both groups’ results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, the less lumi- nous primary component’s position was determined with different Z parameters as the parameters of this star were found to be slightly different in the study of the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' According to the evolutionary models, the Z parameters of both binary components were found similar to solar (As- plund et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2009) within the errors which differs from the results of the groups as we determined that the less luminous component’s atmosphere is somewhat enhanced in metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The results of this analysis are given in Table 7 and a H-R diagram is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The observational borders of the δ Scuti instability strip were taken from Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As can be seen from the H-R diagram, both binary components are placed inside the δ Scuti instability strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 8 DISCUSSION AND CONCLUSIONS In this analysis, we present the results of the detailed anal- ysis of AI Hya carried out by two independent groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The system was observed with different high-resolution spectro- graphs (R≳38000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The radial velocity variations of AI Hya were modelled using the vr measurements of both groups and the orbital parameters such as T0, Porb, e and q were updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The resulting parameters of the analysis of both groups are consistent with each other within the errors and they slightly differ from the results of Popper (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Espe- cially the e value shows a discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Popper (1988) found e to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2301 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0015 while in our study it was determined as 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2419 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0036 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2432 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0050 by group-P and -K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Since our high-resolution spectra are spread over all or- bital phases, we were able to derive the atmospheric param- eters of both binary components by modelling either the composite spectra or the spectra of the individual compo- nents after applying spectral disentangling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' To derive the atmospheric parameters, v sin i and the chemical composi- tion of the binary components, group-K analysed disentan- gled spectra of the components, while group-P performed their analysis using both the composite and disentangled spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As a result, group-K found that the more lumi- nous star is cooler than the less luminous component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' They found the Teff values from the Hβ line fit and Fe lines to be 7500 ± 200 K and 7700 ± 100 K for the primary and 7000 ± 150 K and 7200 ± 100 K for the secondary compo- nent, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Group-P used two different codes in their analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' With the gssp code analysis they found a similar result with group-K even though the resulting Teff values differ from each other, they determined that the more lu- minous star is cooler (7150 ± 250 K) and less luminous one is hotter (7350 ± 300 K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the iSpec analysis of group-P, Teff values of both components were found similar to the © 2021 RAS, MNRAS 000, 1–13 Comprehensive study of AI Hya 11 Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Results obtained from the best-fit evolutionary models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Parameter Group-K Group-P P initial (days) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='34 (1) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='34 (1) einitial 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='242 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='243 (2) Z1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='013 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='016 (2) Z2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='018 (2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='018 (2) Age (Myr) 850 (20) 860 (20) results of the gssp analysis within error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The primary’s temperature is the most significant discrepancy between the values derived by the two groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The exact reason for this temperature inconsistency is not fully understood, although it is still only at a level of ∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the chemical abundance analysis, both groups found the less luminous but hotter binary component to show overabundance while the other component has chemical abundance similar to solar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Both groups determined the abundances of some individual elements such as iron (Fe).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' They derived Fe abundances as 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='23 (group-K) and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='16 (group-P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These values are consistent with each other within their 1σ errors, and both demonstrate that the hotter component has a slightly metal-rich chemical abun- dance compared to solar values (see Table 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This comes somewhat to a surprise, as this binary system should have been formed in the same interstellar environment and hence its components should have the same chemical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The difference could be due to the consequences of the evolu- tion of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' If AI Hya had a very eccentric orbit when the system was formed, there could be some material flows from one component to another that could have changed the diffusion in one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Another explanation was given by Yushchenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2015) and they pointed out that pos- sible gas and dust accretion from the circumstellar envelope could alter the atmospheric composition of one component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' After the determination of the atmospheric parameters, they were used as input in the binary modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Overall, even though both working groups used different approaches to estimate the parameters of the binary component of AI Hya, the values determined by both groups are found to be consistent with each other within the error bars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The two groups obtained very similar M and R values with a ⩽1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7% and ∼0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5% accuracy, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' When we compare these values with the ones found by Lee, Hong, & Kristiansen (2020), we notice that there are slight differences, especially in the R parameters, and there is significant diversity in the calculated distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' These differences could be caused by the different assumptions of the atmospheric parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The evolutionary status of the system was examined and it was found that both binary components are inside the δ Scuti instability strip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The age of the system is determined as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' According to the determined ages, we could say that AI Hya is in an important evolutionary phase in terms of binary evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The rapidly evolving massive component will begin the mass transfer process to the less massive one approximately 20 Myr from now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This situation could cause significant variations in the oscillation properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Increas- ing the number of such bodies is important in terms of ex- amining the pulsating structures before the mass transfer processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The pulsation properties of AI Hya were examined us- Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The positions of the binary components in the H-R diagram according the results of both group-K (g-K) and group-P (g-P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The instability strip (IS) borders of the δ Scuti stars were taken from Murphy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' ing the TESS data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' However, the system has only one sector of SC data, which offers us a poor frequency resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' In the analysis, pulsation frequencies were found between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 and 13 d−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As both binary components are placed in the δ Scuti instability strip, we were unable to say whether one or both pulsate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Apart from that, we could not find pulsa- tions related to the orbital frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' As a result of this study, we thoroughly examined a detached binary system showing oscillations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This kind of objects is particularly important to examine the insta- bility strip of δ Scuti stars since they allow us to deter- mine fundamental astrophysical, atmospheric parameters and the chemical abundances of individual binary compo- nents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Hence an increasing number of analyses of such sys- tems is expected to be essential to deeply understand the nature of pulsations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' ACKNOWLEDGMENTS The authors would like to thank the reviewer for useful comments and suggestions that helped to improve the publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This study has been supported by the Sci- entific and Technological Research Council (TUBITAK) project 120F330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' GH thanks the Polish National Center for Science (NCN) for supporting the study through grants 2015/18/A/ST9/00578 and 2021/43/B/ST9/02972.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' TP’s research is supported through NCN OPUS project number 2017/27/B/ST9/02727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' AM’s acknowledges the support provided by the Polish National Science Centre (NCN) OPUS project number 2017/27/B/ST9/02727 and 2021/41/N/ST9/02746.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Based on observations made with the Mercator Telescope, operated on the island of La Palma by the Flemish Community, at the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof`ısica de Canarias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The TESS data presented in this paper were obtained from the Mikulski Archive for Space Telescopes (MAST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Funding for the TESS mission is provided by © 2021 RAS, MNRAS 000, 1–13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 tAl Hya 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 (L/ Lo) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 Primary (g-K), ☆ Primary (g-P) Secondary (g-K), ☆ Secondary (g-P) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='950 MO track (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='013) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='950 MO track (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='016) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='096 MO track (Z=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='018) IS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='00 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='90 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='80 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='70 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='60 log T (K)12 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Kahraman Ali¸cavu¸s et.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' the NASA Explorer Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This work has made use of data from the European Space Agency (ESA) mission Gaia (http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='int/gaia), processed by the Gaia Data Processing and Analysis Consortium (DPAC, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='cosmos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='esa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='int/web/gaia/dpac/consortium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Funding for the DPAC has been provided by national institutions, in particular the institutions participating in the Gaia Multilateral Agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' This research has made use of the SIMBAD data base, operated at CDS, Strasbourq, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' DATA AVAILABILITY The data underlying this work will be shared at reasonable request to the corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' REFERENCES Aerts C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Christensen-Dalsgaard J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kurtz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2010, aste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='book Antoci V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Cunha M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Houdek G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kjeldsen H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Trampedach R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Handler G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', L¨uftinger T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2014, ApJ, 796, 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1088/0004-637X/796/2/118 Asplund M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Grevesse N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Sauval A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Scott P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2009, ARA&A, 47, 481 Balona L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2014, MNRAS, 439, 3453 Baran, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' & Koen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2021, AcA, 71, 113 Blanco-Cuaresma S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Soubiran C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Heiter U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Jofr´e P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2014, A&A, 569, A111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361/201423945 Borucki W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Koch D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Basri G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Batalha N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Brown T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Caldwell D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Caldwell J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2010, Sci, 327, 977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1185402 Bowman D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kurtz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2018, MNRAS, 476, 3169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/sty449 Breger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2000, in Delta Scuti and Related Stars, ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Breger & M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Montgomery, ASP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 210, 3 Breger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1993, A&A, 271, 482 Catanzaro G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Leone F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Dall T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, A&A, 425, 641 Catanzaro G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Colombo C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ferrara C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Giarrusso M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2022, MNRAS, 515, 4350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/stac1723 Catanzaro G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bus`a I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gangi M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Giarrusso M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Leone F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Munari M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2019, MNRAS, 484, 2530.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/stz080 Claret A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2017, A&A, 600, A30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004- 6361/201629705 Claret A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Torres G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2016, A&A, 592, A15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361/201628779 Coelho P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Barbuy B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Mel´endez J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Schiavon R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Castilho B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2005, A&A, 443, 735.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004- 6361:20053511 Dupret M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Grigahc`ene A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Garrido R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gabriel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Scu- flaire R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, A&A, 414, L17 Dupret M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Grigahc`ene A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Garrido R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gabriel M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Scuflaire R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2005, A&A, 435, 927.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004- 6361:20041817 Eker Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Soydugan F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bilir S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bakı¸s V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', ¨Ozer S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Aslan G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2020, MNRAS, 496, 3887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/staa1659 Fitch, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1981, ApJ 249, 218 Gaia Collaboration, 2022, Gaia DR3: data release content and main properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A&A, in prep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' Gaulme P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Guzik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2019, A&A, 630, A106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361/201935821 Gray R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Corbally C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1994, AJ, 107, 742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1086/116893 Gustafsson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Edvardsson B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Eriksson K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Jørgensen U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Nordlund ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Plez B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2008, A&A, 486, 951.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361:200809724 Grevesse N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Asplund M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Sauval A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2007, SSRv, 130, 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1007/s11214-007-9173-7 Iglesias-Marzoa, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', L´opez-Morales, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', & Jes´us Ar´evalo Morales, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' 2015, pasp, 127,567 Handler G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kurtz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Rappaport S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Saio H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Fuller J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Jones D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Guo Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2020, NatAs, 4, 684.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1038/s41550-020-1035-1 Herwig F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2000, A&A, 360, 952 Ilijic S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Hensberge H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Pavlovski K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Freyhammer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, ASPC, 318, 111 Jayasinghe T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2018, MNRAS, 477, 3145 Joergensen H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gronbech B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1978, A&A, 66, 377 Kahraman Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2016, MNRAS, 458, 2307 Kahraman Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Soydugan E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Smalley B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kub´at J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2017, MNRAS, 470, 915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/stx1241 Kahraman Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2019, MNRAS, 488, 5279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/stz2003 Kahraman Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', G¨um¨u¸s D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kırmızıta¸s ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ekinci ¨O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', C¸avu¸s S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kaya Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ali¸cavu¸s F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2022, RAA, 22, 085003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1088/1674-4527/ac71a4 Kambe E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2013, PASJ, 65, 15 ¸s Konacki M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Muterspaugh M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kulkarni S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', He�lminiak K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2010, ApJ, 719, 1293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1088/0004- 637X/719/2/1293 Kreiner J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, AcA, 54, 207 Kurtz D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Handler G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Rappaport S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Saio H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Fuller J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Jacobs T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Schmitt A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2020, MNRAS, 494, 5118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/staa989 Kurucz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Avrett E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1981, SAOSR, 391, Kurucz R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1993, KurCD, 13, Lampens P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2021, Galax, 9, 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3390/galaxies9020028 Lee J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Hong K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kristiansen M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2020, PASJ, 72, 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/pasj/psaa020 Lenz P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Breger M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2005, CoAst, 146, 53 Leone F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2016, AJ, 151, 116 Liakos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Niarchos P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2017, MNRAS, 465, 1181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1093/mnras/stw2756 Montgomery M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', O’Donoghue D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1999, Delta Scuti Star Newsletter 13, 28 Murphy S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Hey D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Van Reeth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bedding T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2019, MN- RAS 485, 2380 Paxton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bildsten L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Dotter A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Herwig F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Lesaffre P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Timmes F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2011, ApJS, 192, 3 Paxton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2013, ApJS, 208, 4 Paxton B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2015, ApJS, 220, 15 Pepe F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bouchy F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Mayor M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Udry S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2018, haex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='book, 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1007/978-3-319-55333-7 190 Pilecki B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gieren W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Smolec R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Pietrzy´nski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Thomp- son I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Anderson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bono G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2017, ApJ, 842, 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3847/1538-4357/aa6ff7 Pojma´nski G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2002, AcA, 52, 397 Popper D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Etzel P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1981, AJ, 86, 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' © 2021 RAS, MNRAS 000, 1–13 Comprehensive study of AI Hya 13 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1086/112862 Popper D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1988, AJ, 95, 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1086/114627 Raskin G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2011, A&A, 526, A69 Ricker G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Winn J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Vanderspek R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Latham D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bakos G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' ´A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bean J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Berta-Thompson Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2014, SPIE, 9143, 914320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2063489 Rogers F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Nayfonov A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2002, ApJ, 576, 1064.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1086/341894 Ruci´nski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1969, AcA, 19, 245 Smalley B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gardiner R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kupka F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bessell M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2002, A&A, 395, 601 Southworth J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Zucker S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Maxted P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Smal- ley B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, MNRAS, 355, 986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1365- 2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='08389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='x Southworth J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Maxted P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Smalley B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, MNRAS, 351, 1277 Southworth J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2013, A&A, 557, A119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004- 6361/201322195 Southworth J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2021, Univ, 7, 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3390/universe7100369 Stancliffe R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Fossati L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Passy J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Schneider F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2015, A&A, 575, A117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361/201425126 Stassun K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Oelkers R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Paegert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Torres G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Pep- per J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', De Lee N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Collins K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2019, AJ, 158, 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3847/1538-3881/ab3467 Shulyak D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Tsymbal V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ryabchikova T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', St¨utz C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Weiss W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, A&A, 428, 993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004- 6361:20034169 Torres G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Andersen J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gim´enez A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2010, A&ARv, 18, 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1007/s00159-009-0025-1 Tsymbal V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1996, ASPC, 108, 198 Uytterhoeven K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Moya A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Grigahc`ene A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Guzik J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Guti´errez-Soto J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Smalley B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Handler G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2011, A&A, 534, A125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361/201117368 van Hamme W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1993, AJ, 106, 2096 von Zeipel H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1924, MNRAS, 84, 665 Wilson R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Devinney E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1971, ApJ, 166, 605 Yushchenko A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gopka V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kang Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kim C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Lee B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Yushchenko V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Dorokhova T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2015, AJ, 149, 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1088/0004-6256/149/2/59 Zucker S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Mazeh T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1994, ApJ, 420, 806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1086/173605 Zola S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2004, AcA, 54, 299 Zola S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Gazeas K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Kreiner J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Ogloza W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Siwak M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Koziel-Wierzbowska D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Winiarski M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2010, MNRAS, 408, 464 Rucinski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1999, ASPC, 185, 82 Gray D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2005, oasp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='book Claret A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', Bloemen S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2011, A&A, 529, A75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004-6361/201116451 Rucinski S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 1999, TJPh, 23, 271 Tkachenko A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=', 2015, A&A, 581, A129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1051/0004- 6361/201526513 Table A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The vr measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' The subscripts “1” and “2” represent the more and the less luminous components, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' HJD vr,1 vr,2 Instrument +2450000 (km s−1) (km s−1) 9263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='45270 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 CAOS 9161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='65803 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 HERMES 9162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='64306 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 HERMES 9230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='65226 121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 HERMES 9231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='66393 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 HERMES 9233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='62648 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 HERMES 9234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='55784 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 HERMES 9237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='61273 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 HERMES 9235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='43315 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 HERMES 9257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='49195 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 HERMES 9260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='61123 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 HERMES 9276.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='55613 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 HERMES 9296.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='42427 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 HERMES 9297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='44747 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 HERMES 9298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='45846 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 HERMES 9299.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='46357 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 HERMES 7075.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='62231 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 CORALIE 7076.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63954 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 CORALIE 7109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='63123 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 CORALIE 7022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='31643 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 HIDES 7060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='09414 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 HIDES 7109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='96513 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 HIDES 7114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='92732 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 HIDES 7146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='98403 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 HIDES 7147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='96084 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 HIDES 7363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='28986 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 HIDES 7755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='22744 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 HIDES 7813.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='13416 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 HIDES 7814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='08321 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 HIDES 7846.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='01822 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='0 HIDES 8035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='34461 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 HIDES 8066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='24908 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='1 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content='3 HIDES This paper has been typeset from a TEX/ LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} +page_content=' © 2021 RAS, MNRAS 000, 1–13' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/4NE3T4oBgHgl3EQfQAnJ/content/2301.04409v1.pdf'} diff --git a/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf b/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..95b8deafefec214f29d5111a35acd6894ccb636c --- /dev/null +++ b/4tFIT4oBgHgl3EQf7Ctv/content/2301.11396v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7c34e9fdcac538452e1b0b8fed87ac7cf9cfab239f87c3b0eb8dc377a1293062 +size 1727682 diff --git a/4tFIT4oBgHgl3EQf7Ctv/vector_store/index.faiss b/4tFIT4oBgHgl3EQf7Ctv/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..f7c1fccb4608ce9cae7277feedeba6bf753ec52b --- /dev/null +++ b/4tFIT4oBgHgl3EQf7Ctv/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:50def715f9ed1d2ca9d3a8a54901cca3900bdd97f2beb1c71ed2cebc8587880e +size 4259885 diff --git a/4tFIT4oBgHgl3EQf7Ctv/vector_store/index.pkl b/4tFIT4oBgHgl3EQf7Ctv/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..355fe5a18133447c0e010a332006469e00a37080 --- /dev/null +++ b/4tFIT4oBgHgl3EQf7Ctv/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:436080ac86a600c01de70ef846d0b00b89e5010bbdb65254ef928d7a2791292c +size 157212 diff --git a/5dE1T4oBgHgl3EQf6gWk/vector_store/index.faiss b/5dE1T4oBgHgl3EQf6gWk/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..6916d9bc631b2c14762710a0c62926b85c346dbd --- /dev/null +++ b/5dE1T4oBgHgl3EQf6gWk/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8a72fca7fdf7da2bf0fbd049f445fcb360035c4113e190fbbdc1330feb584cb0 +size 5963821 diff --git a/6NE1T4oBgHgl3EQfTQM9/content/tmp_files/2301.03075v1.pdf.txt b/6NE1T4oBgHgl3EQfTQM9/content/tmp_files/2301.03075v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a7c7bd9a7659f2992fe8192872b3bb62267d95a2 --- /dev/null +++ b/6NE1T4oBgHgl3EQfTQM9/content/tmp_files/2301.03075v1.pdf.txt @@ -0,0 +1,1491 @@ +arXiv:2301.03075v1 [math.DS] 8 Jan 2023 +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN +MAPPINGS AND SMOOTHNESS ANALYSIS +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +Abstract. Let T be a self-map on a metric space (X, d). Then T is called +Kannan map if there exists α, 0 < α < 1 +2, such that +d(T(x), T(y)) ≤ α[d(x, T(x)) + d(y, T(y))], for all x, y ∈ X. +This paper aims to introduce a new method to construct fractal functions +using Kannan mappings. First, we give the rigorous construction of fractal +functions with the help of the Kannan iterated function system (IFS). We also +show the existence of a Borel probability measure supported on the attractor +of the Kannan IFS satisfying the strong separation condition. Moreover, we +study the smoothness of the constructed fractal functions. We end the paper +with some examples and graphical illustrations. +1. INTRODUCTION +The concept of fractal interpolation function (FIF) was introduced by Barnsley +[2, 3] through iterated function system (IFS), and their construction is rooted in the +theory of IFS [9]. The FIF is an interpolation function whose graph is an invariant +set of an IFS. The pioneering research on fractal interpolation has gotten much at- +tention in the literature, and it continues to flourish. The concept of FIF has been +extended and generalized in several ways given in the literature. Wang and Yu [27] +gave the construction of new class IFSs with variable parameters and generated as- +sociated FIFs. Also, they studied the smoothness and stability of FIFs under some +conditions on data points. The construction of nonlinear FIF using Matkowski and +the Rakotch fixed point theorems is given in [20]. In this order, Songli [21] gave +the construction of nonlinear FIF on Sierpi´nski gasket. The reader may refer to +books [3, 16] for the details on fractal functions. The fractal dimension is one of +the major themes in fractal geometry. Many works on the fractal dimensions of +fractals functions are in the literature. There are various approaches, such as the +mass-distribution principle, potential theory, Fourier transform, positive operators, +etc., to compute or estimate the Hausdorff dimension of a set [11, 26]. Using the +potential theoretic approach, Barnsley gave results on the Hausdorff dimension of +an affine FIF in [3]. Falconer [11] also gave the estimate of the Hausdorff dimension +of an affine FIF. The results on the Hausdorff dimension using the positive oper- +ators approach are given in [26]. Priyadarshi [19] gave an algorithm to determine +lower bounds for the Hausdorff dimension of a set of complex continued fractions +and estimated the best lower bound. Jha and Verma [12] established very inter- +esting results for fractal dimensions of fractal functions and some invariant sets. +2020 Mathematics Subject Classification. +28A80, 47H10, 28A33, 28A78. +Key words and phrases. Kannan IFS, Fractal Functions, Borel Probability Measure, Fractal +Dimension. +1 + +2 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +They estimated fractal dimensions for a class of FIFs, widely known as α-fractal +functions, by using function spaces such as H¨older space, oscillation space, and +space of bounded variation. Ruan et al. [22] estimated the box dimension of the +new class of linear FIFs by using the δ-covering method. Additionally, they have +established a relationship between the order of fractional integral and box dimen- +sions of two linear FIFs. As we know, recurrent FIF is the generalization of linear +FIF, and the graph of recurrent FIF is the invariant set of recurrent IFS. Barnsley +and Massopust [4] gave results on the bilinear FIFs and their box dimension. Few +recent developments on fractal dimensions can be seen in [6, 24, 25]. Cheng et +al. [6] introduced the notion of upper metric mean dimension with potential on +any subset via Carath´eodory-Pesin structures. Selmi [24] studied the multifractal +Hausdorff and packing dimensions of Borel probability measures and studied their +behaviors under orthogonal projections. In this order, Selimi estimated the multi- +fractal Hausdorff and the packing dimensions of product measures in [25]. +Barnsley [2, 3] considered the collection of self-contraction mappings and used the +Hutchinson operator and Banach fixed point principle to construct fractal functions. +Kannan [13, 14] introduced a new fixed point theorem widely known as Kannan +fixed point theorem. +Other related results on Kannan mapping can be seen in +[8, 10]. By using the concept of Kannan mapping, Sahu et al. [23] introduced the +notion of the Kannan iterated function system. +Theorem 1.1. [14] Let T is a map of the complete metric space X into itself and +if +d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))], ∀ x, y ∈ X, 0 < α < 1 +2. +Then T has the unique fixed point in X. +A natural question arises can we construct fractal functions using the concept +of Kannan fixed point theory? This question motivates us to conduct the current +study. In this study, we use the concept of Kannan IFS and Kannan fixed point +theorem and derive very interesting results. +This paper is organized as follows: Section 2 is devoted to preliminaries and required +terminologies related to this article. Section 3 presents the construction of fractal +functions and the existence of self-similar measures. In Section 4, the smoothness +result of the Kannan fractal function is given. The graphical illustration of the +Kannan fractal functions is given in Section 5. +2. Background and preliminaries +This section aims to provide some basic definitions and results that act as prelude +to this article. Let F ̸= ∅ be a subset of Rn. The diameter of F is given by +diamd(F) = sup {d(x, y) : x, y ∈ F} . +If {Fi} is a countable (or finite) collection of sets having a diameter at most δ +which cover set E ⊆ Rn, then we say that {Fi} is a δ-cover of E. For δ > 0 and a +non-negative real number s, we define +Hs +δ,d(E) = inf +� ∞ +� +i=1 +diamd(Fi)s : {Fi} is a δ − cover of E +� +. +Definition 2.1. The s-dimensional Hausdorff measure of set E is given by Hs(E) = +limδ→0 Hs +δ (E). + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +3 +Definition 2.2. ( Hausdorff dimension) Let s ≥ 0 and E ⊆ Rn. The Hausdorff +dimension of E is defined as +dimH(E) = inf{s : Hs(E) = 0} = sup{s : Hs(E) = ∞}. +Definition 2.3. (Box Dimension) Let E ⊆ Rn be bounded and non-empty and let +Nδ(E) be the smallest number of sets of diameter at most δ which cover E. The +lower box dimension of E is +dimB(E) = lim +δ→0 +log Nδ(E) +− log δ +, +and the upper box dimension of E is +dimB(E) = lim +δ→0 +log Nδ(E) +− log δ +, +If both lower and upper box dimensions are the same, then that quantity is called +the box dimension of E and it is given by +dimB(E) = lim +δ→0 +log Nδ(E) +− log δ +. +For the details on the Hausdorff and box dimensions, the reader may be referred +to [11]. +Definition 2.4. Let d1 and d2 are two matrices on X, then d1 and d2 are topo- +logically equivalent if and only if +d1(xn, x) → 0 ⇐⇒ d2(xn, x) → 0, +for {xn} ⊂ X and x ∈ X. +Let d1 and d2 are two matrices on X, then d1 and d2 are metrically equivalent if +and only if there exists c1, c2 > 0 and x, y ∈ X such that +c1d1(x, y) ≤ d2(x, y) ≤ c2d1(x, y). +Fractal Interpolation Function. Now, we introduce FIF in brief. +Here, we +consider a set for interpolation as {(xn, yn) : n = 1, 2, . . . , N}. +We set J = +{1, 2, ..., N − 1}, I = [x1, xN] and for j ∈ J, let Ij = [xj, xj+1]. For j ∈ J, let +Lj : I → Ij be a contractive homomorphism such that +Lj(x1) = xj, Lj(xN) = xj+1, j ∈ J. +Now, define Fj : K = I × R → R, j ∈ J, which is a contraction in the second +variable, that is, |Fj(x, y) − Fj(x, y′)|≤ rj|y − y′|, for all x ∈ I, rj ∈ [0, 1) and +y, y′ ∈ R and satisfying Fj(x1, y1) = yj, Fj(xN, yN) = yj+1, j ∈ J. We shall take +(2.1) +Lj(x) = ajx + bj Fj(x, y) = αjy + qj(x), +In the above expression aj and bj are determined by using conditions Lj(x1) = +xj, Lj(xN) = xj+1. Here, αj is the scaling factor with |αj|< 1 and continuous +functions qj : I → R, j ∈ J satisfy “join-up conditions” imposed for the bivariate +maps Fj. That is, qj(x1) = yj − αjy1 and qj(xN) = yj+1 − αjyN for all j ∈ J. Now +define functions Wj : I × R → I × R for j ∈ J by +Wj(x, y) = (Lj(x), Fj(x, y)). + +4 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +Theorem 1 in [3] says that the IFS I := {I × R; W1, W2, . . . , WN−1} defined above +has a unique attractor which is the graph of a function f which satisfies the following +functional equation reflects self-referentiality: +f(x) = αjf(L−1 +j (x)) + qj(L−1 +j (x)), x ∈ Ij, j ∈ J. +The above function f is known as the fractal interpolation function. +Kannan mapping. In 1969, Kannan [13] introduced a mapping, which was an +improvement over the contraction mapping, known as Kannan mapping, defined as +follows: +If there exists a number α, 0 < α < 1 +2, such that, for all x, y ∈ X, +d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))]. +Then T is called a Kannan mapping and α is called Kannan-contractivity factor +of T . Let Tn : X → X are Kannan mappings having contractivity factor αn, for +n = 1, 2, . . ., N and (X, d) be a complete metric space. Then, the set {X; Tn, n = +1, 2, . . ., N} is said to be Kannan IFS. +Remark 2.5. Let f : [0, 1] → [0, 1] be defined by f(x) = x +3. Then this function f is +a contraction mapping with contraction factor 1 +3, but it is not a Kannan mapping. +On the other hand, the function g : [0, 1] → [0, 1] defined by +g(x) = +� +x +4, if 0 ≤ x < 1 +2 +x +5, +if 1 +2 ≤ x ≤ 1. +is a Kannan mapping with β = 4 +9 but it is not a contraction mapping. The concepts +of the Kannan operator and contraction are independent. The self-map T given in +the previous example is Kannan, but it is not a contraction due to its discontinuity. +The following simple note can be seen in [10]. However, we include its details +for the reader’s convenience. +Note 2.6. Let (X, d) be a metric space and T : X → X is contraction with constant +c < 1 +3. Then T is Kannan contractive with respect to metric d. +Because of the contractivity of T , we have +d(T (x1), T (x2)) ≤ cd(x1, x2) ≤ cd(x1, T x1)+cd(T x1, T x2)+c(T x2, x2), ∀ x1, x2 ∈ X. +This turns +d(T (x1), T (x2)) ≤ α[d(x1, T (x1)) + d(x2, T (x2))], ∀ x1, x2 ∈ X. +Since 0 < α := +c +1−c < 1 +2, T is a Kannan mapping. +Proposition 2.7. Let X be a complete metric space and d1 and d2 are equivalent +metrics on X, i.e., there exist positive constants c1, c2 such that +c1d1(x1, x2) ≤ d2(x1, x2) ≤ c2d1(x1, x2), x1, x2 ∈ X. +If T is a contraction on X with respect to the metric d1 then there exists an m ∈ N +such that T m is a Kannan contraction with respect to the metric d2. +Proof. Since T is contraction on (X, d1), there exits 0 ≤ k < 1 such that +d1(T x1, T x2) ≤ kd1(x1, x2), x1, x2 ∈ X. + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +5 +Whence d2(T x1, T x2) ≤ c2d1(T x1, T x2) ≤ c2kd1(x1, x2) ≤ +� +c2 +c1 k +� +d2(x1, x2). Take +m ∈ N such that c2 +c1 km < 1 +3. Then +d2(T mx1, T mx2) ≤ c2d1(T mx1, T mx2) ≤ c2kmd1(x1, x2) ≤ +� +c2 +c1 +km +� +d2(x1, x2). +Hence, T m is a contraction with respect to the metric d2. +From Note 2.6, we +conclude that T m is a Kannan contraction with respect to the metric d2. Thus, the +proof is completed. +□ +Theorem 2.8. [14] Let T is a map of the complete metric space X into itself and +if +d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))], ∀ x, y ∈ X, 0 < α < 1 +2. +Then T has the unique fixed point in X. +The Hausdorff distance from the set A to the set B is defined as +h(A, B) = max{sup +a∈A +inf +b∈B d(a, b), sup +b∈B +inf +a∈A d(a, b)}. +Note 2.9. In [23, Lemma 3.5] the authors claimed that for all B, C ∈ H(X), +h(T (B), T (C)) ≤ β[h(B, T (B)) + h(C, T (C))]. +The above claim is not true, for instance, see the following example, which is +borrowed from [8]. +Example 2.10. Let X = {0, 1, 2}, and the function d : X × X → R and the map +f : X → X be given by +d(0, 0) = d(1, 1) = d(2, 2) = 0 +d(0, 1) = d(1, 0) = 5, d(1, 2) = d(2, 1) = 2, d(0, 2) = d(2, 0) = 3 +f(1) = f(2) = 2, f(0) = 1. +Then the map f : X → X is Kannan on (X, d) with contractivity factor α ∈ [ 2 +5, 1 +2) +but the map T : H(X) → H(X) given by T (B) = ∪x∈Bf(x) for all B ∈ H(X) is +not a Kannan map on (H(X), h(d)) for any contractivity factor α ∈ [0, 1 +2). +Remark 2.11. The above example does not work for the following lemma due to +different contractive factors. Here, we correct Lemma 3.5 of [23], and we show that +it holds under certain additional conditions. +Lemma 2.12. Suppose T : X → X be a continuous Kannan mapping on the metric +space (X, d) with contractivity factor 0 < β < 1 +6. Then T : H(X) → H(X) given by +T (B) = {T (x) : x ∈ B} for every B ∈ H(X) is Kannan mapping on (H(X), h(d)) +with contractivity factor 0 < γ = +β +(1−4β) < 1 +2. +Proof. Let us first recall a basic real-analysis result that the image of a compact +set under a continuous map is compact. Since T is continuous, it maps H(X) into +itself. Now, since T is Kannan mapping on (X, d), for x, y ∈ X, we have +d(T (x), T (y)) ≤ β[d(x, T (x)) + d(y, T (y))] +≤ β[d(x, T (y)) + d(T (y), T (x)) + d(y, T (x)) + d(T (x), T (y))] += β[d(x, T (y)) + d(y, T (x))] + 2βd(T (x), T (y)). + +6 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +So, we obtain +d(T (x), T (y)) ≤ +β +(1 − 2β)[d(x, T (y)) + d(y, T (x))]. +Now, for B, C ∈ H(X) +sup +x∈B +inf +y∈C d(T (x), T (y)) ≤ +β +(1 − 2β)[sup +x∈B +inf +y∈C d(x, T (y)) + sup +x∈B +inf +y∈C d(y, T (x))]. +That is, +sup +x∈B +inf +y∈C d(T (x), T (y)) ≤ +β +(1 − 2β)[h(B, T (C)) + h(C, T (B))]. +Thanks to the triangle inequality, +h(T (B), T (C)) ≤ +β +(1 − 2β)[h(B, T (B)) + h(T (B), T (C)) ++h(C, T (C)) + h(T (C), T (B))]. +Consequently, +� +1 − +2β +(1 − 2β) +� +h(T (B), T (C)) ≤ +β +(1 − 2β)[h(B, T (B)) + h(C, T (C))]. +Therefore, +h(T (B), T (C)) ≤ γ[h(B, T (B)) + h(C, T (C))], +where γ = +β +(1−4β) < 1 +2 for 0 < β < 1 +6. This completes the proof. +□ +Theorem 2.13. For a complete metric space (X, d), let Tn : n = 1, 2, ..., N are +continuous Kannan mappings on (H(X), h) with contractivity factor 0 < βn < +1 +6, for all n. Define T : H(X) → H(X) by T (B) = ∪N +n=1Tn(B) for each B ∈ +H(X). Then T is a Kannan mapping with contractivity factor γ = max{γn : n = +1, 2, ..., N}. +Proof. For B, C ∈ H(X), we have +h(T (B), T (C)) = h(T1(B) ∪ T2(B) ∪ . . . ∪ Tn(B), T1(C) ∪ T2(C) ∪ . . . ∪ Tn(C)) +≤ max{h(T1(B), T1(C)), h(T2(B), T2(C)), . . . , h(Tn(B), Tn(C))}. +By using the above lemma, we obtain +h(T (B), T (C)) += max +� +β1 +1 − 4β1 +[h(B, T1(B)) + h(C, T1(C))], +β2 +1 − 4β2 +[h(B, T2(B)) ++ h(C, T2(C))], . . . , +βn +1 − 4βn +[h(B, Tn(B)) + h(C, Tn(C))] +� +≤ max +1≤i≤n +� +βi +1 − 4βi +� +[max{h(B, T1(B)), h(B, T2(B)), ..., h(B, Tn(B))} ++ max{h(C, T1(C)), h(C, T2(C)), ..., h(C, Tn(C))}] +≤ max +1≤i≤n{γi}[h(B, T1(B) ∪ T2(B)) . . . ∪ Tn(B) + h(C, T1(C) ∪ T2(C)) . . . ∪ Tn(C)] +≤ γ[h(B, T (B)) + h(C, T (C))], + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +7 +where γ = max1≤i≤n{γi} = max1≤i≤n +� +βi +1−4βi +� +< 1 +2. Hence, T is Kannan with +contractivity factor γ. +□ +Remark 2.14. In [8] Dung et al. proposed a question, whether their results are true +or not for n ≥ 3. In this order, in the above theorem, we show that T is Kannan +for all n. Moreover, from Theorem 2.8, T has a unique fixed point. +We use the following notations throughout the article: C(I) denotes the set of +continuous functions f : I = [x0, xN] → [a, b]. Let C∗(I) ⊂ C(I) and given by +C∗ = {f ∈ C(I) : f(x0) = y0, f(xN) = yn}. K = I × R. +Let us define a metric dθ on K as follows +dθ((x, y), (z, w)) = |x − z|+θ|y − w|, θ > 0. +Note that (C∗(I), Hθ) is a complete metric space with respect to Hausdorff metric +Hθ, where +Hθ(f, g) = Hθ(Gf, Gg) = max{ sup +x∈Gf +inf +y∈Gg dθ(x, y), sup +y∈Gg +inf +x∈Gf dθ(x, y)}. +In the following section, we give the construction of fractal functions using Kannan +IFS and the existence of self-similar measures. +3. Construction of fractal functions via Kannan Iterated function +systems +Let Fi : K → [a, b] be continuous mappings and satisfying for some k ≥ 0, and +0 ≤ βi < 1 +2 +|Fi(x, y)−Fi(w, y)|≤ k|x−w|, |Fi(x, y)−Fi(x, z)|≤ βi +� +|y −Fi(., y)|+|z −Fi(., z)| +� +for all x, w ∈ I, y, z ∈ [a, b], and i = 1, 2, . . . , N. Now, let {K; Wi, i = 1, 2, . . ., N} +be an IFS with +Wi(x, y) = (Li(x), Fi(x, y)) = (aix + bi, Fi(x, y)), +where transformations are constrained by the data according to +Wi(x0, y0) = (xi−1, yi−1), Wi(xN, yN) = (xi, yi) +for i = 1, 2, . . . , N. For all i = 1, 2, ..., N, Wi : K → K are Kannan mappings. Then +{K; Wi : i = 1, 2, ..., N} is the Kannan IFS. +Theorem 3.1. Let N > 1, and {K; Wi, i = 1, 2, . . . , N} denote the IFS defined as +above, associated with the set of data +{(xi, yi) : i = 1, 2, ..., N} such that amax = max +i (xi+1 − xi) < 1 +3. +Then, there is a metric dθ on K = I × R, equivalent to the Euclidean metric such +that for all i = 1, 2, ..., N, Wi are Kannan maps with respect to dθ. + +8 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +Proof. For all (x, y), (w, z) ∈ K, we have +dθ(Wi(x, y), Wi(w, z)) = dθ +� +(Li(x), Fi(x, y)), (Li(w), Fi(w, z)) +� += |Li(x) − Li(w)|+θ|Fi(x, y) − Fi(w, z)| +≤ |ai||x − w|+θ|Fi(x, y) − Fi(w, z)| +≤ amax|x − w|+θ|Fi(x, y) − Fi(w, z)|. +Now, thanks to the triangle inequality, we have +|x − w|≤ |x − Li(x)|+|Li(x) − Li(w)|+|Li(w) − w|, +this further yields +|x − w|≤ +1 +1 − amax +(|x − Li(x)|+|Li(w) − w|). +We now estimate +|Fi(x, y) − Fi(w, z)|≤ |Fi(x, y) − Fi(w, y)|+|Fi(w, y) − Fi(w, z)| +≤ k|x − w|+βi +� +|y − Fi(w, y)|+|z − Fi(w, z)| +� +≤ k|x − w|+βi +� +|y − Fi(x, y)|+|Fi(x, y) − Fi(w, y)|+|z − Fi(w, z)| +� +≤ k|x − w|+βi +� +|y − Fi(x, y)|+k|x − w|+|z − Fi(w, z)| +� +≤ (k + kβmax)|x − w|+βmax +� +|y − Fi(x, y)|+|z − Fi(w, z)| +� +. +With the help of the above estimates, we obtain +dθ(Wi(x, y), Wi(w, z)) += amax + (k + kβmax)θ +1 − amax +(|x +− Li(x)|+|Li(w) − w|) + βmaxθ +� +|y − Fi(x, y)|+|z − Fi(w, z)| +� +≤ γ +� +dθ((x, y), Wi(x, y)) + dθ((w, z), Wi(w, z)) +� +, +where γ = max +� +amax+(k+kβmax)θ +1−amax +, βmaxθ +� +. Using the condition amax < 1 +3, we may +choose a suitable (sufficiently small) θ > 0 such that γ < 1 +2. For this value of θ, the +mapping Wi is a Kannan mapping, completing the proof. +□ +Remark 3.2. From the above proof, if amax < 1 +5 then we may choose a suitable +(sufficiently small) θ > 0 such that the Kannan contractivity factor γ < 1 +4. +Theorem 3.3. Let N > 1 and {K; Wi, i = 1, 2, . . ., N} denote the IFS defined as +above, associated with the set of data +{(xi, yi) : i = 1, 2, ..., N} such that amax = max +i (xi+1 − xi) < 1 +7. +Then, there exists a unique non empty compact set G ⊂ K = I × [a, b] such that +G = ∪N +i=1Wi(G). + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +9 +Proof. On similar lines of the proof of Theorem 3.1, we have +dθ(Wi(x, y), Wi(w, z)) ≤ γ +� +dθ((x, y), Wi(x, y)) + dθ((w, z), Wi(w, z)) +� +, +where γ = max +� +amax+(k+kβmax)θ +1−amax +, βmaxθ +� +. Using the condition amax < 1 +7, we may +choose a suitable (sufficiently small) θ > 0 such that γ < 1 +6. For this value of θ, +the mapping Wi is a Kannan mapping with contractivity factor γ < 1 +6. Now, using +Theorem 2.13 and Remark 2.14, we obtain a unique compact set G satisfying +G = ∪N +i=1Wi(G). +This completes the proof. +□ +Theorem 3.4. Let the IFS {K; Wi, i = 1, 2, ..., N} defined as above associated with +the set of data +{(xi, yi) : i = 1, 2, ..., N} such that amax = max +i (xi+1 − xi) < 1 +5. +Let G denote the attractor of the IFS. Then, G is the graph Gf of continuous +function f : [x0, xN] → [a, b] satisfying f(xi) = yi for all i = 0, 1, ..., N. That is, +Gf = {(x, f(x)) : x ∈ [x0, xN]}, +where f(xi) = yi for all i = 0, 1, ..., N. +Proof. Let C∗(I) = {f ∈ C(I) : f(x1) = y1, f(xN) = yN}. Here, C∗(I) is a +closed subset of C(I) and (C∗(I), Hθ) is a complete metric space. Define Read- +Bajraktarevi´c (RB) operator T : C∗(I) → C∗(I) by +(3.1) +(T g)(x) = Fi(L−1 +i (x), g(L−1 +i (x))). +Now, we show that T is a Kannan mapping w.r.t. Hθ. We will proceed as follows. +Here, graphs of T g and T h are given by +GT g = {(x, T g(x)) : x ∈ I} +and +GT h = {(y, T h(y)) : y ∈ I}. +Let (x, T g(x)) ∈ GT g and (y, T h(y)) ∈ GT h. Since Wi is Kannan with contractivity +factor γ, from Theorem 3.1, we get +dθ +� +(x, T g(x)), (y, T h(y)) +� += dθ +� +(x, Fi(L−1 +i (x), g(L−1 +i +(x))), (y, Fi(L−1 +i (y), h(L−1 +i (y)))) +� += dθ +� +(Li(w), Fi(w, g(w)), (Li(z), Fi(z, g(z)) +� += dθ +� +Wi(w, g(w)), Wi(z, h(z)) +� +≤ γ +� +dθ +� +(w, g(w)), Wi(w, g(w)) +� ++ dθ +� +(z, h(z)), Wi(z, h(z)) +�� += γ +� +dθ +� +L−1 +i (x), g(L−1 +i (x)), (x, T g(x)) +� ++ dθ +� +L−1 +i (y), h(L−1 +i (y)), (y, T h(y)) +�� +, + +10 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +where w = L−1 +i (x) and z = L−1 +i (y). Thanks to the triangle inequality, +dθ +� +(x, T g(x)), (y, T h(y)) +� +≤ γ +� +dθ +� +(L−1 +i (x), g(L−1 +i +(x))), (y, T g(y)) +� ++ dθ +� +(x, T g(x)), (y, T h(y)) +� ++ dθ +� +(L−1 +i (y), h(L−1 +i (y))), (x, T h(x)) +� ++ dθ +� +(x, T g(x)), (y, T h(y)) +�� +. +On taking infimum both sides, we have +inf +y ∈I dθ +� +(x, T g(x)), (y, T h(y)) +� +≤ γ +� +inf +y∈I dθ +� +(L−1 +i (x), g(L−1 +i +(x))), (y, T g(y)) +� ++ inf +y∈I dθ +� +(L−1 +i (y), h(L−1 +i (y))), (x, T h(x)) +� ++ 2 inf +y∈I dθ +� +(x, T g(x)), (y, T h(y)) +�� +. +That is, +(1 − 2γ) inf +y ∈I dθ +� +(x, T g(x)), (y, T h(y)) +� +≤ γ +� +inf +y∈I dθ +� +(L−1 +i (x), g(L−1 +i (x))), (y, T g(y)) +� ++ inf +y∈I dθ +� +(L−1 +i (y), h(L−1 +i (y))), (x, T h(x)) +�� +. +By taking supremum over x ∈ I, we have +Hθ(GT g, GT h) ≤ +γ +1 − 2γ [Hθ(Gg, GT g) + Hθ(Gh, GT h)]. +That is, +Hθ(T g, T h) ≤ +γ +1 − 2γ +� +Hθ(g, T g) + Hθ(h, T h) +� +. +Since amax < 1 +5, Remark 3.2 yields that β := +γ +1−2γ < 1 +2. Therefore, T is Kannan +w.r.t. Hθ. By Theorem 2.8 , T has a unique fixed point f ∈ C∗(I). Further, it is +easy to check that f interpolates the data set. Now, we show that the graph Gf of +f is an attractor of the IFS. Since Wi(x, y) = (Li(x), Fi(x, y)) for all i = 1, 2, ..., N, +I = ∪j∈JLj(I), and from the functional Equation 2.1, we get that +Wi(Gf) = Wi({(x, f(x)) : x ∈ [x0, xN]}) += {(Li(x), Fi(x, f(x))) : x ∈ [x0, xN]} += {(Li(x), f(Li(x))) : x ∈ [x0, xN]} += {(x, f(x)) : x ∈ [xi−1, xi]}. +Hence +Gf = {(x, f(x)) : x ∈ [x0, xN]} += ∪N +i=1{(x, f(x)) : x ∈ [xi−1, xi]} += ∪N +i=1Wi(Gf). +By Theorem 3.3, G is the unique attractor of the IFS {K; Wi, i = 1, 2, ..., N}. Thus, +G = Gf. This completes the proof. +□ + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +11 +Example 3.5. The continuous function h : [−1, 21 +10] → [−1, 21 +10] defined by +h(x) = +� +x2 +4 − x +8 , if − 1 ≤ x < 1 +2 +x2 +5 − x +10, +if 1 +2 ≤ x ≤ 21 +10. +is Kannan mapping with β = 10 +21 but it is not a contraction mapping. +First, we show that T is Kannan contraction, and we choose β = 10 +21 < 1 +2. +(i) For the range −1 ≤ x, y < 1 +2, we have +|T (x) − T (y)|= 1 +8|x(2x − 1) − y(2y − 1)| +and +|x − T (x)|+|y − T (y)|= 1 +8 +� +|x|9 − 2x|+|y||9 − 2y| +� +. +For β = 10 +21, we can see that +|T (x) − T (y)|≤ β +� +|x − T (x)|+|y − T (y)| +� +. +(ii) For 1 +2 ≤ x, y < 21 +10, we have +|T (x) − T (y)|= 1 +10|x(2x − 1) − y(2y − 1)| +and +|x − T (x)|+|y − T (y)|= 1 +10 +� +|x|11 − 2x|+|y||11 − 2y| +� +. +For β = 10 +21, we can see that +|T (x) − T (y)|≤ β +� +|x − T (x)|+|y − T (y)| +� +. +(iii) For −1 ≤ x < 1 +2 and 1 +2 ≤ y < 21 +10, we have +|T (x) − T (y)|= |x(2x − 1) +8 +− y(2y − 1) +10 +| +and +|x − T (x)|+|y − T (y)|= +�1 +8|x|9 − 2x|+ 1 +10|y||11 − 2y| +� +. +For β = 10 +21, we can see that +|T (x) − T (y)|≤ β +� +|x − T (x)|+|y − T (y)| +� +. +Now, one can see that T is not a contraction because for x = −1, y = −0.99, we +have +|T (x) − T (y)|= 0.2537 > |x − y|= 0.01. +Remark 3.6. In this order, we can construct many different Kannan mappings with +the help of the functional Equation 2.1 +Fj(x, y) = αjy + qj(x), j ∈ J. +For, instance +Fj(x, y) = T (y) + qj(x), + +12 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +where +T (y) = +� +y2 +4 − y +8, if − 1 ≤ y < 1 +2 +y2 +5 − y +10, +if 1 +2 ≤ y ≤ 21 +10, +and qj : I → R, j ∈ J are suitable continuous functions satisfying qj(x1) = yj −αjy1 +and qj(xN) = yj+1 − αjyN for all j ∈ J. +3.1. Existence of Self-Similar Measures. +Definition 3.7. Let I = {K; Wi : i = 1, 2, ..., N} be an IFS and A be the attarctor +of the IFS I. Then, we say that I satisfies strong separation condition (SSC) if +Wi(A) ∩ Wj(A) = ∅ whenever i ̸= j. +Note that there are several separation conditions are available for any IFS, for +instance, [9, 11]. +Hutchinson [9] computed the Hausdorff dimension of self-similar sets under the +open set condition. Assuming the SSC, Priyadarshi and his collaborators [26] gave +a formula for the Hausdorff dimension of the invariant set of generalized graph- +directed systems. +Theorem 3.8. Let I = {K; Ti : i = 1, 2, ..., N} be an IFS consisting of Kannan +mappings satisfies the SSC. Let (p1, p2, · · · , pN) be a probability vector. Then, there +exists a Borel probability measure µ∗ supported on the attractor A of the IFS such +that +µ∗ = +N +� +i=1 +piµ∗ ◦ T −1 +i +. +Proof. Since the Kannan IFS {K; Ti : i = 1, 2, ..., N} satisfies the SSC. That is, +Ti(A) ∩ Tj(A) = φ ∀ i ̸= j. Let E0 = A. We have +A = +N +� +i=1 +Ti(A) = +N +� +i,j=1 +Tij(A) = +N +� +i1,i2,...,in +Ti1,i2,...,in(A). +For k = 1, 2, ..., we define Ek as follows: +Ek = +� +Ti1i2···ik(A) : ij ∈ {1, 2, ..., N}, j = 1, 2, ..., k +� +, +where Ek denotes the collection of disjoint Borel subsets of A. Let B ∈ Ek. Note +that each B is contained in one of the sets in Ek−1 and contains a finite number +of the sets in Ek+1. We can see that |Ti1i2···ik(A)|→ 0 as k → ∞ as follows. Let +Ti : A → A and |A|= supx,y∈A d(x, y) = d(x0, y0), x0, y0 ∈ A. Since Ti is Kannan +contraction, we have +d(Ti(x), Ti(y)) ≤ βi[d(x, Ti(x)) + d(y, Ti(y))] +≤ βi[d(x0, y0) + d(x0, y0)] +≤ 2βid(x0, y0). +By taking supremum of both side, we have +sup +x,y∈A +d(Ti(x), Ti(y)) ≤ 2βid(x0, y0), +and 2βi < 1. Now, +|Ti(A)|≤ 2βid(x0, y0) = ci|A|, ci = 2βi. + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +13 +In a similar way, we get +|Ti1i2···ik(A)|≤ ck +max|A|→ 0 when k → ∞, +where cmax = 2βmax = 2 max{β1, β2, ..., βk}. +Let a probability vector p = (p1, p2, ..., pN) satisfying pi > 0 for all i and �N +i=1 pi = +1. We assign µ(A) with µ(A) = 1 = �N +i=1 pi. Let B ∈ Ek such that B = Ti1i2···ik(A). +Let Ek = � +B∈Ek B = � +ij,j=1,2,...,k Ti1i2···ik(A) = A. Hence, we have µ(C) = 0 ∀ C +with C ∩ A = ∅ and E = � Ek +� Rn \Ek and µ(Rn) = 1. It follows that (Cf. [11, +Proposition 1.7]) the definition of µ may be extended to all subsets of Rn so that +µ becomes a measure. Now, we show that µ = �N +i=1 piµ ◦ T −1 +i +. +Let Tj(A) be an arbitrary cylinder in the first stage. Then +N +� +i=1 +piµ ◦ T −1 +i +(Tj(A)) = pjµ(A) = pj, +and from the construction of measure µ, we have µ(Tj(A)) = pj. +Therefore, +µ(Tj(A)) = �N +i=1 piµ ◦ T −1 +i +(Tj(A)) = pj for all j. +Similarly, we have µ(B) = +�N +i=1 piµ ◦ T −1 +i +(B) for all cylinders B ∈ Ek at any stage k. Thus, the proof is +complete. +□ +Remark 3.9. Recall that the collection of all Borel probability measures on Rn, de- +noted by P(Rn), is a complete metric space with respect to the Monge-Kantorovich +metric dH defined as +dH(µ, ν) = sup +����� +� +fdµ(x) − +� +fdν(x) +���� : where f : Rn → R, Lip(f) ≤ 1 +� +. +Define a mapping M : P(Rn) → P(Rn) by M(µ) = �N +i=1 piµ ◦ f −1 +i +. Now, we have +dH(M(µ), M(ν)) = sup +���� +� +fdM(µ)(x) − +� +fdM(ν)(x) +��� : Lip(f) ≤ 1 +� += sup +���� +N +� +i=1 +pi +� +fdµ ◦ f −1 +i +(x) − +N +� +i=1 +pi +� +fdν ◦ f −1 +i +(x) +��� : Lip(f) ≤ 1 +� +. +Hutchinson [9] showed that if all fi are contractions, then M is the contraction +with respect to the Monge-Kantorovich metric. +Here, a natural question arises +whether M is Kannan with respect to the Monge-Kantorovich metric when all fi +are Kannan. It is open for further investigation. +4. Smooth fractal functions +Let us denote the space of m-times continuously differentiable functions by +Cm(I). +Now, we define a new metric with the help of Hausdorff distance such +as +D(g, h) := max +0≤k≤m Hθ(Gg(k), Gh(k)), +where Hθ(Gg, Gh) denotes the Hausdorff distance induced from the metric dθ be- +tween the graphs of f and g. +Since Hθ(Gg, Gh) and ∥g − h∥∞ are equivalent, +(Cm(I), D) will be a complete metric space. + +14 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +Theorem 4.1. Let g ∈ Cm(I), where m ∈ N. Suppose that Lj : I → Ij is affine +map defined by Lj(x) = ajx + bj satisfying Lj(x1) = xj, Lj(xN) = xj+1, j ∈ J and +Fj(x, y) = αjy + qj(x). Let q(k) +j +(x1) = g(k)(x1), q(k) +j +(xN) = g(k)(xN), j ∈ J, 0 ≤ +k ≤ m, and scaling factor αj satisfying αj < ak +5 , where ak = min{ak +j : j ∈ J}. +Then T has a unique fractal function f ∗ +∆ ∈ Cm +∗ (I). Moreover, dimH(Gr(f ∗ +∆)) = +dimB(Gr(f ∗ +∆)) = 1. +Proof. Let Cm +∗ (I) = {g ∈ Cm(I) : h(k)(x1) = g(k)(x1), h(k)(xN) = g(k)(xN), 0 ≤ +k ≤ m}. Here, Cm +∗ (I) is a closed subset of Cm(I). It can be seen that (Cm +∗ (I), D) is +a complete metric space. Define the RB operator T : Cm +∗ (I) → Cm +∗ (I) by +(T g)(x) = αjf(L−1 +j (x)) + qj(L−1 +j (x)), x ∈ Ij, j ∈ J. +It can be observed that T is well-defined. Let g, h ∈ Cm +∗ (I), +D(g, h) := max +0≤k≤m Hθ(Gg(k), Gh(k)) +for each 0 ≤ k ≤ m, we have +(T g)(k)(x) = a−k +j [αjg(k)(L−1 +j (x)) + q(k) +j +(L−1 +j (x))]. +ˆF k +j (x, y) = a−k +j αjy + a−k +j q(k) +j +(x) and ˆW k +j (x, y) = (Lj(x), ˆF k +j (x, y)). It can be seen +that ˆ +W k +j are Kannan contractions as follows. +D( ˆW k +j (x1, y1), ˆW k +j (x2, y2)) +≤ αj +ak D((x1, y1), (x2, y2)) +≤ αj +ak D((x1, y1), ˆ +W k +j (x1, y1)) + αj +ak D( ˆ +W k +j (x1, y1), ˆ +W k +j (x2, y2)) + αj +ak D( ˆ +W k +j (x2, y2), (x2, y2)). +Hence, we obtain +D( ˆW k +j (x1, y1), ˆW k +j (x2, y2)) ≤ +αj +ak − αj +[D((x1, y1), ˆW k +j (x1, y1))+D((x2, y2), ˆ +W k +j (x2, y2))]. +Hence, ˆ +W k +j are Kannan contractions with contractivity factor +αj +ak−αj . +Now, we show that T is Kannan w.r.t. metric D. That is +D(T g, T h) ≤ γ′[D(g, T g) + D(h, T h)]. +Let (x, (T g)(k)(x)) ∈ G(T g)(k) and (y, (T h)(k)(y)) ∈ G(T h)(k). We have +dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +� += dθ +�� +x, ˆF k +j (L−1 +j (x), g(k)(L−1 +j (x))) +� +, +� +y, ˆF k +j (L−1 +j (y), h(k)(L−1 +j (y))) +�� += dθ +�� +Lj(w), ˆF k +j (w, g(k)(w)) +� +, +� +Lj(z), ˆF k +j (z, h(k)(z)) +�� += dθ +� +ˆW k +j (w, g(k)(w)), ˆ +W k +j (z, h(k)(z)) +� +. + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +15 +Since ˆ +W k +j +are Kannan contraction and by substituting again w = L−1 +j (x) and +z = L−1 +j (y), we have +dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +� +≤ +αj +ak − αj +� +dθ +� +(L−1 +j (x), g(k)(L−1 +j (x))), ˆ +W k +j (L−1 +j (x), g(k)(L−1 +j (x))) +� ++ dθ +� +(L−1 +j (y), h(k)(L−1 +j (y))), ˆW k +j (L−1 +j (y), h(k)(L−1 +j (y))) +�� += +αj +ak − αj +� +dθ +� +L−1 +j (x), g(k)(L−1 +j (x)), (x, (T g)(k)(x)) +� ++ dθ +� +L−1 +j (y), h(k)(L−1 +j (y)), (y, (T h)(k)(y)) +�� +≤ +αj +ak − αj +� +dθ +� +(L−1 +j (x), g(k)(L−1 +j (x))), (y, (T g)(k)(y)) +� ++ dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +� ++ dθ +� +(L−1 +j (y), h(k)(L−1 +j (y))), (x, (T h)(k)(x)) +� ++ dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +�� +. +On taking infimum both sides, we have +inf +y∈I dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +� +≤ +αj +ak − αj +� +inf +y∈I dθ +� +(L−1 +j (x), g(k)(L−1 +j (x))), (y, (T g)(k)(y)) +� ++ inf +y∈I dθ +� +(L−1 +j (y), (h)(k)(L−1 +j (y))), (x, (T h)(k)(x)) +� ++ 2 inf +y∈I dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +�� +. +That is, +(1 − +2αj +ak − αj +) inf +y∈I dθ +� +(x, (T g)(k)(x)), (y, (T h)(k)(y)) +� +≤ +αj +ak − αj +� +inf +y∈I dθ +� +(L−1 +j (x), g(k)(L−1 +j (x))), (y, (T g)(k)(y)) +� ++ inf +y∈I dθ +� +(L−1 +j (y), h(k)(L−1 +j (y))), (x, (T h)(k)(x)) +�� +. +By taking supremum over x ∈ I, we have +max +0≤k≤m Hθ(G(T g)(k), G(T h)(k)) ≤ +αj +ak − 3αj +[ max +0≤k≤m Hθ(Gg(k), G(T g)(k)) ++ max +0≤k≤m Hθ(Gh(k), G(T h)(k))] +That is +D(T g, T h) ≤ γ′[D(g, T g) + D(h, T h)], +where γ′ = +αj +ak−3αj < 1 +2. Hence, T is Kannan contrcation with contractivity factor +γ′ = +αj +ak−3αj < 1 +2. By Theorem 2.8 , T has a unique fractal function f ∗ +∆ ∈ Cm +∗ (I) and +obeys the equation 3.1. We know that any continuous function with the bounded +derivative is of bounded variation. This result with [15, Theorem 1.3] yields +dimH(Gr(f ∗ +∆)) = dimB(Gr(f ∗ +∆)) = 1. + +16 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +Hence, the proof is complete. +□ +Remark 4.2. [5] The relationship between the Heisenberg and Euclidean geometry +on H = R3 is rather intricate. +The Heisenberg-Hausdorff dimension is always +greater than or equal to its Euclidean counterpart. The Hausdorff dimension of +(R3, dH) is equal to 4 (in fact, balls in the metric dH have a measure proportional +to the fourth power of their radius). This implies, for instance, that the Heisenberg +metric dH cannot be locally bi-Lipschitz equivalent with any Riemannian metric, +particularly with the Euclidean metric dE. +Remark 4.3. From the above remark, we can conclude that if two metrics are +topologically equivalent, that does not imply that the dimension of the graph of any +function can be equal, but if metrics are metrically equivalent, then the Hausdorff +dimension will be equal, but the Hausdorff measure need not be equal; see the +following +Let δ > 0. Then +Hs +δ,d2(E) = inf +� ∞ +� +i=1 +diamd2(Fi)s : {Fi} is a δ − cover of E +� +≤ inf +� ∞ +� +i=1 +cs +2diamd1(Fi)s : {Fi} is a δ − cover of E +� += cs +2Hs +δ,d1(E). +Similarly, cs +1Hs +δ,d1(E) ≤ Hs +δ,d2(E). Therefore, +cs +1Hs +δ,d1(E) ≤ Hs +δ,d2(E) ≤ cs +2Hs +δ,d1(E) holds for all δ > 0. +As δ → 0+, we have +cs +1Hs +d1(E) ≤ Hs +d2(E) ≤ cs +2Hs +d1(E). +Now, using the definition of the Hausdorff dimension and the above inequality, we +get +dimH,d1(E) = inf{s : Hs +δ,d1(E) = 0} = inf{s : Hs +δ,d2(E) = 0} = dimH,d2(E), +completing the reamark. +5. Graph of Kannan fractal functions +Here, we have the functional equation +Fj(x, y) = T (y) + qj(x), j ∈ J, +and the self-referential equation is +f(Lj(x)) = T (f(x)) + qj(x), x ∈ Ij, j ∈ J. +We choose qj(x) = cjx + dj satisfying the join-up conditions such as +T (y1) + cjx1 + dj = yj and T (yN) + cjxN + dj = yj+1, j ∈ J. +So, we have +cj = (yj − yj+1) − (T (y1) − T (yN)) +(x1 − xN) + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +17 +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Figure 1. +0 +0.2 +0.4 +0.6 +0.8 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +1.8 +Figure 2. + +18 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +0 +0.2 +0.4 +0.6 +0.8 +1 +-1 +-0.5 +0 +0.5 +1 +1.5 +Figure 3. +0 +0.2 +0.4 +0.6 +0.8 +1 +-1 +-0.5 +0 +0.5 +1 +1.5 +Figure 4. + +CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS +19 +and +dj = yj − T (y1) − (yj − yj+1) − (T (y1) − T (yN)) +(x1 − xN) +x1. +The initial data is taken as follows for Figure 1 and Figure 2, respectively. +{(0, 0.9), (0.1, 0.5), (0.2, 0.75), (0.3, 0.95), (0.4, 1.25), (0.5, 1.5), (0.6, 1.6), (0.7, 1.2), +(0.8, 1.8), (0.9, 1.9), (1.0, 2.0)} and {(0, 0.6), (0.1, 1.4), (0.2, 0.9), (0.3, 1.2), (0.4, 1.8), +(0.5, 1.3), (0.6, 0.9), (0.7, 1.75), (0.8, 0.85), (0.9, 1.75), (1.0, 2.0)}. +For Figure 3 and Figure 4, we consider qj(x) = x2 + cjx + dj satisfying the join-up +conditions such as +T (y1) + x2 +1 + cjx1 + dj = yj and T (yN) + x2 +n + cjxN + dj = yj+1, j ∈ J. +So, we have +cj = (yj − yj+1) − (T (y1) − T (yN)) − (x2 +1 − x2 +N) +(x1 − xN) +and +dj = yj − T (y1) − (yj − yj+1) − (T (y1) − T (yN)) − (x2 +1 − x2 +N) +(x1 − xN) +x1. +The initial data is taken as follows for Figure 3 and Figure 4, respectively. +{(0, 0.75), (0.1, 1.4), (0.2, 0.65), (0.3, 1.55), (0.4, 1.25), (0.5, 1.0), (0.6, 1.75), (0.7, 1.3), +(0.8, 2.0), (0.9, 1.15), (1.0, 0.95)} and {(0, 0.5), (0.1, 1.5), (0.2, 1.75), (0.3, 0.95), (0.4, 1.0), +(0.5, 1.8), (0.6, 1.2), (0.7, 1.6), (0.8, 1.4), (0.9, 0.85), (1.0, 1.4)}. +Acknowledgements. The first author’s work is financially supported by the CSIR, +India, with grant number 09/1058(0012)/2018-EMR-I. +References +1. G. Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff +distance, Proc. Amer. Math. Soc. 95 (1985) 653-658. +2. M. F. Barnsley, Fractal functions and interpolation, Constr. Approx. 2 (1986) 303-329. +3. M. F. Barnsley, Fractal Everywhere, Academic Press, Orlando, Florida, 1988. SIAM J. Math. +Anal. 20(5) (1989) 1218–1242. +4. M. F. Barnsley and P. R. Massopust, Bilinear fractal interpolation and box dimension, J. +Approx. Theory 192 (2015) 362–378. +5. Z. M. Balogh and J. T. Tyson, Hausdorff dimension of self-similar and self-affine fractals the +Heisenberg group, Proc. London Math. Soc. (3) 91 (2005) 153-183. +6. D. Cheng, Z. Liand and B. Selmi, Upper metric mean dimensions with potential on subsets. +Nonlinearity. 34 (2021) 852–867. +7. S. Chandra and S. Abbas, On fractal dimensions of fractal functions using functions spaces, +Bull. Aust. Math. Soc. (2022) 1-11. +8. Van Dung, Nguyen and Adrian Petru¸sel, On iterated function systems consisting of Kannan +maps, Reich maps, Chatterjea type maps, and related results, Journal of Fixed Point Theory +and Applications 19, no. 4 (2017), 2271-2285. +9. J. E. Hutchinson, Fractals and self-similarity, Indian Univ. Math. J. 30 (1981) 713-747. +10. Janos Ludvik, On Mappings Contractive in the Sense of Kannan, Proceedings of the American +Mathematical Society 61, no. 1 (1976): 171-75. +11. K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley +Sons Inc., New York, 1999. +12. S. Jha and S. Verma, Dimensional analysis of α-fractal functions, Results in Mathematics 76 +(4) (2021) 1-24. +13. R. Kannan, Some results on fixed points, Bull. Calcutta Math. Soc. 60 (1968) 71-76. +14. R. Kannan, Some results on fixed points II, Amer. Math. Monthly 76 (1969) 405-408. + +20 +SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS +15. Y. S. Liang, Box dimensions of Riemann-Liouville fractional integrals of continuous functions +of bounded variation, Nonlin. Anal. 72(11) (2010) 4304-4306. +16. P. R. Massopust, Fractal Functions, Fractal Surfaces, and Wavelets. 2nd ed., Academic Press, +San Diego, 2016. +17. M. A. Navascu´es, Fractal polynomial interpolation, Z. Anal. Anwend. 25(2) (2005) 401-418. +18. M. A. Navascu´es, Fractal approximation, Complex Anal. Oper. Theory 4(4) (2010) 953-974. +19. A. Priyadarshi, Lower bound on the Hausdorf dimension of a set of complex continued frac- +tions. J. Math. Anal. Appl. 449(1), (2017) 91–95. +20. S.I. Ri, A new idea to construct the fractal interpolation function, Indag. Math. 29(3) (2018) +962-971. +21. S.I. Ri, Fractal functions on the Sierpinski gasket, Chaos, Solitons Fractals 138 (2020) 110142. +22. H.J. Ruan, W.-Y. Sub and K. Yao, Box dimension and fractional integral of linear fractal +interpolation functions, J. Approx. Theory 161 (2009) 187-197. +23. D. R. Sahu, A. Chakraborty and R. P. Dubey, K-Iterated function system, Fractals 18 (2010) +139–144. +24. B. Selmi, Multifractal dimensions for projections of measures. Bol. Soc. Paran. Mat. 40 (2022) +1-15. +25. B. Selmi, On the multifractal dimensions of product measures. Nonlinear Studies, 29(1) (2022). +26. R. D. Nussbaum, A. Priyadarshi and S. V. Lunel, Positive operators and Hausdorff dimension +of invariant sets, Trans. Amer. Math. Soc. 364(2) (2012) 1029-1066. +27. H. Y. Wang and J. S. Yu, Fractal interpolation functions with variable parameters and their +analytical properties, J. Approx. Theory 175 (2013) 1-18. +School of Mathematical and Statistical Sciences, Indian Institute of Technology +Mandi, Kamand (H.P.) - 175005, India +Email address: sahusubhash77@gmail.com +Department of Applied Sciences, IIIT Allahabad, Prayagraj-211015, India +Email address: saurabh331146@gmail.com +School of Mathematical and Statistical Sciences, Indian Institute of Technology +Mandi, Kamand (H.P.)- 175005, India +Email address: sabbas.iitk@gmail.com + diff --git a/6NE1T4oBgHgl3EQfTQM9/content/tmp_files/load_file.txt b/6NE1T4oBgHgl3EQfTQM9/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..60453b73f2b802e0a2533f49adb71c74ac2e5afc --- /dev/null +++ b/6NE1T4oBgHgl3EQfTQM9/content/tmp_files/load_file.txt @@ -0,0 +1,887 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf,len=886 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='03075v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='DS] 8 Jan 2023 CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS AND SMOOTHNESS ANALYSIS SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let T be a self-map on a metric space (X, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T is called Kannan map if there exists α, 0 < α < 1 2, such that d(T(x), T(y)) ≤ α[d(x, T(x)) + d(y, T(y))], for all x, y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This paper aims to introduce a new method to construct fractal functions using Kannan mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' First, we give the rigorous construction of fractal functions with the help of the Kannan iterated function system (IFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We also show the existence of a Borel probability measure supported on the attractor of the Kannan IFS satisfying the strong separation condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Moreover, we study the smoothness of the constructed fractal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We end the paper with some examples and graphical illustrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' INTRODUCTION The concept of fractal interpolation function (FIF) was introduced by Barnsley [2, 3] through iterated function system (IFS), and their construction is rooted in the theory of IFS [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The FIF is an interpolation function whose graph is an invariant set of an IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The pioneering research on fractal interpolation has gotten much at- tention in the literature, and it continues to flourish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The concept of FIF has been extended and generalized in several ways given in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wang and Yu [27] gave the construction of new class IFSs with variable parameters and generated as- sociated FIFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Also, they studied the smoothness and stability of FIFs under some conditions on data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The construction of nonlinear FIF using Matkowski and the Rakotch fixed point theorems is given in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In this order, Songli [21] gave the construction of nonlinear FIF on Sierpi´nski gasket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The reader may refer to books [3, 16] for the details on fractal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The fractal dimension is one of the major themes in fractal geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Many works on the fractal dimensions of fractals functions are in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' There are various approaches, such as the mass-distribution principle, potential theory, Fourier transform, positive operators, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', to compute or estimate the Hausdorff dimension of a set [11, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Using the potential theoretic approach, Barnsley gave results on the Hausdorff dimension of an affine FIF in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Falconer [11] also gave the estimate of the Hausdorff dimension of an affine FIF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The results on the Hausdorff dimension using the positive oper- ators approach are given in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Priyadarshi [19] gave an algorithm to determine lower bounds for the Hausdorff dimension of a set of complex continued fractions and estimated the best lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Jha and Verma [12] established very inter- esting results for fractal dimensions of fractal functions and some invariant sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 28A80, 47H10, 28A33, 28A78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Kannan IFS, Fractal Functions, Borel Probability Measure, Fractal Dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 1 2 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS They estimated fractal dimensions for a class of FIFs, widely known as α-fractal functions, by using function spaces such as H¨older space, oscillation space, and space of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Ruan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [22] estimated the box dimension of the new class of linear FIFs by using the δ-covering method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Additionally, they have established a relationship between the order of fractional integral and box dimen- sions of two linear FIFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' As we know, recurrent FIF is the generalization of linear FIF, and the graph of recurrent FIF is the invariant set of recurrent IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Barnsley and Massopust [4] gave results on the bilinear FIFs and their box dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Few recent developments on fractal dimensions can be seen in [6, 24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [6] introduced the notion of upper metric mean dimension with potential on any subset via Carath´eodory-Pesin structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Selmi [24] studied the multifractal Hausdorff and packing dimensions of Borel probability measures and studied their behaviors under orthogonal projections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In this order, Selimi estimated the multi- fractal Hausdorff and the packing dimensions of product measures in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Barnsley [2, 3] considered the collection of self-contraction mappings and used the Hutchinson operator and Banach fixed point principle to construct fractal functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Kannan [13, 14] introduced a new fixed point theorem widely known as Kannan fixed point theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Other related results on Kannan mapping can be seen in [8, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By using the concept of Kannan mapping, Sahu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [23] introduced the notion of the Kannan iterated function system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [14] Let T is a map of the complete metric space X into itself and if d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))], ∀ x, y ∈ X, 0 < α < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T has the unique fixed point in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' A natural question arises can we construct fractal functions using the concept of Kannan fixed point theory?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This question motivates us to conduct the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In this study, we use the concept of Kannan IFS and Kannan fixed point theorem and derive very interesting results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This paper is organized as follows: Section 2 is devoted to preliminaries and required terminologies related to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Section 3 presents the construction of fractal functions and the existence of self-similar measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In Section 4, the smoothness result of the Kannan fractal function is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The graphical illustration of the Kannan fractal functions is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Background and preliminaries This section aims to provide some basic definitions and results that act as prelude to this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let F ̸= ∅ be a subset of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The diameter of F is given by diamd(F) = sup {d(x, y) : x, y ∈ F} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' If {Fi} is a countable (or finite) collection of sets having a diameter at most δ which cover set E ⊆ Rn, then we say that {Fi} is a δ-cover of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For δ > 0 and a non-negative real number s, we define Hs δ,d(E) = inf � ∞ � i=1 diamd(Fi)s : {Fi} is a δ − cover of E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The s-dimensional Hausdorff measure of set E is given by Hs(E) = limδ→0 Hs δ (E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 3 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ( Hausdorff dimension) Let s ≥ 0 and E ⊆ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The Hausdorff dimension of E is defined as dimH(E) = inf{s : Hs(E) = 0} = sup{s : Hs(E) = ∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (Box Dimension) Let E ⊆ Rn be bounded and non-empty and let Nδ(E) be the smallest number of sets of diameter at most δ which cover E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The lower box dimension of E is dimB(E) = lim δ→0 log Nδ(E) − log δ , and the upper box dimension of E is dimB(E) = lim δ→0 log Nδ(E) − log δ , If both lower and upper box dimensions are the same, then that quantity is called the box dimension of E and it is given by dimB(E) = lim δ→0 log Nδ(E) − log δ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For the details on the Hausdorff and box dimensions, the reader may be referred to [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let d1 and d2 are two matrices on X, then d1 and d2 are topo- logically equivalent if and only if d1(xn, x) → 0 ⇐⇒ d2(xn, x) → 0, for {xn} ⊂ X and x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let d1 and d2 are two matrices on X, then d1 and d2 are metrically equivalent if and only if there exists c1, c2 > 0 and x, y ∈ X such that c1d1(x, y) ≤ d2(x, y) ≤ c2d1(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Fractal Interpolation Function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we introduce FIF in brief.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, we consider a set for interpolation as {(xn, yn) : n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We set J = {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N − 1}, I = [x1, xN] and for j ∈ J, let Ij = [xj, xj+1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For j ∈ J, let Lj : I → Ij be a contractive homomorphism such that Lj(x1) = xj, Lj(xN) = xj+1, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, define Fj : K = I × R → R, j ∈ J, which is a contraction in the second variable, that is, |Fj(x, y) − Fj(x, y′)|≤ rj|y − y′|, for all x ∈ I, rj ∈ [0, 1) and y, y′ ∈ R and satisfying Fj(x1, y1) = yj, Fj(xN, yN) = yj+1, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We shall take (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1) Lj(x) = ajx + bj Fj(x, y) = αjy + qj(x), In the above expression aj and bj are determined by using conditions Lj(x1) = xj, Lj(xN) = xj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, αj is the scaling factor with |αj|< 1 and continuous functions qj : I → R, j ∈ J satisfy “join-up conditions” imposed for the bivariate maps Fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, qj(x1) = yj − αjy1 and qj(xN) = yj+1 − αjyN for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now define functions Wj : I × R → I × R for j ∈ J by Wj(x, y) = (Lj(x), Fj(x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 4 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS Theorem 1 in [3] says that the IFS I := {I × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' W1, W2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , WN−1} defined above has a unique attractor which is the graph of a function f which satisfies the following functional equation reflects self-referentiality: f(x) = αjf(L−1 j (x)) + qj(L−1 j (x)), x ∈ Ij, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The above function f is known as the fractal interpolation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Kannan mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In 1969, Kannan [13] introduced a mapping, which was an improvement over the contraction mapping, known as Kannan mapping, defined as follows: If there exists a number α, 0 < α < 1 2, such that, for all x, y ∈ X, d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T is called a Kannan mapping and α is called Kannan-contractivity factor of T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let Tn : X → X are Kannan mappings having contractivity factor αn, for n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N and (X, d) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then, the set {X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Tn, n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} is said to be Kannan IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let f : [0, 1] → [0, 1] be defined by f(x) = x 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then this function f is a contraction mapping with contraction factor 1 3, but it is not a Kannan mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' On the other hand, the function g : [0, 1] → [0, 1] defined by g(x) = � x 4, if 0 ≤ x < 1 2 x 5, if 1 2 ≤ x ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' is a Kannan mapping with β = 4 9 but it is not a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The concepts of the Kannan operator and contraction are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The self-map T given in the previous example is Kannan, but it is not a contraction due to its discontinuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The following simple note can be seen in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' However, we include its details for the reader’s convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let (X, d) be a metric space and T : X → X is contraction with constant c < 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T is Kannan contractive with respect to metric d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Because of the contractivity of T , we have d(T (x1), T (x2)) ≤ cd(x1, x2) ≤ cd(x1, T x1)+cd(T x1, T x2)+c(T x2, x2), ∀ x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This turns d(T (x1), T (x2)) ≤ α[d(x1, T (x1)) + d(x2, T (x2))], ∀ x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since 0 < α := c 1−c < 1 2, T is a Kannan mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let X be a complete metric space and d1 and d2 are equivalent metrics on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', there exist positive constants c1, c2 such that c1d1(x1, x2) ≤ d2(x1, x2) ≤ c2d1(x1, x2), x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' If T is a contraction on X with respect to the metric d1 then there exists an m ∈ N such that T m is a Kannan contraction with respect to the metric d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since T is contraction on (X, d1), there exits 0 ≤ k < 1 such that d1(T x1, T x2) ≤ kd1(x1, x2), x1, x2 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 5 Whence d2(T x1, T x2) ≤ c2d1(T x1, T x2) ≤ c2kd1(x1, x2) ≤ � c2 c1 k � d2(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Take m ∈ N such that c2 c1 km < 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then d2(T mx1, T mx2) ≤ c2d1(T mx1, T mx2) ≤ c2kmd1(x1, x2) ≤ � c2 c1 km � d2(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence, T m is a contraction with respect to the metric d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' From Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6, we conclude that T m is a Kannan contraction with respect to the metric d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Thus, the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [14] Let T is a map of the complete metric space X into itself and if d(T (x), T (y)) ≤ α[d(x, T (x)) + d(y, T (y))], ∀ x, y ∈ X, 0 < α < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T has the unique fixed point in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The Hausdorff distance from the set A to the set B is defined as h(A, B) = max{sup a∈A inf b∈B d(a, b), sup b∈B inf a∈A d(a, b)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Note 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In [23, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5] the authors claimed that for all B, C ∈ H(X), h(T (B), T (C)) ≤ β[h(B, T (B)) + h(C, T (C))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The above claim is not true, for instance, see the following example, which is borrowed from [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let X = {0, 1, 2}, and the function d : X × X → R and the map f : X → X be given by d(0, 0) = d(1, 1) = d(2, 2) = 0 d(0, 1) = d(1, 0) = 5, d(1, 2) = d(2, 1) = 2, d(0, 2) = d(2, 0) = 3 f(1) = f(2) = 2, f(0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then the map f : X → X is Kannan on (X, d) with contractivity factor α ∈ [ 2 5, 1 2) but the map T : H(X) → H(X) given by T (B) = ∪x∈Bf(x) for all B ∈ H(X) is not a Kannan map on (H(X), h(d)) for any contractivity factor α ∈ [0, 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The above example does not work for the following lemma due to different contractive factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, we correct Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 of [23], and we show that it holds under certain additional conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Suppose T : X → X be a continuous Kannan mapping on the metric space (X, d) with contractivity factor 0 < β < 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T : H(X) → H(X) given by T (B) = {T (x) : x ∈ B} for every B ∈ H(X) is Kannan mapping on (H(X), h(d)) with contractivity factor 0 < γ = β (1−4β) < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let us first recall a basic real-analysis result that the image of a compact set under a continuous map is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since T is continuous, it maps H(X) into itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, since T is Kannan mapping on (X, d), for x, y ∈ X, we have d(T (x), T (y)) ≤ β[d(x, T (x)) + d(y, T (y))] ≤ β[d(x, T (y)) + d(T (y), T (x)) + d(y, T (x)) + d(T (x), T (y))] = β[d(x, T (y)) + d(y, T (x))] + 2βd(T (x), T (y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 6 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS So, we obtain d(T (x), T (y)) ≤ β (1 − 2β)[d(x, T (y)) + d(y, T (x))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, for B, C ∈ H(X) sup x∈B inf y∈C d(T (x), T (y)) ≤ β (1 − 2β)[sup x∈B inf y∈C d(x, T (y)) + sup x∈B inf y∈C d(y, T (x))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, sup x∈B inf y∈C d(T (x), T (y)) ≤ β (1 − 2β)[h(B, T (C)) + h(C, T (B))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Thanks to the triangle inequality, h(T (B), T (C)) ≤ β (1 − 2β)[h(B, T (B)) + h(T (B), T (C)) +h(C, T (C)) + h(T (C), T (B))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Consequently, � 1 − 2β (1 − 2β) � h(T (B), T (C)) ≤ β (1 − 2β)[h(B, T (B)) + h(C, T (C))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Therefore, h(T (B), T (C)) ≤ γ[h(B, T (B)) + h(C, T (C))], where γ = β (1−4β) < 1 2 for 0 < β < 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For a complete metric space (X, d), let Tn : n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N are continuous Kannan mappings on (H(X), h) with contractivity factor 0 < βn < 1 6, for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Define T : H(X) → H(X) by T (B) = ∪N n=1Tn(B) for each B ∈ H(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T is a Kannan mapping with contractivity factor γ = max{γn : n = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For B, C ∈ H(X), we have h(T (B), T (C)) = h(T1(B) ∪ T2(B) ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ∪ Tn(B), T1(C) ∪ T2(C) ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ∪ Tn(C)) ≤ max{h(T1(B), T1(C)), h(T2(B), T2(C)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , h(Tn(B), Tn(C))}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By using the above lemma, we obtain h(T (B), T (C)) = max � β1 1 − 4β1 [h(B, T1(B)) + h(C, T1(C))], β2 1 − 4β2 [h(B, T2(B)) + h(C, T2(C))], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , βn 1 − 4βn [h(B, Tn(B)) + h(C, Tn(C))] � ≤ max 1≤i≤n � βi 1 − 4βi � [max{h(B, T1(B)), h(B, T2(B)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', h(B, Tn(B))} + max{h(C, T1(C)), h(C, T2(C)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', h(C, Tn(C))}] ≤ max 1≤i≤n{γi}[h(B, T1(B) ∪ T2(B)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ∪ Tn(B) + h(C, T1(C) ∪ T2(C)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ∪ Tn(C)] ≤ γ[h(B, T (B)) + h(C, T (C))], CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 7 where γ = max1≤i≤n{γi} = max1≤i≤n � βi 1−4βi � < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence, T is Kannan with contractivity factor γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In [8] Dung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' proposed a question, whether their results are true or not for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In this order, in the above theorem, we show that T is Kannan for all n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Moreover, from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8, T has a unique fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We use the following notations throughout the article: C(I) denotes the set of continuous functions f : I = [x0, xN] → [a, b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let C∗(I) ⊂ C(I) and given by C∗ = {f ∈ C(I) : f(x0) = y0, f(xN) = yn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' K = I × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let us define a metric dθ on K as follows dθ((x, y), (z, w)) = |x − z|+θ|y − w|, θ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Note that (C∗(I), Hθ) is a complete metric space with respect to Hausdorff metric Hθ, where Hθ(f, g) = Hθ(Gf, Gg) = max{ sup x∈Gf inf y∈Gg dθ(x, y), sup y∈Gg inf x∈Gf dθ(x, y)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In the following section, we give the construction of fractal functions using Kannan IFS and the existence of self-similar measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Construction of fractal functions via Kannan Iterated function systems Let Fi : K → [a, b] be continuous mappings and satisfying for some k ≥ 0, and 0 ≤ βi < 1 2 |Fi(x, y)−Fi(w, y)|≤ k|x−w|, |Fi(x, y)−Fi(x, z)|≤ βi � |y −Fi(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', y)|+|z −Fi(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', z)| � for all x, w ∈ I, y, z ∈ [a, b], and i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, let {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} be an IFS with Wi(x, y) = (Li(x), Fi(x, y)) = (aix + bi, Fi(x, y)), where transformations are constrained by the data according to Wi(x0, y0) = (xi−1, yi−1), Wi(xN, yN) = (xi, yi) for i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N, Wi : K → K are Kannan mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} is the Kannan IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let N > 1, and {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' , N} denote the IFS defined as above, associated with the set of data {(xi, yi) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} such that amax = max i (xi+1 − xi) < 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then, there is a metric dθ on K = I × R, equivalent to the Euclidean metric such that for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N, Wi are Kannan maps with respect to dθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 8 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For all (x, y), (w, z) ∈ K, we have dθ(Wi(x, y), Wi(w, z)) = dθ � (Li(x), Fi(x, y)), (Li(w), Fi(w, z)) � = |Li(x) − Li(w)|+θ|Fi(x, y) − Fi(w, z)| ≤ |ai||x − w|+θ|Fi(x, y) − Fi(w, z)| ≤ amax|x − w|+θ|Fi(x, y) − Fi(w, z)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, thanks to the triangle inequality, we have |x − w|≤ |x − Li(x)|+|Li(x) − Li(w)|+|Li(w) − w|, this further yields |x − w|≤ 1 1 − amax (|x − Li(x)|+|Li(w) − w|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We now estimate |Fi(x, y) − Fi(w, z)|≤ |Fi(x, y) − Fi(w, y)|+|Fi(w, y) − Fi(w, z)| ≤ k|x − w|+βi � |y − Fi(w, y)|+|z − Fi(w, z)| � ≤ k|x − w|+βi � |y − Fi(x, y)|+|Fi(x, y) − Fi(w, y)|+|z − Fi(w, z)| � ≤ k|x − w|+βi � |y − Fi(x, y)|+k|x − w|+|z − Fi(w, z)| � ≤ (k + kβmax)|x − w|+βmax � |y − Fi(x, y)|+|z − Fi(w, z)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' With the help of the above estimates, we obtain dθ(Wi(x, y), Wi(w, z)) = amax + (k + kβmax)θ 1 − amax (|x − Li(x)|+|Li(w) − w|) + βmaxθ � |y − Fi(x, y)|+|z − Fi(w, z)| � ≤ γ � dθ((x, y), Wi(x, y)) + dθ((w, z), Wi(w, z)) � , where γ = max � amax+(k+kβmax)θ 1−amax , βmaxθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Using the condition amax < 1 3, we may choose a suitable (sufficiently small) θ > 0 such that γ < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For this value of θ, the mapping Wi is a Kannan mapping, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' From the above proof, if amax < 1 5 then we may choose a suitable (sufficiently small) θ > 0 such that the Kannan contractivity factor γ < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let N > 1 and {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} denote the IFS defined as above, associated with the set of data {(xi, yi) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} such that amax = max i (xi+1 − xi) < 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then, there exists a unique non empty compact set G ⊂ K = I × [a, b] such that G = ∪N i=1Wi(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 9 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' On similar lines of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1, we have dθ(Wi(x, y), Wi(w, z)) ≤ γ � dθ((x, y), Wi(x, y)) + dθ((w, z), Wi(w, z)) � , where γ = max � amax+(k+kβmax)θ 1−amax , βmaxθ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Using the condition amax < 1 7, we may choose a suitable (sufficiently small) θ > 0 such that γ < 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For this value of θ, the mapping Wi is a Kannan mapping with contractivity factor γ < 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, using Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='13 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='14, we obtain a unique compact set G satisfying G = ∪N i=1Wi(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let the IFS {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} defined as above associated with the set of data {(xi, yi) : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} such that amax = max i (xi+1 − xi) < 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let G denote the attractor of the IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then, G is the graph Gf of continuous function f : [x0, xN] → [a, b] satisfying f(xi) = yi for all i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, Gf = {(x, f(x)) : x ∈ [x0, xN]}, where f(xi) = yi for all i = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let C∗(I) = {f ∈ C(I) : f(x1) = y1, f(xN) = yN}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, C∗(I) is a closed subset of C(I) and (C∗(I), Hθ) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Define Read- Bajraktarevi´c (RB) operator T : C∗(I) → C∗(I) by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1) (T g)(x) = Fi(L−1 i (x), g(L−1 i (x))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we show that T is a Kannan mapping w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We will proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, graphs of T g and T h are given by GT g = {(x, T g(x)) : x ∈ I} and GT h = {(y, T h(y)) : y ∈ I}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let (x, T g(x)) ∈ GT g and (y, T h(y)) ∈ GT h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since Wi is Kannan with contractivity factor γ, from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' we get dθ � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' T g(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' T h(y)) � = dθ � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Fi(L−1 i (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(L−1 i (x))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Fi(L−1 i (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(L−1 i (y)))) � = dθ � (Li(w),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Fi(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(w)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (Li(z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Fi(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(z)) � = dθ � Wi(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(w)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(z)) � ≤ γ � dθ � (w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(w)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi(w,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(w)) � + dθ � (z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(z)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(z)) �� = γ � dθ � L−1 i (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(L−1 i (x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' T g(x)) � + dθ � L−1 i (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(L−1 i (y)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' T h(y)) �� ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 10 SUBHASH CHANDRA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' SAURABH VERMA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' AND SYED ABBAS where w = L−1 i (x) and z = L−1 i (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Thanks to the triangle inequality, dθ � (x, T g(x)), (y, T h(y)) � ≤ γ � dθ � (L−1 i (x), g(L−1 i (x))), (y, T g(y)) � + dθ � (x, T g(x)), (y, T h(y)) � + dθ � (L−1 i (y), h(L−1 i (y))), (x, T h(x)) � + dθ � (x, T g(x)), (y, T h(y)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' On taking infimum both sides, we have inf y ∈I dθ � (x, T g(x)), (y, T h(y)) � ≤ γ � inf y∈I dθ � (L−1 i (x), g(L−1 i (x))), (y, T g(y)) � + inf y∈I dθ � (L−1 i (y), h(L−1 i (y))), (x, T h(x)) � + 2 inf y∈I dθ � (x, T g(x)), (y, T h(y)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, (1 − 2γ) inf y ∈I dθ � (x, T g(x)), (y, T h(y)) � ≤ γ � inf y∈I dθ � (L−1 i (x), g(L−1 i (x))), (y, T g(y)) � + inf y∈I dθ � (L−1 i (y), h(L−1 i (y))), (x, T h(x)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By taking supremum over x ∈ I, we have Hθ(GT g, GT h) ≤ γ 1 − 2γ [Hθ(Gg, GT g) + Hθ(Gh, GT h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, Hθ(T g, T h) ≤ γ 1 − 2γ � Hθ(g, T g) + Hθ(h, T h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since amax < 1 5, Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 yields that β := γ 1−2γ < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Therefore, T is Kannan w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 , T has a unique fixed point f ∈ C∗(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Further, it is easy to check that f interpolates the data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we show that the graph Gf of f is an attractor of the IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since Wi(x, y) = (Li(x), Fi(x, y)) for all i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N, I = ∪j∈JLj(I), and from the functional Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1, we get that Wi(Gf) = Wi({(x, f(x)) : x ∈ [x0, xN]}) = {(Li(x), Fi(x, f(x))) : x ∈ [x0, xN]} = {(Li(x), f(Li(x))) : x ∈ [x0, xN]} = {(x, f(x)) : x ∈ [xi−1, xi]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence Gf = {(x, f(x)) : x ∈ [x0, xN]} = ∪N i=1{(x, f(x)) : x ∈ [xi−1, xi]} = ∪N i=1Wi(Gf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3, G is the unique attractor of the IFS {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Thus, G = Gf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 11 Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The continuous function h : [−1, 21 10] → [−1, 21 10] defined by h(x) = � x2 4 − x 8 , if − 1 ≤ x < 1 2 x2 5 − x 10, if 1 2 ≤ x ≤ 21 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' is Kannan mapping with β = 10 21 but it is not a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' First, we show that T is Kannan contraction, and we choose β = 10 21 < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (i) For the range −1 ≤ x, y < 1 2, we have |T (x) − T (y)|= 1 8|x(2x − 1) − y(2y − 1)| and |x − T (x)|+|y − T (y)|= 1 8 � |x|9 − 2x|+|y||9 − 2y| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For β = 10 21, we can see that |T (x) − T (y)|≤ β � |x − T (x)|+|y − T (y)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (ii) For 1 2 ≤ x, y < 21 10, we have |T (x) − T (y)|= 1 10|x(2x − 1) − y(2y − 1)| and |x − T (x)|+|y − T (y)|= 1 10 � |x|11 − 2x|+|y||11 − 2y| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For β = 10 21, we can see that |T (x) − T (y)|≤ β � |x − T (x)|+|y − T (y)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (iii) For −1 ≤ x < 1 2 and 1 2 ≤ y < 21 10, we have |T (x) − T (y)|= |x(2x − 1) 8 − y(2y − 1) 10 | and |x − T (x)|+|y − T (y)|= �1 8|x|9 − 2x|+ 1 10|y||11 − 2y| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For β = 10 21, we can see that |T (x) − T (y)|≤ β � |x − T (x)|+|y − T (y)| � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, one can see that T is not a contraction because for x = −1, y = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='99, we have |T (x) − T (y)|= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2537 > |x − y|= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' In this order, we can construct many different Kannan mappings with the help of the functional Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1 Fj(x, y) = αjy + qj(x), j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For, instance Fj(x, y) = T (y) + qj(x), 12 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS where T (y) = � y2 4 − y 8, if − 1 ≤ y < 1 2 y2 5 − y 10, if 1 2 ≤ y ≤ 21 10, and qj : I → R, j ∈ J are suitable continuous functions satisfying qj(x1) = yj −αjy1 and qj(xN) = yj+1 − αjyN for all j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Existence of Self-Similar Measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let I = {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wi : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} be an IFS and A be the attarctor of the IFS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then, we say that I satisfies strong separation condition (SSC) if Wi(A) ∩ Wj(A) = ∅ whenever i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Note that there are several separation conditions are available for any IFS, for instance, [9, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hutchinson [9] computed the Hausdorff dimension of self-similar sets under the open set condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Assuming the SSC, Priyadarshi and his collaborators [26] gave a formula for the Hausdorff dimension of the invariant set of generalized graph- directed systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let I = {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Ti : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} be an IFS consisting of Kannan mappings satisfies the SSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let (p1, p2, · · · , pN) be a probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then, there exists a Borel probability measure µ∗ supported on the attractor A of the IFS such that µ∗ = N � i=1 piµ∗ ◦ T −1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since the Kannan IFS {K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Ti : i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N} satisfies the SSC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, Ti(A) ∩ Tj(A) = φ ∀ i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let E0 = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We have A = N � i=1 Ti(A) = N � i,j=1 Tij(A) = N � i1,i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=',in Ti1,i2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=',in(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', we define Ek as follows: Ek = � Ti1i2···ik(A) : ij ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', N}, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', k � , where Ek denotes the collection of disjoint Borel subsets of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let B ∈ Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Note that each B is contained in one of the sets in Ek−1 and contains a finite number of the sets in Ek+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We can see that |Ti1i2···ik(A)|→ 0 as k → ∞ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let Ti : A → A and |A|= supx,y∈A d(x, y) = d(x0, y0), x0, y0 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since Ti is Kannan contraction, we have d(Ti(x), Ti(y)) ≤ βi[d(x, Ti(x)) + d(y, Ti(y))] ≤ βi[d(x0, y0) + d(x0, y0)] ≤ 2βid(x0, y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By taking supremum of both side, we have sup x,y∈A d(Ti(x), Ti(y)) ≤ 2βid(x0, y0), and 2βi < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, |Ti(A)|≤ 2βid(x0, y0) = ci|A|, ci = 2βi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 13 In a similar way, we get |Ti1i2···ik(A)|≤ ck max|A|→ 0 when k → ∞, where cmax = 2βmax = 2 max{β1, β2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', βk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let a probability vector p = (p1, p2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', pN) satisfying pi > 0 for all i and �N i=1 pi = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We assign µ(A) with µ(A) = 1 = �N i=1 pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let B ∈ Ek such that B = Ti1i2···ik(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let Ek = � B∈Ek B = � ij,j=1,2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=',k Ti1i2···ik(A) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence, we have µ(C) = 0 ∀ C with C ∩ A = ∅ and E = � Ek � Rn \\Ek and µ(Rn) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' It follows that (Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [11, Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7]) the definition of µ may be extended to all subsets of Rn so that µ becomes a measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we show that µ = �N i=1 piµ ◦ T −1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let Tj(A) be an arbitrary cylinder in the first stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then N � i=1 piµ ◦ T −1 i (Tj(A)) = pjµ(A) = pj, and from the construction of measure µ, we have µ(Tj(A)) = pj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Therefore, µ(Tj(A)) = �N i=1 piµ ◦ T −1 i (Tj(A)) = pj for all j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Similarly, we have µ(B) = �N i=1 piµ ◦ T −1 i (B) for all cylinders B ∈ Ek at any stage k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Thus, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Recall that the collection of all Borel probability measures on Rn, de- noted by P(Rn), is a complete metric space with respect to the Monge-Kantorovich metric dH defined as dH(µ, ν) = sup ����� � fdµ(x) − � fdν(x) ���� : where f : Rn → R, Lip(f) ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Define a mapping M : P(Rn) → P(Rn) by M(µ) = �N i=1 piµ ◦ f −1 i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we have dH(M(µ), M(ν)) = sup ���� � fdM(µ)(x) − � fdM(ν)(x) ��� : Lip(f) ≤ 1 � = sup ���� N � i=1 pi � fdµ ◦ f −1 i (x) − N � i=1 pi � fdν ◦ f −1 i (x) ��� : Lip(f) ≤ 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hutchinson [9] showed that if all fi are contractions, then M is the contraction with respect to the Monge-Kantorovich metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, a natural question arises whether M is Kannan with respect to the Monge-Kantorovich metric when all fi are Kannan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' It is open for further investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Smooth fractal functions Let us denote the space of m-times continuously differentiable functions by Cm(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we define a new metric with the help of Hausdorff distance such as D(g, h) := max 0≤k≤m Hθ(Gg(k), Gh(k)), where Hθ(Gg, Gh) denotes the Hausdorff distance induced from the metric dθ be- tween the graphs of f and g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Since Hθ(Gg, Gh) and ∥g − h∥∞ are equivalent, (Cm(I), D) will be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 14 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let g ∈ Cm(I), where m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Suppose that Lj : I → Ij is affine map defined by Lj(x) = ajx + bj satisfying Lj(x1) = xj, Lj(xN) = xj+1, j ∈ J and Fj(x, y) = αjy + qj(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let q(k) j (x1) = g(k)(x1), q(k) j (xN) = g(k)(xN), j ∈ J, 0 ≤ k ≤ m, and scaling factor αj satisfying αj < ak 5 , where ak = min{ak j : j ∈ J}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then T has a unique fractal function f ∗ ∆ ∈ Cm ∗ (I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Moreover, dimH(Gr(f ∗ ∆)) = dimB(Gr(f ∗ ∆)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let Cm ∗ (I) = {g ∈ Cm(I) : h(k)(x1) = g(k)(x1), h(k)(xN) = g(k)(xN), 0 ≤ k ≤ m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Here, Cm ∗ (I) is a closed subset of Cm(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' It can be seen that (Cm ∗ (I), D) is a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Define the RB operator T : Cm ∗ (I) → Cm ∗ (I) by (T g)(x) = αjf(L−1 j (x)) + qj(L−1 j (x)), x ∈ Ij, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' It can be observed that T is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let g, h ∈ Cm ∗ (I), D(g, h) := max 0≤k≤m Hθ(Gg(k), Gh(k)) for each 0 ≤ k ≤ m, we have (T g)(k)(x) = a−k j [αjg(k)(L−1 j (x)) + q(k) j (L−1 j (x))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ˆF k j (x, y) = a−k j αjy + a−k j q(k) j (x) and ˆW k j (x, y) = (Lj(x), ˆF k j (x, y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' It can be seen that ˆ W k j are Kannan contractions as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' D( ˆW k j (x1, y1), ˆW k j (x2, y2)) ≤ αj ak D((x1, y1), (x2, y2)) ≤ αj ak D((x1, y1), ˆ W k j (x1, y1)) + αj ak D( ˆ W k j (x1, y1), ˆ W k j (x2, y2)) + αj ak D( ˆ W k j (x2, y2), (x2, y2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence, we obtain D( ˆW k j (x1, y1), ˆW k j (x2, y2)) ≤ αj ak − αj [D((x1, y1), ˆW k j (x1, y1))+D((x2, y2), ˆ W k j (x2, y2))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence, ˆ W k j are Kannan contractions with contractivity factor αj ak−αj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, we show that T is Kannan w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' metric D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is D(T g, T h) ≤ γ′[D(g, T g) + D(h, T h)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Let (x, (T g)(k)(x)) ∈ G(T g)(k) and (y, (T h)(k)(y)) ∈ G(T h)(k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We have dθ � (x, (T g)(k)(x)), (y, (T h)(k)(y)) � = dθ �� x, ˆF k j (L−1 j (x), g(k)(L−1 j (x))) � , � y, ˆF k j (L−1 j (y), h(k)(L−1 j (y))) �� = dθ �� Lj(w), ˆF k j (w, g(k)(w)) � , � Lj(z), ˆF k j (z, h(k)(z)) �� = dθ � ˆW k j (w, g(k)(w)), ˆ W k j (z, h(k)(z)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 15 Since ˆ W k j are Kannan contraction and by substituting again w = L−1 j (x) and z = L−1 j (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' we have dθ � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T g)(k)(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T h)(k)(y)) � ≤ αj ak − αj � dθ � (L−1 j (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(k)(L−1 j (x))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ˆ W k j (L−1 j (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(k)(L−1 j (x))) � + dθ � (L−1 j (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(k)(L−1 j (y))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' ˆW k j (L−1 j (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(k)(L−1 j (y))) �� = αj ak − αj � dθ � L−1 j (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(k)(L−1 j (x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T g)(k)(x)) � + dθ � L−1 j (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(k)(L−1 j (y)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T h)(k)(y)) �� ≤ αj ak − αj � dθ � (L−1 j (x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' g(k)(L−1 j (x))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T g)(k)(y)) � + dθ � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T g)(k)(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T h)(k)(y)) � + dθ � (L−1 j (y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' h(k)(L−1 j (y))),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T h)(k)(x)) � + dθ � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T g)(k)(x)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (T h)(k)(y)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' On taking infimum both sides, we have inf y∈I dθ � (x, (T g)(k)(x)), (y, (T h)(k)(y)) � ≤ αj ak − αj � inf y∈I dθ � (L−1 j (x), g(k)(L−1 j (x))), (y, (T g)(k)(y)) � + inf y∈I dθ � (L−1 j (y), (h)(k)(L−1 j (y))), (x, (T h)(k)(x)) � + 2 inf y∈I dθ � (x, (T g)(k)(x)), (y, (T h)(k)(y)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' That is, (1 − 2αj ak − αj ) inf y∈I dθ � (x, (T g)(k)(x)), (y, (T h)(k)(y)) � ≤ αj ak − αj � inf y∈I dθ � (L−1 j (x), g(k)(L−1 j (x))), (y, (T g)(k)(y)) � + inf y∈I dθ � (L−1 j (y), h(k)(L−1 j (y))), (x, (T h)(k)(x)) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By taking supremum over x ∈ I, we have max 0≤k≤m Hθ(G(T g)(k), G(T h)(k)) ≤ αj ak − 3αj [ max 0≤k≤m Hθ(Gg(k), G(T g)(k)) + max 0≤k≤m Hθ(Gh(k), G(T h)(k))] That is D(T g, T h) ≤ γ′[D(g, T g) + D(h, T h)], where γ′ = αj ak−3αj < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hence, T is Kannan contrcation with contractivity factor γ′ = αj ak−3αj < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 , T has a unique fractal function f ∗ ∆ ∈ Cm ∗ (I) and obeys the equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We know that any continuous function with the bounded derivative is of bounded variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This result with [15, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3] yields dimH(Gr(f ∗ ∆)) = dimB(Gr(f ∗ ∆)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 16 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS Hence, the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' [5] The relationship between the Heisenberg and Euclidean geometry on H = R3 is rather intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The Heisenberg-Hausdorff dimension is always greater than or equal to its Euclidean counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The Hausdorff dimension of (R3, dH) is equal to 4 (in fact, balls in the metric dH have a measure proportional to the fourth power of their radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' This implies, for instance, that the Heisenberg metric dH cannot be locally bi-Lipschitz equivalent with any Riemannian metric, particularly with the Euclidean metric dE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' From the above remark, we can conclude that if two metrics are topologically equivalent, that does not imply that the dimension of the graph of any function can be equal, but if metrics are metrically equivalent, then the Hausdorff dimension will be equal, but the Hausdorff measure need not be equal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' see the following Let δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Then Hs δ,d2(E) = inf � ∞ � i=1 diamd2(Fi)s : {Fi} is a δ − cover of E � ≤ inf � ∞ � i=1 cs 2diamd1(Fi)s : {Fi} is a δ − cover of E � = cs 2Hs δ,d1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Similarly, cs 1Hs δ,d1(E) ≤ Hs δ,d2(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Therefore, cs 1Hs δ,d1(E) ≤ Hs δ,d2(E) ≤ cs 2Hs δ,d1(E) holds for all δ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' As δ → 0+, we have cs 1Hs d1(E) ≤ Hs d2(E) ≤ cs 2Hs d1(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Now, using the definition of the Hausdorff dimension and the above inequality, we get dimH,d1(E) = inf{s : Hs δ,d1(E) = 0} = inf{s : Hs δ,d2(E) = 0} = dimH,d2(E), completing the reamark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Graph of Kannan fractal functions Here, we have the functional equation Fj(x, y) = T (y) + qj(x), j ∈ J, and the self-referential equation is f(Lj(x)) = T (f(x)) + qj(x), x ∈ Ij, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' We choose qj(x) = cjx + dj satisfying the join-up conditions such as T (y1) + cjx1 + dj = yj and T (yN) + cjxN + dj = yj+1, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' So, we have cj = (yj − yj+1) − (T (y1) − T (yN)) (x1 − xN) CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 17 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9 1 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 18 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' CONSTRUCTION OF FRACTAL FUNCTIONS USING KANNAN MAPPINGS 19 and dj = yj − T (y1) − (yj − yj+1) − (T (y1) − T (yN)) (x1 − xN) x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The initial data is taken as follows for Figure 1 and Figure 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' {(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='95), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='25), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0)} and {(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='85), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='75), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' For Figure 3 and Figure 4, we consider qj(x) = x2 + cjx + dj satisfying the join-up conditions such as T (y1) + x2 1 + cjx1 + dj = yj and T (yN) + x2 n + cjxN + dj = yj+1, j ∈ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' So, we have cj = (yj − yj+1) − (T (y1) − T (yN)) − (x2 1 − x2 N) (x1 − xN) and dj = yj − T (y1) − (yj − yj+1) − (T (y1) − T (yN)) − (x2 1 − x2 N) (x1 − xN) x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The initial data is taken as follows for Figure 3 and Figure 4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' {(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='65), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='55), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='25), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='15), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='95)} and {(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='1, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='75), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='95), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='2), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='7, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='6), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='8, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4), (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='9, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='85), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='4)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' The first author’s work is financially supported by the CSIR, India, with grant number 09/1058(0012)/2018-EMR-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Beer, Metric spaces on which continuous functions are uniformly continuous and Hausdorff distance, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 95 (1985) 653-658.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Barnsley, Fractal functions and interpolation, Constr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 2 (1986) 303-329.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Barnsley, Fractal Everywhere, Academic Press, Orlando, Florida, 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 20(5) (1989) 1218–1242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Barnsley and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Massopust, Bilinear fractal interpolation and box dimension, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theory 192 (2015) 362–378.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Balogh and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Tyson, Hausdorff dimension of self-similar and self-affine fractals the Heisenberg group, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (3) 91 (2005) 153-183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Cheng, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Liand and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Selmi, Upper metric mean dimensions with potential on subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 34 (2021) 852–867.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Chandra and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Abbas, On fractal dimensions of fractal functions using functions spaces, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Aust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' (2022) 1-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Van Dung, Nguyen and Adrian Petru¸sel, On iterated function systems consisting of Kannan maps, Reich maps, Chatterjea type maps, and related results, Journal of Fixed Point Theory and Applications 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 4 (2017), 2271-2285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Hutchinson, Fractals and self-similarity, Indian Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 30 (1981) 713-747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Janos Ludvik, On Mappings Contractive in the Sense of Kannan, Proceedings of the American Mathematical Society 61, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 1 (1976): 171-75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley Sons Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', New York, 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Jha and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Verma, Dimensional analysis of α-fractal functions, Results in Mathematics 76 (4) (2021) 1-24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Kannan, Some results on fixed points, Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Calcutta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 60 (1968) 71-76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Kannan, Some results on fixed points II, Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Monthly 76 (1969) 405-408.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 20 SUBHASH CHANDRA, SAURABH VERMA, AND SYED ABBAS 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Liang, Box dimensions of Riemann-Liouville fractional integrals of continuous functions of bounded variation, Nonlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 72(11) (2010) 4304-4306.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Massopust, Fractal Functions, Fractal Surfaces, and Wavelets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=', Academic Press, San Diego, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Navascu´es, Fractal polynomial interpolation, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Anwend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 25(2) (2005) 401-418.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Navascu´es, Fractal approximation, Complex Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Oper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theory 4(4) (2010) 953-974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Priyadarshi, Lower bound on the Hausdorf dimension of a set of complex continued frac- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 449(1), (2017) 91–95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Ri, A new idea to construct the fractal interpolation function, Indag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 29(3) (2018) 962-971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Ri, Fractal functions on the Sierpinski gasket, Chaos, Solitons Fractals 138 (2020) 110142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Ruan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Sub and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Yao, Box dimension and fractional integral of linear fractal interpolation functions, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theory 161 (2009) 187-197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Sahu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Chakraborty and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Dubey, K-Iterated function system, Fractals 18 (2010) 139–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Selmi, Multifractal dimensions for projections of measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Bol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Paran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 40 (2022) 1-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Selmi, On the multifractal dimensions of product measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Nonlinear Studies, 29(1) (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Nussbaum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Priyadarshi and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Lunel, Positive operators and Hausdorff dimension of invariant sets, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 364(2) (2012) 1029-1066.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Wang and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Yu, Fractal interpolation functions with variable parameters and their analytical properties, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Approx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' Theory 175 (2013) 1-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Kamand (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=') - 175005, India Email address: sahusubhash77@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='com Department of Applied Sciences, IIIT Allahabad, Prayagraj-211015, India Email address: saurabh331146@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='com School of Mathematical and Statistical Sciences, Indian Institute of Technology Mandi, Kamand (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content=' )- 175005, India Email address: sabbas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='iitk@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} +page_content='com' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6NE1T4oBgHgl3EQfTQM9/content/2301.03075v1.pdf'} diff --git a/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf b/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c0e52a7bf7e1b376c4b3cbfd0034869104e4725a --- /dev/null +++ b/6tE2T4oBgHgl3EQf7Qi3/content/2301.04208v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:29dc35c6593eeb204573f5570cb3d0d2119ead335bcb37a09330a9ae07801c3b +size 1683591 diff --git a/6tE2T4oBgHgl3EQf7Qi3/vector_store/index.pkl b/6tE2T4oBgHgl3EQf7Qi3/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..00931108ff5c8ecff4a7c47224d032b5f3af94fe --- /dev/null +++ b/6tE2T4oBgHgl3EQf7Qi3/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d6ffbe234da9af4bb6de1cd45cebad106728b7d07baf2b186d7769acd94581ac +size 104372 diff --git a/89E1T4oBgHgl3EQfUAPq/vector_store/index.pkl b/89E1T4oBgHgl3EQfUAPq/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..a4151aed7c0c1f014fb10e63cc6eef863de60bac --- /dev/null +++ b/89E1T4oBgHgl3EQfUAPq/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:691d8c6e8e67a0b23d7648fedd28338293b6b4599cda38bd05d72dbdbaec2d9c +size 87198 diff --git a/99A0T4oBgHgl3EQfO__U/vector_store/index.pkl b/99A0T4oBgHgl3EQfO__U/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e03dd664fd3e8a43ad952a9725ae0a041721d38a --- /dev/null +++ b/99A0T4oBgHgl3EQfO__U/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7d402fd86b0d953efb2e18c0b560be9792cf2de9e3499c19f9bda80a7c96c095 +size 280526 diff --git a/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf b/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d61dc3c3ea8411967a68a383e48d6ba36e6ad7f8 --- /dev/null +++ b/99E0T4oBgHgl3EQfwwHR/content/2301.02638v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b661a59d5af76170bd86cf7cb1fa0d1d7ffe4e1746909cd6d07188d7f78e9c7c +size 1147221 diff --git a/99E0T4oBgHgl3EQfwwHR/vector_store/index.faiss b/99E0T4oBgHgl3EQfwwHR/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b611d7469d0f8d045c491c8a50bfe783245d2e51 --- /dev/null +++ b/99E0T4oBgHgl3EQfwwHR/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:12d777ac03966851b55ac66e2bd3b8681eba8c8b680340c533d73b67302e9721 +size 7733293 diff --git a/99E0T4oBgHgl3EQfwwHR/vector_store/index.pkl b/99E0T4oBgHgl3EQfwwHR/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b911b9ab6975cd85f645be466f53de883a2669d4 --- /dev/null +++ b/99E0T4oBgHgl3EQfwwHR/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0ba7d7b0ec18d35951cfb9f9e360bccb7ae2980cab27630adcf25f07846ac248 +size 312003 diff --git a/9NFLT4oBgHgl3EQfty_-/content/tmp_files/2301.12153v1.pdf.txt b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/2301.12153v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..623b22dbabaf2db1df09e89b2cb9cf1a589acba3 --- /dev/null +++ b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/2301.12153v1.pdf.txt @@ -0,0 +1,9826 @@ +arXiv:2301.12153v1 [math.AP] 28 Jan 2023 +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +EDUARDO GARC´IA-JU´AREZ˚, PO-CHUN KUO:,;, YOICHIRO MORI:,§, +AND ROBERT M. STRAIN:,¶ +Abstract. This paper introduces the 3D Peskin problem: a two-dimensional +elastic membrane immersed in a three-dimensional steady Stokes flow. +We +obtain the equations that model this free boundary problem and show that +they admit a boundary integral reduction, providing an evolution equation +for the elastic interface. We consider general nonlinear elastic laws, i.e., the +fully nonlinear Peskin problem, and prove that the problem is well-posed in +low-regularity H¨older spaces. Moreover, we prove that the elastic membrane +becomes smooth instantly in time. +Contents +1. +Introduction +2 +2. +Formulation and Boundary Integral Reduction +9 +3. +Preliminaries +12 +4. +Nonlinear decomposition +18 +5. +Calculus estimates +24 +6. +Frozen-coefficient Semigroup +39 +7. +Local well-posedness +56 +8. +Higher Regularity +66 +Appendix A. +Besov Spaces and Fourier Multiplier Theorems +75 +Appendix B. +Estimates for the semigroup e´tLApξq +77 +References +90 +˚Departamento de An´alisis Matem´atico, Universidad de Sevilla, C/Tarfia s/n, Cam- +pus Reina Mercedes, 41012, Sevilla, Spain. egarciajuarez@ub.edu +:Department +of +Mathematics, +University +of +Pennsylvania, +David +Rittenhouse +Lab., +209 +South +33rd +St., +Philadelphia, +PA +19104, +USA. +;kuopo@sas.upenn.edu +§y1mori@math.upenn.edu ¶strain@math.upenn.edu +Date: January 31, 2023. +2020 Mathematics Subject Classification. 35Q35, 35C15, 35R11, 35R35, 76D07. +Key words and phrases. Peskin problem, 3D, Fluid-Structure Interaction, immersed boundary +problem, Stokes flow. +˚supported by the European Union’s Horizon 2020 research and innovation programme under +the Marie Sk�lodowska-Curie grant agreement CAMINFLOW No 101031111, and the AEI project +PID2021-125021NAI00 (Spain). +;partially supported by NSF grant DMS-2042144 (USA) awarded to YM. +§partially supported by the NSF grant DMS-1907583, 2042144 (USA) and the Math+X award +from the Simons Foundation. +¶partially supported by the NSF grants DMS-1764177 and DMS-2055271 (USA). + +2 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +1. Introduction +The immersed boundary method, introduced by Peskin [40, 41] to study the +blood flow around heart valves, has been widely applied to numerically study fluid- +structure interaction (FSI) problems. These FSI problems, in which a fluid interacts +with elastic structures, appear naturally in many engineering and biophysics appli- +cations [44,46]. Despite their importance, both the computational methods and the +FSI problems themselves are poorly understood from an analytical standpoint. A +major impediment has been the lack of analytical understanding of the underlying +PDEs, which are typically nonlinear and nonlocal. Results are particularly scarce in +the more realistic three-dimensional settings, where the coupling of nonlocal effects +with non-trivial geometry substantially increases the complexity of the problem. +Since the recent breakthrough works [34] and [37], which provided the strong +solution theory for the problem of an immersed elastic string in a two-dimensional +fluid, the so-called 2D Peskin problem has attracted a lot of attention [8,9,23,25, +33,51,52]. In this paper, we initiate the study of its three-dimensional counterpart. +We introduce the formulation and develop the well-posedness theory for the three- +dimensional (fully nonlinear) Peskin problem of an elastic membrane immersed in +a fluid. +1.1. Description of the problem. We consider the following problem in which +a three-dimensional incompressible Stokes fluid interacts with an elastic membrane +in R3. A closed elastic interface Γ encloses a simply connected bounded domain +Ω Ă R3 filled with a Stokes fluid with viscosity µ. The outside region R3zΩ is filled +with a Stokes fluid of viscosity 1. The equations satisfied are: +µ∆u ´ ∇p “ 0 in Ω, +(1.1) +∆u ´ ∇p “ 0 in R3zΩ, +(1.2) +∇ ¨ u “ 0 in R3zΓ. +(1.3) +Here u is the velocity field and p is the pressure. We impose the following condition +in the far field: +(1.4) +u Ñ 0 as |x| Ñ 8. +We supplement the above with interface conditions on the time-evolving surface Γ. +For any quantity w defined on Ω and R3zΩ, we set: +�w� “ w|Γi ´ w|Γe +where w|Γi,e are the trace values of w at Γ evaluated from the Ω (interior) and +R3zΩ (exterior) sides of Γ. Let n be the outward pointing unit normal vector on +Γ. The interface conditions are: +�u� “ 0, +(1.5) +�Σn� “ F el, Σ “ +# +µ +` +∇u ` p∇uqT˘ +´ pI +in Ω +∇u ` p∇uqT ´ pI +in R3zΩ , +(1.6) +BX +Bt “ upX, tq, +(1.7) +where I is the 3 ˆ 3 identity matrix and X : S2 ÞÑ Γptq the map that describes the +evolving membrane. This map gives the deformation of the reference configuration +S2, the standard embedding of the sphere of radius 1 in R3. The first condition + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +3 +is the no-slip boundary condition and the second is the stress balance condition +where Σ is the fluid stress and F el is the elastic force exerted by the interface Γ. +The last condition states that the membrane evolves with the fluid flow. Note that +the elastic surface Γ “ Γptq and hence Ω “ Ωptq changes with time. Once given +the constitutive equation for the elastic force F el, equations (1.1)-(1.7) form the +so-called jump formulation of the 3D Peskin problem. Let pg and g denote the metric +tensors on S2 and Γ respectively. A natural choice for the elastic stretching force +is given by [19,28] +(1.8) +F el “ +a +detppg´1gq∇S2 ¨ T p∇S2Xq, +where +T p∇S2Xq :“ T p|∇S2X|q +|∇S2X| +∇S2X “: T p|∇S2X|q∇S2X, +∇S2 denotes the surface gradient on S2, |A| denotes the Frobenius norm of matrix +A, and T has to satisfy T ą 0, dT {dλ ě 0 (see Section 3.1 for further notation). +In Section 2 more details about the derivation of the elastic force are given. For a +Hookean material, T is linear and hence the elastic force is linear in X. We will +consider general T , i.e., the fully nonlinear Peskin problem. +Compared to fluid interface problems, such as a drop of liquid surrounded by +another fluid or vacuum [17,43,45,49,50], where only the shape of the interface mat- +ters, here it is not expected that Eulerian methods on their own should suffice. Due +to the elastic nature of the membrane, the stretching, given by the parametriza- +tion, has a strong influence on the evolution. Thus, one needs to keep track of +the membrane configuration. Lagrangian methods are needed, making it harder +to work in higher dimensions. In particular, one cannot freely reparametrize the +surface, an idea frequently used to obtain extra cancellations in the study of fluid +interfaces [13,22,30]. +An important feature of the Peskin problem is that it admits a Boundary Integral +formulation, whose derivation is given in Section 2. When µ “ 1, the problem (1.1)- +(1.8) is equivalent to the following evolution equation for X: +(1.9) +BX +Bt ppxq “ +ż +S2GpXppxq ´ Xppyqq∇S2 ¨ +´ +T p|∇S2Xppyq|q ∇S2Xppyq +|∇S2Xppyq| +¯ +dµS2ppyq, +Xppxq|t“0 “ X0ppxq, +where Gpxq is the Stokeslet tensor in R3: +(1.10) +Gpxq “ 1 +8π +´ 1 +|x|I3 ` x b x +|x|3 +¯ +. +We have suppressed the dependence of X on t to avoid cluttered notation. Hence- +forth, we will assume µ “ 1. +It will be sometimes convenient in the analysis +to work with coordinates. +Let θ “ pθ1, θ2q be a (local) coordinate system on +S2 and let px “ x +Xpθq P S2 Ă R3 be the point on S2 corresponding to θ. +Let +Xpθq “ Xpx +Xpθqq P Γ Ă R3 be the position on Γ corresponding to the coordinate +point θ (see Figure 1). If px “ x +Xpθq, we will write Xppxq and Xpθq in an abuse of +notation. Then, after integration by parts and choosing an isothermal coordinate +system, equation (1.9) becomes +BX +Bt pθq “ ´p.v. +ż +R2 +B +Bηi +GpXpθq´Xpηqq ˜F el,ipXqpηqdη1dη2, +(1.11) + +4 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where we denote +˜F el,ipXqpηq “ T pλpηqq +λpηq +B +Bηi +Xpηq, +λpηq “ +a +trppg´1pηqgpηqq. +Above we use the explicit definitions of pg and g given in (2.1). +Figure 1. Deformation map Xp¨, tq : S2 Ñ Γptq. +Some important properties of the solutions to the Peskin problem (1.9) are easier +to deduce from the jump formulation (1.1)-(1.7). The incompressibility condition +(1.3), together with (1.7), implies the conservation of the volume of the enclosed +region Ω: +d +dt|Ωptq| “ 0, +|Ωptq| “ 1 +3 +ż +S2 Xppxq ¨ nppxqdµΓptqppxq. +Moreover, the elastic energy defined as follows +EpXq “ +ż +S2 AEp|∇S2Xppxq|qdµS2ppxq, +A1 +Epλq “ T pλq, +satisfies the balance +d +dtEpXq “ ´ +ż +R3 |∇u|2dx, +which shows that the elastic energy is dissipated due to the viscosity of the fluid. +This relation follows from (1.7), integration by parts, and using conditions (1.6), +(1.3), (1.1)-(1.2), and (1.5), consecutively. For a linear elasticity law, the elastic +energy is the 9H1pS2q norm of the interface, +EpXq “ 1 +2 +ż +S2 |∇S2Xppxq|2dµS2ppxq. +A third important property of the Peskin problem is that it satisfies a scaling +invariance. We must first mention that the definition of solutions requires that the + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +5 +interface is non-degenerate and does not self-intersect. This is typically enforced +through the arc-chord condition: +|X|˚ :“ +inf +px‰py +px, pyPS2 +|X ppxq ´ X ppyq| +|px ´ py| +ą 0. +If Xppx, tq solves (1.9), then, for any λ ą 0, Xλppx, tq :“ λ´1Xpλpx, λtq also solves +the equation, and |Xλ|˚ “ |X|˚. Hence, 9C1pS2q and spaces with the same scaling, +such as 9H2pS2q, are critical spaces for 3D Peskin problem. Notice that the energy +balance above only gives control of the 9H1pS2q norm, hence the Peskin problem is +supercritical. +1.2. Main results. The formulation of the problem, both in jump and Boundary +Integral forms, is derived in Section 2. Once the formulation is provided, the main +objective of the paper is to show that the problem is well-posed. More specifically, +we will first show the existence and uniqueness of strong solutions with initial data +in little H¨older spaces, h1,γpS2q, γ P p0, 1q, defined as the completion of the set of +smooth functions in C1,γpS2q. +Definition 1.1 (Strong solution). Let X P Cpr0, T s; C1,γpS2qqXC1pr0, T s; CγpS2qq, +γ P p0, 1q, and |Xptq|˚ ą 0 for t P r0, T s. Then, X is a strong solution to the +3D Peskin problem with initial data Xp0q “ X0 if it satisfies equation (1.9) for +t P p0, T s and Xptq Ñ X0 in C1,γpS2q as t Ñ 0. +The choice of little H¨older spaces will be needed to obtain the convergence to +the initial data. In Section 7 we will prove the following: +Theorem 1.2. Consider the 3D Peskin problem (1.9) with initial data satisfying +X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0. Then, there +exists some time T ą 0 such that (1.9) has a unique strong solution X, +X P Cpr0, T s; h1,γpS2qq X C1pr0, T s; hγpS2qq. +It is instructive to briefly recall the idea of the proof for the 2D linear Peskin +problem [37]. For X a non-degenerate, closed simple plane curve, the boundary +integral formulation in 2D is given by +BtXpθ, tq “ ´ +ż +S1 BηGpXpθq ´ XpηqqBηXpηqdη, +where G is the Stokeslet in R2. It turns out that one can perform a small-scale +decomposition [30,37] to write it as follows +BtX “ 1 +4ΛX ` RpXq, +ΛX “ HBθX, +with RpXq a lower order operator compared to Λ. Then, it is natural to construct +the solution as a fixed point of the equation written in Duhamel form: +Xptq “ etΛX0 ` +ż t +0 +ept´τqΛRpXpτqqdτ +We notice two important facts: the semigroup is explicit, both in space and Fourier +variables, and the equation is semilinear. Even for nonlinear elastic law, the lead- +ing term has a kernel not depending on the curve itself, ´ 1 +4HpT p|∇S2X|q∇S2Xq, +making it possible to use the Λ-like structure via energy methods [8]. + +6 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Here, we first consider the strategy adapted to nonlinear equations in [35] (see +also [47] for 2D Peskin). Let us write equation (1.9) as follows +BX +Bt “ FpXq, +X|t“0 “ X0. +Then, at least formally, linearization around the initial data would give +(1.12) +Xptq “ eLpX0qtX0 ` +ż t +0 +eLpX0qpt´τqEpXpτqqdτ, +with LpX0q “ BXFpX0qX the Gateaux derivative of F at X0 and EpXq “ +FpXq ´ LpX0q. Hence, while EpXq is not expected to be smoother than LpX0q, +it should be small for short time. However, one first need to make sense of the +above expression (1.12) by showing that LpX0q generates an analytic semigroup, +which amounts to proving that the operator is sectorial in adequate spaces. This +is the core of the abstract Theorem 7.1, whose proof encompasses a fixed point +argument. The application of this theorem to our problem soon becomes highly +involved. This is done in Propositions 7.3-7.6. Since the equation is not semilinear, +the process will require to further decompose the operator LpX0q and then freeze +the coefficients at a given point. The decomposition must be done maintaining +a derivative structure for the kernel that allows extra cancellations, required to +control the singular integral operators that appear, and so that we can invert the +frozen-coefficient operator (the study of this part is done separately in Section 6). +Schematically, we decompose the kernel in (1.11) as follows +B +Bηi +` +GpXpθq´Xpηqq +˘ +« ´ B +Bxj +G p∇Xpηqpθ ´ ηqq BXj +Bηi +pηq ` RpXqpθq +« 1 +8π +BXpηq +Bηi +¨ p∇Xpηqpθ´ηqq +|∇Xpηqpθ´ηq|3 +` ¨ ¨ ¨ ` RpXqpθq, +(1.13) +where one expects RpXq to be lower order and the dots represent additional terms +of high order coming from the second term in Gppxq (1.10) (we note that in 2D +these additional high-order terms cancel each other). These leading kernels are +not of convolution type and cannot be written as a derivative. For this purpose, +one could be tempted to use ∇Xpθq in the approximation instead of ∇Xpηq. +However, higher derivatives of X would appear later in the proof and the argument +would not close. Thus, to take advantage of the derivative structure, we will be +forced to estimate together the leading and remainder terms. In a second step, +we approximate ∇Xpηq in the leading kernels above by its value at a given point +(see Lemma 5.10 for more details), which requires the introduction of a partition +of unity for the sphere. Due to the geometry of the problem, we need to work +with charts, and due to the nonlocal character of the equation a second localization +procedure will be needed. A fine implementation of these localization procedures +will be crucial to avoid transition maps that would otherwise overcomplicate the +proof. +For the fully nonlinear Peskin problem, we must linearize and freeze the coeffi- +cient of the elastic force as well. In Section 4.2, we show that the frozen-coefficient +linear operator in the general force case is given by +pLAY qkpθq “ ´ +ż +R2 +B +Bηi +pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +7 +where A is a constant matrix and TFpAq a tensor (4.31). Thus, in the general case, +the multiplier for the frozen-coefficient linear operator becomes: +LApξq “ I ` vpξq b vpξq +4detpBq |B´1ξ| +ˆT p∥A∥F q +∥A∥F +ˆ +|ξ|2 I´ Aξ b Aξ +∥A∥F +˙ +` dT +dλ p∥A∥F qAξ b Aξ +∥A∥F +˙ +, +where || ¨ ||F denotes the Frobenius norm. It is not difficult to see that, if T ą 0 +and dT {dλ ě 0, then the above is coercive in |ξ|2. Moreover, in contrast to the +2D case, dT {dλ “ 0 is allowed. In fact, if T satisfies T ą 0 and pdT {dλq{T ą ´1, +then the problem is expected to be locally well-posed if the initial condition is +sufficiently close to the uniform sphere (pg´1g is close to a multiple of the identity +matrix). This is an interesting difference between 2D and 3D Peskin. We will use +this operator (in conjunction with the localization procedures) to show that the full +operator LpX0q “ BXFpX0qX is sectorial. The approximation in (1.13) is done +on the equation written in coordinates partly to obtain a linear leading operator +given by a Fourier multiplier. +Next, we notice that the regularity obtained in Theorem 1.2 for the strong solu- +tions is not enough to satisfy equation (1.9) in a classical sense. Obtaining higher +regularity for the solutions is also important since this further regularity is needed +for the equivalence between different formulations to hold. The abstract theory for +nonlinear equations in [35] does not yield gain of smoothness for the solution, and +in fact this important point is left open in the 2D results in [47]. Nevertheless, we +are able to prove that initial data in little H¨older spaces become smooth for positive +times. +Theorem 1.3. Let X be the solution to the Peskin problem with initial data X0 P +h1,γpS2q constructed in Theorem 1.2. Then, for any α P p0, 1q, it holds that X P +C1pp0, T s; C3,αpS2qq. Moreover, for any 3 ď n P N and α P p0, 1q, assuming that +T P Cn,α, it holds that X P C1pp0, T s; Cn`1,βpS2qq, for any β ă α. +We use the solutions constructed in the previous theorem and Duhamel formula +(1.12) to perform a bootstrapping argument. We build on the properties of the +semigroup e´tLA (see Section 6 and Appendix B) to first gain regularity in mixed- +type spaces Lpp0, T ; Cn,αpS2qq and then transfer this higher regularity in space to +show regularity in time as well. A key point is that, while the kernels are not of +convolution type, we find that it is possible to move derivatives in θ to derivatives +in η at the expense of new terms of the same order, but not higher (see (8.12)). As +explained above, we must work with the equation localized around a given point +and later deal with the corresponding commutators and combine the estimates (see +Section 8 for more details). However, the bootstrapping argument cannot be done +on (1.12) directly, because the right-hand side contains terms of highest regularity. +We combine this process with a regularization argument (see (8.14)), where the use +of little H¨older spaces becomes crucial. +1.3. Related results. The first analytical results for the 2D Peskin problem ap- +peared recently in [34,37]. In [34], energy arguments are used to prove local well- +posedness for H +5 +2 initial data and also exponential convergence to steady states for +sufficiently close to equilibrium initial data is shown. The authors in [37] lowered +the required initial regularity to barely subcritical spaces, h1,γ, γ P p0, 1q, showed +instant smoothing, and provided a blow up criterion. + +8 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +After these works, many improvements for the 2D Peskin problem have appeared. +The work [25] deals with the setting in which the enclosed fluid is different to the +exterior one, and shows asymptotic stability for small data in Wiener algebra critical +spaces. The result [8] shows the local well-posedness and smoothing for general data +in the critical Besov space B +3 +2 +2,1, including the case of nonlinear elastic law. The +sharpest result in terms of regularity appeared in [9], where the semilinear 2D Peskin +problem is shown to be well-posed in B1 +8,8, and thus with possibly non-Lipschitz +curves. +In relation to the Peskin problem, the article [51] introduces a regularization of +the problem inspired by the immersed boundary method and studies its conver- +gence. Filaments that resist both bending and stretching are considered in [33]. +Finally, we mention two works that introduce simplified models of the 2D Peskin +problem. The work [23] considers a model for the normal component and shows +the existence of global solutions for Lipschitz data near the equilibrium. Very re- +cently, [52] derives a PDE to model the tangential effects of the Peskin problem in +the case of an infinitely long and straight string and obtains global solutions with +initial data in the energy class. Moreover, the author presents many connections of +the model with well-known one-dimensional PDEs. +From a mathematical point of view, there are remarkable similarities between +the 2D Peskin problem and the so-called Muskat problem. +In particular, both +problems have the same leading linear operator, they can be written in Boundary +Integral form [13,30], they have the same scaling and satisfy an energy balance [12, +29]. The Muskat problem, which describes the movement of the interface between +incompressible fluids in a porous medium, has been intensively studied in the last +two decades [1,2,4,7,12,13,18,36,39], and some of the techniques developed there +have been successfully extended in the last years to lower the required regularity +for the well-posedness of the 2D Peskin problem [8,9,23,25]. However, while there +are also results for the 3D Muskat problem [3,5,10,14,24], in all these results the +interface is a surface given by graph, hence the geometry does not play a major role. +Even in 2D, in the recent non-graph setting [22] that considers a bubble of fluid +surrounded by another in a porous medium, a change of parametrization becomes +crucial, which is not allowed in the Peskin problem. +We finally mention some results with more complex elastic interactions [6,11,16, +32,38,42,53,54], mostly dedicated to more qualitative results and weak solutions. +Part of the interest generated by the Peskin problem is due to its relative simplic- +ity, which makes it possible to initiate the analytical study of the rich variety of +behaviors in FSI problems, including longtime dynamics. +1.4. Outline. The rest of the paper is structured as follows. In Section 2, we ob- +tain the expression for the elastic law and show the Boundary Integral formulation +for the 3D Peskin problem. Section 3.1 contains the notation used along the paper +as well as some definitions and standard results concerning the stereographic pro- +jection. Next, in Section 4, we introduce the operators that will be used later in +the paper, we decompose the equation and compute the multiplier of the leading +term. Section 5 is dedicated to study the operators previously defined, to show the +needed commutators estimates, and to prove Lemma 5.10. These lemmas will be +repeatedly used in the proof of the main theorems. In Section 6, we show that the +frozen-coefficient operator generates an analytic semigroup (for which we need the + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +9 +multiplier results contained in Appendix A, with further properties studied in Ap- +pendix B). Finally, Sections 7 and Section 8 contain the proofs of the main results: +Theorems 1.2 and 1.3. +2. Formulation and Boundary Integral Reduction +The formulation of the problem (1.1)-(1.6) is closed once that the expression for +F el is given. To specify the elastic force F el in (1.6), we consider the elastic energy +of the interface Γ. We consider an elastic energy EpXq of the form: +EpXq “ +ż +S2 E +ˆBX +Bθ , θ +˙ +dµS2, +where µS2 is the standard measure on the unit sphere. From this, we may com- +pute the elastic force by taking the variational derivative as follows. +Let X “ +pX1, X2, X3qT. Define the following metric tensors pg and g on S2 and Γ respec- +tively, whose i, j components are given by: +(2.1) +pgij “ Bx +X +Bθi +¨ Bx +X +Bθj +, +gij “ BX +Bθi +¨ BX +Bθj +. +We write the energy density as follows: +(2.2) +E “ AEpsij, θq, sij “ BXi +Bθj +, i “ 1, ¨ ¨ ¨ 3, j “ 1, 2. +Let Y “ pY1, Y2, Y3qT be a perturbation of the configuration that is compactly +supported on the open set U on which the coordinate system θ is defined. We have: +d +dτ EpX ` τY q +ˇˇˇˇ +τ“0 +“ +ż +U +BAE +Bsij +BYi +Bθj +a +detpgdθ1dθ2 +“ ´ +ż +U +B +Bθj +ˆBAE +Bsij +a +detpg +˙ +Yidθ1dθ2, +(2.3) +where the summation convention is in effect. We set: +Fel,i “ +1 +?detg +B +Bθj +ˆBAE +Bsij +a +detpg +˙ +, +where Fel,i are the components of the elastic force F el of equation (1.6). With this +prescription of the elastic force, the solutions satisfy the following energy relation: +(2.4) dE +dt “ ´ +˜ż +Ω +2µ |∇Su|2 dx ` +ż +R3zΩ +2 |∇Su|2 +¸ +dx, ∇Su “ 1 +2 +` +∇u ` p∇uqT˘ +. +We will now impose symmetry conditions to determine the explicit form of AE and +hence E. Let θ be a (local) orthogonal coordinate system on S2 so that the two +coordinate tangent vectors are orthogonal: +Bx +X +Bθ1 +¨ Bx +X +Bθ2 +“ 0. +We thus have an orthonormal frame on (a neighborhood of) S2 given by the two +vectors: +pei “ Bx +X +Bθi +����� +Bx +X +Bθi +����� +´1 +, i “ 1, 2. + +10 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +The deformation map X maps the above unit orthogonal vectors to the following +two vectors: +ei “ BX +Bθi +����� +Bx +X +Bθi +����� +´1 +, i “ 1, 2. +Consider the matrix 3 ˆ 2 matrix B “ pe1, e2q whose column vectors are given by +ei. We may say that the energy density E is a function of B and θ: +E “ AEpB, θq, +where we have continued to use the notation AE as in (2.2). By homogeneity of +the unit sphere, we impose that AE does not have an explicit dependence on θ. +Furthermore, the value of AE should not depend on the choice of orthonormal frame +pei or the coordinate system in which X resides. This implies the following. +(2.5) +AEpBq “ AEpR3BR2q for all R3 P SOp3q and R2 P SOp2q +where SOp2q and SOp3q are the group on rotation matrices in 2 and 3 dimensions +respectively. Let: +H “ +ˆ +e1 ¨ e1 +e1 ¨ e2 +e1 ¨ e2 +e2 ¨ e2 +˙ +. +The invariance condition (2.5) implies that AE can only be a function of the trace +and determinants of H, +(2.6) +E “ AEpλ, γq, λ “ +a +trpHq, γ “ +a +detpHq. +In terms of the metric tensors g and pg, we can write λ and γ as: +(2.7) +λ2 “ trppg´1gq, γ2 “ detppg´1gq. +The above expressions for λ and γ are valid even when θ is not an orthogonal +coordinate system. We may substitute (2.6) into (2.3) to obtain: +Fel,k “ Fλ,k ` Fγ,k, +Fλ,k “ +1 +?detg +B +Bθi +ˆ 1 +λ +BAE +Bλ +a +detpg pgij BXk +Bθj +˙ +, +Fγ,k “ +1 +?detg +B +Bθi +ˆBAE +Bγ +a +detg gij BXk +Bθj +˙ +, +(2.8) +where we use the standard notation aij to denote the inverse tensor pa´1qij. Note +that the expressions Fλ,k and Fγ,k are similar but differ crucially in whether pg or +g features inside the force expressions. This is most clearly seen in the following +simple cases. If we let AE “ λ2{2, we have: +(2.9) +Fel,k “ Fλ,k “ γ∆S2Xk, ∆S2Xk “ +1 +a +detpg +B +Bθi +ˆa +detpg pgij BXk +Bθj +˙ +, +where ∆S2 is the Laplace-Beltrami operator on the unit sphere. If we let AE “ γ, +we have: +Fel,k “ Fγ,k “ ∆ΓXk “ +1 +?detg +B +Bθi +ˆa +detg gij BXk +Bθj +˙ +“ ´2κΓnk, +where ∆Γ is the Laplace-Beltrami operator of the closed elastic surface Γ, κΓ is the +mean curvature of Γ and nk is the k-th component of the outward normal vector +n of Γ. This is just the well-known statement on the variation of surface area. We +see from the above expressions that the Fλ,k expresses an elastic force that depends + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +11 +strongly on the stretching of the spherical reference configuration whereas Fγ,k is +a surface tension force. +The prescription of interfacial elastic energy density as in (2.2) or (2.6) has its +origins the classical work of [19], and may be called the membrane neo-Hookean +model. Specific forms for this energy have been used extensively in the modeling +and simulation of fluid-structure interaction problems [20,28,31]. +We now rewrite our evolution equation in a form suitable for our analysis. Hence- +forth we focus on the case when AE is only a function of λ, and the viscosity µ of +the interior fluid is equal to 1. +Let us rewrite the equations of motion. Let G be the Stokeslet tensor in R3: +(2.10) +Gi,jpxq “ 1 +8π +˜ +δi,j +|x| ` xixj +|x|3 +¸ +, x “ px1, x2, x3q. +Let +pFel,k “ 1 +γ Fel,k “ +1 +a +detpg +B +Bθi +ˆ +λ´1T pλq +a +detpg pgij BXk +Bθj +˙ +“ ∇S2 ¨ +` +λ´1T pλq∇S2Xk +˘ +, +where T pλq “ BAE{Bλ (see (2.8)). Let pF el “ pFel,1, Fel,2, Fel,3qT. Notice that we +can write λ in terms of X, +(2.11) +λppxq2 “ |∇S2Xppxq|2, +which can be seen from their definitions +λ2 “ trppg´1gq “ pgijgji, +|∇S2X|2 “ ∇S2Xk ¨ ∇S2Xk “ BXk +Bxi +BXk +Bxl +pgijpglmpgjm “ gilpgil. +When µ “ 1, we may write the evolution of X as +BX +Bt ppxq “ +ż +S2 GpXppxq ´ XppyqqpF elppyqdµS2ppyq +“ +ż +S2GpXppxq ´ Xppyqq∇S2 ¨ +´ +T p|∇S2Xppyq|q ∇S2Xppyq +|∇S2Xppyq| +¯ +dµS2ppyq, +and integrating by parts, we obtain +(2.12) +BX +Bt ppxq“´p.v. +ż +S2∇S2GpXppxq´Xppyqq¨T p|∇S2Xppyq|q ∇S2Xppyq +|∇S2Xppyq|dµS2ppyq. +In the following, we will suppress the principal value notation. Introducing a smooth +partition of unity tρnu, subordinate to a finite atlas of the sphere tUnu, we may +write our problem as follows +BX +Bt ppxq“´ +ÿ +n +ż +S2∇S2GpXppxq´Xppyqq¨ T p|∇S2Xppyq|q +|∇S2Xppyq| +∇S2` +ρnppyqXppyq +˘ +dµS2ppyq. +This can be rewritten using the local charts: +BX +Bt ppxq “ ´ +ÿ +n +ż +Un +pgijpηq B +Bηi +GpXppxq´Xnpηqq +ˆ pgjmpηqT p +a +pgqrgrqpηqq +a +pgqrgrqpηq +pgpmpηq B +Bηp +` +ρnpηqXnpηq +˘a +detpgpηqdη1dη2, + +12 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where Xnpηq is the coordinate map on the n-th coordinate chart and ρnpηq “ +ρnpx +Xpηqq (see Section 3.1 for details of notation). We may take an isothermal +coordinate system (the stereographic projection gives such a system, for example) +on each chart Un, which yields: +BX +Bt ppxq “ ´ +ÿ +n +ż +Un +B +Bηi +GpXppxq´XnpηqqT pλnpηqq +λnpηq +B +Bηi +` +ρnpηqXnpηq +˘ +dη1dη2, +(2.13) +where we denote +(2.14) +λnpηq “ +a +trppg´1pηqgpηqq “ +? +2}∇Xnpηq}F }∇x +Xnpηq}´1 +F , +and }A}F :“ +a +trpAT Aq is the Frobenius norm. +3. Preliminaries +In this section we introduce the notations that will be used in the rest of the +paper and summarize some standard results about stereographic projection charts +for the sphere. +3.1. Notations. Einstein notation over repeated indices will be of constant use. +Given vectors v, w and matrices A, B, C with the same size, we denote +|v| :“ ∥v∥ “ ?vivi, +∥A∥ :“ sup +|v|ą0 +|Av| +|v| “ sup +|v|“1 +|Av| , +A : B :“tr +` +ATB +˘ +“ AijBij, +|A| :“ ∥A∥F :“ +? +A : A, +v b w :“vwT , +pA b Bqijkl :“AijBkl, +ppA b Bq Cqij :“AijBklCkl “ pB : Cq Aij. +We will denote C‚,j to the vector given by the jth column of C, and Cj,‚ to the +one given by the jth row. +We will denote µS2 the standard measure on the unit sphere, and for simplicity +we will write dpy instead of dµS2ppyq. +We will write high partial derivatives in Rk by multi-index α, where multi-index +α is a sequence of k nonegative integers. i.e. α “ pα1, α2, ¨ ¨ ¨ , αkq P Nk +0, where +N0 “ t0u Ş N. +Definition 3.1 (Multi-index). Given α, β P Nk +0, we have the following arithmetic +about the multi-index. +(i) +|α| “α1 ` α2 ` ¨ ¨ ¨ ` αk +α! “α1!α2! . . . αk! +α ` β “ pα1 ` β1, α2 ` β2, ¨ ¨ ¨ , αk ` βkq + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +13 +(ii) We set α ď β, which is αi ď βi for all i “ 1, 2, ¨ ¨ ¨ , k. Then, we have +α ´ β “ pα1 ´ β1, α2 ´ β2, ¨ ¨ ¨ , αk ´ βkq +ˆ +α +β +˙ +“ +α! +pα ´ βq!β! “ +α1! +pα1 ´ β1q!β1! ¨ ¨ ¨ +αk! +pαk ´ βkq!βk! +High partial derivatives can be written as +Bα +x f pxq :“ Bα1 +Bxα1 +1 +Bα2 +Bxα2 +2 +¨ ¨ ¨ Bαk +Bxαk +k +f pxq +(3.1) +where α :“ pα1, α2, ¨ ¨ ¨ , αkq and |α| “ α1 ` ¨ ¨ ¨ ` αk is the total number of deriva- +tives. +We will use the following set of non-singular matrices +(3.2) +DAσ1,σ2 :“ tA : @ξ ‰ 0, σ2 |ξ| ď |Aξ| ď σ1 |ξ|u +where σ1 ě σ2 ą 0. +Euclidean balls of radius R centered at px P Rn will be denoted by Bpx,R, and for +balls centered at the origin we will also denote BpRq :“ B0,R. +We will denote Xppx; tq : S2 ÞÑ Γptq the deformation map that describes the +evolving membrane, and we will omit the dependence on time for simplicity of +notation, Xppxq. We will consider a finite atlas tUn, x +Xnu of the sphere with 0 P Un, +such that the coordinate functions x +Xnpθq : Un Ă R2 Ťt8u ÞÑ S2 satisfy +(3.3) +Bx +Xnpθq +Bθ1 +¨ Bx +Xnpθq +Bθ2 +“ 0, +����� +Bx +Xnpθq +Bθ1 +����� “ +����� +Bx +Xnpθq +Bθ2 +����� . +In particular, we will choose the standard stereographic coordinates. We set tρnu +to be a smooth partition of unity subordinate to the coordinate patches tUnu. For +convenience in the definition of H¨older continuity, we take our system tUn, x +Xn, ρnu +satisfying the following properties with some R, δ ą 0. +Definition 3.2 (System tUn, x +Xn, ρnu). Given R ą 2δ ą 0, we set our isothermal +coordinate charts tUnu with the coordinate functions tx +Xnpθqu and the partition +tρnu to have the following properties: +i) Set pxn “ x +Xn p0q, then S2 Ă Ť +n Bpxn,R, and there exists 0 ă Rn ă 8 s.t +Bpxn,4R X S2 Ă x +Xn pB0,Rnq Ă x +XnpUnq +ii) @px P S2, Dn s.t. +x +Xn pθq “ px +for some θ P B0,Rn, and pBpx,2δ X S2q Ă x +Xn pB0,Rnq . +iii) 0 ď ρn ď 1, +ρn P C8pS2q, +supp pρnq Ă Bpxn,2R X S2. +iv) @px P S2, ř +n ρn ppxq “ 1. +Remark 3.3. If |x +Xn pθq´ x +Xn pηq | ě C |θ ´ η| on Un, then Un is totally bounded. +Given f ppxq : S2 Ñ R, we will denote fn pθq : Un Ă R2 Ñ R with fnpθq :“ +fpx +Xnpθqq. Analogously, we will denote Xnpθq “ Xpx +Xnpθqq. +Remark 3.4. If x +Xn pθq “ x +Xm pηq, then Xn pθq “ Xm pηq. + +14 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Definition 3.5 (H¨older semi-norm). +�f ppxq�Cγ +δ pS2q :“ sup +0ă|px´py|ăδ +|f ppxq´f ppyq| +|px ´ py|γ +“ sup +n +sup +0ă|x +Xnpθq´x +Xnpηq|ăδ +|fn pθq ´ fn pηq| +|x +Xn pθq´x +Xn pηq |γ , +�f ppxq�CγpS2q :“ +sup +0ă|px´py| +|f ppxq ´ f ppyq| +|px ´ py|γ +“ +sup +x +Xnpθq‰x +Xmpηq +|fn pθq ´ fm pηq| +|x +Xn pθq ´ x +Xm pηq |γ . +Definition 3.6 (Arc-chord condition). +|f|˚ :“ inf +px‰py +|f ppxq ´ f ppyq| +|px ´ py| +“ +inf +x +Xnpθq‰x +Xmpηq +|fn pθq ´ fm pηq| +|x +Xn pθq ´ x +Xm pηq | +. +Definition 3.7 (Locally Arc-chord condition in the Charts). Given Vn Ă Un, +|f|˝,n :“ +inf +θ‰η,θ,ηPVn +|fnpθq ´ fnpηq| +|θ ´ η| +, +|f|˝ :“ inf +n |f|˝,n . +Definition 3.8 (Lp norms). +∥f ppxq∥p +LppS2q :“ +ÿ +n +ż +Un +ρn pθq |fn pθq|p a +det pgndθ +“ +ÿ +n +���pρnq +1 +p fn +��� +p +LppUnq ď +ÿ +n +∥fn pθq∥p +LppUnq . +3.2. Standard Stereographic Projection. We will see the properties of the +standard stereographic projection (i.e. the projection point is p0, 0, 1q). For the +other projection points, because S2 is centrosymmetric, we just need to rotate the +coordinates of S2. +Hence, most properties among the projection charts are the +same. +Definition 3.9 (Standard Stereographic Projection). We set x +X : R2 Ťt8u Ñ S2 +with +x +X pθq “ +˜ +2θ1 +1 ` |θ|2 , +2θ2 +1 ` |θ|2 , ´1 ` |θ|2 +1 ` |θ|2 +¸ +, +x +X p8q :“ lim +|θ|Ñ8 +x +X pθq “ p0, 0, 1q. +Then, +θ ppxq “ +ˆ +px1 +1 ´ px3 +, +px2 +1 ´ px3 +˙ +, θ p0, 0, 1q “ 8. +We call the parameterization x +X the standard stereographic projection. +We will denote VR the coordinate balls in R2, +(3.4) +VR :“ B +´ +R +? +4 ´ R2 +¯ +Ă R2. +Proposition 3.10. The standard stereographic projection has the following prop- +erties: + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +15 +i) +Bx +Xpθq +Bθ1 +¨ Bx +Xpθq +Bθ2 +“ 0, +����� +Bx +Xpθq +Bθ1 +����� “ +����� +Bx +Xpθq +Bθ2 +����� “ +2 +1 ` |θ|2 . +ii) For all R ą 0 and all VR (3.4), x +XpVRq “ Bp0,0,´1q,R X S2. +iii) For θ, η P R2, +|x +X pθq ´ x +X pηq | ď 2 |θ ´ η| , +(3.5) +and if θ, η P VR with R ď +? +2, +|x +X pθq ´ x +X pηq | ě 2 +π |θ ´ η| . +(3.6) +Proof. For iii), set pξ “ +θ´η +|θ´η|, then +|x +X pθq´x +X pηq | “ +ˇˇˇ +ż |θ´η| +0 +B +Bs +x +Xpη`spξqds +ˇˇˇ ď +ż |θ´η| +0 +|∇x +Xpη`spξq ¨ pξ|ds +“ +ż |θ´η| +0 +2 +1`|η ` spξ|2 ds ď 2 |θ ´ η| . +Set distppx, py; S2q as the length of shortest curve connecting px and py on S2. When +θ, η P VR with R ď +? +2, the shortest curve ℓ for distpx +X pθq , x +X pηq ; S2q is on +Bp0,0,´1q,R X S2 and +2 +π ď +R +? +4 ´ R2 +2 cos´1 2´R2 +2 +ď +|x +X pθq ´ x +X pηq | +distpx +X pθq , x +X pηq ; S2q +ď 1, +where the above function of R is a decreasing function. Then, since x +X is isothermal +and x +X +´1 pℓq is in VR, +distpx +X pθq , x +X pηq ; S2q “ +ż +ℓ +dl ppxq “ +ż +x +X +´1pℓq +2 +1 ` |θ|2 dl pθq +ě4 ´ R2 +2 +ż +x +X +´1pℓq +dl pθq ě 4 ´ R2 +2 +|θ ´ η| . +Therefore +|x +X pθq ´ x +X pηq | ě 2 +π dist +´ +x +X pθq , x +X pηq ; S2¯ +ě 2 +π |θ ´ η| . +□ +3.3. The Quantitative Relationships between S2 and R2 on the Standard +Stereographic Projection Chart. Given a function f ppxq on S2, we may define +f pθq :“ fpx +X pθqq on R2. x +X pθq is isothermal, but the chart is neither isometric nor +area-preserving. Therefore, some quantities of f between S2 and R2 are different. +We have to check their quantitative relationships. +First, let see the H¨older continuous seminorm Cγ and the arc-chord condition. +Proposition 3.11. Given f on S2, it holds that �f�CγpR2q ď 2γ�f�CγpS2q. + +16 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Proof. +�f�CγpR2q “ sup +θ‰η +´|x +X pθq ´ x +X pηq | +|θ ´ η| +¯γ |fpx +X pθqq ´ fpx +X pηqq| +|x +X pθq ´ x +X pηq |γ +ď 2γ�f�CγpS2q. +□ +Notice that |f|˝ for R2 is zero, and the other sided inequality between Cγ pUq +and Cγ pUq cannot hold with some constant C ą 0, thus we only can find local +inequalities. Set BR “ Bp0,0,´1q,R X S2, VR as in (3.4), and ρ smooth on S2 and +supported in BR. +Proposition 3.12. Given R ď +? +2, it holds that +�ρf�CγpS2q ď +`π +2 +˘γ�ρf�CγpR2q, +|f|˚ ďπ +2 |f|˝ . +Proof. For �ρf�CγpS2q, when px “ x +X pθq , py “ x +X pηq P BR, θ, η P VR, so +|ρf ppxq ´ ρf ppyq| +|px ´ py|γ +“ +´ +|θ ´ η| +|x +X pθq ´ x +X pηq | +¯γ |ρf pθq ´ ρf pηq| +|θ ´ η|γ +ď +`π +2 +˘γ�ρf�CγpR2q. +Next, when px “ x +X pθq , py “ x +X pηq P Bc +R, |ρfppxq´ρfp pyq| +|px´ py|γ +“ 0. Finally, when px “ +x +X pθq P BR, py “ x +X pηq P Bc +R, θ P VR, η P V c +R, set pz “ x +X pξq P BBR s.t. |px ´ pz| “ +dist ppx, BBRq. Since ρ is smooth on S2 and supported in BR, ρ ppzq “ 0 “ ρ ppyq. +Then, +|θ ´ ξ| ď π +2 |px ´ pz| ď π +2 |px ´ py| , +so +|ρf ppxq ´ ρf ppyq| +|px ´ py|γ +“ |ρf ppxq ´ ρf ppzq| +|px ´ py|γ +“ +´ +|θ ´ ξ| +|x +X pθq ´ x +X pηq | +¯γ |ρf pθq ´ ρf pξq| +|θ ´ ξ|γ +ď +`π +2 +˘γ�ρf�CγpR2q. +Now, for |f|˝, +|f|˝ “ +inf +θ‰η,θ,ηPV +|x +X pθq ´ x +X pηq | +|θ ´ η| +|fpx +X pθqq ´ fpx +X pηqq| +|x +X pθq ´ x +X pηq | +ě 2 +π |f|˚ . +□ +Next, let us discuss the relationship between C1,γ ` +S2˘ +and C1,γ ` +R2˘ +on the +standard stereographic projection chart. In the standard stereographic projection +chart px “ x +X pθq, the surface gradient of f, ∇S2f ppxq, is +∇S2f ppxq “ +ÿ +i,j +pgi,j B +Bθi +f pθq B +Bθj +x +X pθq “ +´1 ` |θ|2 +2 +¯2 ÿ +i +B +Bθi +f pθq B +Bθi +x +X pθq , +where pgij denotes the inverse tensor of pg. Hence, +2 +1 ` |θ|2 +���∇S2fpx +Xpθqq +��� “ |∇f pθq| . +We may use the above expressions to obtain the following proposition. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +17 +Proposition 3.13. There exist R0 ą 0 and C ě 21`γ s.t. for all R ă R0 +∥f∥C1,γpR2q ďC ∥f∥C1,γpS2q , +∥ρf∥C1,γpS2q ď ∥ρf∥C1,γpR2q , +|f|˚ ď |f|˝ , +where ρ is smooth on S2 and supported in BR. +3.4. Stereographic Projection Charts x +Xn Covering S2. We consider a finite +cover of the sphere consisting of balls of radius R ă R0 and center pxn, tBpxn,RXS2u, +and a smooth partition of unity subordinated to it, tρnu, such that ρnppxq “ 0 if +|px´ pxn| ě 2R. For convenience, we set px0 “ p0, 0, ´1q. Then we take stereographic +projection charts x +Xn covering S2 (see Figure 2). +Figure 2. Stereographic projection charts, x +Xnpθq. +Definition 3.14 (Stereographic Projection Charts covering S2). x +Xn are standard +stereographic projection charts with +x +Xn : R2 Ñ S2 +x +Xnpθq “ Θnx +Xpθq ++ +, +where Θn is the rotation matrix with pxn “ Θnpx0. +Proposition 3.15. In each chart x +Xn, given R ă R0 +4 and R0, C from Proposition +3.13, since ρ are supported in Bn, +∥f∥C1,γpR2q ď C ∥f∥C1,γpS2q , +∥ρnf∥C1,γpS2q ď ∥ρnf∥C1,γpR2q , +|f|˚ ď |f|˝,n . + +18 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +As a consequence, |f|˚ ď |f|˝. +4. Nonlinear decomposition +In this section we extract the leading structure of the equation and compute +its symbol, introducing the notation for the operators that will appear along the +paper. +4.1. Nonlinear decomposition. We will usually consider separately the two terms +involved in the Stokeslet kernel (2.10), +(4.1) +Gk,lpxq “ G1 +k,lpxq ` G2 +k,lpxq, +G1 +k,lpxq “ 1 +8π +δk,l +|x| , +G2 +k,lpxq “ 1 +8π +xkxl +|x|3 , +and thus we will write our equation (2.12) as follows: +(4.2) +BX +Bt ppxq “ FpXqppxq “ F 1pXqppxq ` F 2pXqppxq, +where +(4.3) +FpXqppxq “ ´ +ż +S2 ∇S2GpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy, +F jpXqppxq “ ´ +ż +S2 ∇S2GjpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy. +and we introduced the notation +(4.4) +T p|∇S2X|q “ T p|∇S2X|q +|∇S2X| +. +Above, we use the shorter notation dpy “ dµS2ppyq. We define the following associate +linear operators, +(4.5) +pNpXqZqkppxq “ ´ +ż +S2 ∇S2GklpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy, +pN jpXqZqkppxq “ ´ +ż +S2 ∇S2Gj +klpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy. +Then, we compute the kernels: +(4.6) +B +Bxi +G1 +k,lpxq “ ´1 +8π +xi +|x|3 δk,l, +B +Bxi +G2 +k,lpxq “ 1 +8π +δk,ixl ` xkδi,l +|x|3 +´ 3 +8π +xkxlxi +|x|5 , +and by the chain rule, +(4.7) +qj +k,lppx, pyq : “ ∇S2Gj +k,lpXppxq ´ Xppyqq +“ ´ B +Bxi +Gj +k,lpXppxq ´ Xppyqq∇S2Xippyq. +so that we write +(4.8) +pN jpXqZqkppxq “ ´ +ż +S2 qj +k,lppx, pyq ¨ Zl,‚ppyqdpy. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +19 +The explicit expression for qj +k,l is given by +q1 +k,lppx, pyq “ 1 +8π +∆ pyXjppxq∇S2Xjppyq +|∆ pyXppxq|3 +δk,l +|px ´ py|2 , +and +q2 +k,lppx, pyq “ ´ 1 +8π +∆ pyXlppxq∇S2Xkppyq ` ∆ pyXkppxq∇S2Xlppyq +|∆ pyXppxq|3 +1 +|px ´ py|2 +` 3 +8π +∆ pyXkppxq∆ pyXlppxq∆ pyXjppxq∇S2Xjppyq +|∆ pyXppxq|5 +1 +|px ´ py|2 . +Using the standard stereographic projection (see Section 3.1) and the notation +Xpθq “ Xpx +Xpθqq, the equation for each component of Xpθq becomes +BXk +Bt pθq “ pFpXqqkpθq +“ ´ +ż +R2 +B +Bηi +Gk,lpXpθq´XpηqqT pλpηqqBXl +Bηi +pηqdη1dη2, +where λpηq is given in (2.14) and we denote accordingly F jpXqpθq, N jpXqZpθq. If +we use the stereographic projection centered at pxn, then we denote N jpXqZnpθq. +Then, we take the derivative in Gk,l (4.1) to obtain +B +Bηi +Gk,lpXpθq´Xpηqq “ qi,k,lpθ, ηq +“ q1 +i,k,lpθ, ηq ` q2 +i,k,lpθ, ηq, +where +(4.9) +q1 +i,k,lpθ, ηq “ +B +Bηi +G1 +k,lpXpθq´Xpηqq “ 1 +8πδk,l +δηXpθq ¨ BX +Bηi pηq +|δηXpθq|3 +, +q2 +i,k,lpθ, ηq “ +B +Bηi +G2 +k,lpXpθq´Xpηqq +“ ´ 1 +8π +BXk +Bηi pηqδηXlpθq ` δηXkpθq BXl +Bηi +|δηXpθq|3 +` 3 +8π +δηXkpθqδηXlpθq +|δηXpθq|5 +δηXpθq ¨ BXpηq +Bηi +, +so that we can write +(4.10) +pN jpXqZqkpθq “ ´ +ż +R2 qj +m,k,lpθ, ηq B p +Xi +Bηm +pηqZl,ipηqdη1dη2. +We notice that the kernels in (4.9) are given by +(4.11) +qj +i,k,lpθ, ηq “ ´ B +Bxm +Gj +k,lpXpθq ´ XpηqqBXm +Bηi +pηq. +We introduce the following notation for finite differences, +(4.12) +δηgpθq “ gpθq ´ gpηq, +∆ηgpθq “ δηgpθq +|θ ´ η|, +and we extract the expected leading terms by replacing +δηXpθq « ∇Xpηqpθ ´ ηq, +p∇Xqp,qpηq “ BXn,p +Bηq +pηq. + +20 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Hence, we define the associate kernels +(4.13) +mi,k,lpθ, ηq “ ´ B +Bxj +Gk,l p∇Xpηqpθ ´ ηqq BXj +Bηi +pηq +“ m1 +i,k,lpθ, ηq ` m2 +i,k,lpθ, ηq, +and define the linear operators Mp∇Xqz as follows: +(4.14) +pMp∇XqZqkpθq “ ´ +ż +R2 mm,k,lpθ, ηq B p +Xi +Bηm +pηqZl,ipηqdη1dη2 +“ pM1p∇XqZqkpθq ` pM2p∇XqZqkpθq, +with +pMjp∇XqZqkpθq “ ´ +ż +R2 mj +m,k,lpθ, ηq B p +Xi +Bηm +pηqZl,ipηqdη1dη2. +We compute the explicit expression of these kernels mj +i,k,l (4.13), +m1 +i,k,lpθ, ηq “ 1 +8π +BXpηq +Bηi +¨ p∇Xpηqpθ´ηqq +|∇Xpηqpθ´ηq|3 +, +m2 +i,k,lpθ, ηq “ ´ 1 +8π +BXkpηq +Bηi +p∇Xpηqpθ´ηqql ` BXlpηq +Bηi +p∇Xpηqpθ´ηqqk +|∇Xpηqpθ´ηq|3 +` 3 +8π +p∇Xpηqpθ´ηqqkp∇Xpηqpθ´ηqql +|∇Xpηqpθ´ηq|5 +p∇Xpηqpθ´ηqq ¨ BXpηq +Bηi +. +We will use the notation +∇Xpηqpθ ´ ηq “ pθ ´ ηq ¨ ∇Xpηq, +pz “ z +|z|, +and we define +(4.15) +EηXpθq :“ p{ +θ ´ ηq ¨ ∇Xpηq ´ ∆ηXpθq, +for which we have that, +(4.16) +|EηpXpθqq| +|θ ´ η|γ +ď �∇X�CγpR2q. +Thus, we can write +(4.17) +NpXqZpθq “ Mp∇XqZpθq ` RpXqZpθq, +where the remainder term +RpXqZpθq “ +2ÿ +j“1 +RjpXqZpθq +is given by +(4.18) +pRjpXqZqqkpθq “ pN jpXqZqkpθq ´ pMjp∇XqpZqqkpθq +“ +ż +R2 Kj +m,k,lpθ, ηq B p +Xi +Bηm +pηqZl,ipηqdη1dη2, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +21 +with kernels +K1 +i,k,lpθ, ηq “ ´q1 +i,k,l pθ, ηq ` m1 +i,k,l pθ, ηq +“ 1 +8π +δk,l +|θ ´ η|2 +BXpηq +Bηi +¨ +ˆ EηXpθq +|∆ηXpθq|3 +´ pp{ +θ ´ ηq ¨ ∇Xpηqq +´ +1 +|∆ηXpθq|3 ´ +1 +|p{ +θ ´ ηq ¨ ∇Xpηq|3 +¯˙ +, +and +K2 +i,k,lpθ, ηq “ ´q2 +i,k,l pθ, ηq ` m2 +i,k,l pθ, ηq “ K2,1 +i,k,lpθ, ηq ` K2,2 +i,k,lpθ, ηq, +K2,1 +i,k,lpθ, ηq “ 1 +8π +1 +|θ ´ η|2 +ˆ ´ BXkpηq +Bηi +|∆ηXpθq|3 EηXlpθq +` BXkpηq +Bηi +pp{ +θ ´ ηq ¨ ∇Xpηqql +´ +1 +|∆ηXpθq|3 ´ +1 +|p{ +θ ´ ηq ¨ ∇Xpηq|3 +¯ +´ +BXlpηq +Bηi +|∆ηXpθq|3 EηXkpθq +` BXlpηq +Bηi +pp{ +θ ´ ηq ¨ ∇Xpηqqk +´ +1 +|∆ηXpθq|3 ´ +1 +|p{ +θ ´ ηq ¨ ∇Xpηq|3 +¯˙ +, +K2,2 +i,k,lpθ, ηq “ 1 +8π +3 +|θ ´ η|2 +ˆ∆ηXkpθq∆ηXlpθq +|∆ηXpθq|5 +BXpηq +Bηi +¨ EηXpθq +` +BXpηq +Bηi +¨ pp{ +θ ´ ηq ¨ ∇Xpηqq +|∆ηXpθq|5 +∆ηXkpθqEηXlpθq +` +BXpηq +Bηi +¨ pp{ +θ ´ ηq ¨ ∇Xpηqq +|∆ηXpθq|5 +pp{ +θ ´ ηq ¨ ∇XpηqqlEηXkpθq +´ +BXpηq +Bηi +¨ pp{ +θ ´ ηq ¨ ∇Xpηqq +|∆ηXpθq|5 +pp{ +θ ´ ηq ¨ ∇Xpηqqlpp{ +θ ´ ηq ¨ ∇Xpηqqk +ˆ +´ +1 +|∆ηXpθq|5 ´ +1 +|p{ +θ ´ ηq ¨ ∇Xpηq|5 +¯˙ +. +Remark 4.1. Note that for all positive odd integers k, it holds that +1 +|u|k ´ +1 +|v|k “ pv ´ uq ¨ pu ` vq +|u|k ` |v|k +řk +i“1 |u|2pi´1q|v|2pk´iq +|u|k|v|k +(4.19) +and +���� +1 +|u|k ´ +1 +|v|k +���� “ ||v| ´ |u|| řk +i“1 |u|i´1 |v|k´i +|u|k |v|k +ď |v ´ u| řk +i“1 |u|i´1 |v|k´i +|u|k |v|k +(4.20) +In particular, formula (4.19) with u “ ∆ηXpθq and v “ p{ +θ ´ ηq ¨ ∇Xpηq, +together with (4.15)-(4.16), makes it clear that there is an extra cancellation in the +kernels of (4.18), + +22 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +In summary, our equation (4.2) is given by +BX +Bt pθq “ FpXqpθq +“ NpXqpT p|∇S2X|q∇S2Xqpθq +“ Mp∇XqpT p|∇S2X|q∇S2Xqpθq ` RpXqpT p|∇S2X|q∇S2Xqpθq. +4.2. Symbol of the leading term. As a preliminary step towards studying the +leading term M (4.14), let us consider its frozen-coefficient counterpart, i.e., re- +placing ∇Xpηq by a constant matrix A and letting pg “ I2. We start with the case +T “ Id, that is, T ” 1: +(4.21) +pLL +AY qkpθq “ p ˜ +MpAq∇Y qkpθq, +where we define +˜ +MpAq “ ˜ +M1pAq ` ˜ +M2pAq and +(4.22) +p ˜ +MjpAqZqkpθq “ ´ +ż +R2 +B +Bηi +pGj +k,lpA pθ ´ ηqqqZl,ipηqdη1dη2. +Let Fθ be the 2D Fourier transform in θ and ξ “ pξ1, ξ2qT: +Fθwpξq “ +ż +R2 wpθq expp´iθ ¨ ξqdθ. +We now compute the Fourier transform of the function GA: +GApθq “ GpAθq “ 1 +8π +˜ +I +|Aθ| ` Aθ b Aθ +|Aθ|3 +¸ +, +where I3 is the 3 ˆ 3 identity matrix. Given that θ P R2 and A is a 3 ˆ 2 matrix, +it is convenient to rewrite GA as follows. First, note that: +|Aθ|2 “ Aθ ¨ Aθ “ θ ¨ +` +ATAθ +˘ +“ |Bθ|2 , B “ +? +ATA. +Notice that B is a 2 ˆ 2 symmetric positive definite matrix. Using this B, we have: +(4.23) +GApθq “ 1 +8π +˜ +I +|Bθ| ` Q +˜ +Bθ b Bθ +|Bθ|3 +¸ +QT +¸ +, Q “ AB´1. +We note that Q is an isometry in the sense that QTQ “ I2 where I2 is the 2 ˆ 2 +identity matrix. We are now ready to compute the Fourier transform of GA. First, +note that: +(4.24) +Fθ +ˆ 1 +|θ| +˙ +“ 2π +|ξ|. +Thus, a simple change of variable yields: +(4.25) +Fθ +ˆ +1 +|Bθ| +˙ +“ +2π +detpBq |B´1ξ|. +Next, note that: +Fθ +˜ +θiθj +|θ|3 +¸ +“ Fθ +ˆ +θiθj∆θ +ˆ 1 +|θ| +˙˙ +“ +B2 +BξiBξj +ˆ +|ξ|2 Fθ +ˆ 1 +|θ| +˙˙ +“ 2π +B2 +BξiBξj +|ξ| “ 2π +˜ +δij +|ξ| ´ ξiξj +|ξ|3 +¸ +, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +23 +where ∆θ is the Laplacian in R2, δij is the Kronecker delta and we used (4.24) in +the third equality. In matrix notation, the above can be written as: +Fθ +˜ +θ b θ +|θ|3 +¸ +“ 2π +˜ +I2 +|ξ| ´ ξ b ξ +|ξ|3 +¸ +. +Again, by changing variables, we see that: +(4.26) +Fθ +˜ +Bθ b Bθ +|Bθ|3 +¸ +“ +2π +detpBq +˜ +I2 +|B´1ξ| ´ B´1ξ b B´1ξ +|B´1ξ|3 +¸ +. +Using (4.26), (4.24) and (4.23), we obtain: +FθGA “ +1 +4detpBq +˜ +I ` QQT +|B´1ξ| +´ QB´1ξ b QB´1ξ +|B´1ξ|3 +¸ +“ +1 +4 +a +detpATAq +˜ +I ` ApATAq´1AT +pξ ¨ pATAq´1ξq1{2 ´ ApATAq´1ξ b ApATAq´1ξ +pξ ¨ pATAq´1ξq3{2 +¸ +This implies that the Fourier symbol of LL +A is given by: +LL +AY “ ´F´1 +ξ LL +ApξqFθY , LL +Apξq “ |ξ|2 pFθGAq pξq. +To better understand the properties of Fourier multiplier LL +Apξq, we first note that: +QQT ´ QB´1ξ b QB´1ξ +|B´1ξ|2 +“ Q +˜ +I2 ´ B´1ξ b B´1ξ +|B´1ξ|2 +¸ +QT “ vpξq b vpξq, +vpξq “ QRπ{2 +B´1ξ +|B´1ξ|, Rπ{2 “ +ˆ +0 +´1 +1 +0 +˙ +. +(4.27) +Note that vpξq P R3 is a unit vector, and hence, the above matrix 3 ˆ 3 matrix is +an orthogonal projection on to the subspace spanned by vpξq. We see that: +(4.28) +LL +Apξq “ +|ξ|2 +4detpBq |B´1ξ| pI ` vpξq b vpξqq . +It is now immediate that LL +Apξq is a symmetric positive definite matrix for each +ξ ‰ 0 with eigenvalues: +(4.29) +λ “ µ +4 |ξ|2 and µ +2 |ξ|2 , +µ “ +1 +detpBq |B´1ξ|, +where the eigenspace for µ{2 is spanned by vpξq and the two-dimensional eigenspace +of µ{4 is spanned by the orthogonal complement of vpξq. We also have: +(4.30) +µ +4 |ξ|2 |w|2 ď w ¨ Lpξqw ď µ +2 |ξ|2 |w|2 +for any w P R3. +In the case of general T , the frozen coefficient linear operator can be obtained +by a further linearization of the force function. Consider the expression: +T pλτq +λτ +BpXl ` τYlq +Bθi +, λτ “ λpX ` τY q + +24 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where λ is here viewed as a function of X through its dependence on g (see (2.7)). +Now, +d +dτ +ˆT pλτq +λτ +BpXl ` τYlq +Bθi +˙ˇˇˇˇ +τ“0 +“ +ˆ 1 +λ +dT +dλ ´ T +λ2 +˙ dλτ +dτ +ˇˇˇˇ +τ“0 +BXl +Bθi +` T +λ +BYl +Bθi +“ +ˆ 1 +λ +dT +dλ ´ T +λ2 +˙ 1 +λppg´1qm,n +BXq +Bθm +BYq +Bθn +BXl +Bθi +` T +λ +BYl +Bθi +. +Now, the frozen coefficient approximation amounts to taking pg “ I2, BXl{Bθi “ Al,i +and λ “ ∥A∥F . Thus, +d +dτ +ˆT pλτq +λτ +BpXl ` τYlq +Bθi +˙ˇˇˇˇ +τ“0 +« pTF pAqqi,l,m,q +BYq +Bθm +, +with +(4.31) +TFpAq “ T p∥A∥F q +∥A∥F +I2 b I2 ´ +ˆT p∥A∥F q +∥A∥F +´ dT +dλ p∥A∥F q +˙ A b A +∥A∥2 +F +. +Thus, the frozen-coefficient linear operator in the general force case is given by +(4.32) +pLAY qkpθq “ ´ +ż +R2 +B +Bηi +pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2. +Let us now take the Fourier transform of the divergence of the above: +Fp∇ ¨ pTF pAq∇Y qqpξq “ ´MApξqFY pξq, +where +(4.33) +MApξq “ T p∥A∥F q +∥A∥F +˜ +|ξ|2 I ´ Aξ b Aξ +∥A∥2 +F +¸ +` dT +dλ p∥A∥F qAξ b Aξ +∥A∥2 +F +. +Note that, if we set T “ Id, then TF “ Id and the above reduces to Mpξq “ |ξ|2. +Thus, in the general case, the multiplier in LApξq of (4.32) becomes: +LApξq “ pFθGAq pξqMApξq +“ I ` vpξq b vpξq +4detpBq |B´1ξ| +ˆT p∥A∥F +∥A∥F +ˆ +|ξ|2 I ´ Aξ b Aξ +∥A∥F +˙ +` dT +dλ p∥A∥F qAξ b Aξ +∥A∥F +˙ +. +(4.34) +It is not difficult to see that, if T ą 0 and dT {dλ ě 0, then the above is coercive in +|ξ|2. +5. Calculus estimates +In this section we include some estimates of the operators that will be frequently +used in later sections. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +25 +Lemma 5.1. Let X P C1pS2q, such that |X|˚ ą 0. Then, the kernels Gj +k,lpxq (4.1) +and qj +k,lppx, pyq (4.7) satisfy the following bounds +(5.1) +���Bα +x Gj +k,lpxq +��� ď +C +|x|1`|α| , +���∆y +´ +Bα +x Gj +k,lpxq +¯��� ď C M 1`|α| +m3`2|α| , +|qj +k,lppx, pyq| ď C |∇S2Xppyq| +|∆ pyXppxq|2 +1 +|px ´ py|2 ď C }∇S2X}C0pS2q +|X|2˚ +1 +|px ´ py|2 , +where |α| defined in (3.1), M “ max p|x| , |y|q, and m “ min p|x| , |y|q. +For the sake of completeness, we include a version of the divergence theorem +that will be used. Notice that, following standard convention, we will not explicitly +write the principal values elsewhere. +Lemma 5.2. Given a matrix A and a compact set D Ă R2 containing 0, then +p.v. +ż +Dc ∇GpAηqdη : “ lim +LÑ8 +ż +DcXBpLq +∇GpAηqdη +“ ´ +ż +BD +GpAηqn pηq dl pηq , +where B pLq Ă R2 is the ball centered at 0 of radius L. In particular, +p.v. +ż +R2 ∇GpAηqdη :“ +lim +LÑ8,εÑ0 +ż +BpLqzBpεq +∇GpAηqdη “ 0. +Proof. Since D is compact and contains 0, D Ă B pLq when L is large enough. +Then, by integration by parts +ż +DcXBpLq +∇GpAηqdη “ ´ +ż +BD +GpAηqn pηq dl pηq ` +ż +BBpLq +GpAηqn pηq dl pηq . +Since GpAηq is even, the boundary term vanishes. Therefore, +ż +Dc ∇GpAηqdη “ lim +LÑ8 +ż +DcXBpLq +∇GpAηqdη “ ´ +ż +BD +GpAηqn pηq dl pηq . +Next, set D “ B pεq, +ż +R2 ∇GpAηqdη “ +lim +LÑ8,εÑ0 +ż +BpLqXBpεqc ∇GpAηqdη “ 0. +□ +Lemma 5.3. Let A be a matrix in the set DAσ1,σ2. Then, the linear operator +˜ +MpAq (4.22) maps CγpR2q X L2pR2q to CγpR2q X L2pR2q for any any γ P p0, 1q. +Moreover, +} ˜ +MjpAqZ}CγpR2q ď C +σ2 +´ +1 ` +´σ1 +σ2 +¯2¯ +}Z}CγpR2qXL2pR2q. +And given A1, A2 P DAσ1,σ2 +} ˜ +MjpA1qZ ´ ˜ +MjpA2qZ}CγpR2q ď C +σ2 +2 +´ +1 ` +´σ1 +σ2 +¯5¯ +}Z}CγpR2qXL2pR2q ∥A1 ´ A2∥ . + +26 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Proof. Taking into account Lemma 5.2, we have +p ˜ +MjpAqZqkpθq “ ´ +ż +R2 +B +Bηm +Gj +k,lpApθ ´ ηqq +` +Zl,mpηq ´ Cl,m +˘ +dη1dη2, +where Cl,m is an arbitrary constant, that we will take to be zero or Zl,mpθq. Then, +the estimate for |p ˜ +MjpAqZqkpθq| follows by splitting the integral in two terms, +p ˜ +MjpAqZqkpθq “ I1pθq ` I2pθq, +with +I1pθq “ ´ +ż +|θ´η|ď1 +B +Bηm +Gj +k,lpApθ ´ ηqq +` +Zl,mpηq ´ Zl,mpθq +˘ +dη1dη2, +I2pθq “ ´ +ż +|θ´η|ě1 +B +Bηm +Gj +k,lpApθ ´ ηqqZl,mpηqdη1dη2. +Since +B +Bηm +Gj +k,lpApθ ´ ηqq “ ´ +BGj +k,l +Bxi +pApθ ´ ηqqAi,m, +(5.2) +the kernel bounds (5.1) and the fact that A P DAσ1,σ2 provide that +|I1pθq| ď C σ1 +σ2 +2 +�Z�CγpR2q, +|I2pθq| ď C σ1 +σ2 +2 +}Z}L2pR2q, +hence +|p ˜ +MjpAqZqkpθq| ď C σ1 +σ2 +2 +}Z}CγpR2qXL2pR2q. +We proceed with the seminorm. Let h P R2, |h| ď 1, and perform the following +splitting +p ˜ +MjpAqZqkpθq ´ p ˜ +MjpAqZqkpθ ` hq “ J1 ` J2 ` J3 ` J4, +where +J1 “ ´ +ż +|θ´η|ď2|h| +B +Bηm +Gj +k,lpApθ ´ ηqqδηZl,mpθqdη, +J2 “ +ż +|θ´η|ď2|h| +B +Bηm +Gj +k,lpApθ ` h ´ ηqqδηZl,mpθ ` hqdη, +J3 “ δθZl,mpθ ` hq +ż +|θ´η|ą2|h| +B +Bηm +Gj +k,lpθ ´ ηqdη, +J4 “ +ż +|θ´η|ą2|h| +´ B +Bηm +Gj +k,l pApθ`h´ηqq´ +B +Bηm +Gj +k,lpApθ´ηqq +¯ +δηZl,mpθ`hqdη. +Absolutely, +|J1| ` |J2| ď C σ1 +σ2 +2 +�Z�CγpR2q |h|γ . +Then, by Lemma 5.2, +J3 “ pZl,mpθ ` hq ´ Zl,mpθqq +ż +|θ´η|“2|h| +Gj +k,l pApθ ´ ηqq nmpηqdlpηq, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +27 +and so +|J3| ď C�Z�CγpR2q |h|γ 1 +σ2 +ż +|θ´η|“2|h| +1 +|θ ´ η|dl pηq +ď C 1 +σ2 +�Z�CγpR2q |h|γ . +Finally, since +B +Bθp +B +Bηm +Gj +k,lpApθ ´ ηqq “ ´ B +Bxq +B +Bxi +Gj +k,lpApθ ´ ηqqAi,mAq,p, +(5.3) +it follows that +|J4| “ +ˇˇˇ +ż +|θ´η|ą2|h| +ż 1 +0 +hp +B +Bθp +B +Bηm +Gj +k,l pApθ ` sh ´ ηqq δηpZl,mpθ ` hqdsdη +ˇˇˇ +ď C σ2 +1 +σ3 +2 +�Z�CγpR2q |h| +ż +|θ´η|ą2|h| +ż 1 +0 +|θ ` h ´ η|γ +|θ ` sh ´ η|3 dsdη. +In the domain where |θ ´ η| ą 2 |h|, it holds that for s P r0, 1s, +1 +2 |θ ` h ´ η| ď |θ ` sh ´ η| ď 3 +2 |θ ` h ´ η| . +Hence, +|J4| ď C σ2 +1 +σ3 +2 +�Z�CγpR2q |h|γ . +Therefore, we obtain +��� ˜ +MjpAqZ +��� +CγpRq ď C +σ2 +´ +1 ` σ2 +1 +σ2 +2 +¯ +}Z}CγpR2qXL2pR2q. +For the L2 norm, since ξmFθ +” +Gj +k,l pAθq +ı +pξq is bounded by σ1 and σ2, we have +���p ˜ +MjpAqZqk +��� +L2pRq “ +���ξmFθ +” +Gj +k,l pAθq +ı +pξq FrZl,ms pξq +��� +L2pRq +ď C pσ1, σ2q ∥FrZs∥L2pRq “ C pσ1, σ2q ∥Z∥L2pRq +(5.4) +Next, through (5.1), (5.2), and (5.3), we have +���Gj +k,lpA1pθ ´ ηqq´Gj +k,lpA2pθ ´ ηqq +��� ď C σ1 |pA1´A2q pθ ´ ηq| +σ3 +2 |θ ´ η|3 +ď C σ1 +σ3 +2 +∥pA1´A2q∥ +|θ ´ η| +, +ˇˇˇ B +Bηm +Gj +k,lpA1pθ´ηqq´ B +Bηm +Gj +k,lpA2pθ´ηqq +ˇˇˇ +ď +ˇˇˇ +BGj +k,l +Bxi +pA1pθ´ηqq´ +BGj +k,l +Bxi +pA2pθ´ηqq +ˇˇˇ|A1,i,m| +` +ˇˇˇ +BGj +k,l +Bxi +pA2pθ´ηqq pA1,i,m´A2,i,mq +ˇˇˇ +ď C +´ 1 +σ2 +2 +` σ3 +1 +σ5 +2 +¯∥pA1 ´ A2q∥ +|θ ´ η|2 +, + +28 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +and +ˇˇˇ B +Bθp +B +Bηm +Gj +k,lpA1pθ´ηqq´ B +Bθp +B +Bηm +Gj +k,lpA2pθ´ηqq +ˇˇˇ +ď +ˇˇˇ B +Bxq +B +Bxi +Gj +k,lpA1pθ´ηqq´ B +Bxq +B +Bxi +Gj +k,lpA2pθ´ηqq +ˇˇˇ|A1,i,mA1,q,p| +` +ˇˇˇ B +Bxq +B +Bxi +Gj +k,lpA2pθ´ηqq pA1,i,mA1,q,p´A2,i,mA2,q,pq +ˇˇˇ +ď C +´σ1 +σ3 +2 +` σ5 +1 +σ7 +2 +¯∥pA1´A2q∥ +|θ ´ η|3 +. +Hence, we obtain +} ˜ +MjpA1qZ´ ˜ +MjpA2qZ}CγpR2q ď C +σ2 +2 +´ +1` +´σ1 +σ2 +¯5¯ +}Z}CγpR2qXL2pR2q ∥A1´A2∥ . +□ +As an immediate consequence of the previous lemma with ˜Zl,m “ Bx +Xi +Bηm pηqZl,ipηq, +we obtain the following lemma for MpAq: +Lemma 5.4. Let A be a matrix in the set DAσ1,σ2. Then, the linear operators +MjpAq (4.14) map CγpS2q to CγpR2q for any any γ P p0, 1q. Moreover, +}MjpAqZ}CγpR2q ď C +σ2 +´ +1 ` +´σ1 +σ2 +¯2¯ +sup +l,m +} Bx +X +Bηm +pηq ¨ Zl,‚pηq}CγpR2qXL1pR2q +ď Cpσ1, σ2q}Z}CγpS2q. +Remark 5.5. In most cases, Proposition 5.4 will be used with Z compactly sup- +ported and given by a multiple of a gradient. Notice that in that case, for Z “ +λ∇S2X, λ : R2 ÞÑ R, +Bx +X +Bηm +pηq ¨ Zl,‚pηq “ λpηq BXl +Bηm +pηq, +and therefore +}MjpAqpλ∇S2Xq}CγpR2q ď Cpσ1, σ2q}λ∇X}CγpR2q. +Lemma 5.6. Let X P C1pS2q such that |X|˚ ą 0. Then, the linear operators +N jpXq (4.5) map CγpS2q to CγpS2q for any γ P p0, 1q. Moreover, +(5.5) +}N jpXqZ}CγpS2q ď +C +|X|˚ +´ +1 ` +´}∇S2X}C0pS2q +|X˚| +¯2¯ +}Z}CγpS2q. +Proof. We first notice that we can introduce an arbitrary constant (matrix), +(5.6) +NpXqZppxq “ ´ +ż +S2 ∇S2GpXppxq ´ Xppyqq ¨ pZppyq ´ Cqdpy. +We will usually take C “ 0 or C “ Zppxq. Recalling the kernels (4.7) and (4.8), we +write the equation for each component +pN jpXqZqkppxq “ ´ +ż +S2 qj +k,lppx, pyq ¨ pZl,‚ppyq ´ Cl,‚qdpy, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +29 +where Cl,‚ “ 0 or Cl,‚ “ Zl,‚ppxq. We first perform the estimate for |NpXqZppxq|, +px P S2, +(5.7) +|NpXqZppxq| ď +2ÿ +j“1 +|N jpXqZppxq|. +Using the bound (5.1) for the kernel, we have +(5.8) +|NpXqZppxq| ď C }∇S2X}C0pS2q +|X|2˚ +ż +S2 +|Zl,‚ppxq ´ Zl,‚ppyq| +|px ´ py|2 +dpy +ď C }∇S2X}C0pS2q +|X|2˚ +}Z}CγpS2q. +We proceed to estimate the H¨older seminorm. Let px, pxh P S2, and denote h “ +|px ´ pxh|. We write +pN jpXqZqkppxq´pN jpXqZqkppxhq “ +ż +S2qj +k,lppx, pyq ¨ pZl,‚ppxq´Zl,‚ppyqqdpy +´ +ż +S2qj +k,lppxh, pyq¨pZl,‚ppxhq´Zl,‚ppyqqdpy, +and perform the following splitting +(5.9) +pN jpXqZqkppxq ´ pN jpXqZqkppxhq “ I1 ` I2 ` I3 ` I4, +where +I1 “ +ż +t|px´ py|ď2huXS2 qj +k,lppx, pyq ¨ pZl,‚ppxq ´ Zl,‚ppyqqdpy, +I2 “ ´ +ż +tpx´ py|ď2huXS2 qj +k,lppxh, pyq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy, +I3 “ pZl,‚ppxq ´ Zl,‚ppxhqq ¨ +ż +t|px´ py|ě2huXS2 qj +k,lppx, pyqdpy, +I4 “ +ż +t|px´ py|ě2huXS2pqj +k,lppx, pyq ´ qj +k,lppxh, pyqq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy. +The first two terms are estimated directly +(5.10) +|I1| ` |I2| ď C }∇S2X}C0pS2q +|X|2˚ +}Z}CγpS2qhγ. +For the third, we use that the kernel is a derivative to integrate by parts and obtain +that +(5.11) +|I3| “ |pZl,‚ppxq ´ Zl,‚ppxhqq ¨ +ż +t|px´ py|“2huXS2 GjpXppxq ´ XppyqqnppyqdlS2ppyq| +ď C }Z}CγpS2q +|X|˚ +|h|γ. +Finally, we use the mean-value theorem on the kernel to estimate I4. Set ℓpsq the +shortest path function from pxh to px respect to arc-length s variable, and L “ + +30 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +dist +` +pxh, px; S2˘ +. Then, we have +Gj +k,lpXppxq´Xppyqq ´ Gj +k,lpXppxhq´Xppyqq “ +ż L +0 +B +BsGj +k,lpXpℓ psqq´Xppyqqds +“ +ż L +0 +∇S2Gj +k,lpXpℓ psqq´Xppyqq ¨ B +Bsℓ psq ds. +Hence, for qj +k,l (4.7), +|qj +k,lppx, pyq ´ qj +k,lppxh, pyq| “ +ˇˇˇ∇S2 +ż L +0 +∇S2Gk,lpXpℓ psqq´Xppyqq ¨ B +Bsℓ psq ds +ˇˇˇ +“ +ˇˇˇ∇S2 +ż L +0 +B +Bxi +Gk,lpXpℓ psqq´Xppyqq +ˆ +∇S2Xi pℓ psqq ¨ B +Bsℓ psq +˙ +ds +ˇˇˇ +“ +ˇˇˇ +ż L +0 +B +Bxj +B +Bxi +Gk,lpXpℓ psqq´Xppyqq +ˆ +∇S2Xi pℓ psqq¨ B +Bsℓ psq +˙ +∇S2Xj ppyq ds +ˇˇˇ, +and recalling (5.1) we obtain the bound +|qj +k,lppx, pyq ´ qj +k,lppxh, pyq| ďC +∥∇S2X∥2 +C0pS2q +|X|3 +˚ +ż L +0 +1 +|ℓ psq ´ py|3 ds. +Then, we have that +|I4| ď C +}∇S2X}2 +C0pS2q}Z}CγpS2q +|X|3 +˚ +ż +t|px´ py|ě2huXS2 +|pxh´ py|γ +ż L +0 +ds +|ℓ psq´ py|3 dpy. +We notice that since ℓpsq is the shortest path function from pxh to px on S2, it holds +that |ℓ psq ´ px| ď |px ´ pxh|. Thus, +|ℓ psq ´ py| ě |px ´ py| ´ |ℓ psq ´ px| ě |px ´ py| ´ |pxh ´ px| ě 1 +2 |px ´ py| . +In addition, +|pxh ´ py| ď |px ´ py| ` |pxh ´ px| ď 3 +2 |px ´ py| . +Finally, since L ď Ch, we conclude that +(5.12) +|I4| ď C +}∇S2X}2 +C0pS2q}Z}CγpS2q +|X|3 +˚ +h +ż +tpx´ py|ě2huXS2 +dpy +|px ´ py|3´γ +ď C +}∇S2X}2 +C0pS2q}Z}CγpS2q +|X|3 +˚ +hγ. +Joining the bounds (5.10), (5.11), and (5.12) back in (5.9), we conclude that +(5.13) +rNpXqZsCγpS2q ď +C +|X|˚ +´ +1 ` +´}∇S2X}C0pS2q +|X˚| +¯2¯ +}Z}CγpS2q, +and, with (5.7), the same bound holds for the H¨older norm. +□ + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +31 +We will need to localize the operators above. For that purpose, let us define the +cutoff function pρn, +(5.14) +pρnppxq “ +# +1 +if +|px ´ pxn| ď 3R, +0 +if +|px ´ pxn| ě 4R, +and recall the partition of unity tρnu based on the points pxn (see Subsection 3.4). +Lemma 5.7. Let X P C1pS2q such that |X|˚ ą 0, NpXq the linear operator +defined by (4.5), and pρn the cutoff function (5.14). Then, for Z P C0pS2q compactly +supported on Bpxn,2R X S2, it holds that, +}p1 ´ pρnqN jpXqZ}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}Z}C0pS2q. +Proof. Let +Ippxq “ p1 ´ pρnppxqqN jpXqZppxq. +Since 1 ´ pρppxq “ 0 when px P Bpxn,3R, let px P S2zBpxn,3R. +Then, recalling the +condition on the support of Z, +Ippxq“ppρnppxq´1q +ż +B pxn,2RXS2 +∇S2GpXppxq´Xppyqq¨Zppyqdpy, +and using the bound (5.1) for the kernel, +|Ippxq| ď C }∇S2X}C0pS2q +|X|2˚ +}Z}C0pS2q +ż +B pxn,2RXS2 +|px´ py|´2dpy. +Since we have that |px ´ py| ě R, we obtain +(5.15) +|Ippxq| ď Cp|X|˚, }∇S2X}C0pS2qq}Z}C0pS2q. +To estimate the H¨older seminorm, consider two points px, pxh P S2, h “ |px ´ pxh|. +Due to the cut-off function pρn, the only non-trivial case is px, pxh P S2zBpxn,3R: +|Ippxhq ´ Ippxq| “ ppρnppxq ´ pρnppxhqq +ż +B pxn,2RXS2qk,lppx, pyq ¨ Zl,‚ppyqdpy +` p1´pρnppxhqq +ż +B pxn,2RXS2 +` +qk,lppx, pyq´qk,lppxh, pyq +˘ +¨ Zl,‚ppyqdpy +“ J1 ` J2. +The first term is bounded as (5.15), +|J1| ď C}pρn}C1pS2q +}∇S2X}C0pS2q +|X|2˚ +}Z}C0pS2qh. +Recalling the expression of qk,l (4.7), we can check that, since |px´py| ě R, |pxh´py| ě +R, +|qk,lppx, pyq ´ qk,lppxh, pyq| ď C +}∇S2X}2 +C0pS2q +|X|3˚R3 +h, +hence +|J2| ď C +R +}∇S2X}2 +C0pS2q +|X|3˚ +}Z}C0pS2qh. +Therefore, +(5.16) +}I}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2q}C1q}Z}C0pS2q. +□ + +32 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +The previous lemma holds analogously for the operators ˜ +MpAq: +Lemma 5.8. Let A P DAσ1,σ2, +˜ +MjpAq the linear operator defined by (4.22), and +pρn the cutoff function (5.14). Then, for Z P C0pR2q compactly supported on V2R, +it holds that, +}p1 ´ pρnq ˜ +MjpAqZ}C1pR2q ď CpR, σ1, σ2q}Z}C0pR2q. +Lemma 5.9. Let pxn P S2, X P C1pS2q, |X|˚ ą 0, and ρn P C8pBpxn,2R X S2q, +R ă 1{10. Then, the commutator +rρn,NpXqsZppxq “ ´ +ż +S2∇S2GpXppxq´Xppyqq¨ Zppyqpρnppyq´ρnppxqqdpy, +satisfies that, for any γ P p0, 1q, +(5.17) +∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q. +Proof. Recalling the kernel bound (5.1), +|rρn, NpXqsZppxq| ď C }∇S2ρn}C0pS2q}Z}C0pS2q}∇S2X}C0pS2q +|X|2˚ +ż +S2 +1 +|px ´ py|dpy +ď Cp|X˚, }∇S2X|C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q. +Next, we study the H¨older seminorm. +We take two points px, pxh, denote h “ +|px ´ pxh|, and perform a splitting analogous to (5.9). Using the kernel notation +(4.7), +rρn, NpXqsZppxq´rρn, NpXqsZppxhq“ +ż +S2 qk,lppx, pyq pρnppyq´ρnppxqq¨Zl,‚ppyqdpy +´ +ż +S2qk,lppxh, pyq pρnppyq´ρnppxhqq¨Zl,‚ppyqdpy +“ I1 +n ` I2 +n ` I3 +n ` I4 +n, +where +I1 +n “ +ż +t|px´ py|ď2huXS2 qk,lppx, pyq pρnppyq ´ ρnppxqq ¨ Zl,‚ppyqdpy +I2 +n “ ´ +ż +t|px´ py|ď2huXS2 qk,lppxh, pyq pρnppyq ´ ρnppxhqq ¨ Zl,‚ppyqdpy +I3 +n “ +ż +t|px´ py|ě2huXS2 qk,lppx, pyq pρnppxhq ´ ρnppxqq ¨ Zl,‚ppyqdpy +and +I4 +n “ +ż +t|px´ py|ě2huXS2 +pqk,lppx, pyq´qk,lppxh, pyqq pρnppyq´ρnppxhqq ¨ Zl,‚ppyqdpy. +Then, for I1 +n and I2 +n, +|I1 +n| ` |I2 +n| ď C +∥∇S2ρn∥C0pS2q ∥∇S2X∥C0pS2q +|X|2 +˚ +}Z}C0pS2qh. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +33 +Next, for I3 +n, integration by parts gives that +��I3 +n +�� ď C |ρnppxhq´ρnppxq| +ż +t|px´ py|ě2huXS2 +}Z}C0pS2q ∥∇S2X∥C0pS2q +|X|2 +˚ |px ´ py|2 +dpy +ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2qh log h´1. +Finally, in I4 +n, the use of the mean-value theorem provides that +��I4 +n +�� ď C +∥∇S2ρn∥C0pS2q }Z}C0pS2q ∥∇S2X∥2 +C0pS2q +|X|3 +˚ +ˆ +ż +t|px´ py|ď2huXS2 +ż L +0 +|pxh ´ py| +|ℓ psq ´ py|3 dsdpy +ď Cp|X˚, ∥∇S2X∥C0pS2qq ∥∇S2ρn∥C0pS2q ∥Z∥C0pS2q h. +Thus, +∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q. +□ +Lemma 5.10. Let pxn P S2 and x +Xn : R2 Y t8u Ñ S2 the stereographic projection +centered at pxn. Let X P C1,γpS2q, |X|˚ ą 0, and pρn given by (5.14). Let the linear +operators NpXq and MpAq be defined by in (4.5) and (4.14), with A “ ∇Xnp0q, +where we are using the notation Xnpθq “ Xpx +Xnpθqq. Denote +Inpθq “ pρnpθqrMpAq ´ NpXqsZnpθq, +Then, for Z P CγpS2q compactly supported on Bpxn,2R X S2, the following estimates +hold: +(5.18) +}In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq +` +p1 ` }∇S2X}CγpB pxn,5RXS2qq}Z}C0pS2q +` εpRq}Z}CγpS2q +˘ +, +and +(5.19) +}In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq +` +}Z}C0pS2q`}∇S2X}C +γ +2 pB pxn,5RXS2q}Z}C +γ +2 pS2q +` εpRq}Z}CγpS2q +˘ +, +with εpRq Ñ 0 as R Ñ 0 given by the modulus of continuity of ∇S2X. +Proof. First, we write +(5.20) +In “ I1 +n ` I2 +n +“ +2ÿ +j“1 +pρnpθqrMjpAq ´ N jpXqsZnpθq +“ +2ÿ +j“1 +pρnpθqrMjpAq ´ Mjp∇XnqsZnpθq ´ RjpXnqZnpθq, +and focus on the term I1 +n, as the other I2 +n will follow similarly. We then write +I1 +n “ ´pρnpθq +ż +R2 P 1 +m,k,lpθ, ηq B p +Xi +Bηm +pηqZn,lipηqdη1dη2, + +34 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where we used the notation (3.4) and +(5.21) +P 1 +m,k,lpθ, ηq “δk,l +8π +B +Bηm +´ +1 +|Apθ ´ ηq| ´ +1 +|Xnpθq ´ Xnpηq| +¯ +“ ´ 1 +8π +pBpθ´ηqqm +|Apθ´ηq|3 δk,l ` q1 +m,k,lpθ, ηq +“ ´ 1 +8π +pBpθ´ηqqm +|Apθ´ηq|3 δk,l ` m1 +m,k,lpθ, ηq ´ K1 +m,k,lpθ, ηq +“ ´ 1 +8π +ˆpBpθ´ηqqm +|Apθ´ηq|3 ´ pBnpηqpθ´ηqqm +|Anpηqpθ´ηq|3 +˙ +δk,l ´ K1 +m,k,lpθ, ηq +:“P1 +m,k,lpθ, ηq ´ K1 +m,k,lpθ, ηq. +Above, we are denoting B “ AT A, A “ ∇Xnp0q, Anpηq “ ∇Xnpηq, and K1 +m,k,l +was given in (4.18). We denote +˜Zl,mpηq “ B p +Xi +Bηm +pηqZn,lipηq, +and note that +} ˜Z}CγpR2q ď C}Z}CγpS2q. +We start with a bound for |I1 +n|. We split it as follows +(5.22) +I1 +n “ O1 ` O2, +with +O1 “ pρnpθq +ż +V5R +P 1 +mklpθ, ηq +` ˜Zl,mpηq ´ ˜Zl,mpθq +˘ +dη1dη2 +O2 “ pρnpθq ˜Zl,mpθq +ż +V5R +P 1 +m,k,lpθ, ηqdη1dη2. +Then, we split O1 further +|O1| ď O1,1 ` O1,2, +where +O1,1 “ } ˜Z}CγpV5Rq +ż +V5R +|P1 +m,k,lpθ, ηq||θ ´ η|γdη1dη2, +O1,2 “ pρnpθq +ż +V5R +|K1 +m,k,lpθ, ηq||δη ˜Zl,mpθq|dη1dη2. +For θ P V4R, η P V5R, we have the following bound for P1 +m,k,l (5.21), +(5.23) +|P1 +m,k,l| ď 1 +8π |pBnpηq ´ Bqpθ ´ ηq +|Apθ ´ ηq|3 +| +` 1 +8π |Bnpηqpθ ´ ηq|| +1 +|Apθ ´ ηq|3 ´ +1 +|Anpηqpθ ´ ηq|3 | +ď C }Bnp¨q ´ B}C0pV5Rq +|X|3˝,n|θ ´ η|2 +` C +}Anp¨q ´ A}C0pV5Rq}∇Xn}7 +C0pV5Rq +|X|9˝,n|θ ´ η|2 +, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +35 +where (4.19) has been used for the last term. Then, the first term O1,1 is high-order +but with small coefficients, +O1,1 ď C +´}Bnp¨q ´ B}C0pV5Rq +|X|3˝,n +` +}Anp¨q ´ A}C0pV5Rq}∇Xn}7 +C0pV5Rq +|X|9˝,n +¯ +Rγ +ˆ } ˜Z}CγpV5Rq, +since we have that +}Anp¨q ´ A}C0pV5Rq ď εpRqCp}∇X}C0pV5Rqq, +with εpRq Ñ 0 as R Ñ 0. The kernel bound, with θ P V4R, η P V5R, +|K1 +m,k,lpθ, ηq| ď C }∇Xn}C0pV5Rq +|X|3˝,n +r∇XnsCγpV5Rq +1 +|θ ´ η|2´γ , +(5.24) +gives that +O1,2 ď C }∇Xn}C0pV5Rq}∇Xn}CγpV5Rq +|X|3˝,n +Rγ} ˜Z}C0pV5Rq. +But one also has the bound +|K1 +m,k,lpθ, ηq| ď C +}∇Xn}C0pV5Rq}∇Xn}C +γ +2 pV5Rq +|X|3˝,n +1 +|θ ´ η|2´ γ +2 , +which gives that +O1,2 ď C +}∇Xn}C0pV5Rq}∇Xn}C +γ +2 +|X|3˝,n +Rγ} ˜Z}C +γ +2 pV5Rq. +Joining the two bounds, we obtain +(5.25) +|O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` +} ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q +` εpRq} ˜Z}CγpV5Rq +˘ +, +and +|O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` +}∇S2X}C +γ +2 pB pxn,5RXS2q} ˜Z}C +γ +2 pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +. +To estimate O2, we use (5.21) to integrate by parts, +|O2| ď C|pρnpθq|| ˜Zl,mpθq| +ż +BV5R +ˇˇˇ +1 +|Apθ ´ ηq| ´ +1 +|Xnpθq ´ Xnpηq| +ˇˇˇdlpηq. +Next, we note that since θ P V4R, we have that for η P BV5R, |θ ´ η| ě R +2 . We +compute the difference +(5.26) +1 +|Apθ ´ ηq| ´ +1 +|Xnpθq ´ Xnpηq| +“ +1 +|θ ´ η| +´ +1 +|Ap{ +θ ´ ηq| +´ +1 +|∆ηXnpθq| +¯ +“ +1 +|θ´η| +` +∆ηXnpθq´Ap{ +θ ´ ηq +˘ +¨ +` +∆ηXnpθq`Apz +θ´ηq +˘ +|Apz +θ´ηq||∆ηXnpθq| +` +|Apz +θ´ηq|`|∆ηXnpθq| +˘ , + +36 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +and +∆ηXnpθq´Ap{ +θ ´ ηq “ ∆ηXnpθq´∇Xnpηqp{ +θ ´ ηq`p∇Xnpηq ´ Aqp{ +θ ´ ηq +“ ´EηXnpθq ` pAnpηq ´ Aqp{ +θ ´ ηq, +where EηXnpθq is given in (4.15). Therefore, +|O2| ď C} ˜Z}C0pV5Rq +}∇Xn}C0pV5Rq +|X|3˝,n +ż +BV5R +|EηXnpθq|`}Anp¨q´A}C0pV5Rq +|θ´η| +dlpηq, +hence +|O2| ď Cp|X|˚, }∇S2X}C0pS2qq} ˜Z}C0pV5RqpR +γ +2 }∇S2X}C +γ +2 pB pxn,5RXS2q ` εpRqq, +and +|O2| ď Cp|X|˚, }∇S2X}C0pS2qqRγ}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq. +Together with (5.25) back in (5.22), we conclude that +(5.27) +|I1 +n| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` +} ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q +` εpRq} ˜Z}CγpV5Rq +˘ +, +and also +|I1 +n| ď Cp|X|˚, }∇S2X}C0pS2qq +` +R +γ +2 }∇S2X}C +γ +2 pB pxn,5RXS2q} ˜Z}C +γ +2 pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +. +We proceed to estimate the H¨older seminorm. Take θ, θ ` h P V4R. We use the +splitting (5.22), and start with the estimate for O1: +(5.28) +|O1pθ ` hq ´ O1pθq| +“ |Q1 ` Q2 ` Q3 ` Q4 ` Q5|, +where, using the notation (4.12), +Q1 “ ´pρnpθ ` hq +ż +t|θ´η|ď2|h|uXV5R +P 1 +m,k,lpθ ` h, ηqδη ˜Zl,mpθ ` hqdη1dη2, +Q2 “ ´pρnpθ ` hq +ż +t|θ´η|ď2|h|uXV5R +P 1 +m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2, +Q3 “ pρnpθ ` hqδθ ˜Zl,mpθ ` hq +ż +t|θ´η|ě2|h|uXV5R +P 1 +m,k,lpθ, ηqdη1dη2, +Q4 “ pρnpθ ` hq +ż +t|θ´η|ě2|h|uXV5R +pP 1 +m,k,lpθ, ηq ´ P 1 +m,k,lpθ ` h, ηqqδη ˜Zl,mpθ ` hqdη1dη2, +Q5 “ ppρnpθq ´ pρnpθ ` hqq +ż +V5R +P 1 +m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2. +Recalling (5.21) and the bounds for P1 +m,k,l (5.23) and K1 +m,k,l (5.24), we obtain +|Q1| ` |Q2| ď C +´}Bnp¨q ´ B}C0pV5Rq +|X|3˝,n +` +}Anp¨q ´ A}C0pV5Rq}∇Xn}7 +C0pV5Rq +|X|9˝,n +¯ +ˆ +ż +t|θ´η|ď2|h|uXV5R +r ˜ZsCγpV5Rq +|θ ´ η|2´γ dη1dη2 +` C }∇Xn}C0pV5Rq +|X|3˝,n +ż +t|θ´η|ď2|h|uXV5R +r∇XnsCγpV5Rq} ˜Z}C0pV5Rq +|θ ´ η|2´γ +dη1dη2, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +37 +thus +(5.29) +|Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq +` +}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +|h|γ. +It is clear that we could also obtain the estimate +|Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq|h|γ` +}∇S2X}C +γ +2 pB pxn,5RXS2q} ˜Z}C +γ +2 pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +. +We see that the difference between those two type of estimate comes only from the +kernel K, where we can distribute half a derivative. The same idea propagates along +the lines below, hence we only show the first estimate (5.18). Then, integration by +parts in Q3 gives +|Q3| ď C|pρnpθ ` hq||δθ ˜Zpθ ` hq| +ˆ +ż +t|θ´η|“2|h|uYBV5R +ˇˇˇ +1 +|Apθ ´ ηq| ´ +1 +|Xnpθq ´ Xnpηq| +ˇˇˇdlpηq. +Using (5.26), |h| ď 8R and that θ P V4R, we obtain that +ż +t|θ´η|“2|h|uYBV5R +ˇˇˇ +1 +|Apθ ´ ηq| ´ +1 +|Xnpθq ´ Xnpηq| +ˇˇˇdlpηq +ď C }∇X}C0pV5Rq +|X|3˝,n +ż +t|θ´η|“2|h|uYBV5R +|EηXnpθq| +|θ´η| +` }Anp¨q´A}C0pV5Rq +|θ´η| +dlpηq +ď C }∇X}C0pV5Rq +|X|3˝,n +´ +}∇X}C0pV5Rqpεp|h|q ` εpRqq ` }∇X}C0pV5RqεpRq +¯ +ď C +}∇X}2 +C0pV5Rq +|X|3˝,n +εpRq, +hence +(5.30) +|Q3| ď εpRqCp|X|˚, }∇S2X}C0pS2qq} ˜Z}CγpV5Rq|h|γ. +The term Q4 in (5.28) is estimated by applying the mean-value theorem. As in the +previous terms, we need to consider separately the two kernels in P 1 +m,k,l (5.21). We +take a derivative on P1 +m,k,l, +B +Bθj +P1 +m,k,lpθ, ηq “ δk,l +8π +´ +´ +Bm,j +|Apθ ´ ηq|3 ` +pBnpηqqm,j +|Anpηqpθ ´ ηq|3 +¯ +` δk,l +8π +´ +3pBpθ ´ ηqqmpBpθ ´ ηqqj +|Apθ ´ ηq|5 +´ 3pBnpηqpθ ´ ηqqmpBnpηqpθ ´ ηqqj +|Anpηqpθ ´ ηq|5 +¯ +, +and thus +| B +Bθj +P1 +m,k,lpθ, ηq| ď Cp}∇Xn}C0pV4Rq, |X|˝,nq +εpRq +|θ ´ η|3 , + +38 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +while the derivative of K1 +m,k,l is computed and bounded below +B +Bθj +K1,1 +m,k,lpθ, ηq “ ´δk,l +BXnpηq +Bηm +¨ +ˆ +¨ +˝3 +∆ηXnpθq ¨ BXnpθq +Bθj +|∆ηXnpθq|5 +EηXnpθq +|θ ´ η|3 +` +δη +BXnpθq +Bθj +|∆ηXnpθq|3|θ ´ η|3 +˛ +‚, +| B +Bθj +K1,1pθ, ηq| ď }∇Xn}C0pV4Rq +|X|3˝,n +r∇XnsCγpV4Rq +|θ ´ η|3´γ +ˆ +1`3}∇Xn}C0pV4Rq +|X|˝,n +˙ +ď Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpB pxn,2RXS2q +|θ ´ η|3´γ +. +Therefore, we obtain +(5.31) +|Q4| ď Cp|X|˚, }∇S2X}C0pS2qq +` +}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +|h|γ. +Finally, the estimate for Q5 (5.28) follows from the bounds (5.23) and (5.24), +(5.32) +|Q5| ď Cp}∇Xn}C0pV5Rq, |X|˝,nq}pρn}CγpV5Rq|h|γ +ˆ +´ +εpRq} ˜Z}CγpV5Rq`}∇Xn}CγpV5Rq} ˜Z}C0pV5Rq +¯ +ď Cp|X|˚, }∇S2X}C0pS2qq +` +}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +|h|γ. +We combine the bounds (5.29), (5.30), (5.31), and (5.32) into (5.28) to conclude +that +(5.33) +|O1pθ ` hq ´ O1pθq| ď Cp|X|˚, }∇S2X}C0pS2qq +` +}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +|h|γ. +We continue with the H¨older seminorm for O2 in (5.22). Integrating by parts +O2pθ ` hq ´ O2pθq +“ δk,l +8π pρnpθ ` hq ˜Zl,mpθ ` hq +ˆ +ż +BV5R +´ +1 +|Apθ ` h ´ ηq| ´ +1 +|Xpθ ` hq ´ Xnpηq| +¯ +nmpηqdlpηq +´ δk,l +8π pρnpθq ˜Zl,mpθq +ż +BV5R +´ +1 +|Apθ´ηq| ´ +1 +|Xnpθq´Xnpηq| +¯ +nmpηqdlpηq. +Therefore, +(5.34) +|O2pθ ` hq ´ O2pθq| “ |Q6 ` Q7|, +with +Q6 “ δk,l +8π +´ +pρnpθ ` hq ˜Zl,mpθ ` hq ´ pρnpθq ˜Zl,mpθq +¯ +ˆ +ż +BV5R +| +1 +|Apθ ` h ´ ηq| ´ +1 +|Xpθ ` hq ´ Xnpηq||nmpηqdlpηq, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +39 +Q7 “ δk,l +8π pρnpθq ˜Zl,mpθq +ˆ +ˆ ż +BV5R +´ +1 +|Apθ ` h ´ ηq| ´ +1 +|Xnpθ ` hq ´ Xnpηq| +¯ +nmpηqdlpηq +´ +ż +BV5R +´ +1 +|Apθ ´ ηq| ´ +1 +|Xnpθq ´ Xnpηq| +¯ +nmpηqdlpηq +˙ +. +Using (5.26) and that θ P V4R, we obtain +|Q6| ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq +|X|3˝,n +ˆ +ż +BV5R +|EηXnpθq|`}Anp¨q´A}C0pV5Rq +|θ´η| +dlpηq +ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq +|X|3˝,n +}∇Xn}C0pV5RqεpRq. +Finally, we estimate Q7, +|Q7| ď C} ˜Z}C0pV5Rq +ż +BV5R +| +1 +|Apθ ` h ´ ηq| ´ +1 +|Apθ ´ ηq||dlpηq +` C} ˜Z}C0pV5Rq +ż +BV5R +| +1 +|Xnpθ ` hq ´ Xnpηq| ´ +1 +|Xnpθq ´ Xnpηq||dlpηq. +Performing the differences we obtain that +|Q7| ď Cp|X|˝,n, }∇Xn}C0pV5Rqqp|h| ` |h|1´γRγq +R +} ˜Z}C0pV5Rq|h|γ +ď Cp|X|˝,n, }∇Xn}C0pV5Rqq} ˜Z}C0pV5Rq|h|γ. +Thus, going back to (5.34), we conclude that +(5.35) +|O2pθ`hq´O2pθq|ďCp|X|˚, }∇S2X}C0pS2qq +` +} ˜Z}C0pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +|h|γ, +and together with (5.33) and (5.27) back into (5.22) we obtain the H¨older norm +estimate for I1 +n. Since the kernel in I2 +n (5.20) satisfy the same estimates, we conclude +that +}In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq +` +p1 ` }∇S2X}CγpB pxn,5RXS2qq} ˜Z}C0pV5Rq +` εpRq} ˜Z}CγpV5Rq +˘ +|h|γ. +□ +6. Frozen-coefficient Semigroup +We will need later in the proof (see Lemma 7.6) to deal with the following kernels, +with 0 ď α ď 1: +GαpAθq “ G1pAθq ` αG2pAθq, + +40 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where G1, G2 are given in (4.1). From Section 4.2, we have that +pFθGα,Aq pξq “ +` +FθG1pAθq +˘ +pξq ` α +` +FθG2pAθq +˘ +pξq, +` +FθG1pAθq +˘ +pξq “ +1 +4 detpBq |Uξ|, +` +FθG2pAθq +˘ +pξq “ +1 +4 detpBq |Uξ| +ˆ +P ´ Uξ +|Uξ| b Uξ +|Uξ| +˙ +, +where λ “ ∥A∥F , B “ +? +ATA, P “ ApATAq´1AT, U “ ApATAq´1. +Let us +consider the operator defined in (4.14). In preparation for this, we consider the +operator with ∇X given by a constant matrix and with pg “ I2, as defined in +(4.32): +Lα,AY :“ ˜ +Mα pAq pTFpAq∇Y q “ +´ +˜ +M1pAq ` α ˜ +M2pAq +¯ +pTF pAq∇Y q +where TFpAq is defined in (4.31). The parameter α is useful in Section 7. We +will prove Lα,A is a sectorial operator first. That is to say, we have to estimate +pz ` Lα,Aq´1 where z is in a set with some ω P R, 0 ă δ ă π +2 in the complex plane: +Sω,δ “ tz P C : |argpz ´ ωq| ď π ´ δu. +Since +˜ +MjpAq is a singular integral operator, it is difficult to compute its inverse +operator. However, ˜ +MjpAq is a convolution with kernel Gj pAθq, so we may use its +Fourier multiplier to obtain +pz ` Lα,Aq´1 Y “F´1ppz ´ Lα,Apξqq´1 FY pξqqpθq +where Lα,A is defined in (4.34) (in this section, we will write LA instead of Lα,A). +Then, we will use the Fourier multiplier to estimate the original operator. +In +harmonic analysis, the Fourier multiplier theorem in Lp norms is well-known [26]. +We will use a Fourier multiplier theorem in semi-norms �¨�CγpR2q. +Theorem 6.1. If T is a Fourier multiplier operator with multiplier +m pξq P Cs pRnzt0uq X L8 pRnq , +s ą n +2 , +and +��Bα +ξ m pξq +�� ď Cα |ξ|´|α| +for all |α| ď s, then for all u P Cγ pRnq X L2 pRnq where 0 ă γ ă 1, +�T u�CγpRnq ď Cγ,s,nDm�u�CγpRnq, +where Dm “ max|α|ďs Cα. +Remark 6.2. The proof of Theorem 6.1 may be split into two parts. The first part +is the well-known equivalence between the homogeneous Besov norm ∥¨∥ 9Bγ +8,8pRnq +and H¨older seminorm �¨�CγpRnq [27]. The second part is proving the Fourier mul- +tiplier theorem in homogeneous Besov norms ∥¨∥ 9Bγ +8,8pRnq. Although these results +are classical, we include the proof of the version we need in Appendix A for the +covenience of the reader. +We will compute pz ´ Lα,Apξqq´1 in Section 6.1. Next, for the norm ∥¨∥C0pR2q, +we may expand the result in [47, Section 3.1] in R2. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +41 +Lemma 6.3 ( [47, Proposition 3.1.2]). If �u�CγpR2q ă 8 for some γ P p0, 1q and +Fu pξq “ 0 in a neighborhood of ξ “ 0, then ∥u∥C0pR2q ď C�u�CγpR2q, where C +depends on the neighborhood. +Therefore, we may choose a suitable cutting function ϕ pξq with ϕ pξq “ 1 in a +neighborhood of ξ “ 0. Then, the rest of the work for +���pz ` Lα,Aq´1 Y +��� +C0pR2q is +only +���F´1ppz ´ Lα,Apξqq´1 ϕ pξq FY pξqqpθq +��� +C0pR2q. +6.1. Fundamental estimates. We need some elementary estimates on operators +LA and LA. To achieve it, we first compute the estimate of TFpAq. +Lemma 6.4. Given matrice A1, A2 in DAσ1,σ2, we have σ2 ď ∥A1∥F , ∥A2∥F ď +? +2σ1. Then, we have the following estimates for TF pAq: +|TF pA1qijkl| ďzp0q +M , +|TFpA1qijkl ´ TF pA2qijkl| ďC +ˆ +zp0q +M +σ1 +σ2 +2 +` zp1q +M +˙ +∥A1 ´ A2∥ , +where +zp0q +M “ +max +σ2ďλď +? +2σ1 +|f1 pλq| ` |f2 pλq| , +zp1q +M “ +max +σ2ďλď +? +2σ1 +���� +df1 +dλ +���� ` +���� +df2 +dλ +���� , +f1 pλq “ T +λ , +f2 pλq “ T +λ ´ dT +dλ . +More specific, given Z P Cγ ` +R2˘ Ş L2 ` +R2˘ +with the size of A1, +∥TFpA1qZ∥CγpR2q Ş L2pR2q ďCzp0q +M ∥Z∥CγpR2q Ş L2pR2q +(6.1) +∥pTF pA1q ´ TFpA2qq Z∥CγpR2q Ş L2pR2q ďC +ˆ +zp0q +M +σ1 +σ2 +2 +` zp1q +M +˙ +∥A1 ´ A2∥ ∥Z∥CγpR2q Ş L2pR2q +(6.2) +Proof. Set λi “ ∥Ai∥F . Since +TF pA1qijkl “ f1 pλ1q δikδjl ´ f2 pλ1q +pA1qij pA1qkl +λ2 +1 +, +It is obvious to obtain the result of TF pAqijkl. +Next, +TF pA1qijkl ´ TF pA2qijkl “ +pf1 pλ1q ´ f1 pλ2qq δikδjl +´ pf2 pλ1q ´ f2 pλ2qq +pA1qij pA1qkl +λ2 +1 +´ f2 pλ2q +ˆpA1qij pA1qkl +λ2 +1 +´ +pA2qij pA2qkl +λ2 +2 +˙ +Since +|λ1 ´ λ2| ď ∥A1 ´ A2∥F ď C ∥A1 ´ A2∥ , +we obtain +|fi pλ1q ´ fi pλ2q| ď zp1q +M |λ1 ´ λ2| ď Czp1q +M ∥A1 ´ A2∥ . + +42 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +pA1qij pA1qkl +λ2 +1 +´ +pA2qij pA2qkl +λ2 +2 +“ +pA1 ´ A2qij pA1qkl ` pA2qij pA1 ´ A2qkl +λ2 +1 +` +pA2qij pA2qkl pλ2 ` λ1q pλ2 ´ λ1q +λ2 +1λ2 +2 +, +so +���� +pA1qij pA1qkl +λ2 +1 +´ +pA2qij pA2qkl +λ2 +2 +���� ď C σ1 +σ2 +2 +∥A1 ´ A2∥ +Therefore, +|TF pA1qijkl ´ TF pA2qijkl| ď C +ˆ +zp0q +M +σ1 +σ2 +2 +` zp1q +M +˙ +∥A1 ´ A2∥ +Since TF pA1q, TF pA2q are linear operators, we may obtain the estimates in Cγ ` +R2˘ +X +L2 ` +R2˘ +of TFpA1qZ and pTF pA1q ´ TFpA2qq Z. +□ +Then, because we have estimated ˜ +Mα in Thoerem 5.3, we can obtain the bounds +of Lα,A and its difference Lα,A1 ´ Lα,A2. +Theorem 6.5. Given a matrix A1, A2 P DAσ1,σ2, then for all Y P C1,γ ` +R2˘ +compactly supported, Lα,AY P Cγ ` +R2˘ +X L2 ` +R2˘ +, and +∥Lα,AY ∥CγpRq ď zp0q +M +σ2 +´ +1 ` +´σ1 +σ2 +¯2¯ +∥∇Y ∥CγpR2q Ş L2pR2q , +∥Lα,A1Y ´ Lα,A2Y ∥CγpRq +ď C +σ2 +´zp0q +M +σ2 +´ +1 ` σ1 +σ2 +¯ +` zp1q +M +¯´ +1 ` +´σ1 +σ2 +¯5¯ +∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ +Proof. Given Y +P C1,γ ` +R2˘ +compactly supported, ∇Y is also in L2 ` +R2˘ +, so +pTF pAq∇Y q P Cγ ` +R2˘ +X L2 ` +R2˘ +by Lemma 6.4. By Theorem 5.3 and Lemma +6.4, Lα,AY P Cγ ` +R2˘ +X L2 ` +R2˘ +and +∥Lα,AY ∥CγpRq ď C +σ2 +´ +1 ` +´σ1 +σ2 +¯2¯ +∥TF pAq∇Y ∥CγpR2q Ş L2pR2q +ďzp0q +M +σ2 +´ +1 ` +´σ1 +σ2 +¯2¯ +∥∇Y ∥CγpR2q Ş L2pR2q . +Next, since +Lα,A1Y ´ Lα,A2Y “ +´ +˜ +Mα pA1q ´ ˜ +Mα pA2q +¯ +pTF pA1q∇Y q +´ ˜ +Mα pA2q ppTF pA1q ´ TF pA2qq ∇Y q , +by Theorem 5.3 and Lemma 6.4, +∥Lα,A1Y ´ Lα,A2Y ∥CγpRq +ď +C zp0q +M +σ2 +2 +´ +1 ` +´σ1 +σ2 +¯5¯ +∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ +` C +σ2 +ˆ +zp0q +M +σ1 +σ2 +2 +` zp1q +M +˙ ´ +1 ` +´σ1 +σ2 +¯2¯ +∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ +□ + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +43 +Next, we compute some elementary estimates on the symbol LA. We denote +Gα,A pθq :“ G1 pAθq ` αG2 pAθq. The pairs of eigenvalues and respective eigenvec- +tors of pFθGα,Aq pξq are +´ +p1 ` αq µ +4 , v pξq +¯ +, +´µ +4 , vK pξq +¯ +, +´µ +4 , v0 +¯ +, +(6.3) +and the pairs of MApξq (4.33) are +˜ +T +λ +˜ +|ξ|2 ´ |Aξ|2 +λ2 +¸ +` dT +dλ +|Aξ|2 +λ2 +, Aξ +¸ +, +ˆT +λ |ξ|2 , URπ{2ξ +˙ +, +ˆT +λ |ξ|2 , v0 +˙ +. +(6.4) +Since pFθGα,Aq pξq and MApξq are symmetric positive definite (s.p.d.), LA pξq is +diagonalizable and p.d. Then, we have some estimates of LA and its derivatives. +Lemma 6.6. Given A P DAσ1,σ2, LA and its derivatives satisfy +(i) +σ2 +σ2 +1 +|ξ|´1 ď µ ď σ1 +σ2 +2 +|ξ|´1 , +(6.5) +zm |ξ|2 ď ∥MApξq∥ ď zp0q +M |ξ|2 , +(6.6) +σ2zm +4σ2 +1 +|ξ| ď ∥LApξq∥ ď σ1zp0q +M +2σ2 +2 +|ξ| , +(6.7) +where zm “ minσ1ďλď +? +2σ1 +´ +T +λ , +` T +λ ` dT +dλ +˘ σ2 +2 +λ2 +¯ +, zp0q +M “ maxσ1ďλď +? +2σ1 +` T +λ ` dT +dλ +˘ +. +(ii) +���� +BLA +Bξk +���� ď C1 +σ2 +1 +σ3 +2 +zp0q +M , +(6.8) +���� +B2LA +BξkBξl +���� ď C1 +σ3 +1 +σ4 +2 +zp0q +M |ξ|´1 , +(6.9) +where C1 is a constant that does not depend on α or A. +(iii) +B +Bξj ∆ξLApξq, p∆ξq2 LApξq and +B +Bξj p∆ξq2 LApξq may be written as +B +Bξj +∆ξLApξq “ 1 +|ξ|2 Φp3q +A,jpˆξq, +(6.10) +p∆ξq2 LApξq “ 1 +|ξ|3 Φp4q +A pˆξq, +(6.11) +B +Bξj +p∆ξq2 LApξq “ 1 +|ξ|4 Φp5q +A,jpˆξq, +(6.12) +where Φp3q +A,j, Φp4q +A , Φp5q +A,j are bounded on +���ˆξ +��� “ 1. +Proof. (i) We first note the following inequalities: +(6.13) +σ2 ď ∥B∥ ď σ1, +1 +σ1 +ď +��B´1�� “ ∥U∥ ď 1 +σ2 +. +We thus have: +(6.14) +σ2 +2 ď detpBq “ ∥B∥ +��B´1��´1 ď σ2 +1, + +44 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where we used the fact that B is a 2 ˆ 2 symmetric positive definite matrix. From +(6.13), we immediately have: +(6.15) +|Uξ| “ +��B´1ξ +�� ě 1 +σ1 +|ξ| . +Therefore, +σ2 +σ2 +1 +|ξ|´1 ď µ “ +1 +detpBq |B´1ξ| ď σ1 +σ2 +2 +|ξ|´1 . +Next, through (6.4), one of the eigenvalues of MA is bounded by +ˆT +λ ` dT +dλ +˙ σ2 +2 +λ2 |ξ|2 ď T +λ +˜ +|ξ|2 ´ |Aξ|2 +λ2 +¸ +` dT +dλ +|Aξ|2 +λ2 +ď +ˆT +λ ` dT +dλ +˙ +|ξ|2 , +so we may obtain (6.6). Finally, since LA is diagonalizable and p.d. with LA “ +FθGα,AMA, the eigenvalues of LA are between µ +4 zm |ξ|2 and µ +2 zp0q +M |ξ|2. Hence, we +get the bound (6.7). +(ii) We now turn to (6.8). Note that: +B +Bξk +ˆ +1 +|Uξ| +˙ +“ ´ +` +U TUξ +˘ +k +|Uξ|3 +, +(6.16) +B +Bξk +B +Bξl +ˆ +1 +|Uξ| +˙ +“ ´ +` +U TU +˘ +k,l +|Uξ|3 +` 3 +` +U TUξ +˘ +k +` +U TUξ +˘ +l +|Uξ|5 +. +(6.17) +Likewise, we have: +B +Bξk +ˆpUξqj +|Uξ| +˙ +“ Ujk +|Uξ| ´ pUξqj +` +U TUξ +˘ +k +|Uξ|3 +, +(6.18) +B +Bξk +B +Bξl +ˆpUξqj +|Uξ| +˙ +“ ´Ujk +` +U TUξ +˘ +l +|Uξ|3 +´ Ujl +` +U TUξ +˘ +k +|Uξ|3 +´ +pUξqj +` +U TU +˘ +k,l +|Uξ|3 +` 3pUξqj +` +U TUξ +˘ +k +` +U TUξ +˘ +l +|Uξ|5 +. +(6.19) +The above relations, together with (6.13), show that: +���� +B +Bξk +ˆ +1 +|Uξ| +˙���� ď σ2 +1 +σ2 +1 +|ξ|2 , +���� +B +Bξk +B +Bξl +ˆ +1 +|Uξ| +˙���� ď 4σ3 +1 +σ2 +2 +1 +|ξ|3 , +���� +B +Bξk +ˆpUξqj +|Uξ| +˙���� ď 2σ1 +σ2 +1 +|ξ|, +���� +B +Bξk +B +Bξl +ˆpUξqj +|Uξ| +˙���� ď 6σ2 +1 +σ2 +2 +1 +|ξ|2 . +Thus, we obtain +���� +BFθGα,A +Bξk +���� ď C σ2 +1 +σ3 +2 +|ξ|´2 , +���� +B2FθGα,A +BξkBξl +���� ď C σ3 +1 +σ4 +2 +|ξ|´3 , +Next, set A “ rA1A2s, since +���� +B +Bξk +Aξ +���� “ |Ak| ď λ, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +45 +we have +���� +B +Bξk +pAξ b Aξq +���� ď Cσ1λ |ξ| , +���� +B +Bξk +B +Bξl +pAξ b Aξq +���� ď Cλ2. +Therefore, +���� +B +Bξk +MA +���� ď C +ˆT +λ ` dT +dλ +˙ +|ξ| , +���� +B +Bξk +B +Bξl +MA +���� ď C +ˆT +λ ` dT +dλ +˙ +. +The desired bound (6.8) now follows easily by combining the above estimates and +1 ď σ1 +σ2 . +(iii) By lemma B.1, we may obtain +B +Bξj +∆ξLApξq “ +ÿ +i“1,2 +B +Bξjii +LApξq “ +ÿ +i“1,2 +1 +|ξ|2 +Pjii +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +9 +. +Since σ2 ď +���Uˆξ +��� ď σ1 and Pjii +´ +ˆξ1, ˆξ2 +¯ +is a matrix of polynomials on the domain +���ˆξ +��� “ 1, +Pjiipˆξ1,ˆξ2q +|Uˆξ| +9 +is bounded. Therefore, we have +Φp3q +A,jpˆξq “ +ÿ +i“1,2 +Pjii +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +9 +, +B +Bξj +∆ξLApξq “ 1 +|ξ|2 Φp3q +A,jpˆξq, +p∆ξq2 LApξq “ 1 +|ξ|3 Φp4q +A pˆξq, +B +Bξj +p∆ξq2 LApξq “ 1 +|ξ|4 Φp5q +A,jpˆξq. +Similarly, +Φp4q +A pˆξq “ +ÿ +i,k“1,2 +Pkkii +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +11 +, +Φp5q +A,jpˆξq “ +ÿ +i,k“1,2 +Pjkkii +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +13 +. +□ + +46 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +6.2. Some estimates for z ` Lα,A and pz ` Lα,Aq´1. Since LA pξq is p.d. and +diagonalizable, P ´1 pξq LA pξq P pξq “ D pξq where D is a positive diagonal matrix. +Then, +P ´1 pξq pz ` LA pξqq P pξq “ z ` D pξq , +P ´1 pξq pz ` LA pξqq´1 P pξq “ pz ` D pξqq´1 , +and the eigenvalues of pz ` LA pξqq´1 are on a curve +# +1 +z ` a +ˇˇˇ a P +”σ1zp0q +M +2σ2 +2 +|ξ| , σ2zm +4σ2 +1 +|ξ| +ı+ +. +Remark 6.7. Let λ ą 0 and z P Sω,δ with ω ą 0, β :“ π´argpz ` λ ´ ωq ą δ. +Then, we obtain the following inequality +|z ` λ|2 “λ2 ` |z|2 ´ 2 |z| λ cos β +ěλ2 ` |z|2 ´ 2 |z| λ cos δ +“ cos δ pλ ´ |z|q2 ` p1 ´ cos δq +´ +λ2 ` |z|2¯ +ě p1 ´ cos δq +´ +λ2 ` |z|2¯ +. +Now, we estimate +���Bα +ξ pz ` LA pξqq´1��� with |α| ď 2. +Lemma 6.8. Given Sω,δ with ω ą 0 and A P DAσ1,σ2, we have the following +estimates for all z P Sω,δ, +1 +|z| ` σ1zp0q +M +2σ2 +2 +|ξ| +ď +���pz ` LA pξqq´1��� ď +2 +b +p1 ´ cos δq +` +p σ2zm +4σ2 +1 |ξ|q2 ` |z|2˘, +(6.20) +���� +B +Bξk +pz ` LA pξqq´1 +���� ď C1 +σ2 +1 +σ3 +2 +zp0q +M +4 +p1 ´ cos δq +` +p σ2zm +4σ2 +1 |ξ|q2 ` |z|2 ˘, +(6.21) +and +���� +B2 +BξlBξk +pz ` LA pξqq´1 +���� ď C2 +1 +σ4 +1 +σ6 +2 +zp0q +M +2 +16 +ˆ +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙˙ 3 +2 +` C1 +σ3 +1 +σ4 +2 +zp0q +M +4 +p1 ´ cos δq |ξ| +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙. +(6.22) +Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.t. for +all |α| ď 2, +���Bα +ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T +|z| +|ξ|´|α| , +(6.23) +and +���Bα +ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T +|z|2 +|ξ|1´|α| . +(6.24) + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +47 +Proof. Since the eigenvalues of pz ` LA pξqq´1 are between +´ +z ` σ1zp0q +M +2σ2 +2 +|ξ| +¯´1 +and +´ +z ` σ2zm +4σ2 +1 |ξ| +¯´1 +, it follows that +1 +|z| ` σ1zp0q +M +2σ2 +2 +|ξ| +ď +���pz ` LA pξqq´1��� ď +2 +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙. +Next, +B +Bξk +pz ` LA pξqq´1 “ ´ pz ` LA pξqq´1 +B +Bξk +LA pξq pz ` LA pξqq´1 , +so by (6.8), +���� +B +Bξk +pz ` LA pξqq´1 +���� ď C1 +σ2 +1 +σ3 +2 +zp0q +M +4 +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙. +Finally, +B2 +BξlBξk +pz ` LA pξqq´1 +“ +pz ` LA pξqq´1 B +Bξl +LA pξq pz ` LA pξqq´1 B +Bξk +LA pξq pz ` LA pξqq´1 +` pz ` LA pξqq´1 B +Bξl +LA pξq pz ` LA pξqq´1 +B +Bξk +LA pξq pz ` LA pξqq´1 +´ pz ` LA pξqq´1 +B2 +BξlBξk +LA pξq pz ` LA pξqq´1 . +Therefore, +���� +B2 +BξlBξk +pz ` LA pξqq´1 +���� ď C2 +1 +σ4 +1 +σ6 +2 +zp0q +M +2 +16 +ˆ +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙˙ 3 +2 +` C1 +σ3 +1 +σ4 +2 +zp0q +M +4 +p1 ´ cos δq |ξ| +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙. +From the inequalities +1 +dˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ ď 1 +|z| and +1 +dˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ ď +1 +σ2zm +4σ2 +1 |ξ|, +we obtain +���Bα +ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2 +|z| +|ξ|´|α| , +���Bα +ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2 +|z|2 +|ξ|1´|α| +for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T . +□ +Next, let us consider +���Bα +ξ |ξ| pz ` LA pξqq´1��� with |α| ď 2. + +48 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Lemma 6.9. Given Sω,δ with ω ą 0 and A P DAσ1,σ2, the following estimates hold +for all z P Sω,δ +���|ξ| pz ` LA pξqq´1��� ď +2 |ξ| +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙, +(6.25) +���� +B +Bξk +´ +|ξ| pz ` LA pξqq´1¯���� ď +2 +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ +` C1 +σ2 +1 +σ3 +2 +zp0q +M +4 |ξ| +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙, +(6.26) +���� +B2 +BξlBξk +´ +|ξ| pz ` LA pξqq´1¯���� ď +4 +|ξ| +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ +` C1 +σ3 +1 +σ4 +2 +zp0q +M +12 +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ +` C2 +1 +σ4 +1 +σ6 +2 +zp0q +M +2 +16 |ξ| +ˆ +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙˙ 3 +2 . +(6.27) +Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.t. for +all |α| ď 2, +���Bα +ξ |ξ| pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| . +(6.28) +Proof. By (6.20), it is easy to obtain +���|ξ| pz ` LA pξqq´1��� ď +2 |ξ| +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙. +Next, +B +Bξk +´ +|ξ| pz ` LA pξqq´1¯ +“B |ξ| +Bξk +pz ` LA pξqq´1 ` |ξ| B +Bξk +pz ` LA pξqq´1 . + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +49 +Therefore, with (6.20) and (6.21), +���� +B +Bξk +´ +|ξ| pz ` LA pξqq´1¯���� ď +2 +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ +` C1 +σ2 +1 +σ3 +2 +zp0q +M +4 |ξ| +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙. +Finally, +B2 +BξlBξk +´ +|ξ| pz ` LA pξqq´1¯ +“ +B2 +BξlBξk +|ξ| pz ` LA pξqq´1 ` +B +Bξk +|ξ| B +Bξl +pz ` LA pξqq´1 +` B +Bξl +|ξ| B +Bξk +pz ` LA pξqq´1 ` |ξ| +B2 +BξlBξk +pz ` LA pξqq´1 . +Hence, +���� +B2 +BξlBξk +´ +|ξ| pz ` LA pξqq´1¯���� ď +4 +|ξ| +d +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ +` C1 +σ3 +1 +σ4 +2 +zp0q +M +12 +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙ +` C2 +1 +σ4 +1 +σ6 +2 +zp0q +M +2 +16 |ξ| +ˆ +p1 ´ cos δq +ˆ´ +σ2zm +4σ2 +1 |ξ| +¯2 +` |z|2 +˙˙ 3 +2 . +With +1 +dˆˆ +σ2zm +4σ2 +1 +|ξ| +˙2 +`|z|2 +˙ ď +1 +σ2zm +4σ2 +1 +|ξ|, we obtain +���Bα +ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| +for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T . +□ +Now, we may prove Lα,A is a sectorial operator and obtain the estimate of +pz ´ Lα,Aq´1. +Theorem 6.10. Given a matrix A satisfying the condition (3.2) and a constant +K ą 0 , there exists Sω,δ with ω, δ ą 0 s.t. for all z P Sω,δ +���pz ´ Lα,Aq´1 Y +��� +CγpR2q ď Cω,δ,σ1,σ2,T +|z| +∥Y ∥CγpR2q , +and +���pz ´ Lα,Aq´1 Y +��� +C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q , +for all Y P CγpR2q X LppR2q with 1 ď p ď 2. + +50 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Proof. Since +Lα,AY pθq “ ´F´1LApξqFY , +the Fourier multiplier of pz ´ Lα,Aq´1 is pz ` LAq´1 and of +B +Bθi pz ´ Lα,Aq´1 is +ξi pz ` LAq´1. With (6.23) and (6.28), we obtain there exists Cω,δ,σ1,σ2,T s.t. +�pz ´ Lα,Aq´1 Y �CγpR2q ďCω,δ,σ1,σ2,T +|z| +�Y �CγpR2q, +(6.29) +�pz ´ Lα,Aq´1 Y �C1,γpR2q ďCω,δ,σ1,σ2,T �Y �CγpR2q. +(6.30) +Next, for +���pz ´ Lα,Aq´1 Y +��� +C0pR2q, set ϕpξq to be a smooth and radial cutting func- +tion with a compact support in B p1q and ϕpξq “ 1 in a neighborhood of ξ “ 0. +Then, +���pz ´ Lα,Aq´1 Y +��� +C0pR2q “ +���F´1 pz ` LAq´1 pξqFY +��� +C0pR2q +ď +���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY +��� +C0pR2q +` +���F´1 pz ` LAq´1 pξqϕpξqFY +��� +C0pR2q . +For the first term, since 1´ϕpξq “ 0 in a neighborhood of ξ “ 0 and |1 ´ ϕpξq| ď 1, +by Lemma 6.3, we obtain +���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY +��� +C0pR2q +ďC�F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY �CγpR2q ď Cω,δ,σ1,σ2,T +|z| +�Y �CγpR2q. +For the second term, define the kernel +K0 pθq :“ F´1 pz ` LAq´1 pξqϕpξq, +so +���F´1 pz ` LAq´1 pξqϕpξqFY +��� +C0pR2q +“ ∥K0 ˚ Y ∥C0pR2q ď ∥K0∥L1pR2q ∥Y ∥C0pR2q . +Then, we estimate ∥K0∥L1, by Lemma B.2, we have +∥K0 pθq∥ ďCω,δ,σ1,σ2,T +|z| +1 +1 ` |θ|3 , +so +∥K0∥L1 ď Cω,δ,σ1,σ2,T +|z| +ż +R2 +1 +1 ` |θ|3 dθ ď Cω,δ,σ1,σ2,T +|z| +. +Therefore, +���F´1 pz ` LAq´1 pξqϕpξqFY +��� +C0pR2q ď Cω,δ,σ1,σ2,T +|z| +∥Y ∥C0pR2q , +so +���pz ´ Lα,Aq´1 Y +��� +CγpR2q ď Cω,δ,σ1,σ2,T +|z| +∥Y ∥CγpR2q . + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +51 +Similarly, for +��� B +Bθi pz ´ Lα,Aq´1 Y +��� +C0pR2q, we may use the above technique with the +kernel K1,i +K1,i pθq :“F´1ξi pz ` LAq´1 pξqϕpξq, +∥K1,j pθq∥ ďCω,δ,σ1,σ2,T +1 +1 ` |θ|4 +to obtain +���� +B +Bθi +pz ´ Lα,Aq´1 Y +���� +C0pR2q +ď Cω,δ,σ1,σ2,T ∥Y ∥C0pR2q . +Thus, +���pz ´ Lα,Aq´1 Y +��� +C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q . +□ +Theorem 6.11. Given a matrix A in DAσ1,σ2, there exists Sω,δ with ω, δ ą 0 s.t. +for all z P Sω,δ +∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q , +and +∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q , +for all compactly supported Y P C1,γ ` +R2˘ +. +Proof. Given Y P C1,γ with a compact support, by Theorem 6.5, W “ pz ´ Lα,Aq Y P +Cγ ` +R2˘ +X L2 ` +R2˘ +. Through Theorem 6.10, +∥W ∥CγpR2q ě Cp1q +ω,δ,σ1,σ2,T |z| +���pz ´ Lα,Aq´1 W +��� +CγpR2q “ Cp1q +ω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q +(6.31) +and +∥W ∥CγpR2q ě Cp2q +ω,δ,σ1,σ2,T �pz ´ Lα,Aq´1 W �C1,γpR2q “ Cp2q +ω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q +(6.32) +□ +In each chart x +Xn pθq and Y n pθq, A is ∇Xn p0q and ρnY n is supported in V4R +(3.4). Lα,A will be used in Proposition 6.13 and 7.6. +Remark 6.12. Since +Apξ “ lim +hÑ0 +Xnphpξq ´ Xnp0q +h +, +it is clear that +|X|˚ ď |X|˝,n ď lim inf +ξÑ0 +|Xn pξq ´ Xn p0q| +|ξ| +ď +���Apξ +��� , +C ∥X∥C1,γpS2q ě ∥Xn∥C1pV4Rq ě lim sup +ξÑ0 +|Xn pξq ´ Xn p0q| +|ξ| +ě +���Apξ +��� . +Thus, we may set σ1 “ C ∥X∥C1,γpS2q , σ2 “ |X|˚ . + +52 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Next, set A0 “ ∇x +Xn p0q, so +A0 “ +» +– +2 +0 +0 +2 +0 +0 +fi +fl , +and +L0,A0Y pθq “ ´ +ż +R2 +B +Bηk +1 +4π |θ|W kdη, L0,A0 pξq “ |ξ| +8 I +Set Lpσq +A +“ p1 ´ σq L0,A0 ` σL0,A, which will be used in Proposition 7.6 , then +Lpσq +A Y pθq “ ´ 1 +8π +ż +R2 +B +Bηk +ˆ2 p1 ´ σq +|θ| +` +σ +|Aθ| +˙ +W kdη, +Lpσq +A pξq “|ξ| +4 +ˆ1 ´ σ +2 +` σ +|ξ| +det pBq |Uξ| +˙ +We just have to adapt the above theorems and their proofs. Finally, we obtain +Proposition 6.13. Given X P C1,γpS2q, and our Stereoghraphic projection charts +and the partition functions tx +Xn, ρnu with the radius R, set σ1 “ C ∥X∥C1,γpS2q, +σ2 “ |X|˚. There exists Sω,δ with ω, δ ą 0 s.t. in each chart x +Xn, for Y P C1,γpS2q, +we have the following inequalities: +(6.33) +∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp1q +ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q , +∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp2q +ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q , +��� +´ +z ´ Lpσq +A +¯ +ρnY n +��� +CγpR2q ě Cp3q +ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q , +��� +´ +z ´ Lpσq +A +¯ +ρnY n +��� +CγpR2q ě Cp4q +ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q , +where A “ ∇Xn p0q, and σ, α P p0, 1q. +6.3. Some estimates for etLA. We have two ways of representing the semigroup +e´tLA. One is by the Dunford integral +etLA “ +1 +2πi +ż +ω`γr,η +etz pz ` LAq´1 dz, +(6.34) +where r ą 0, δ ă η ă π +2 and the curve γr,η “ tz P C : |argz| “ π ´ η, |z| ě ru X tz P +C : |argz| ď π ´ η, |z| “ ru. The other is through Fourier transform, +etLAf pθq :“ F´1 ” +e´tLApξqFrfs pξq +ı +pθq “ +ż +R2 KA pt, θ ´ ηq f pηq dη, +where KA pt, θq “ F´1 “ +e´tLApξq‰ +pθq. +Proposition 6.14. Given 0 ď t0 ď t ď T and 0 ď β´α ă 1 +2, we have the following +estimates: +}ept´t0qLAfpt0q}CβpR2q ď +C +pt ´ t0qβ´α }fpt0q}CαpR2q, +}ept´t0qLAfpt0q}L2pt0,T ;CβpR2qq ď C}fpt0q}CαpR2q, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +53 +Proof. By (6.34), Theorem 6.10 and 0 ď t ´ t0 ď T , we have +���ept´t0qLAfpt0q +��� +CαpR2q “ +����� +1 +2πi +ż +ω`γr,η +ept´t0qz pz ` LAq´1 fpt0qdz +����� +CαpR2q +ď Cω,δ,σ1,σ2,T ∥fpt0q∥CαpR2q +ż +ω`γr,η +���ept´t0qz��� 1 +|z|d |z| +ď Cω,δ,σ1,σ2,T ∥fpt0q∥CαpR2q +ż +pt´t0qpω`γr,ηq +|ez| 1 +|z|d |z| +ď Cω,δ,σ1,σ2,T ,T ∥fpt0q∥CαpR2q , +and +���ept´t0qLAfpt0q +��� +C1,αpR2q ď Cω,δ,σ1,σ2,T ∥fpt0q∥CαpR2q +ż +ω`γr,η +���ept´t0qz��� d |z| +ď Cω,δ,σ1,σ2,T +t ´ t0 +∥fpt0q∥CαpR2q +ż +pt´t0qpω`γr,ηq +|ez| d |z| +ď Cω,δ,σ1,σ2,T ,T +t ´ t0 +∥fpt0q∥CαpR2q . +Then, by interpolation theorem, we obtain +���ept´t0qLAfpt0q +��� +CβpR2q ď Cω,δ,σ1,σ2,T ,T,β´α +pt ´ t0qβ´α +∥fpt0q∥CαpR2q , +so +}ept´t0qLAfpt0q}L2pt0,T ;CβpR2qq ď Cω,δ,σ1,σ2,T ,T,β´α}fpt0q}CαpR2q. +□ +Proposition 6.15. Given 0 ď t0 ď t ď T and 0 ď α ă 1, we have the following +estimates: +���� +ż t +t0 +ept´sqLAfpsqds +���� +C1,αpR2q +ď C sup +t0ďsďt }fpsq}CαpR2q, +���� +ż t +t0 +ept´sqLAfpsqds +���� +L2pt0,T ;C1,αpR2qq +ď C}f}L2pt0,T ;CαpR2qq. +Proof. First, define u as +upt, t0, θq :“ urfs pθq“ +ż t +t0 +ept´sqLAfps, θqds “ +ż t +t0 +ż +R2KApt´s, θ´ηqfps, ηqdηds. +Since tLA pξq “ LA ptξq, +KApt ´ s, x ´ yq “ +1 +pt ´ sq2 KA +ˆ +1, θ ´ η +t ´ s +˙ +. + +54 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +By (B.48) in Lemma B.3, +|upt, t0, θq| “ +����� +ż t +t0 +ż +R2 +1 +pt ´ sq2 KA +ˆ +1, θ ´ η +t ´ s +˙ +fps, ηqdηds +����� +ď +ż t +t0 +ż +R2 +1 +pt ´ sq2 +C +1 ` +��� θ´η +t´s +��� +3 |fps, ηq| dηds +ďC ∥f∥C0 +ż t +t0 +ż +R2 +1 +1 ` |θ ´ η|3 dηds ď C ∥f∥C0 T. +Next, we assume f ps, θq “ 0 when s ă t0, so +Bu +Bθi +“ +ż t +t0 +ż +R2 +1 +pt ´ sq3 +BKA +Bθi +ˆ +1, θ ´ η +t ´ s +˙ +fps, ηqdηds +“ +ż t +´8 +ż +R2 +1 +pt ´ sq3 +BKA +Bθi +ˆ +1, θ ´ η +t ´ s +˙ +fps, ηqdηds. +Through (B.49), +ż +R2 +1 +pt ´ sq3 +BKA +Bθi +ˆ +1, θ ´ η +t ´ s +˙ +fps, ηqdη +“ +ż +R2 +1 +pt ´ sq3 +BKA +Bθi +ˆ +1, θ ´ η +t ´ s +˙ +pfps, ηq ´ f ps, θqq dη, +and +����� +ż +R2 +1 +pt ´ sq3 +BKA +Bθi +ˆ +1, θ ´ η +t ´ s +˙ +pfps, ηq ´ f ps, θqq dη +����� +ď +1 +pt ´ sq3 +ż +R2 +|fps, ηq ´ f ps, θq| +1 ` +��� θ´η +t´s +��� +4 +dη +ď C�f ps, ¨q�Cα +1 +pt ´ sq3 +ż +R2 +|θ ´ η|α +1 ` +��� θ´η +t´s +��� +4 dη +“ C�f ps, ¨q�Cα +1 +pt ´ sq1´α +ż +R2 +|θ ´ η|α +1 ` |θ ´ η|4 dη +“ C�f ps, ¨q�Cα +1 +pt ´ sq1´α . +By Lemma B.5, for all m ď t, we obtain +����� +ż t +m +ż +R2 +1 +pt ´ sq3 +BKA +Bθi +ˆ +1, θ ´ η +t ´ s +˙ +fps, ηqdηds +����� +ďC +ż t +m +�f ps, ¨q�Cα +1 +pt ´ sq1´α ds ď C pt ´ mqα Ml r�f ps, ¨q�Cαs ptq , +so set m “ 0, +���� +Bu +Bθi +pt, t0, θq +���� ď CT αMl r�f ps, ¨q�Cαs ptq . + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +55 +For � Bu +Bθi �Cα, we first compute +B2 +BθiBθj +ż M +´8 +ż +R2KApt ´ s, θ ´ ηqfps, ηqdηds +“ +ż M +8 +ż +R2 +1 +pt ´ sq4 +B2KA +BθiBj +ˆ +1, θ ´ η +t ´ s +˙ +fps, ηqdηds, +where M ă t. Then, by (B.50) and Lemma B.5, we have +����� +ż +R2 +1 +pt´sq4 +B2KA +BθiBj +´ +1, θ ´ η +t´s +¯ +fps, ηqdη +����� ď C�f ps, ¨q�Cα +1 +pt´sq2´α +ż +R2 +|θ´η|α +1`|θ´η|4 dη +ď C�f ps, ¨q�Cα +1 +pt ´ sq2´α . +and +����� +B2 +BθiBθj +ż M +´8 +ż +R2 KApt ´ s, θ ´ ηqfps, ηqdηds +����� ď C pt ´ Mqα´1 Ml r�f ps, ¨q�Cαs ptq . +When |θ ´ η| “ 1, we set a cutting function φpsq s.t. φpsq “ 1 on r´8, ´2s and +φpsq “ 0 on r´1, 8s, and define φptqpsq “ φps ´ tq. We obtain +���� +Bu +Bθi +pt, t0, θq ´ Bu +Bηi +pt, t0, ηq +���� “ +���� +Bu +Bθi +rfspθq ´ Bu +Bηi +rfspηq +���� +ď +���� +Bu +Bθi +rφptqfspθq ´ Bu +Bηi +rφptqfspηq +����` +���� +Bu +Bθi +rp1´φptqqfspθq +����` +���� +Bu +Bηi +rp1´φptqqfspηq +���� +ď Cpt´Mqα´1Ml +“ +�φptqfps, ¨q�Cα‰ +ptq` Cpt´mqαMl +“ +�p1´φptqqfps, ¨q�Cα‰ +ptq, +where M “ t ´ 1 and m “ t ´ 2. Therefore, since 0 ď φ ď 1 and �f ps, ¨q�Cα ě 0, +���� +Bu +Bθi +pt, t0, θq ´ Bu +Bηi +pt, t0, ηq +���� ď CMl r�f ps, ¨q�Cαs ptq . +When ρ “ |θ ´ η| ‰ 1, we define uρ pt, t0, θq “ 1 +ρu pρt, ρt0, ρθq , f ρ pt, θq “ f pρt, ρθq +and ¯θ “ θ +ρ , ¯η “ η +ρ , ¯t “ t +ρ, ¯t0 “ t0 +ρ . We have +Buρ +Bt pt, θq “LAuρ pt, θq ` f ρ pt, θq , +Buρ +Bθi +pt, θq “ Bu +Bθi +pρt, ρθq , +so +ˇˇˇ Bu +Bθi +pt, t0, θq´ Bu +Bηi +pt, t0, ηq +ˇˇˇ“ +ˇˇˇBuρ +B¯θi +p¯t, ¯t0, ¯θq´ Buρ +B¯ηi +p¯t, ¯t0, ¯ηq +ˇˇˇďCMlr�f ρ p¯s, ¨q�Cαsptq. +Since �f ρ�Cα p¯sq “ ρα�f�Cα pρ¯sq, +Ml +“ +�f ρ p¯s, ¨q�Cα‰ +p¯tq “ sup +¯rą0 +1 +¯r +ż ¯t +¯t´¯r +�f ρ�Cα p¯sq d¯s “ ρα sup +¯rą0 +1 +¯r +ż ¯t +¯t´¯r +�f�Cα pρ¯sq d¯s +“ρα sup +rą0 +1 +r +ż t +t´r +�f�Cα psq ds “ ραMl r�f ps, ¨q�Cαs ptq . + +56 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Hence, +���� +Bu +Bθi +pt, t0, θq ´ Bu +Bηi +pt, t0, ηq +���� ď CMl r�f ps, ¨q�Cαs ptq |θ ´ η|α , +and by the Hardy-Littlewood maximal function theorem, for all 1 ď p ď 8 +���∥u∥CγpR2q +��� +Lppt0,T q ď +���C ∥f∥C0pR2q ` CMl +“ +�f ps, ¨q�CαpR2q +‰ +ptq +��� +Lppt0,T q +ďC +���∥f∥C0pR2q +��� +Lppt0,T q ` C +��Ml +“ +�f ps, ¨q�CαpR2q +‰ +ptq +�� +Lppt0,T q +ďC +���∥f∥C0pR2q +��� +Lppt0,T q ` C +���f pt, ¨q�CαpR2q +�� +Lppt0,T q +ďC +���∥f∥CγpR2q +��� +Lppt0,T q . +□ +7. Local well-posedness +We write the Peskin problem as an evolution equation +(7.1) +BX +Bt “ FpXq, +t ą 0, +Xp0q “ X0, +where FpXq is given in (4.2). We will make use of Theorem 8.4.1 in [35]: +Theorem 7.1. Let E1 Ă E0 Ă E be Banach spaces and let 0 ă σ ă 1. Given +T ą 0, open set O1 Ă E1 and a function +F : r0, T s ˆ O1 ÞÑ E0, +pt, uq ÞÑ Fpt, uq +such that F and Fu are continuous in r0, T s ˆ O1. If for every p¯t, ¯uq P r0, T s ˆ O1 +we have Fup¯t, ¯uq : E1 ÞÑ E0 is the part of a sectorial operator S : DpSq Ă E ÞÑ E +with DSpσq » E0 and DSpσ ` 1q » E1, then for every ¯t P r0, ts and ¯u P O1 there +are δ ą 0, r ą 0 such that if t0 P r0, T q, |t0 ´ ¯t| ď δ, and }u0 ´ ¯u} ď r then the +problem +v1ptq “ Fpt, vptqq, +t0 ď t ď t0 ` δ, +vpt0q “ u0, +has a unique solution v P Cprt0, t0 ` δs; E1q X C1prt0; t0 ` δs; E0q. +Then, our main result is the following Theorem: +Theorem 7.2. Consider the 3D Peskin problem (7.1) with initial data satisfying +X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0. Then, there +exists some time T ą 0 such that (7.1) has a unique solution X, +X P Cpr0, T s; h1,γpS2qq X C1pr0, T s; hγpS2qq. +Proof. Let Om “ tY P h1,γpS2q : |Y |˚ ě m ą 0u, E1 “ h1,γpS2q, E0 “ hγpS2q, and +E “ hαpS2q, with 0 ă α ă γ. Define the operator S as the linearization of F (4.3) +around X0: +SpX0qY :“ BXFpX0qY “ d +dεFpX0 ` εY q|ε“0. +Since X0 P Om is arbitrary, we can study the Gˆateaux derivative of F at any +X P Om, which is given by +(7.2) +SpXqY “ S1pXqY ` S2pXqY , + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +57 +with +S1pXqY “ ´ +ż +S2 ∇S2GpXppxq ´ Xppyqq¨ +ˆ d +dε +´ +T p|∇S2pXppyq ` εY ppyqq|qp∇S2pX ` εY qqppyq +¯ +|ε“0dpy, +S2pXqY “ ´ +ż +S2∇S2 d +dε +´ +GpXppxq´Xppyq`εpY ppxq ´ Y ppyqqq +¯ˇˇˇ +ε“0¨ +ˆ T p|∇S2Xppyq|q∇S2Xppyqdpy. +It remains to check that the hypothesis of Theorem 7.1 are satisfied, which follow +from Propositions 7.3, 7.4, and 7.6 below. +□ +Proposition 7.3. If m ą 0 and γ P p0, 1q, T P C2, then F (4.3) is a continuous +map from Om Ă h1,γpS2q to hγpS2q. +Proof. Given that FpXq “ NpXqpT p|∇S2X|q∇S2Xq (4.5), we apply Proposition +5.5 to obtain that +}FpXq}CγpS2q ď C +1 +|X|˚ +´ +1 ` +´}∇S2X}C0pS2q +|X˚| +¯2¯ +}T p|∇S2Xq|q∇S2Xq}CγpS2q, +hence recalling the expression for T (4.4), the bound above yields that +(7.3) +}FpXq}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q. +We have thus proved that F maps C1,γpS2q to CγpS2q. We need to show that +it also maps h1,γpS2q to hγpS2q. Having the estimate (7.3), it suffices to show that +if X P h1,γpS2q, then FpXq P hγpS2q. Since hk,γpS2q is the completion of Ck,γpS2q +in any Ck,αpS2q with 0 ă γ ă α ă 1, k ě 0, let X P h1,γpS2q, and tXmum a +sequence Xm P C1,αpS2q, α ą γ, such that Xm Ñ X in C1,γpS2q. It is clear that +the previous estimate (7.3) also holds replacing γ by α, thus FpXmq P Cα. We +conclude that FpXq P hγpS2q by showing that +(7.4) +}FpXmq ´ FpXq}CγpS2q ď C}Xm ´ X}C1,γpS2q. +The estimate will follow from the previous ones by writing FpXmq ´ FpXq as +follows: +(7.5) +FpXmqppxq ´ FpXqppxq “ ∆1ppxq ` ∆2ppxq, +with +∆1ppxq “ ´ +ż +S2∇S2GpXmppxq´Xmppyqq¨ +ˆ +` +T p∇S2Xmppyqq∇S2Xmppyq´T p∇S2Xppyqq∇S2Xppyq +˘ +dpy, +∆2ppxq “ +ż +S2∇S2 +´ +GpXmppxq´Xmppyqq´GpXppxq´Xppyqq +¯ +¨ +ˆ T p∇S2Xppyqq∇S2Xppyqdpy, + +58 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where both terms have kernels given by a derivative. The first term has thus already +been treated, +(7.6) +}∆1}CγpS2q +ď Cp}∇S2Xm}C0pS2q, |Xm|˚q}T p∇S2Xmq∇S2Xm´T p∇S2Xq∇S2X}CγpS2q +ďCp}∇S2Xm}C0pS2q, |Xm|˚, }∇S2X}C0pS2q, |X|˚, }T }C2q +´ +}∇S2pXm´Xq}CγpS2q +` p}∇S2X}CγpS2q ` }∇S2Xm}CγpS2qq}∇S2pXm ´ Xq}C0pS2q +¯ +, +while the second one can be estimated in a similar manner by noticing that one +can always extract Xm ´ X from the difference of kernels. Consider for example +the kernel q1 +k,l (4.7), +q1 +k,lppx, pyq “ ´ 1 +8π +δ pyXippxq +|δ pyXppxq|3 ∇S2Xippyqδk,l. +We can write +1 +8π +δ pyXm +i ppxq +|δ pyXmppxq|3 ∇S2Xm +i ppyqδk,l ´ 1 +8π +δ pyXippxq +|δ pyXppxq|3 ∇S2Xippyqδk,l +“ 1 +8π +δ pypXm +i ´ Xiqppxq∇S2Xm +i ppyq +|δ pyXmppxq|3 +` 1 +8π +δ pyXippxq∇S2pXm +i ´ Xiqppyq +|δ pyXmppxq|3 +` 1 +8π δ pyXippxq∇S2Xippyq +´ +1 +|δ pyXmppxq|3 ´ +1 +|δ pyXppxq|3 +¯ +. +Therefore, it holds that +}∆2}CγpS2q ď Cp}∇S2X}C0pS2q, |X|˚, }∇S2Xm}C0pS2q, |Xm|˚, }T }C1q +ˆ }∇S2X}CγpS2q}∇S2pXm ´ Xq}C0pS2q, +which together with (7.6) proves (7.4). +□ +Proposition 7.4. If m ą 0 and γ P p0, 1q, T P C3, then the Gˆateaux derivative of +F at any X P Om Ă h1,γpS2q (7.2) is continuous and maps h1,γpS2q to hγpS2q. +Proof. The first term S1pXqY in the Gˆateaux derivative of F (7.2) is given in +terms of the operator NpXq (4.5), +(7.7) +S1pXqY ppxq “ NpXqpTSp∇S2Xq∇S2Y qppxq, +with TS given by +(7.8) +TSp∇S2Xq“ T p|∇S2X|q +|∇S2X| +` +´ +T 1p|∇S2X|q´ T p|∇S2X|q +|∇S2X| +¯∇S2X b ∇S2X +|∇S2X|2 +, +and, in index notation, +pTSp∇S2Xq∇S2Y ql,i “ T p|∇S2X|q +|∇S2X| +p∇S2Y ql,i +` +´ +T 1p|∇S2X|q´ T p|∇S2X|q +|∇S2X| +¯p∇S2Xql,ip∇S2Xqq,m +|∇S2X|2 +p∇S2Y qq,m. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +59 +Proposition 5.5 then gives that +(7.9) +}S1pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2Y }CγpSq +ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq +` Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pSq. +We proceed with S2pXqY , +S2pXqY “ S2,1pXqY ` S2,2pXqY , +pS2,jpXqY qkppxq “ +ż +S2 Qj +k,lppx, pyq ¨ pT p∇S2Xq∇S2Xlppyq ´ Clqdpy, +where we define the kernels +Qj +k,lppx, pyq “ ∇S2 d +dε +´ +GjpXppxq ´ Xppyq ` εpY ppxq ´ Y ppyqqq +¯ˇˇˇ +ε“0. +Taking the derivatives we see that +Qj +k,lppx, pyq “ ∇S2 +´ B +Bxi +GjpXppxq ´ XppyqqpYippxq ´ Yippyqq +¯ +“ ´ B +Bxl +B +Bxi +GjpXppxq ´ Xppyqq∇S2XlppyqpYippxq ´ Yippyqq +´ B +Bxi +GjpXppxq ´ Xppyqq∇S2Yippyq, +hence we have the following bound, similarly as in (5.1) +|Qj +k,lppx, pyq| ď C +´ |∇S2Y ppyq| +|∆ pyXppxq|2 ` |∇S2Xppyq||∆ pyY ppxq| +|∆ pyXppxq|3 +¯ +ď C }∇S2Y }C0pS2q +|X|2˚ +´ +1 ` }∇S2X}C0pS2q +|X|˚ +¯ +1 +|px ´ py|2 . +Therefore, +|S2,jpXqY ppxq|ďC }T p∇S2Xq∇S2X}CγpS2q +|X|2˚ +´ +1` }∇S2X}C0pS2q +|X|˚ +¯ +}∇S2Y }C0pS2q, +hence +|S2,jpXqY ppxq| ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q. +Given that the kernels Qj +k,l are also a derivative, the estimate of the H¨older semi- +norm follows the same steps as in Proposition 5.5. In fact, performing the splitting +as in (5.9), we find that +rS2,jpXqY sCγpS2q +ď C }T p∇S2Xq∇S2X}CγpS2q +|X|2˚ +´ +1 ` +´}∇S2X}C0pS2q +|X˚| +¯2¯ +}∇S2Y }C0pS2q, +thus +(7.10) +}S2pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q. + +60 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Together with (7.9), this shows that SpXq maps C1,γpS2q to CγpS2q, +}SpXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pS2q +` Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq +ď Cp|X|˚ , }∇S2X}CγpS2q, }T }C2q ∥∇S2Y ∥CγpS2q . +(7.11) +We are left to show that SpXq also maps h1,γpS2q to hγpS2q and that it is +continuous with respect to X. +We follow the lines below (7.3). +It suffices to +show that if Y P h1,γpS2q, then SpXqY P hγpS2q. Let Y P h1,γpS2q, and tY mum a +sequence Y m P C1,αpS2q, α ą γ, such that Y m Ñ Y in C1,γpS2q. Since SpXqY m P +CαpS2q, we conclude that SpXqY P hγpS2q by showing that +}SpXqY m ´ SpXqY }CγpS2q ď C}Y m ´ Y }C1,γpS2q. +But since we are dealing with a linear operator, the estimate is trivially satisfied +from (7.11). That the Gˆateaux derivative is continuous in X follows along the lines +below (7.4). In fact, +` +SpX1q ´ SpX2q +˘ +Y “ pS1pX1q ´ S1pX2qqY ` pS2pX1q ´ S2pX2qqY , +and we decompose each Sj as in (7.5). Then, it is not hard to see that the following +bound holds +}pSpX2q ´ SpX1qqY }CγpS2q +ď Cp}∇S2X1}C0pS2q, |X1|˚, }∇S2X2}C0pS2q, |X2|˚, }T }C3q +ˆ +´ +}∇S2Y }CγpS2q}∇S2pX1 ´ X2q}CγpS2q +` }∇S2Y }C0pS2q}∇S2pX1 ´ X2q}C0pS2qp}∇S2X1}CγpS2q ` }∇S2X2}CγpS2qq +¯ +. +□ +Proposition 7.5. Consider the linear operator SpXq : C1,γpS2q Ñ CγpS2q defined +in (7.2) with X P C1,γpS2q, T P C2, T ą 0, dT {dλ ě 0. Then, there exists a sector +such that for all z in the sector +}z ´ SpXqY }CγpS2q ě Cp}Y }C1,γpS2q ` |z|}Y }CγpS2qq, +where the constant C depends only on the sector, γ, the norms }X}C1,γpS2q and +}T }C2, and the arc-chord condition |X|˚. +Proof. From (7.2), we have +(7.12) +}pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ }S2pXqY }CγpS2q, +and using (7.10) we obtain that +(7.13) +}pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ C ∥∇S2Y ∥C0pS2q . +We use the notation (7.8). Then, we can write S1pXq (7.7) as +(7.14) +S1pXqY ppxq “ ´ +ż +S2 ∇S2GpXppxq ´ Xppyqq ¨ pTSp∇S2Xppyqq∇S2Y ppyq ´ Cqdpy. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +61 +Next, we introduce the partition of unity ρn (see Section 3.4) to write +S1pXqY ppxq “ +ÿ +n +S1pXq pρnY q ppxq +“´ +ÿ +n +ż +S2∇S2GpXppxq´Xppyqq¨pTSp∇S2Xppyqq∇S2pρnY qppyq´Cnqdpy, +where now we will choose Cn “ 0 or Cn “ TSp∇S2Xppxqq∇S2pρnY qppxq. We will +extensively use that +(7.15) +supp ρn Ă Bpxn,2R X S2. +We notice that +ρnppxqzY ppxq ´ S1pXqpρnY qppxq “ ρnppxq +´ +zY ppxq ´ S1pXqY ppxq +¯ +` +´ +ρnppxqS1pXqY ´ S1pXqpρnY qppxq +¯ +, +hence +2}ρn}CγpS2q}pz ´ S1pXqqY }CγpS2q ě }ρn +´ +zY ´ S1pXqY +¯ +}CγpS2q +ě }zρnY ´ S1pXqpρnY q}CγpS2q ´ }ρnS1pXqY ´ S1pXqpρnY q}CγpS2q, +and summing in n we obtain +(7.16) +}pz ´ S1pXqqY }CγpS2q ě C +ÿ +n +p}I1 +n}CγpS2q ´ }I2 +n}CγpS2qq, +where +(7.17) +I1 +n “ zρnY ´ S1pXqpρnY q, +I2 +n “ ρnS1pXqY ´ S1pXqpρnY q. +Recalling that S1pXq is given in terms of NpXq (7.7), we split I2 +n further: +I2 +n “ rρn, NpXqspTSp∇S2Xq∇S2Y q ` NpXq +` +Tsp∇S2XqY b ∇S2ρn +˘ +, +and by Proposition 5.5 and Lemma 5.9, +}I2 +n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq +´ +}∇S2ρn}C0pS2q}TSp∇S2Xq∇S2Y }C0pS2q +` }Tsp∇S2XqY b ∇S2ρn}CγpS2q +¯ +. +Therefore, +}I2 +n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2ρn}CγpS2q +ˆ p}∇S2X}CγpS2q}Y }CγpS2q`}∇S2Y }C0pS2qq. +We proceed to deal with the term I1 +n (7.17). We introduce the cutoff (5.14) so that +(7.18) +}I1 +n}CγpS2q ě }zρnY ´pρnS1pXqpρnY q}CγpS2q´ }p1 ´ pρnqS1pXqpρnY q}CγpS2q +“ }I1,1 +n }CγpS2q ´ }I1,2 +n }CγpS2q. +The last term will be smoother because the integral is not singular. In fact, recalling +again the expression of S1pXq in terms of NpXq (7.7), we use Proposition 5.7 to +obtain +}I1,2 +n }CγpS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q +ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q. + +62 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Although the constant in the bound above (5.16) becomes large for R small, it will +suffice since it is lower order in terms of regularity for Y . +Next, we proceed to estimate I1,1 +n +(7.18). We can decompose further by intro- +ducing the frozen-coefficient linear operator. We denote by x +Xn the stereographic +projection centered at pxn, i.e. pxn “ x +Xnp0q, and Xnpθq “ Xpx +Xnpθqq (see Section +3.4). Recalling (7.7) and (4.10), (4.17), we have +(7.19) +}I1,1 +n }CγpS2q ě C}zρnY n ´ pρnNpXqpTSp∇S2Xq∇S2pρnY qqn}CγpR2q, +and +I1,1 +n pθq “ zρnpθqY npθq ´ pρnpθqNpXqpTSp∇S2Xq∇S2pρnY qqnpθq +“ J3 ` J4 ` J5 ` J6, +with +J3 “ zρnpθqY npθq ´ LApρnY nqpθq, +J4 “ pρnpθqrMpAq ´ NpXqspTSp∇S2Xq∇S2pρnY qqnpθq +“ pρnpθqrMpAq ´ Mp∇Xnq ´ RnpXnqspTSp∇S2Xq∇S2pρnY qqnpθq, +J5 “ pρnpθq +´ +LApρnY nqpθq ´ MpAqpTSp∇S2Xqq∇S2pρnY q +˘ +npθq +¯ +, +J6 “ p1 ´ pρnpθqqLApρnY nqpθq, +where we denote A the constant matrix A “ ∇Xnp0q, Mp∇Xnq is defined in +(4.14), RpXnq in (4.18), LA in (4.32), pxn “ x +Xnp0q. The bound for J6 follows from +Lemma 5.8 together with Remark 6.12, +}J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TFpAq∇pρnY nq}C0pR2q, +thus, +(7.20) +}J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q. +Next, Lemma 5.10 with Z “ TSp∇S2Xq∇S2pρnY q provides the following bound for +J4: +}J4}CγpR2q +ď Cp|X|˚, }∇S2X}C0pS2qq +` +p1 ` }∇S2X}CγpS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q +` εpRq}TSp∇S2Xq∇S2pρnY q}CγpS2q +˘ +, +where εpRq Ñ 0 as R Ñ 0. Thus, +}J4}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q +´ +εpRq}∇S2pρnY q}CγpS2q +` p1 ` }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q +¯ +. +We proceed with J5. Recalling the expression for TS (7.8), we have +pTSp∇S2Xq∇S2pρnY qqn,lipηq “ pTSp∇S2Xq∇S2pρnY q ˝ x +Xnql,ipηq +“ pdetppgpηqqq´ 1 +2 B +Bηr +pρnYn,qqpηqB p +Xm +Bηr +pηq +´T pλnpηqq +λnpηq +δlqδim +` +` +T 1pλnpηqq ´ T pλnpηq +λnpηq +˘ +BXn,l +Bηj pηq Bx +Xi +Bηj pηq BXn,q +Bηp pηq Bx +Xm +Bηp pηq +pλnpηqq2detppgpηqq +¯ +, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +63 +with λnpηq given in (2.14). Substituting into (4.14), +(7.21) +pMpAqpTSp∇S2Xq∇S2pρnY qqnqkpθq +“ ´ +ż +R2 mm,k,lpθ, ηq B p +Xi +Bηm +pηqpTSp∇S2Xpηqq∇S2pρnY nqpηqql,idη1dη2 +“ ´ +ż +R2 mi,k,lpθ, ηqp ˜TSqipqlpηq B +Bηp +pρnYn,qqpηqdη1dη2 +“ ˜ +MpAqp ˜TS∇pρnY nqqpθq, +where we denote +(7.22) +p ˜TSp∇Xqqipqlpηq“ T pλnpηqq +λnpηq +δpiδql` +` +T 1pλnpηqq´ T pλnpηq +λnpηqq +˘ +BXn,l +Bηi pηq BXl,q +Bηp pηq +pλnpηqq2a +detppgpηqq +. +Thus we can write +J5 “ pρnpθq ˜ +MpAqppTF pAq ´ ˜TSp∇Xqq∇pρnY nqqpθq. +Proposition 5.3 with Lemma 6.12 gives that +}J5}CγpR2q ďCp|X˚, }∇S2X}C0pS2qq}pTF pAq ´ ˜TSp∇Xqq∇pρnY nq}CγpR2q, +and since TF pAq “ ˜TSp∇Xp0qqp0q, we obtain +}J5}CγpR2q ď Cp|X˚, }∇S2X}C0pS2q, }T }C2q +´ +εpRq}∇S2pρnY q}CγpS2q +` }∇S2X}CγpS2q}∇S2pρnY q}C0pS2q +¯ +. +Then, we continue from (7.19), +}I1,1 +n }CγpS2q ě }J3 +n}CγpR2q ´ }J4 +n}CγpR2q ´ }J5 +n}CγpR2q ´ }J6 +n}CγpR2q, +so inserting back the bounds for J4 +n, J5 +n, and J6 +n, we have that +}I1,1 +n }CγpS2q ě }J3 +n}CγpR2q +´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q +´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q +´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q. +Then, we use the frozen-coefficient estimate in Proposition 6.13 for J3 +n (7.19). We +first interpolate the inequalities in Theorem 6.11 to control the lower-order terms, +J3 +n “ pz ´ LAqpρnY nqpθq, +(7.23) +}J3 +n}CγpR2q ě C|z|}ρnY n}CγpR2q ` C}ρnY n}C1,γpR2q ` C|z|1´σ}ρnY n}Cγ`σpR2q, +where σ P r0, 1s is chosen so that 1 ă γ ` σ ă 1 ` γ. Therefore, we have +}I1,1 +n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q +´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q +´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q +´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q. + +64 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +and taking R small enough, +(7.24) +}I1,1 +n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q +´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q. +Next, we go back to (7.16) and substitute the above bound together with (5.17) +and (7.24), +}pz ´ S1pXqqY }CγpS2q +ě +ÿ +n +´ +C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q +¯ +´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2Y }C0pS2q +´ Cp|X|˚, }∇S2X}CγpS2qq}∇S2ρn}CγpS2qp}Y }CγpS2q ` }∇S2Y }C0pS2qq. +Plugging this inequality in (7.13), and then using the triangle inequality and the +fact that Cα ãÑ Cβ for α ě β, we obtain +}pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q ` C|z|1´σ}Y }Cγ`σpS2q +´ CpR, |X|˚, }∇S2X}CγpS2qq}Y }Cγ`σpS2q. +Finally, by moving the sector if necessary to make |z| big, we conclude the result +}pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q. +□ +Proposition 7.6. The Gˆateaux derivative of F at any X P Om, SpXq (7.2), +generates an analytic semigroup on the space h0,γpS2q. +Proof. We need to prove that the operator SpXq is sectorial, i.e., that there exists +a sector such that for any z in the sector +}pz ´ SpXqq´1Y }hγpS2q ď C +|z|}Y }hγpS2q. +Since the norm on little H¨older spaces hγpS2q is the same as in the usual H¨older +spaces CγpS2q, from the previous Proposition 7.5 we are left to prove that the +operator pz ´ SpXqq is invertible from hγpS2q to h1,γpS2q for any z in the sector. +Similarly as we did in Section 6, define the following family of operators SαpXq, +α P r0, 1s, +SαpXqY ppxq +“ ´α +ż +S2∇S2 +´ +G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq +¯ +¨pTSp∇S2Xq∇S2Y ppyqqdpy +´p1 ´ αq +ż +S2∇S2 +´ +G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq +¯ +¨ ∇S2Y ppyqdpy +` αS2pXqY ppxq, +with +Gαpxq “ 1 +8π pG1pxq ` αG2pxqq , x “ px1, x2, x3q, +pG1qi,jpxq “ δij +|x|, +pG2qi,jpxq “ xixj +|x|3 , + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +65 +and TS given in (7.8). In particular, SpXq “ S1pXq. Propositions 7.4 and 7.5 +hold analogously for SαpXq, as all the remainder estimates were always done inde- +pendently for each part of the kernel G and Proposition 6.13 already included the +parameter α. In particular, for all α P r0, 1s, it holds that +1 +C }Y }C1,γpS2q ě }pz ´ SαpXqqY }CγpS2q ě C}Y }C1,γpS2q. +Then, by the method of continuity, it suffices to show that the inverse of pz´S0pXqq +exists. Additionally, define a new family of operators S0,σpXq, σ P r0, 1s, as follows +S0,σpXqY ppxq +“ ´ +ż +S2∇S2 +´ +p1´σqG1ppx´ pyq`σG1pXppxq´Xppyqq +¯ +¨ ∇S2Y ppyqdpy, +so that S0,1pXq “ S0pXq. Then, taking into account (6.33), it is clear that the +following bound holds for all σ P r0, 1s, +1 +C }Y }C1,γpS2q ě }pz ´ S0,σpXqqY }CγpS2q ě C}Y }C1,γpS2q. +Hence, by the method of continuity again we just need to show that pz ´ S0,0pXqq +is invertible. Since the range is closed, it suffices to show that it is also dense. The +operator S0,0pXq is linear and explicit, so we can compute its eigenspace. Since +S0,0pXqY “ 1 +8π +ż +S2 +1 +|px ´ py|∆S2Y ppyqdpy, +we only have to check a component. From [21], a single layer potential u pxq of +g ppyq with +u pxq “ 1 +4π +ż +S2 +1 +|x ´ py|g ppyq dpy +can be transformed into a harmonic problem with +∆u “ 0 +in R3zS2 +�u� “ 0, +�∇u ¨ n� “ g +on S2 +(7.25) +If we denote the standard spherical coordinate system pr, θ, ϕq, where r is the +radial coordinate, θ is the polar angle, and ϕ is the azimuthal angle, then for +the harmonic equation on R3zS2, by separation of variables [15], we obtain some +solutions uℓm pr, θ, ϕq with l ě 0 and |m| ď ℓ : +uℓm pr, θ, ϕq “ +" +ArℓYℓm, +|r| ă 1 +Br´pℓ`1qYℓm, +|r| ą 1 +where Yl,m pθ, ϕq is the usual spherical harmonic function of degree l and order m, +which satisfies the following equation: +(7.26) +∆S2Yℓm “ +1 +sin θ +B +Bθ +ˆ +sin θBYℓm +Bθ +˙ +` +1 +sin2 θ +B2Yℓm +Bϕ2 +“ ´ℓpℓ ` 1qYℓm. +By plugging uℓm into (7.25), we obtain +uℓm “ +1 +2ℓ ` 1Yℓm. +Therefore, combining (7.26), +S0,0pXqYℓ,m “ ´ ℓpℓ ` 1q +2p2ℓ ` 1qYℓ,m, +ℓ ě 0, +|m| ď ℓ. + +66 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Finally, since finite linear combinations of Yl,m are dense in C8pS2q, we conclude +the existence of the inverse pz ´ S0,0pXqq´1 : hγpS2q Ñ h1,γpS2q. +□ +8. Higher Regularity +Following the notation in Section 4.1, we recall that +BX +Bt ppxq “ NpXqpT p∇S2Xqqppxq, +where we denote, with T given in (4.4), +T p∇S2Xq “ T p|∇S2X|q∇S2X. +We localize using the partition tρnu (see Section 3.4), and linearize T at Xppxnq, +B +BtpρnXqppxq “ ρnppxqNpXqpTSp∇S2Xppxnqq∇S2Xqqppxq +` ρnppxqNpXq +´ +T p∇S2Xq´TSp∇S2Xppxnqq∇S2X +¯ +ppxq, +where we recall that TSp∇S2Xq∇S2Y “ +d +dsT p∇S2pX `sY qq|s“0 was given in (7.8). +Next, we introduce the commutators, +(8.1) +B +BtpρnXqppxq “ NpXqpTSp∇S2Xppxnqq∇S2pρnXqqppxq +` NpXq +´ +ρn +` +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +˘¯ +ppxq +` rρn, NpXqspTSp∇S2Xppxnqq∇S2Xqppxq +´ NpXqpTSp∇S2XppxnqqX∇S2ρnqppxq +` rρn, NpXqs +´ +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +¯ +ppxq, +and we move to stereographic coordinates to introduce the frozen-coefficient (at +t “ 0, px “ pxn) operator and the cutoff pρn (5.14), +(8.2) +B +BtpρnXnqpθq “ LA0pρnXnqpθq ` +7ÿ +j“1 +f jpXqpθq, +with +f 1pXqpθq “ NpXq +` +ρn +` +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +˘˘ +npθq, +f 2pXqpθq “ pρnpθq +“ +NpXq ´ MpAq +‰ +pTSp∇S2Xppxnqq∇S2pρnXqqnpθq, +f 3pXqpθq “ pρnpθq +` +MpAqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq´LApρnXnqpθq +˘ +, +f 4pXqpθq “ p1 ´ pρnpθqqNpXqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq, +f 5pXqpθq “ ´p1 ´ pρnpθqqLApρnXnqpθq, +f 6pXqpθq “ rLA ´ LA0spρnXnqpθq, +and +f 7pXqpθq “ ´NpXqpTSp∇S2XppxnqqX∇S2ρnqpx +Xnpθqq +` rρn, NpXqsT p∇S2Xqpx +Xnpθqq, +where A0 “ ∇X0,np0q and A “ ∇Xnp0, tq. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +67 +Proposition 8.1. Let X be the solution to the Peskin problem with initial data +X0 P h1,γpS2q constructed in Theorem 7.2. Then, for any α P p0, 1q, it holds that +X P C1pp0, T s; C3,αpS2qq. Moreover, for any 3 ď n P N and α P p0, 1q, assuming +that T P Cn,α, it holds that X P C1pp0, T s; Cn`1,βpS2qq, for any β ă α. +Proof. The main difficulty is to show the smoothing in space. In fact, assume we +have the higher regularity information X P L8p0, T ; Cn`1,αpS2qq. Then, Theorem +7.2 states that BtX P C0pr0, T s; CγpS2qq, and using the equation together with X P +L8p0, T ; Cn`1,αpS2qq, it is straightforward to see that BtX P L8p0, T ; Cn,αpS2qq. +Finally, to get the continuity in time for the higher regularity, it suffices to inter- +polate taking into account the higher regularity bounds and the continuity in the +lower norm. +We proceed to show the smoothing in space. +We will consider the following +mollified version of the system (8.2), +(8.3) +B +BtpρnXδ +nqpθq “ LA0pρnXδ +nqpθq ` +7ÿ +j“1 +Jδf jpXδqpθq, +with mollified initial data Xδ +0,npθq “ JδX0,npθq, where Jδ is the standard mollifier +by convolution with a Gaussian. +Our main goal is to obtain uniform in δ bounds for Xδ in L8p0, T ; Cn,αpS2qq. +In fact, by construction, Xδ is smooth, and it is not hard to show that the +limit of tXδu in L8p0, T ; C1,γpS2qq is given by the solution X in Theorem 7.2. +Hence, by interpolation and using the uniform bounds, we would conclude that +X P L8p0, T ; Cn,βpS2qq for any β ă α. We thus proceed to obtain the uniform +bounds first, and show the convergence Xδ Ñ X at the end. +We use the semigroup etLA0 to write ρnXδ +nptq in Duhamel form: +(8.4) +ρnXδ +nptq “ ept´t0qLA0 pρnXδ +npt0qq ` +7ÿ +j“1 +ż t +t0 +ept´τqLA0Jδf jpXδqpτqdτ. +In the following, we will repeatedly use the estimates in Propositions 6.14-6.15. For +simplicity of notation, we will drop the index δ and the mollifier Jδ. +Improving regularity to C1,αpS2q: We proceed to obtain bounds in Cα, α P p0, 1q, +for the terms f j. We will be denoting C “ Cp|X|˚, }X}C1pS2q, }T }C2q, CpRq “ +Cp|X|˚, }X}C1pS2q, }T }C2, Rq in the bounds that follow. Lemma 5.6 gives that +}f1}CαpR2q ďC}ρnpT p∇S2Xq´TSp∇S2Xppxnqq∇S2Xq}CαpS2q. +We note that +I :“ T p∇S2Xppx1qq´TSp∇S2Xppxnqq∇S2Xppx1q +´ T p∇S2Xppx2qq`TSp∇S2Xppxnqq∇S2Xppx2q +“ T p∇S2Xppx1qq´Tp∇S2Xppx2qq´TSp∇S2Xppxnqqp∇S2Xppx1q´∇S2Xppx2qq, +thus +|I| “ | +ż 1 +0 +` +TSps∇S2Xppx1q ` p1 ´ sq∇S2Xppx2qq ´ TSp∇S2Xppxnqq +˘ +ds +ˆ p∇S2Xppx1q´∇S2Xppx2qq| +ď C maxt|∇S2Xppx1q´∇S2Xppxnq|, |∇S2Xppx2q´∇S2Xppxnqu +ˆ |∇S2Xppx1q´∇S2Xppx2q|. + +68 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Hence, thanks to the presence of ρn, we obtain +(8.5) +}f 1}CαpR2q ďCεpRq}ρn}CαpS2q}∇S2X}CαpB pxn,2RXS2q. +Using Lemma 5.10, +(8.6) +}f 2}CαpS2q ď C +´ +εpRq}∇S2pρnXq}CαpS2q +` }∇S2X}C +α +2 pB5RppxnqXS2q}∇S2pρnXq}C +α +2 pS2q +¯ +, +while f 3 is identically zero (see (7.21) and (4.32)). Lemma 5.7 provides the estimate +for f 4, +(8.7) +}f 4}CαpS2q ď CpRq}∇S2pρnXq}C0pS2q. +Then, by Lemmas 5.6 and 5.9, +(8.8) +}f 7}CαpS2q ď C}∇S2ρn}CαpS2q, +and Lemma 5.8, +(8.9) +}f 5}CαpS2q ď C}∇S2pρnXq}C0pS2q. +Finally, by writing +rLA ´ LA0spρnXnqpθq “ r ˜ +MpAq ´ ˜ +MpA0qspTF pAq∇pρnXnqqpθq +` ˜ +MpA0qppTF pAq ´ TF pA0qq∇pρnXnqqpθq, +Lemma 5.4 yields that +(8.10) +}f 6}CαpS2q ď C}A ´ A0}}∇S2pρnXq}CαpS2q. +We thus see from (8.4) and Propositions 6.14-6.15 that we can bootstrap to get +that X P L8p0, T ; C1,αpS2qq for all α P p0, 1q. In fact, consider the case γ ă 1 +2 and +take α such that γ ă α ` 1 +2 ď 2γ. Then, with 0 ă ǫ ď γ ´ α arbitrarily small, and +substituting the bounds for f j, we obtain +}ρnX}L2p0,T ;C +3 +2 `αpS2qq ď C}ρnp0qX0}C1`α`ǫpS2qq ` C +7ÿ +j“1 +}f j}L2p0,T ;C +1 +2 `αpR2qq +ď C}ρnp0qX0}C1`γpS2q ` CpR, T q ` }X}2 +L4p0,T ;C1` 1 +4 ` α +2 pS2qq +` CεpR, T q +` +}X}L2p0,T ;C +3 +2 `αpB5RppxnqXS2qq ` }ρnX}L2p0,T ;C +3 +2 `αpS2q +˘ +, +and so +(8.11) +}ρnX}L2p0,T ;C +3 +2 `αpS2qq ď C}ρnp0qX0}C1`γpS2q ` CpR, T q +` CεpR, T q}X}L2p0,T ;C +3 +2 `αpB5RppxnqXS2qq. +Now, we can write +}X}L2p0,T ;C +3 +2 `αpB5RppxnqXS2qq ď } +ÿ +mPMn +ρmX}L2p0,T ;C +3 +2 `αpB5RppxnqXS2qq, +where the cardinal number |Mn| can be picked independent of R and n, since the +radius of the support of ρn and B5Rppxnq X S2 are comparable. Therefore, adding +in n in (8.11) we obtain +ÿ +n +}ρnX}L2p0,T ;C +3 +2 `αpS2qq ď CpR, T q ` CεpR, T q|Mn| +ÿ +n +}ρnX}L2p0,T ;C +3 +2 `αpS2qq, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +69 +hence we conclude that +}X}L2p0,T ;C +3 +2 `αpS2qq ď +ÿ +n +}ρnX}L2p0,T ;C +3 +2 `αpS2qq ď CpR, T q. +In particular, choosing α “ 2γ ´ 1 +2, this uniform bound allows us to conclude that +X P L2p0, T ; C1`2γpS2qq, and thus Xptq P C1`2γpS2q for a.e. t P p0, T q. Now, +pick t0 P p0, T q arbitrarily close to 0 and such that Xpt0q P C1`2γpS2q. It is clear +that we can repeat the process to find t1 ą t0 such that Xpt1q P C1,αpS2q for any +α P p0, 1q (the case γ ą 1 +2 follows in one step). Starting at t1, we find that +}ρnXptq}C1,αpS2q ď }ρnXpt1q}C1,αpS2q ` C +sup +t1ďτďt +7ÿ +m“1 +}fm}CαpS2q. +We can thus take the supremum in t P pt1, T q and use the previous estimates on +f m to conclude that X P L8pt1, T ; C1,αpS2qq for any t1 ą 0 and any α P p0, 1q. +Higher regularity: To study further smoothing, we first show that we can move +derivatives in px to derivatives in py. In fact, denoting ∆X “ Xppxq ´ Xppyq, +∇S2NpXqY ppxq “ +“ ´ +ż +S2 ∇S2,px∇S2, pyGp∆Xq ¨ ∆∇S2Y dpy +“ ´ +ż +S2 +´ +´ ∇S2, py∇S2, pyGp∆Xq`∇S2, py +` +∇S2,pxGp∆Xq`∇S2, pyGp∆Xq +˘¯ +¨∆∇S2Y dpy, +so further integration by parts gives that +(8.12) +∇S2NpXqY ppxq “ NpXq∇S2Y ppxq +´ +ż +S2 ∇S2, py +` +∇S2,pxGp∆Xq`∇S2, pyGp∆Xq +˘ +¨ ∆∇S2Y ppyqdpy +“ NpXq∇S2Y ppxq +´ +ż +S2 ∇S2, py +´ B +Bxi +Gp∆Xq +` +∇S2Xippxq ´ ∇S2Xippyq +˘¯ +¨ ∆∇S2Y ppyqdpy. +Therefore, we take a derivative in (8.1) to get +B +Bt∇S2pρnXqppxq “ NpXq +` +∇S2 +` +TSp∇S2Xppxnqq∇S2pρnXq +˘˘ +ppxq +` NpXq +´ +∇S2` +ρn +` +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +˘˘¯ +ppxq +´ +ż +S2 ∇S2, py +´ B +Bxi +Gp∆Xq +` +∇S2Xippxq ´ ∇S2Xippyq +˘¯ +¨ +ˆ ∆ +` +TSp∇S2Xppxnqq∇S2pρnXq +˘ +ppyqdpy +´ +ż +S2 ∇S2, py +´ B +Bxi +Gp∆Xq +` +∇S2Xippxq ´ ∇S2Xippyq +˘¯ +¨ +ˆ ∆ +` +ρn +` +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +˘ +ppyqdpy +` ∇S2rρn, NpXqsT p∇S2Xqppxq ´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqppxqq. + +70 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +We introduce the frozen-coefficient operator and the cutoff pρn, +B +Bt∇S2pρnXnqpθq “ LA1p∇S2pρnXq +˘ +npθq ` +8ÿ +j“1 +f jpθq, +f 1pθq “ NpXq +´ +∇S2` +ρn +` +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +˘˘¯ +npθq, +f 2pθq “ pρnpθqrNpXq ´ MpAqs +`` +TSp∇S2Xppxnqq∇S2∇S2pρnXq +˘˘ +npθq, +f 3pθq“ pρnpθq +´ +MpAqpTSp∇S2Xppxnqq∇S2∇S2pρnXqqnpθq´LAp∇S2pρnXqqnpθq +¯ +, +f 4pθq “ p1´pρnpθqqNpXq +` +TSp∇S2Xppxnqq∇S2∇S2pρnXq +˘ +npθq, +f 5pθq “ p1´pρnpθqqLAp∇S2pρnXqqnpθq, +f 6pθq “ rLA ´ LA1s +` +∇S2pρnXq +˘ +npθq, +f 7pθq “ ∇S2rρn, NpXqsT p∇S2Xqpx +Xnpθqq +´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqpx +Xnpθqq, +and +f 8pθq “ f 8,1pθq ` f 8,2pθq, +with +f 8,1pθq “ ´ +ż +S2 ∇S2, py +´ B +Bxi +GpXpx +Xpθq ´ Xppyqq +` +∇S2Xipx +Xpθqq ´ ∇S2Xippyq +˘¯ +¨ +ˆ ∆ +` +TSp∇S2Xppxnqq∇S2pρnXq +˘ +ppyqdpy, +f 8,2pθq “ ´ +ż +S2 ∇S2, py +´ B +Bxi +GpXpx +Xpθqq ´ Xppyqq +` +∇S2Xipx +Xpθqq ´ ∇S2Xippyq +˘¯ +¨ +ˆ ∆ +` +ρn +` +T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X +˘ +ppyqdpy, +and A1 “ ∇Xnp0, t1q, A “ ∇Xnp0, tq. Thus, we proceed as we previously did in +(8.4), +(8.13) ∇S2pρnXnqptq “ ept´t0qLA1p∇S2pρnXnqpt0qq ` +8ÿ +j“1 +ż t +t0 +ept´τqLA1f jpτqdτ. +Therefore, to bootstrap and get C2,α regularity we need to use Propositions 6.14- +6.15 and obtain Cα estimates for the forced terms above. The estimate (5.18) in +Lemma 5.10 gives that +}f 2}CαpS2q ď C +´ +εpRq}∇2 +S2pρnXq}CαpS2q +` }∇S2X}CαpS2q}∇2 +S2pρnXq}C0pS2q ` }∇2 +S2pρnXq}C0pS2q +¯ +, +while f 3 ” 0, and Lemmas 5.7 and 5.8 provide that +}f 4}CαpS2q ` }f 5}CαpS2q ď CpRq}∇2 +S2X}C0pS2q. +As done before in (8.10), we have that +}f 6}CαpS2q ď C}A ´ A1}}∇2 +S2pρnXq}CαpS2q. +We interpolate the C2pS2q norm followed by Young’s inequality to get a small +coefficient for the higher regularity part: +}∇2 +S2pρnXq}C0pS2q ď Cpεq}∇S2pρnXq}C1´αpS2q ` ε}∇2 +S2pρnXq}CαpS2q, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +71 +so that +6ÿ +j“2 +}f j}CαpS2q ď CεpR, ∆tq}∇2 +S2pρnXq}CαpS2q`CpRq}∇S2pρnXq}C1´αpS2q, +where from now on the constants C and CpR, ∆tq, ∆t “ t ´ t1, also depend on the +controlled norm }X}L8p0,T ;C1,maxtα,1´αupS2qq. Next, +}f 1}CαpS2q ď Cε}∇S2X}CαpS2q +` C}ρn +` +TSp∇S2Xq∇2 +S2X ´ TSp∇S2ppxnqq∇2 +S2X +˘ +}CαpS2q +ď C ` C}∇S2X}CαpS2q}∇2 +S2X}C0pB pxn,2RXS2q +` CεpRq}∇2 +S2X}CαpB pxn,2RXS2q, +so by interpolation again +}f 1}CαpS2q ď CpRq ` CεpRq}∇2 +S2X}CαpB pxn,2RXS2q. +The term f 8 is lower order, and thus we can control it using interpolation once +more. In fact, taking the derivative in the kernel, we have for f 8,1 +f 8,1ppxq “ +ż +S2 +B +Bxj +B +Bxi +Gp∆Xq∇S2Xjppyq∆∇S2Xi ¨ ∆ +` +TSp∇S2Xppxnqq∇S2pρnXq +˘ +dpy +´ +ż +S2 +B +Bxi +Gp∆Xq∇2 +S2Xippyq ¨ ∆ +` +TSp∇S2Xppxnqq∇S2pρnXq +˘ +dpy, +and therefore, proceeding as in Lemma 5.9, we obtain +}f 8,1}CαpS2q ď C ` C}∇2 +S2X}C0pB pxn,2RXS2q +ď CpRq ` CεpRq}∇2 +S2X}CαpB2RppxqXS2qq. +The estimate for f 8,2 follows in the same manner. Next, we estimate the commu- +tator terms, f 7. Using (8.12), we write +f 7ppxq “ f 7,1ppxq ` f 7,2ppxq ` f 7,3ppxq, +with +f 7,1ppxq “ ´rNpXq∇S2, ρnsT p∇S2Xqppxq, +f 7,2ppxq “ +ż +S2 ∇S2, py +´ B +Bxi +Gp∆Xq +` +∇S2Xippxq ´ ∇S2Xippyq +˘¯ +¨ ∆ +` +ρnT p∇S2Xqppyq +˘ +dpy +´ ρnppxq +ż +S2 ∇S2, py +´ B +Bxi +Gp∆Xq +` +∇S2Xippxq ´ ∇S2Xippyq +˘¯ +¨ ∆ +` +T p∇S2Xqppyq +˘ +dpy +` +ż +S2∇S2, py +´ B +Bxi +Gp∆Xq +` +∇S2Xippxq´∇S2Xippyq +˘¯ +¨∆ +` +TSp∇S2XppxnqqX∇S2ρnppyq +˘ +dpy, +and +f 7,3ppxq “ ´NpXq +` +TSp∇S2Xppxnqq∇S2pX∇S2ρnq +˘ +ppxq +` ∇S2ρnNpXqpT p∇S2Xqqppxq. +The term f 7,3 is lower order and it only requires C1,αpS2q regularity for X, while +the estimate for f 7,2 follows taking the derivative of the kernel, as done for f 8. We +get that +}f 7,2}CαpS2q ď C ` C}∇2 +S2X}C0pB pxn,2RXS2q ` C}∇2 +S2X}C0pS2q. +By interpolation, +}f 7,2}CαpS2q ` }f 7,3}CαpS2q ď Cp˜εq ` C˜ε}∇2 +S2X}CαpS2q, + +72 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +with ˜ε ą 0 to be chosen. The term f 7,1 is written as follows: +f 7,1ppxq “ ´ +ż +S2 ∇S2, pyGpXppxq´Xppyqq +¨ +` +ρnppxq∇S2T p∇S2Xppyqq ´ ∇S2 +` +ρnppyqT p∇S2Xppyqq +˘˘ +dpy +“ rρn, NpXqs∇S2T p∇S2Xqppxq +` +ż +S2 ∇S2, pyGpXppxq´Xppyqq ¨ ∇S2ρnppyqT p∇S2Xppyqqdpy, +hence, by Lemmas 5.9 and 5.6, we have that +}f 7,1}CαpS2q ď C}∇S2ρn}C1,αpS2q ` C}∇S2ρn}C0pS2q}∇2 +S2X}C0pS2q, +and, by interpolation, +}f 7,1}CαpS2q ď Cp˜εq ` C˜ε}∇2 +S2X}CαpS2q. +Then, we have that for any t1 ą 0 and α P p0, 1 +2q, +}∇S2pρnXqptq}L2pt1,T ;C1,αpS2qq ď C}ρnpt1qXpt1q}C +3 +2 `α´ǫ +` +7ÿ +m“1 +}f mpτq}L2pt1,T ;CαpS2qq. +Hence, introducing the estimates above for f j and summing in n, we take the +partition so that εpRq is small enough and then choose ˜ε small enough (depending on +the partition ρn), to obtain that X P L2pt1, T ; C2,αpS2qq. Finally, we can take t2 ą +t1 so that Xpt2q P C2,αpS2q and use (8.4) to conclude that X P L8pt2, T ; C2,αpS2qq +for any t2 ą 0, α P p0, 1 +2q. Now, starting with the upgraded regularity and repeating +the same steps with no changes, we conclude that X P L8pt2, T ; C2,αpS2qq for any +t2 ą 0, α P p0, 1q. +It is not difficult to show by induction that an analogous formula to (8.12) holds +for higher derivatives. Then, by repeating the steps above one can continue the +bootstrapping argument, concluding that for any n P N, X P L8p0, T ; Cn,αpS2qq. +Xδ Ñ X in L8p0, T ; C1,γpS2qq: We write the difference ∆δX :“ Xδ´X as follows: +ρn∆δXnptq “ etLA0 ` +ρn∆δX0,n +˘ +` +7ÿ +j“1 +ż t +0 +ept´τqLA0 +´ +pJδ ´ 1qf jpXδqpτq ` f jpXδqpτq ´ f jpXqpτq +¯ +dτ, +with f jpXq given in (8.2). Thus, +(8.14) +}ρn∆δXn}L8p0,T ;C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q +` C +7ÿ +j“1 +}pJδ ´ 1qf jpXδq}L8p0,T ;CγpR2qq +` C sup +tPr0,T s +7ÿ +j“1 +} +ż t +0 +ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q. +Since X0 P h1,γpS2q and f jpXδq P hγpS2q, the first two terms converge to zero as +δ Ñ 0. For the third term, we need to show that it can be absorbed by the left-hand +side. As in the previous arguments, we will show that for the quasilinear terms we + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +73 +can find a small coefficient, while for the lower order ones we will take advantage +of the extra regularity via (6.14) to get a small coefficient for T small enough. +From previous estimates we immediately get that for j “ 4, 5, 7 and 0 ă ǫ ă 1´γ, +}f jpXδq ´ f jpXq}Cγ`ǫpS2q ď C}∆δX}C1pS2q, +hence (6.14) gives that +sup +tPr0,T s +ÿ +j“4,5,7 +} +ż t +0 +ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q ď C T ǫ}∆δX}C1pS2q. +Also, for f 6, +}f 6pXδq ´ f 6pXq}CγpS2q ď C}A ´ A0}}∇S2pρn∆δXq}CγpS2q +` C}∆δX}C1pS2q}ρnXδ}C1,γpS2q, +so that (6.14)-(6.15) give +} +ż t +0 +ept´τqLA0pf 6pXδq ´ f 6pXqqpτq}L8p0,T ;CγpR2qq ď CεpT q}ρn∆δX}C1,γpS2q +` CT ε}∆δX}C1pS2q. +Next, we proceed with the term f 1: +pf 1pXδq ´ f 1pXqqpθq “ I1 ` I2, +with +I1pθq “ pNpXδq ´ NpXqq +` +ρn +` +T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ˘˘ +npθq +I2pθq “ NpXq +´ +ρn +´ +T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ +´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X +¯¯ +npθq. +The first term is estimated easily from Proposition 5.6 by noticing that one can +always extract ∆δX “ Xδ ´X from the difference of the kernels (similarly as done +in the proof of Proposition 7.3), +(8.15) +}I1}CγpR2q ďC}∇S2∆δX}C0pS2q}ρnpT p∇S2Xδq´TSp∇S2Xδppxnqq∇S2Xδq}CγpS2q +ď CεpRq}∇S2∆δX}C0pS2q}∇S2X}CγpB pxn,2RXS2q, +where we have used (8.5) in the second step. Proposition 5.6 gives that +(8.16) +}I2}CγpR2q “ C}ρn +´ +T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ +´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X +¯ +}CγpR2q. +Denote +JpY qppxq “ T p∇S2Y ppxqq ´ TSp∇S2Y ppxnqq∇S2Y ppxq, +so that we can write +JpXδppx1qq ´ JpXδppx2qq “ p∇S2Xδppx1q ´ ∇S2Xδppx2qq +ˆ +ż 1 +0 +` +TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq +˘ +ds1. + +74 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Then, +JpXδppx1qq ´ JpXδppx2qq ´ JpXppx1qq ` JpXppx2qq “ J1 ` J2, +with +J1 “ p∇S2∆δXppx1q ´ ∇S2∆δXppx2qq +ˆ +ż 1 +0 +` +TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq +˘ +ds1, +and +J2 “ p∇S2Xppx1q ´ ∇S2Xppx2qq +ˆ +´ ż 1 +0 +` +TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq +˘ +ds1 +´ +ż 1 +0 +` +TSps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ´ TSp∇S2Xppxnqq +˘ +ds1 +¯ +. +It follows that +|J1| ď C maxt|∇S2Xδppx1q´∇S2Xδppxnq|, |∇S2Xδppx2q´∇S2Xδppxnqu +ˆ |∇S2∆δXppx1q´∇S2∆δXppx2q|. +We apply the mean-value theorem again in J2: +ż 1 +0 +` +TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq +˘ +ds1 +“ +ż 1 +0 +ż 1 +0 +DTS +` +s2ps1∇S2Xδppx1q`p1´s1q∇S2Xδppx2qq`p1´s2q∇S2Xδppxnq +˘ +ds1ds2 +ˆ +` +s1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2q ´ ∇S2Xδppxnq +˘ +, +hence adding and subtracting we obtain +|J2| ď |J2,1| ` |J2,2|, +with +|J2,1| ď |∇S2Xppx1q ´ ∇S2Xppx2q| +ˆ maxt|∇S2Xδppx1q´∇S2Xδppxnq|, |∇S2Xδppx2q´∇S2Xδppxnq|u +ˆ +ˇˇˇDTSps2ps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ` p1 ´ s2q∇S2Xδppxnqq +´ DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq +ˇˇˇ, +|J2,2| ď |∇S2Xppx1q ´ ∇S2Xppx2q| +ˆ |DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq| +ˆ maxt|∇S2∆δXppx1q´∇S2∆δXppxnq|, |∇S2∆δXppx2q´∇S2∆δXppxnqu. +Going back to (8.16), we thus conclude that +}I2}CγpR2q ď CεpRq}∇S2∆δX}CγpB pxn,2RXS2q. +Together with the bound for I1 (8.15), we obtain the following estimate for f 1: +} +ż t +0 +ept´τqLA0pf 1pXδq ´ f 1pXqqpτq}L8p0,T ;CγpR2qq ď C T ǫ}∆δX}C1pS2q +` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q ` CεpRq}∆δX}C1,γpB pxn,2RXS2q. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +75 +The estimate for f 2 follows in the same way than those for f 1 and f 6, from which +we conclude that +}ρn∆δXn}L8p0,T ;C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q +` C +7ÿ +j“1 +}pJδ ´ 1qf jpXδq}L8p0,T ;CγpR2qq +` CT ε}∆δX}C1pS2q ` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q +` CpεpT q ` εpRqq}∆δX}C1,γpB px,2RXS2q. +Taking R and T small enough, the last term is absorbed by the left-hand side. Then, +adding in n, we conclude that, for T and R small enough, the desired estimate holds +}∆δX}L8p0,T ;C1,γpS2qq ďC}∆δX0}C1,γpS2q`C +7ÿ +j“1 +}pJδ´1qfjpXδq}L8p0,T ;CγpR2qq. +□ +Appendix A. Besov Spaces and Fourier Multiplier Theorems +In this section, we will proof Theorem 6.1. First, define Besov Spaces Bγ +p,q by a +dyadic decomposition. Set a function ψ pξq P C8 pRnq s.t. +ψ pξq “ +" +1 +|ξ| ď 1 +0 +|ξ| ě 2 +and define φ pξq :“ ψ pξq ´ ψ p2ξq. Hence, φ pξq P C8 pRnq and +φ pξq “ 0, +|ξ| ď 1 +2, |ξ| ě 2, +8 +ÿ +j“´8 +φ +` +2´jξ +˘ +“ 1, +|ξ| ‰ 0. +Next, the homogeneous dyadic blocks 9∆j are defined by +9∆jf pθq :“ F´1 ` +φ +` +2´jξ +˘ +Ff pξq +˘ +pθq “ Kj ˚ f +(A.1) +where K pθq :“ F´1 pφ pξqq pθq and Kj pθq :“ F´1 ` +φ +` +2´jξ +˘˘ +pθq “ 2jnK +` +2jθ +˘ +. +Now, given γ a real number and p, q ě 1, we may define homogeneous Besov spaces +9Bγ +p,q pRnq with its seminorm ∥¨∥ 9Bγ +p,qpRnq by +∥f∥ 9Bγ +p,qpRnq :“ +˜ +8 +ÿ +j“´8 +ˆ +2jγ ��� 9∆jf +��� +LppRnq +˙q¸ 1 +q +, +(A.2) +∥f∥ 9Bγ +p,8pRnq :“ sup +jPZ +ˆ +2jγ ��� 9∆jf +��� +LppRnq +˙ +. +(A.3) +According to [27, Remark 2.2.2] and [48, Lemma 8.4.2], we know for all 0 ă γ ă +1, ∥¨∥ 9Bγ +p,qpRnq and �¨�CγpRnq are equivalent, so we only need to prove the Fourier +multiplier theorem on 9Bγ +p,q pRnq. The proof is from [48, Theorem 8.4.3]. +Given T a Fourier multiplier operator with multiplier m pξq P Cs pRnzt0uq X +L8 pRnq, for s ą n +2 and for all |α| ď s, such that +��Bα +ξ m pξq +�� ď Cα |ξ|´|α| , +(A.4) + +76 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +we first define a related kernel with λ ą 0 by Kλ pθq :“ F´1 pφ pξq m pλξqq pθq. +Lemma A.1. Given m pξq satisfying (A.4), then Kλ pθq is bounded by +ż +Rn |Kλ pθq| dθ ď Cs,n,φDm, +(A.5) +where Dm “ max|α|ďs Cα +Proof. Since there exist Cs s.t. for all θ P Rn +´ +1 ` |θ|2¯s +ď Cs +ÿ +|α|ďs +|θα|2 , +(A.6) +we obtain +ż +Rn |Kλ pθq|2 ´ +1 ` |θ|2¯s +dθ +ďCs +ÿ +|α|ďs +ż +Rn |θαKλ pθq|2 dθ +“Cn,s +ÿ +|α|ďs +ż +Rn +��Bα +ξ pφ pξq m pλξqq +��2 dξ +“Cn,s +ÿ +|α|ďs +ż +Rn +����� +ÿ +βďα +ˆ +α +β +˙ ´ +Bβ +ξ φ +¯ +pξq λ|α´β| ´ +Bα´β +ξ +m +¯ +pλξq +����� +2 +dξ +ďCn,sD2 +m +ÿ +|α|ďs +ż +Rn +����� +ÿ +βďα +ˆ +α +β +˙ ´ +Bβ +ξ φ +¯ +pξq λ|α´β| |λξ|´|α´β| +����� +2 +dξ. +(A.7) +supp pφq Ă +␣ +ξ| 1 +2 ď |ξ| ď 2 +( +, so +ż +Rn +����� +ÿ +βďα +ˆ +α +β +˙ ´ +Bβ +ξ φ +¯ +pξq λ|α´β| |λξ|´|α´β| +����� +2 +dξ +“ +ż +1 +2 ď|ξ|ď2 +����� +ÿ +βďα +ˆ +α +β +˙ ´ +Bβ +ξ φ +¯ +pξq |ξ|´|α´β| +����� +2 +dξ ď Cφ,α. +(A.8) +Thus, +ż +Rn |Kλ pθq|2 ´ +1 ` |θ|2¯s +dθ ď Cn,sD2 +m +ÿ +|α|ďs +Cφ,α ď Cs,n,φD2 +m, +(A.9) +and by Holder inequality, +ż +Rn |Kλ pθq| dθ ď +ˆż +Rn |Kλ pθq|2 ´ +1 ` |θ|2¯s +dθ +˙ 1 +2 ˆż +Rn +´ +1 ` |θ|2¯´s +dθ +˙ 1 +2 +ď +a +Cs,n,φDm +ˆż +Rn +´ +1 ` |θ|2¯´s +dθ +˙ 1 +2 +ď Cs,n,φDm. +(A.10) +□ +Next, we may use Kλ pθq and homogeneous Besov semi norm ∥¨∥ 9Bγ +p,qpRnq to prove +the Fourier multiplier theorem. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +77 +Proof of Theorem 6.1. First, set +T j +mf pθq :“ 9∆jTmf pθq “ F´1 ` +φ +` +2´jξ +˘ +m pξq Ff pξq +˘ +pθq . +(A.11) +Since +F´1 ` +φ +` +2´jξ +˘ +m pξq +˘ +pθq “ 2njK2j +` +2jθ +˘ +, +(A.12) +��T j +mf pθq +�� “ +��F´1 ` +φ +` +2´jξ +˘ +m pξq +˘ +˚ f pθq +�� +ď +��F´1 ` +φ +` +2´jξ +˘ +m pξq +˘ +pθq +�� +L1pRnq ∥f∥L8pRnq +ď +��2njK2j +` +2jθ +˘�� +L1pRnq ∥f∥L8pRnq +ď ∥K2j pθq∥L1pRnq ∥f∥L8pRnq +ďCs,n,φDm ∥f∥L8pRnq . +(A.13) +Next, supp +` +φ +` +2´jξ +˘˘ +Ă +␣ +ξ +ˇˇ2j´1 ď |ξ| ď 2j`1 ( +, so for all j ` 1 ď k ´ 1 or j ´ 1 ě +k ` 1 +φ +` +2´jξ +˘ +φ +` +2´kξ +˘ +“ 0. +(A.14) +Therefore, +φ +` +2´jξ +˘ +Ff pξq “ φ +` +2´jξ +˘ ´ +F +´ +9∆j´1f +¯ +pξq ` F +´ +9∆jf +¯ +pξq ` F +´ +9∆j`1f +¯ +pξq +¯ +, +(A.15) +so we obtain +T j +mf pθq “ T j +m +´ +9∆j´1f +¯ +pθq ` T j +m +´ +9∆jf +¯ +pθq ` T j +m +´ +9∆j`1f +¯ +pθq . +(A.16) +Finally, +∥Tmf∥ 9Bγ +8,8pRnq “ sup +jPZ +2jγ ��T j +mf +�� +L8pRnq +ď sup +jPZ +2jγ ���T j +m +´ +9∆j´1f +¯��� +L8pRnq ` sup +jPZ +2jγ ���T j +m +´ +9∆jf +¯��� +L8pRnq ` sup +jPZ +2jγ ���T j +m +´ +9∆j`1f +¯��� +L8pRnq +ďCs,nDm +ˆ +sup +jPZ +2jγ ��� 9∆j´1f +��� +L8pRnq ` sup +jPZ +2jγ ��� 9∆jf +��� +L8pRnq ` sup +jPZ +2jγ ��� 9∆j`1f +��� +L8pRnq +˙ +“Cs,nDm +` +2γ ` 1 ` 2´γ˘ +∥f∥ 9Bγ +8,8pRnq . +(A.17) +Since ∥¨∥ 9Bγ +p,qpRnq and �¨�CγpRnq are equivalent, +�Tmu�CγpRnq ď Cγ,s,nDm�u�CγpRnq. +(A.18) +□ +Appendix B. Estimates for the semigroup e´tLApξq +Lemma B.1. For all β “ β1β2 ¨ ¨ ¨ βk, there exists a matrix Pβ +´ +ˆξ1, ˆξ2 +¯ +of polyno- +mials with degree deg pPβq ď 3 |β| ` 4 s.t. +BβLA pξq “ +1 +|ξ||β|´1 +Pβ +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +2|β|`3 . +(B.1) + +78 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +More specifically, Pβ +´ +ˆξ1, ˆξ2 +¯ +can be written as +pPβqi1i2 +´ +ˆξ1, ˆξ2 +¯ +“ +ÿ +j1,j2ě0,j1`j2ď3|β|`4 +cpβ,i1,i2q +j1,j2 +ˆ +A, U, P, +1 +detpBq, T , dT +dλ , 1 +λ2 +˙ +ˆξj1 +1 ˆξj2 +2 , +(B.2) +where +cpβ,i1,i2q +j1,j2 +ˆ +A, U, P, +1 +detpBq, T , dT +dλ , 1 +λ2 +˙ +“cpβ,i1,i2q +j1,j2 +ˆ +A11, ¨ ¨ ¨ , A32, U11, ¨ ¨ ¨ , U32, P11, ¨ ¨ ¨ , P33, +1 +detpBq, T +λ , dT +dλ , 1 +λ2 +˙ +(B.3) +is a polynomial function. +Moreover, +���Pβ +´ +ˆξ1, ˆξ2 +¯��� +C1pDAσ1,σ2q and +���Uˆξ +��� +C1pDAσ1,σ2q are uniformly bounded, +i.e. there exists Cpβq +σ1,σ2,T and Cpβq +σ1,σ2 s.t. for all ˆξ P S1, +���Pβ +´ +ˆξ1, ˆξ2 +¯��� +C1pDAσ1,σ2q ď Cpβq +σ1,σ2,T , +(B.4) +���Uˆξ +��� +C1pDAσ1,σ2q ď Cpβq +σ1,σ2. +(B.5) +Proof. Since +pFθGα,Aq pξq “ 1 +|ξ| pFθGα,Aq pˆξq +(B.6) +“ 1 +|ξ| +pI ` αPq +���Uˆξ +��� +2 +´ αUˆξ b Uˆξ +4 detpBq +���Uˆξ +��� +3 +, +(B.7) +and Zpξq “ |ξ|2 Zpˆξq where Zpˆξq is a matrix of polynomials with degree 2, +LA pξq “ |ξ| +P0 +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +3 +, +(B.8) +where +P0 “ +pI ` αPq +���Uˆξ +��� +2 +´ αUˆξ b Uˆξ +4 detpBq +˜ +T +λ +˜ +I ´ Aˆξ b Aˆξ +λ2 +¸ +` dT +dλ +Aˆξ b Aˆξ +λ2 +¸ +, +(B.9) +where the degree of P0 is 4. Obviously, P0 can be written as (B.2). + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +79 +When |β| “ 1, +BLA +Bξi +pξq “ ξi +|ξ| +P0 +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +3 +` |ξ| +ÿ +j“1,2 +B +Bˆξj +P0 +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +3 +B +Bξi +ξj +|ξ| +“ ξi +|ξ| +P0 +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +3 +` |ξ| +ÿ +j“1,2 +BP0 +Bˆξj +���Uˆξ +��� +2 +´ 3P0 +´ +U T Uˆξ +¯ +j +���Uˆξ +��� +5 +δij ´ ˆξi ˆξj +|ξ| +“ +Pi +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +5 +, +(B.10) +where +Pi +´ +ˆξ1, ˆξ2 +¯ +“ ˆξiP0 +���Uˆξ +��� +2 +` +ÿ +j“1,2 +˜ +BP0 +Bˆξj +���Uˆξ +��� +2 +´ 3P0 +´ +U T Uˆξ +¯ +j +¸ ´ +δij ´ ˆξi ˆξj +¯ +. +(B.11) +The degrees of all terms are at most 1 ` 4 ` 2 “ 3 ` 2 ` 2 “ 4 ` 1 ` 2 “ 7. Since +���Uˆξ +��� +2 +“ +2ÿ +j1,j2“1 +3ÿ +k“1 +Ukj1Ukj2 ˆξj1 ˆξj2, +(B.12) +´ +U T Uˆξ +¯ +j “ +„ř3 +k“1 UkjUk1 ˆξj +ř3 +k“1 UkjUk2 ˆξj + +, +(B.13) +and BP0 +Bˆξj can be written as the form of (B.2), Pi +´ +ˆξ1, ˆξ2 +¯ +can be written as (B.2). +Thus, the case |β| “ 1 holds. +Suppose |β| ď k´1 holds, for +��¯β +�� “ k, we may rewrite ¯β as ββk where |β| “ k´1. +Then, +B ¯βLA pξq “ BβkBβLA pξq “ +B +Bξβk +¨ +˚ +˝ +1 +|ξ||β|´1 +Pβ +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +2|β|`3 +˛ +‹‚ +“ ´ p|β| ´ 1q +1 +|ξ||β| ˆξβk +Pβ +���Uˆξ +��� +2|β|`3 +` +1 +|ξ||β|´1 +ÿ +j“1,2 +BPβ +Bˆξj +���Uˆξ +��� +2 +´ p2 |β| ` 3q Pβ +´ +U T Uˆξ +¯ +j +���Uˆξ +��� +2|β|`5 +δβkj ´ ˆξβk ˆξj +|ξ| +“ +1 +|ξ|| ¯β| +P¯β +´ +ˆξ1, ˆξ2 +¯ +���Uˆξ +��� +¯β`3 . +(B.14) + +80 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where +Pˆβ “ ´ p|β| ´ 1q ˆξiPβ +���Uˆξ +��� +2 +` +ÿ +j“1,2 +˜ +BPβ +Bˆξj +���Uˆξ +��� +2 +´ +` +2 +��¯β +�� ` 1 +˘ +Pβ +´ +U T Uˆξ +¯ +j +¸ ´ +δβkj ´ ˆξβk ˆξj +¯ +. +(B.15) +The degrees of all terms are at most 1 ` p3 |β| ` 4q ` 2 “ p3 |β| ` 3q ` 2 ` 2 “ +p3 |β| ` 4q ` 1 ` 2 “ 3 +��¯β +�� ` 4. Again, BPβ +Bˆξj is still able to written as the form of +(B.2), so the case +��¯β +�� “ k holds. By Induction, for all β, the formulas (B.1) and +(B.2) hold. +Next, in Pβ, since for all square matrix M, ∥M∥ ď ř |Mij|, we just need to +estimate each element pPβqi1i2, +���pPβqi1i2 +´ +ˆξ1, ˆξ2 +¯��� ď +ÿ ����cpβ,i1,i2q +j1,j2 +ˆ +A, U, P, +1 +detpBq, T +λ , dT +dλ , 1 +λ2 +˙���� +(B.16) +and cpβ,i1,i2q +j1,j2 +´ +A, U, P, +1 +detpBq, T +λ , dT +dλ , 1 +λ2 +¯ +is a form of a polynomial, so we may only +check each variable. Given A P DA, +��� +` +AT A +˘´1��� , +1 +detpBq ď +1 +σ2 +2 and σ1 ď λ ď +? +2σ1, +so all variables are bounded by σ1 and σ2. +���� +BAT A +BAij +���� ď 2λ ď 2 +? +2σ1, +(B.17) +����� +B +` +AT A +˘´1 +BAij +����� “ +���� +` +AT A +˘´1 BAT A +BAij +` +AT A +˘´1 +���� ď 2 +? +2σ1 +σ4 +2 +, +(B.18) +so on DA, U, P are C1 functions and their derivatives are bounded by σ1 and σ2. +Absolutely, +���Uˆξ +��� +C1pDAσ1,σ2q ď Cpβq +σ1,σ2. +(B.19) +Since +����� +B det +` +AT A +˘ +BAij +����� ď λ2 ď 8σ2 +1, +(B.20) +���� +B +BAij +1 +detpBq +���� “ +����� +1 +2 detpBq3 +B det +` +AT A +˘ +BAij +����� ď 8σ2 +1 +σ8 +2 +, +(B.21) +and +���� +Bλ +BAij +���� “ +���� +Aij +λ +���� ď 1 +(B.22) +on DA, +1 +detpBq, T +λ , dT +dλ , 1 +λ2 are also C1 functions and their derivatives are bounded +by σ1 and σ2. Therefore, we obtain +���Pβ +´ +ˆξ1, ˆξ2 +¯��� +C1pDAσ1,σ2q ď Cpβq +σ1,σ2,T . +(B.23) +□ + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +81 +Lemma B.2. Given A P DAσ1,σ2 and ϕ pξq “ ϕ p|ξ|q, a cutting and decreasing +respect |ξ| and supported in B p1q, and set +K0 pθq :“ F´1 ” +pz ` LAq´1 pξqϕpξq +ı +, +(B.24) +K1,j pθq :“ F´1 ” +ξj pz ` LAq´1 pξqϕpξq +ı +. +(B.25) +Then, for all z P Sω,δ, we have the following estimates +∥K0 pθq∥ ďCω,δ,σ1,σ2,T +|z| +1 +1 ` |θ|3 , +(B.26) +∥K1,j pθq∥ ďCω,δ,σ1,σ2,T +1 +1 ` |θ|4 . +(B.27) +Proof. For convenience, we define Hpξq :“ pz ` LAq´1 pξq First, since +p1 ` |θ|3q +ż +R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ +“ +ż +R2p1 ´ i θ +|θ| ¨ iθ|θ|2qeiθ¨ξ pz ` LAq´1 pξqϕpξqdξ +“ +ż +R2 +„ +p1 ´ i θ +|θ| ¨ ∇ξ|∇ξ|2qeiθ¨ξ + +pz ` LAq´1 pξqϕpξqdξ +“ +ż +R2 eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ +|θ| ¨ ∇ξ∆ξ +” +pz ` LAq´1 pξqϕpξq +ı +dξ +“ +ż +Bp1q +eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ +|θ| ¨ ∇ξ∆ξ +” +pz ` LAq´1 pξqϕpξq +ı +dξ, +(B.28) +By (6.20), +����� +ż +Bp1q +eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ +����� ď Cδ,σ1,σ2,T ∥ϕ∥C0 +|z| +. +(B.29) +Next, we compute +B +Bξj +∆ξ +” +pz ` LAq´1 ϕ +ı +“ B +Bξj +∆ξ pz ` LAq´1 ϕ ` ∆ξ pz ` LAq´1 B +Bξj +ϕ ` 2 B +Bξj +∇ξ pz ` LAq´1 ¨ ∇ξϕ +`2∇ξ pz ` LAq´1 ¨ B +Bξj +∇ξϕ ` B +Bξj +pz ` LAq´1 ∆ξϕ ` pz ` LAq´1 B +Bξj +∆ξϕ. +(B.30) +Since ϕ is smooth, we may estimate all of the terms except the first by (6.20) and +(6.24). Obviously, for the last term, +�����i θj +|θ| +ż +Bp1q +eiθ¨ξ pz ` LAq´1 +ˆ B +Bξj +∆ξϕ +˙ +dξ +����� ď Cδ,σ1,σ2,T ∥ϕ∥C3 +|z| +. +(B.31) + +82 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Then, for all |α| “ 1, 2, |α| ` |β| “ 3 +�����i θj +|θ| +ż +Bp1q +eiθ¨ξBα +ξ pz ` LAq´1 pξqBβ +ξ ϕpξqdξ +����� ď ∥ϕ∥C2 +ż +Bp1q +���Bα +ξ pz ` LAq´1 pξq +��� dξ +ďCδ,σ1,σ2,T +|z|2 +∥ϕ∥C2 +ż +Bp1q +|ξ|1´|α| dξ ď Cδ,σ1,σ2,T +|z|2 +∥ϕ∥C2 . +(B.32) +Now, for the first term, +B +Bξj +∆ξH “2 +ÿ +k“1,2 +“ +´ pEjkk ` Ekjk ` Ekkjq ` +` +Ekpjkq ` Epjkqk +˘‰ +` +` +Ejpkkq ` Epkkqj +˘ +´ H B +Bξj +∆ξLAH, +(B.33) +where +Eijk “H BLA +Bξi +H BLA +Bξj +H BLA +Bξk +H, +Ekpjkq “H BLA +Bξk +H B2LA +BξjBξk +H, +Ejpkkq “H BLA +Bξj +H∆ξLAH. +Since +��� BLA +Bξj +��� ≲ 1 and +��� B2LA +BξjBξk +��� ≲ |ξ|´1, ∥Eijk∥ ≲ 1 and +��Ekpjkq +�� , +��Ejpkkq +�� ≲ |ξ|´1. +Therefore, we only have to check pz ` LAq´1 +B +Bξj ∆ξLA pz ` LAq´1 ϕ. LA is an even +function, so the term is an odd function. By LA pξq “ |ξ| LA +´ +ˆξ +¯ +and (6.10), +�����i θj +|θ| +ż +Bp1q +eiθ¨ξ pz ` LA pξqq´1 B +Bξj +∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ +����� +“ +�����´ θj +|θ| +ż +Bp1q +sin pθ ¨ ξq pz ` LA pξqq´1 B +Bξj +∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ +����� +ď +������ +ż +Bp1q +sin +´ +θ ¨ ˆξ |ξ| +¯ ´ +z ` |ξ| LA +´ +ˆξ +¯¯´1 Φp3q +A,j +´ +ˆξ +¯ +|ξ|2 +´ +z ` |ξ| LA +´ +ˆξ +¯¯´1 +ϕ p|ξ|q dξ +������ +“ +������ +ż +S1 +ż 1 +0 +´ +z ` rLA +´ +ˆξ +¯¯´1 +Φp3q +A,j +´ +ˆξ +¯ ´ +z ` rLA +´ +ˆξ +¯¯´1 +ϕ prq +sin +´ +θ ¨ ˆξr +¯ +r +drdˆξ +������ +. +(B.34) + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +83 +By Lemma B.4, we obtain for all ˆξ P S1 +������ +ż 1 +0 +´ +z ` rLA +´ +ˆξ +¯¯´1 +Φp3q +A,j +´ +ˆξ +¯ ´ +z ` rLA +´ +ˆξ +¯¯´1 +ϕ prq +sin +´ +θ ¨ ˆξr +¯ +r +dr +������ +ď2 +���� +´ +z ` rLA +´ +ˆξ +¯¯´1 +Φp3q +A,j +´ +ˆξ +¯ ´ +z ` rLA +´ +ˆξ +¯¯´1 +ϕ prq +���� +C1pr0,1sq +ď4 +���Φp3q +A,j +´ +ˆξ +¯��� ∥ϕ prq∥C1pr0,1sq +���� +´ +z ` rLA +´ +ˆξ +¯¯´1���� +2 +C1pr0,1sq +ďCδ,σ1,σ2,T ∥ϕ prq∥C1pr0,1sq +˜ +1 +|z| ` +1 +|z|2 +¸2 +. +(B.35) +Therefore, +������ +ż +S1 +ż 1 +0 +´ +z ` rLA +´ +ˆξ +¯¯´1 +Φp3q +A,j +´ +ˆξ +¯ ´ +z ` rLA +´ +ˆξ +¯¯´1 +ϕ prq +sin +´ +θ ¨ ˆξr +¯ +r +drdˆξ +������ +ďCδ,σ1,σ2,T ∥ϕ prq∥C1 +˜ +1 +|z| ` +1 +|z|2 +¸2 +. +(B.36) +Since +1 +|z| ď Cω,δ if z P Sω,δ, +����� +ż +Bp1q +eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ +|θ| ¨ ∇ξ∆ξ +” +pz ` LAq´1 pξqϕpξq +ı +dξ +����� +ď ∥ϕ prq∥C3 +4ÿ +k“1 +Cpkq +δ,σ1,σ2,T +1 +|z|k +ďCω,δ,σ1,σ2,T ∥ϕ∥C3 +|z| +, +(B.37) +and +���� +ż +R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ +���� ďCω,δ,σ1,σ2,T ∥ϕ∥C3 +|z| +1 +1 ` |θ|3 . +(B.38) +Next, for K1,j, we use the same technique. p1 ` |θ|4qK1,j pθq becomes +p1 ` |θ|4q +ż +R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ +“ +ż +R2 +“ +p1 ` |∇ξ|4qeiθ¨ξ‰ +ξj pz ` LAq´1 pξqϕpξqdξ +“ +ż +Bp1q +eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2 +ξ +” +ξj pz ` LAq´1 pξqϕpξq +ı +dξ, +(B.39) +By (6.20), +����� +ż +Bp1q +eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ +����� ď Cδ,σ1,σ2,T ∥ϕ∥C0 +|z| +. +(B.40) + +84 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Next, +∆2 +ξ +” +ξj pz ` LAq´1 pξqϕpξq +ı +“ 4 B +Bξj +∆ξ +” +pz ` LAq´1 ϕ +ı +` ξj∆2 +ξ +” +pz ` LAq´1 pξqϕpξq +ı +(B.41) +We have estimated the first term, so let us compute the second, +∆2 +ξ +” +pz ` LAq´1 ϕ +ı +“ +∆2 +ξ pz ` LAq´1 ϕ ` 4∇ξ∆ξ pz ` LAq´1 ¨ ∇ξϕ ` 4 +” +∇2 +ξ pz ` LAq´1 : ∇2 +ξϕ +ı +` 2∆ξ pz ` LAq´1 ∆ξϕ ` 4∇ξ pz ` LAq´1 ¨ ∇ξ∆ξϕ ` pz ` LAq´1 ∆2 +ξϕ. +(B.42) +For the last four terms, we may estimate them by (6.20) and (6.24) again. For the +second term, since Bα +ξ LA ≲ |ξ|1´|α|, by (B.33), we may obtain +����� +ż +Bp1q +eiθ¨ξξj∇ξ∆ξ pz ` LAq´1 pξq ¨ ∇ξϕ pξq dξ +����� +ď ∥ϕ∥C1 +4ÿ +k“1 +Cpkq +δ,σ1,σ2,T +1 +|z|k . +(B.43) +In the first term, we have +∆2 +ξH “ +ÿ +j,k“1,2 +r +8 pEjjkk ` Ejkjk ` Ejkkjq +´8 +` +Ejkpjkq ` Ejpjkqk ` Epjkqjk +˘ +` 4H B2LA +BξjBξk +H B2LA +BξjBξk +H + +` +ÿ +j“1,2 +“ +´4 +` +Ejjpkkq ` Ejpkkqj ` Epkkqjj +˘ +` 4 +` +Ejpjkkq ` Epjkkqj +˘‰ +` 2H∆ξLAH∆ξLAH ´ H∆2 +ξLAH, +(B.44) +where +Ejjkk “H BLA +Bξj +H BLA +Bξj +H BLA +Bξk +H BLA +Bξk +H, +Ejkpjkq “H BLA +Bξj +H BLA +Bξk +H B2LA +BξjBξk +H, +Ejjpkkq “H BLA +Bξj +H BLA +Bξj +H∆ξLAH, +Ejpjkkq “H BLA +Bξk +H B∆ξLA +Bξj +H, +and we only have to compute the ´ pz ` LAq´1 ∆2 +ξLA pz ` LAq´1 term. Since LA +is an even function, ´ξj pz ` LAq´1 ∆2 +ξLA pz ` LAq´1 pξq ϕpξq is odd, again, by + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +85 +Lemma 6.6 and B.4 +�����´ +ż +Bp1q +eiθ¨ξξj pz ` LAq´1 ∆2 +ξLA pz ` LAq´1 pξq ϕpξqdξ +����� +“ +����� +ż +Bp1q +ξj sin pθ ¨ ξq pz ` LA pξqq´1 ∆2 +ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ +����� +ď +������ +ż +Bp1q +sin +´ +θ ¨ ˆξ |ξ| +¯ ´ +z ` |ξ| LA +´ +ˆξ +¯¯´1 ξjΦp4q +A +´ +ˆξ +¯ +|ξ|3 +´ +z ` |ξ| LA +´ +ˆξ +¯¯´1 +ϕ p|ξ|q dξ +������ +“ +������ +ż +S1 +ˆξj +ż 1 +0 +´ +z ` rLA +´ +ˆξ +¯¯´1 +Φp4q +A +´ +ˆξ +¯ ´ +z ` rLA +´ +ˆξ +¯¯´1 +ϕ prq +sin +´ +θ ¨ ˆξr +¯ +r +drdˆξ +������ +ď4 +ż +S1 +���ˆξj +��� +���Φp4q +A +´ +ˆξ +¯��� ∥ϕ prq∥C1pr0,1sq +���� +´ +z ` rLA +´ +ˆξ +¯¯´1���� +2 +C1pr0,1sq +dˆξ +ďCδ,σ1,σ2,T ∥ϕ prq∥C1pr0,1sq +˜ +1 +|z| ` +1 +|z|2 +¸2 +. +(B.45) +Hence, +����� +ż +Bp1q +eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2 +ξ +” +ξj pz ` LAq´1 pξqϕpξq +ı +dξ +����� +ď ∥ϕ prq∥C4 +5ÿ +k“1 +Cpkq +δ,σ1,σ2,T +1 +|z|k +ďCω,δ,σ1,σ2,T ∥ϕ∥C4 , +(B.46) +and +���� +ż +R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ +���� ďCω,δ,σ1,σ2,T ∥ϕ∥C4 +1 +1 ` |θ|4 . +(B.47) +□ +Lemma B.3. Given kpxq “ F´1re´LApξqs, then we have the following estimates +∥kpxq∥ ď C +1 +1 ` |x|3 , +(B.48) +���� +B +Bxi +kpxq +���� ď C +1 +1 ` |x|4 , +(B.49) +���� +B +Bxi +B +Bxj +kpxq +���� ď C +1 +1 ` |x|5 . +(B.50) + +86 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Proof. First, +p1 ` |x|3q +ż +R2 eix¨ξe´LApξqdξ “ +ż +R2p1 ´ i x +|x| ¨ ix|x|2qeix¨ξe´LApξqdξ +“ +ż +R2p1 ´ i x +|x| ¨ ∇ξ|∇ξ|2qeix¨ξe´LApξqdξ +“ +ż +R2 eix¨ξe´LApξq ` i x +|x| ¨ ∇ξ∆ξe´LApξqdξ. +Since e´LApξq ≲ e´|ξ|, we obtain +(B.51) +p1 ` |x|3q ∥kpxq∥ ≲ +´ +1 ` +����� +ż +B1p0q +eix¨ξ x +|x| ¨ ∇ξ∆ξe´LApξqdξ +����� +¯ +. +Next, since +B +Bξi +e´LApξq “ ´ +ż 1 +0 +e´p1´tqLApξq B +Bξi +LApξqe´tLApξqdt, +Bjkk +ξ +e´LApξq +“ ´ +ż 1 +0 +e´p1´t1qLApξqBjkk +ξ +LApξqe´t1LApξqdt1 +` H21 pj, kkq ` H21 pk, jkq ` H21 pjk, kq ` H22 pkk, jq ` H22 pjk, kq ` H22 pk, jkq +´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q , j, p1 ´ t1q p1 ´ t2q t3, k, p1 ´ t1q t2, k, t1q +´ H3 pp1 ´ t1q p1 ´ t2q , k, p1 ´ t1q t2 p1 ´ t3q , j, p1 ´ t1q t2t3, k, t1q +´ H3 pp1 ´ t1q p1 ´ t2q , k, p1 ´ t1q t2, k, t1 p1 ´ t3q , j, t1t3q +´ H3 pp1 ´ t1q p1 ´ t3q , j, p1 ´ t1q t3, k, t1 p1 ´ t2q , k, t1t2q +´ H3 pp1 ´ t1q , k, t1 p1 ´ t2q p1 ´ t3q , j, t1 p1 ´ t2q t3, k, t1t2q +´ H3 pp1 ´ t1q , k, t1 p1 ´ t2q t3, k, t1t2 p1 ´ t3q , j, t1t2t3q , +where +H21 pα, βq “ +ż 1 +0 +ż 1 +0 +e´p1´t1qp1´t2qLApξqBα +ξ LApξqe´p1´t1qt2LApξqBβ +ξ LApξqe´t1LApξqdt1dt2, +H22 pα, βq “ +ż 1 +0 +ż 1 +0 +e´t1LApξqBα +ξ LApξqe´p1´t1qt2LApξqBβ +ξ LApξqe´p1´t1qp1´t2qLApξqdt1dt2. +H3 ps1, α, s2, β, s3, γ, s4q +“ +ż 1 +0 +ż 1 +0 +ż 1 +0 +e´s1LApξqBα +ξ LApξqe´t2LApξqBβ +ξ LApξqe´s3LApξqBγ +ξ LApξqe´s4LApξqdt1dt2dt3. +Since LApξq is even and homogeneous of degree one, we have that the third deriva- +tives of LApξq are odd and homogeneous of degree minus two. The other terms are +less singular and thus the corresponding integrals in (B.51) are bounded directly. +That is to say, +���Bα +ξ LApξq +��� ≲ |ξ|1´|ξ| and +��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, +so ∥H21∥ , ∥H22∥ ≲ |ξ|´1 e´C|ξ|,∥H3∥ ≲ e´C|ξ|, and all of them are integrable. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +87 +Therefore, we only have to estimate the first, i.e. +ż +B1p0q +ż 1 +0 +e´p1´t1qLApξqeix¨ξ x +|x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ. +We can write +e´p1´t1qLApξq B +Bξj +∆ξLApξqe´t1LApξq “ +1 +|ξ|2 e´p1´t1qLApξqΦp3q +A,jpˆξqe´t1LApξq, +where ˆξ “ ξ{|ξ| and Φp3q +A,jpˆξq is even and bounded from below and above, thanks +to the arc-chord condition and the C1 regularity (see (3.2) and Remark 6.12). We +then have that +ż +B1p0q +ż 1 +0 +e´p1´t1qLApξqeix¨ξ x +|x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ +“ +ÿ +j“1,2 +ixj +|x| +ż +S1 +ż 1 +0 +ż 1 +0 +sin px ¨ ˆξ rq +r +e´p1´t1qrLApˆξqΦp3q +A,jpˆξqe´t1rLApˆξqdrdt1dˆξ. +By lemma B.4, we obtain for all x P R2, ˆξ P S1 and t1 P r0, 1s +����� +ż 1 +0 +sin px ¨ ˆξ rq +r +e´p1´t1qrLApˆξqΦp3q +A,jpˆξqe´t1rLApˆξqdr +����� +ď2 +���e´p1´t1qrLApˆξqΦp3q +A,jpˆξqe´t1rLApˆξq��� +C1pr0,1s;rq . +(B.52) +LApˆξq is positive definite and diagonalizable and Φp3q +A,jpˆξq is boundned, so for all +ˆξ P S1 and t1 P r0, 1s, +���e´p1´t1qrLApˆξqΦp3q +A,jpˆξqe´t1rLApˆξq��� +C1pr0,1s;rq ď C +´ +1 ` +���LApˆξq +��� +¯ +. +(B.53) +Therefore, since LApˆξq is bounded on S1, +ż +B1p0q +ż 1 +0 +e´p1´t1qLApξqeix¨ξ x +|x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ +(B.54) +is bounded, and we may conclude that +∥kpxq∥ ≲ p1 ` |x|3q´1. +Next, since +B +Bxi +kpxq “ +ż +R2 iξie´LApξqeix¨ξdξ, +we have +���� +B +Bxi +kpxq +���� “ +ż +R2 |ξi| +���e´LApξq��� dξ ď C, +where C only depends on A. Then, +|x|4 B +Bxi +kpxq “ +ż +R2 p∆ξq2 ´ +iξie´LApξq¯ +eix¨ξdξ, + +88 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +where p∆ξq2 ` +ξie´LApξq˘ +“ 4 B +Bξi ∆ξe´LApξq ` ξi p∆ξq2 e´LApξq. The first term is the +i component in ∇ξ∆ξe´LApξq, so we may claim the term is integratable by the +previous techniques. For the second term, again, +Bjjkk +ξ +LApξq +“ ´ +ż 1 +0 +e´p1´t1qLApξqBjjkk +ξ +LApξqe´t1LApξqdt1 +` H2 pp1 ´ t1q p1 ´ t2q , jjk, p1 ´ t1q t2, k, t1q ` ¨ ¨ ¨ +´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q , jj, p1 ´ t1q p1 ´ t2q t3, k, p1 ´ t1q t2, k, t1q ´ ¨ ¨ ¨ +` H4 pp1 ´ t1q p1 ´ t2q p1 ´ t3q p1 ´ t4q , p1 ´ t1q p1 ´ t2q p1 ´ t3q t4, +j, p1 ´ t1q p1 ´ t2q t3, k, p1 ´ t1q t2, k, t1q ` ¨ ¨ ¨ +where +H2 ps1, α, s2, β, s3q +“ +ż 1 +0 +ż 1 +0 +e´s1LApξqBα +ξ LApξqe´s2LApξqBβ +ξ LApξqe´s3LApξqdt1dt2 +H3 ps1, α, s2, β, s3, γ, s4q +“ +ż 1 +0 +ż 1 +0 +ż 1 +0 +e´s1LApξqBα +ξ LApξqe´s2LApξqBβ +ξ LApξqe´s3LApξqBγ +ξ LApξqe´s4LApξq +dt1dt2dt3 +H4 ps1, α, s2, β, s3, γ, s4, δ, s5q +“ +ż 1 +0 +ż 1 +0 +ż 1 +0 +ż 1 +0 +e´s1LApξqBα +ξ LApξqe´s2LApξqBβ +ξ LApξqe´s3LApξqBγ +ξ LApξqe´s4LApξq +Bδ +ξLApξqe´s5LApξqdt1dt2dt3dt4. +There are 14 H2-type terms, 36 H3-type terms, 24 H4-type terms in Bjjkk +ξ +LA. +Since +���Bα +ξ LApξq +��� ≲ |ξ|1´|ξ| and +��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, ∥ξiH2∥ ≲ +|ξ|´1 e´C|ξ|,∥ξiH3∥ ≲ e´C|ξ| and ∥ξiH4∥ ≲ |ξ| e´C|ξ|. Hence, we only have to check +ż +R2 +ż 1 +0 +e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1dξ. +Since +ş1 +0 e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1 is odd, we may use the same tech- +nique in kpxq term to obtain +|x|4 +���� +B +Bxi +kpxq +���� ď C, +so +���� +B +Bxi +kpxq +���� ≲ +1 +1 ` |x|4 . +Finally, since +B +Bxi +kpxq “ +ż +R2 ξie´LApξqeix¨ξdξ, + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +89 +we have +���� +B +Bxi +kpxq +���� “ +ż +R2 |ξi| +���e´LApξq��� dξ ď C, +where C only depends on A. +□ +Lemma B.4. Given a vector function M prq in C1 pr0, 1sq, we have the following +inequality: for all A ě 0, +���� +ż 1 +0 +M prq sin pArq +r +dr +���� ď 2 ∥Mp0q∥ ` +���� +dM prq +dr +���� +C0pr0,1sq +ď 2 ∥M∥C1pr0,1sq . +(B.55) +Proof. +ż 1 +0 +M prq sin pArq +r +dr “ +ż 1 +0 +M p0q sin pArq +r +dr ` +ż 1 +0 +M prq ´ M p0q +r +sin pArqdr. +(B.56) +For the first term, +���� +ż 1 +0 +M p0q sin pArq +r +dr +���� “ +����M p0q +ż 1 +0 +sin pArq +r +dr +���� “ ∥Mp0q∥ +����� +ż A +0 +sin prq +r +dr +����� +ď ∥Mp0q∥ +ż π +0 +sin prq +r +dr « 1.852 ∥Mp0q∥ ď 2 ∥Mp0q∥ . +(B.57) +For the second term, since +���� +M prq ´ M p0q +r +���� “ +��şr +0 +dM +ds psq ds +�� +r +ď +���� +dM prq +dr +���� +C0pr0,1sq +, +(B.58) +���� +ż 1 +0 +M prq ´ M p0q +r +sin pArqdr +���� ď +ż 1 +0 +���� +M prq ´ M p0q +r +���� |sin pArq| dr +ď +���� +dM prq +dr +���� +C0pr0,1sq +. +(B.59) +Therefore, we obtain the result. +□ +Lemma B.5. Given fptq ě 0 is a locally integrable function on R, we have the +following estimates: +(i) If α ą ´1, for all m ă t, +���� +ż t +m +pt ´ sqα fds +���� ď C pt ´ mqα`1 Mlrfsptq. +(B.60) +(ii) If α ă ´1, for all M ă t, +����� +ż M +´8 +pt ´ sqα fds +����� ď C pt ´ Mqα`1 Mlrfsptq. +(B.61) + +90 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +Proof. (i) By integration by part theorem, +ż t +m +pt ´ sqα fds “ ´ +ż t +m +pt ´ sqα +ˆ B +Bs +ż t +s +f prq dr +˙ +ds +“ pt ´ sqα +ż t +s +f prq dr +ˇˇˇˇ +m +s“t +´ α +ż t +m +pt ´ sqα +ˆ +1 +t ´ s +ż t +s +f prq dr +˙ +ds. +Since +lim sup +sÑt´ +����pt ´ sqα +ż t +s +f prq dr +���� “ lim sup +sÑt´ +����pt ´ sqα`1 +1 +t ´ s +ż t +s +f prq dr +���� +ď lim sup +sÑt´ pt ´ sqα`1 Mlrfsptq “ 0, +�����pt ´ sqα +ż t +s +f prq dr +ˇˇˇˇ +m +s“t +����� ď pt ´ mqα`1 Mlrfsptq. +Next, +���� +ż t +m +pt ´ sqα +ˆ +1 +t ´ s +ż t +s +f prq dr +˙ +ds +���� +ďMlrfsptq +ż t +m +pt ´ sqα ds “ +1 +α ` 1 pt ´ mqα`1 Mlrfsptq. +Therefore, +���� +ż t +m +pt ´ sqα fds +���� ď C pt ´ mqα`1 Mlrfsptq. +(ii) The proof is basically the same. It will be +����� +ż M +´8 +pt ´ sqα fds +����� +“ +�����pt ´ sqα +ż t +s +f prq dr +ˇˇˇˇ +M +s“´8 +´ α +ż M +´8 +pt ´ sqα +ˆ +1 +t ´ s +ż t +s +f prq dr +˙ +ds +����� +ď pt ´ Mqα`1 Mlrfsptq ` lim sup +sÑ´8 pt ´ sqα`1 Mlrfsptq ` Mlrfsptq +ż M +´8 +pt ´ sqα ds +“ pt ´ Mqα`1 Mlrfsptq ´ +1 +α ` 1 pt ´ Mqα`1 Mlrfsptq. +since α ` 1 ă 0. Therefore, +����� +ż M +´8 +pt ´ sqα fds +����� ď C pt ´ Mqα`1 Mlrfsptq. +□ +References +[1] Thomas Alazard and Omar Lazar, Paralinearization of the Muskat equation and appli- +cation to the Cauchy problem, Arch. Ration. Mech. Anal. 237 (2020), no. 2, 545–583, +doi:10.1007/s00205-020-01514-6. +[2] Thomas Alazard and Quoc-Hung Nguyen, Endpoint sobolev theory for the Muskat equation, +Commun. Math. Phys. (2022). + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +91 +[3] +, +Quasilinearization +of +the +3D +Muskat +equation, +and +applications +to +the +critical +Cauchy +problem, +Adv. +Math. +399 +(2022), +Paper +No. +108278, +52, +doi:10.1016/j.aim.2022.108278. +[4] David M. Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, +European J. Appl. Math. 15 (2004), no. 5, 597–607, doi:10.1017/S0956792504005662. +[5] +, Well-posedness of two-phase Darcy flow in 3D, Quart. Appl. Math. 65 (2007), no. 1, +189–203, doi:10.1090/S0033-569X-07-01055-3. +[6] David +M. +Ambrose +and +Michael +Siegel, +Well-posedness +of +two-dimensional +hydroe- +lastic +waves, +Proc. +Roy. +Soc. +Edinburgh +Sect. +A +147 +(2017), +no. +3, +529–570, +doi:10.1017/S0308210516000238. +[7] Stephen Cameron, Global well-posedness for the two-dimensional Muskat problem with slope +less than 1, Anal. PDE 12 (2019), no. 4, 997–1022, doi:10.2140/apde.2019.12.997. +[8] Stephen Cameron and Robert M Strain, Critical local well-posedness for the fully nonlinear +Peskin problem, Comm. Pure Appl. Math., arXiv:2112.00692 (2021), Accepted. +[9] Ke +Chen +and +Quoc-Hung +Nguyen, +The +Peskin +problem +with +9B1 +8,8 +initial +data, +arXiv:2107.13854 (2021). +[10] Ke Chen, Quoc-Hung Nguyen, and Yiran Xu, The Muskat problem with C1 data, Trans. Am. +Math. Soc. 375 (2021), 3039–3060. +[11] C. H. Arthur Cheng and Steve Shkoller, The interaction of the 3D Navier-Stokes equations +with a moving nonlinear Koiter elastic shell, SIAM J. Math. Anal. 42 (2010), no. 3, 1094– +1155, doi:10.1137/080741628. +[12] Peter Constantin, Diego C´ordoba, Francisco Gancedo, and Robert M. Strain, On the global +existence for the Muskat problem, J. Eur. Math. Soc. (JEMS) 15 (2013), no. 1, 201–227, +doi:10.4171/JEMS/360. +[13] Antonio C´ordoba, +Diego C´ordoba, +and Francisco Gancedo, +Interface evolution: +the +Hele-Shaw and +Muskat +problems, +Ann. +of +Math. +(2) +173 (2011), +no. +1, +477–542, +doi:10.4007/annals.2011.173.1.10. +[14] Antonio C´ordoba, Diego C´ordoba, and Francisco Gancedo, Porous media: the Muskat prob- +lem in three dimensions, Anal. PDE 6 (2013), no. 2, 447–497, doi:10.2140/apde.2013.6.447. +[15] R. Courant and D. Hilbert, Methods of mathematical physics. Vol. I, Interscience Publishers, +Inc., New York, N.Y., 1953. +[16] Daniel Coutand and Steve Shkoller, The interaction between quasilinear elastodynamics +and the Navier-Stokes equations, Arch. Ration. Mech. Anal. 179 (2006), no. 3, 303–352, +doi:10.1007/s00205-005-0385-2. +[17] I. V. Denisova, Problem of the motion of two viscous incompressible fluids separated by a +closed free interface, Acta Appl. Math. 37 (1994), 31–40, doi:10.1007/BF00995127. +[18] Joachim Escher and Bogdan-Vasile Matioc, On the parabolicity of the Muskat problem: well- +posedness, fingering, and stability results, Z. Anal. Anwend. 30 (2011), no. 2, 193–218, +doi:10.4171/ZAA/1431. +[19] Evan A Evans and Richard Skalak, Mechanics and thermodynamics of biomembranes, CRC +Press, 1980, doi:10.1201/9781351074339. +[20] Thomas G Fai, Boyce E Griffith, Yoichiro Mori, and Charles S Peskin, Immersed boundary +method for variable viscosity and variable density problems using fast constant-coefficient +linear solvers i: Numerical method and results, SIAM Journal on Scientific Computing 35 +(2013), no. 5, B1132–B1161. +[21] Gerald B. Folland, Introduction to partial differential equations, second ed., Princeton Uni- +versity Press, Princeton, NJ, 1995. +[22] F. Gancedo, E. Garc´ıa-Ju´arez, N. Patel, and R. M. Strain, Global regularity for gravity +unstable muskat bubbles, Mem. Amer. Math. Soc., arXiv:1902.02318 (2021), To appear. +[23] Francisco Gancedo, Rafael Granero-Belinch´on, and Stefano Scrobogna, Global existence in +the lipschitz class for the N-peskin problem, Indiana Univ. Math. J., arXiv:2006.01787 (2021), +To appear. +[24] Francisco Gancedo and Omar Lazar, Global well-posedness for the three dimensional Muskat +problem in the critical Sobolev space, Arch. Ration. Mech. Anal. 246 (2022), no. 1, 141–207, +doi:10.1007/s00205-022-01808-x. +[25] Eduardo Garc´ıa-Ju´arez, Yoichiro Mori, and Robert M. Strain, The Peskin Problem with +Viscosity Contrast, Anal. PDE, arXiv:2009.03360 (2020), To appear. + +92 +E. GARC´IA-JU´AREZ, P.-C. KUO, Y. MORI, AND R. M. STRAIN +[26] Loukas Grafakos, Classical fourier analysis, third ed., Graduate Texts in Mathematics, vol. +249, Springer, 2014, doi:10.1007/978-1-4939-1194-3. +[27] +, Modern fourier analysis, third ed., Graduate Texts in Mathematics, vol. 250, +Springer, 2014, doi:10.1007/978-0-387-09434-2. +[28] Boyce E Griffith and Neelesh A Patankar, Immersed methods for fluid–structure interaction, +Annual review of fluid mechanics 52 (2020), 421. +[29] Susanna V. Haziot and Benoˆıt Pausader, Note on the dissipation for the general muskat +problem, Preprint, arXiv:2210.04395 (2022). +[30] Thomas Y. Hou, John S. Lowengrub, and Michael J. Shelley, Removing the stiffness +from interfacial flows with surface tension, J. Comput. Phys. 114 (1994), no. 2, 312–338, +doi:10.1006/jcph.1994.1170. +[31] William Ko and John M Stockie, Parametric resonance in spherical immersed elastic shells, +SIAM Journal on Applied Mathematics 76 (2016), no. 1, 58–86. +[32] Daniel Lengeler and Michael Ruˇziˇcka, Weak solutions for an incompressible Newtonian fluid +interacting with a Koiter type shell, Arch. Ration. Mech. Anal. 211 (2014), no. 1, 205–255, +doi:10.1007/s00205-013-0686-9. +[33] Hui Li, Stability of the Stokes immersed boundary problem with bending and stretching energy, +J. Funct. Anal. 281 (2021), no. 9, Paper No. 109204, 65, doi:10.1016/j.jfa.2021.109204. +[34] Fang-Hua Lin and Jiajun Tong, Solvability of the Stokes immersed boundary problem in two +dimensions, Comm. Pure Appl. Math. 72 (2019), no. 1, 159–226, doi:10.1002/cpa.21764. +[35] Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Mod- +ern Birkh¨auser Classics, Birkh¨auser/Springer Basel AG, Basel, 1995, [2013 reprint of the 1995 +original] [MR1329547]. +[36] Bogdan-Vasile Matioc, The Muskat problem in two dimensions: +equivalence of formu- +lations, well-posedness, and regularity results, Anal. PDE 12 (2019), no. 2, 281–332, +doi:10.2140/apde.2019.12.281. +[37] Yoichiro Mori, Analise Rodenberg, and Daniel Spirn, Well-posedness and global behavior of +the Peskin problem of an immersed elastic filament in Stokes flow, Comm. Pure Appl. Math. +72 (2019), no. 5, 887–980, doi:10.1002/cpa.21802. +[38] Boris Muha and Suncica Cani´c, +Existence of a weak solution to a nonlinear fluid- +structure interaction problem modeling the flow of an incompressible, viscous fluid in a +cylinder with deformable walls, Arch. Ration. Mech. Anal. 207 (2013), no. 3, 919–968, +doi:10.1007/s00205-012-0585-5. +[39] Huy Q. Nguyen and Benoˆıt Pausader, A paradifferential approach for well-posedness +of +the +Muskat +problem, +Arch. +Ration. +Mech. +Anal. +237 +(2020), +no. +1, +35–100, +doi:10.1007/s00205-020-01494-7. +[40] Charles S. Peskin, Numerical analysis of blood flow in the heart, J. Comput. Phys. 25 (1977), +no. 3, 220–252, doi:10.1016/0021-9991(77)90100-0. +[41] +, +The +immersed +boundary +method, +Acta +Numer. +11 +(2002), +479–517, +doi:10.1017/S0962492902000077. +[42] P. +I. +Plotnikov +and +J. +F. +Toland, +Modelling +nonlinear +hydroelastic +waves, +Philos. +Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 369 (2011), no. 1947, 2942–2956, +doi:10.1098/rsta.2011.0104. +[43] C. Pozrikidis, Boundary integral and singularity methods for linearized viscous flow, Cam- +bridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992, +doi:10.1017/CBO9780511624124. +[44] +, Modeling and simulation of capsules and biological cells, Chapman & Hall/CRC +Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL, 2003, +doi:10.1201/9780203503959. +[45] Jan Pr¨uss and Gieri Simonett, +Moving interfaces and quasilinear parabolic evolution +equations, Monographs in Mathematics, vol. 105, Birkh¨auser/Springer, [Cham], 2016, +doi:10.1007/978-3-319-27698-4. +[46] Thomas Richter, Fluid-structure interactions: models, analysis and finite elements, Vol. 118, +Springer, 2017. +[47] Analise Rodenberg, 2D Peskin Problems of an Immersed Elastic Filament in Stokes Flow, +ProQuest LLC, Ann Arbor, MI, 2018, Thesis (Ph.D.)–University of Minnesota. +[48] Yoshihiro Shibata, Theory of the lebesgue integral, Uchida Rokakuho, 2006. + +WELL-POSEDNESS OF THE 3D PESKIN PROBLEM +93 +[49] Senjo Shimizu, Maximal regularity and its application to free boundary problems for the +Navier-Stokes equations [translation of mr2656036], Sugaku Expositions 25 (2012), no. 1, +105–130. +[50] V. A. Solonnikov, Solvability of the problem of the motion of a viscous incompressible fluid +that is bounded by a free surface, Izv. Akad. Nauk SSSR Ser. Mat. 41 (1977), no. 6, 1388–1424, +1448. +[51] Jiajun Tong, Regularized Stokes immersed boundary problems in two dimensions: +Well- +posedness, singular limit, and error estimates, Comm. Pure Appl. Math. 74(2):366–449 +(2021). +[52] +, Global solutions to the tangential Peskin problem in 2-D, Preprint arXiv:2205.14723 +(2022). +[53] Sunˇcica ˇCani´c, Marija Gali´c, and Boris Muha, Analysis of a 3D nonlinear, moving boundary +problem describing fluid-mesh-shell interaction, Trans. Amer. Math. Soc. 373 (2020), no. 9, +6621–6681, doi:10.1090/tran/8125. +[54] Wei Wang, +Pingwen Zhang, +and Zhifei Zhang, +Well-posedness of hydrodynamics on +the moving elastic surface, +Arch. Ration. Mech. Anal. 206 (2012), +no. 3, +953–995, +doi:10.1007/s00205-012-0548-x. + diff --git a/9NFLT4oBgHgl3EQfty_-/content/tmp_files/load_file.txt b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2cf6f117df802e0ab0914ff6ee780547558fc5e2 --- /dev/null +++ b/9NFLT4oBgHgl3EQfty_-/content/tmp_files/load_file.txt @@ -0,0 +1,2752 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf,len=2751 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12153v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='AP] 28 Jan 2023 WELL-POSEDNESS OF THE 3D PESKIN PROBLEM EDUARDO GARC´IA-JU´AREZ˚, PO-CHUN KUO:,;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', YOICHIRO MORI:,§, AND ROBERT M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN:,¶ Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This paper introduces the 3D Peskin problem: a two-dimensional elastic membrane immersed in a three-dimensional steady Stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We obtain the equations that model this free boundary problem and show that they admit a boundary integral reduction, providing an evolution equation for the elastic interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider general nonlinear elastic laws, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', the fully nonlinear Peskin problem, and prove that the problem is well-posed in low-regularity H¨older spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, we prove that the elastic membrane becomes smooth instantly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Introduction 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Formulation and Boundary Integral Reduction 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Preliminaries 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nonlinear decomposition 18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Calculus estimates 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Frozen-coefficient Semigroup 39 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Local well-posedness 56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Higher Regularity 66 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Besov Spaces and Fourier Multiplier Theorems 75 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Estimates for the semigroup e´tLApξq 77 References 90 ˚Departamento de An´alisis Matem´atico, Universidad de Sevilla, C/Tarfia s/n, Cam- pus Reina Mercedes, 41012, Sevilla, Spain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' egarciajuarez@ub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu :Department of Mathematics, University of Pennsylvania, David Rittenhouse Lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', 209 South 33rd St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Philadelphia, PA 19104, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='kuopo@sas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu §y1mori@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu ¶strain@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='upenn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='edu Date: January 31, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 35Q35, 35C15, 35R11, 35R35, 76D07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Peskin problem, 3D, Fluid-Structure Interaction, immersed boundary problem, Stokes flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ˚supported by the European Union’s Horizon 2020 research and innovation programme under the Marie Sk�lodowska-Curie grant agreement CAMINFLOW No 101031111, and the AEI project PID2021-125021NAI00 (Spain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='partially supported by NSF grant DMS-2042144 (USA) awarded to YM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' §partially supported by the NSF grant DMS-1907583, 2042144 (USA) and the Math+X award from the Simons Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ¶partially supported by the NSF grants DMS-1764177 and DMS-2055271 (USA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Introduction The immersed boundary method, introduced by Peskin [40, 41] to study the blood flow around heart valves, has been widely applied to numerically study fluid- structure interaction (FSI) problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' These FSI problems, in which a fluid interacts with elastic structures, appear naturally in many engineering and biophysics appli- cations [44,46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Despite their importance, both the computational methods and the FSI problems themselves are poorly understood from an analytical standpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A major impediment has been the lack of analytical understanding of the underlying PDEs, which are typically nonlinear and nonlocal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Results are particularly scarce in the more realistic three-dimensional settings, where the coupling of nonlocal effects with non-trivial geometry substantially increases the complexity of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the recent breakthrough works [34] and [37], which provided the strong solution theory for the problem of an immersed elastic string in a two-dimensional fluid, the so-called 2D Peskin problem has attracted a lot of attention [8,9,23,25, 33,51,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In this paper, we initiate the study of its three-dimensional counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We introduce the formulation and develop the well-posedness theory for the three- dimensional (fully nonlinear) Peskin problem of an elastic membrane immersed in a fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Description of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider the following problem in which a three-dimensional incompressible Stokes fluid interacts with an elastic membrane in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A closed elastic interface Γ encloses a simply connected bounded domain Ω Ă R3 filled with a Stokes fluid with viscosity µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The outside region R3zΩ is filled with a Stokes fluid of viscosity 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The equations satisfied are: µ∆u ´ ∇p “ 0 in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) ∆u ´ ∇p “ 0 in R3zΩ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) ∇ ¨ u “ 0 in R3zΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) Here u is the velocity field and p is the pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We impose the following condition in the far field: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) u Ñ 0 as |x| Ñ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We supplement the above with interface conditions on the time-evolving surface Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For any quantity w defined on Ω and R3zΩ, we set: �w� “ w|Γi ´ w|Γe where w|Γi,e are the trace values of w at Γ evaluated from the Ω (interior) and R3zΩ (exterior) sides of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let n be the outward pointing unit normal vector on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The interface conditions are: �u� “ 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) �Σn� “ F el, Σ “ # µ ` ∇u ` p∇uqT˘ ´ pI in Ω ∇u ` p∇uqT ´ pI in R3zΩ , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) BX Bt “ upX, tq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) where I is the 3 ˆ 3 identity matrix and X : S2 ÞÑ Γptq the map that describes the evolving membrane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This map gives the deformation of the reference configuration S2, the standard embedding of the sphere of radius 1 in R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first condition WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 3 is the no-slip boundary condition and the second is the stress balance condition where Σ is the fluid stress and F el is the elastic force exerted by the interface Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The last condition states that the membrane evolves with the fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that the elastic surface Γ “ Γptq and hence Ω “ Ωptq changes with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Once given the constitutive equation for the elastic force F el, equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) form the so-called jump formulation of the 3D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pg and g denote the metric tensors on S2 and Γ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A natural choice for the elastic stretching force is given by [19,28] (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) F el “ a detppg´1gq∇S2 ¨ T p∇S2Xq, where T p∇S2Xq :“ T p|∇S2X|q |∇S2X| ∇S2X “: T p|∇S2X|q∇S2X, ∇S2 denotes the surface gradient on S2, |A| denotes the Frobenius norm of matrix A, and T has to satisfy T ą 0, dT {dλ ě 0 (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 for further notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 2 more details about the derivation of the elastic force are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For a Hookean material, T is linear and hence the elastic force is linear in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will consider general T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', the fully nonlinear Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Compared to fluid interface problems, such as a drop of liquid surrounded by another fluid or vacuum [17,43,45,49,50], where only the shape of the interface mat- ters, here it is not expected that Eulerian methods on their own should suffice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Due to the elastic nature of the membrane, the stretching, given by the parametriza- tion, has a strong influence on the evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, one needs to keep track of the membrane configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lagrangian methods are needed, making it harder to work in higher dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, one cannot freely reparametrize the surface, an idea frequently used to obtain extra cancellations in the study of fluid interfaces [13,22,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' An important feature of the Peskin problem is that it admits a Boundary Integral formulation, whose derivation is given in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When µ “ 1, the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)- (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) is equivalent to the following evolution equation for X: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) BX Bt ppxq “ ż S2GpXppxq ´ Xppyqq∇S2 ¨ ´ T p|∇S2Xppyq|q ∇S2Xppyq |∇S2Xppyq| ¯ dµS2ppyq, Xppxq|t“0 “ X0ppxq, where Gpxq is the Stokeslet tensor in R3: (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) Gpxq “ 1 8π ´ 1 |x|I3 ` x b x |x|3 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have suppressed the dependence of X on t to avoid cluttered notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence- forth, we will assume µ “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It will be sometimes convenient in the analysis to work with coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let θ “ pθ1, θ2q be a (local) coordinate system on S2 and let px “ x Xpθq P S2 Ă R3 be the point on S2 corresponding to θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Xpθq “ Xpx Xpθqq P Γ Ă R3 be the position on Γ corresponding to the coordinate point θ (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If px “ x Xpθq, we will write Xppxq and Xpθq in an abuse of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, after integration by parts and choosing an isothermal coordinate system, equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) becomes BX Bt pθq “ ´p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż R2 B Bηi GpXpθq´Xpηqq ˜F el,ipXqpηqdη1dη2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) 4 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where we denote ˜F el,ipXqpηq “ T pλpηqq λpηq B Bηi Xpηq, λpηq “ a trppg´1pηqgpηqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Above we use the explicit definitions of pg and g given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Deformation map Xp¨, tq : S2 Ñ Γptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Some important properties of the solutions to the Peskin problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) are easier to deduce from the jump formulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The incompressibility condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), together with (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), implies the conservation of the volume of the enclosed region Ω: d dt|Ωptq| “ 0, |Ωptq| “ 1 3 ż S2 Xppxq ¨ nppxqdµΓptqppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, the elastic energy defined as follows EpXq “ ż S2 AEp|∇S2Xppxq|qdµS2ppxq, A1 Epλq “ T pλq, satisfies the balance d dtEpXq “ ´ ż R3 |∇u|2dx, which shows that the elastic energy is dissipated due to the viscosity of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This relation follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), integration by parts, and using conditions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), consecutively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For a linear elasticity law, the elastic energy is the 9H1pS2q norm of the interface, EpXq “ 1 2 ż S2 |∇S2Xppxq|2dµS2ppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A third important property of the Peskin problem is that it satisfies a scaling invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We must first mention that the definition of solutions requires that the WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 5 interface is non-degenerate and does not self-intersect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is typically enforced through the arc-chord condition: |X|˚ :“ inf px‰py px, pyPS2 |X ppxq ´ X ppyq| |px ´ py| ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If Xppx, tq solves (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), then, for any λ ą 0, Xλppx, tq :“ λ´1Xpλpx, λtq also solves the equation, and |Xλ|˚ “ |X|˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, 9C1pS2q and spaces with the same scaling, such as 9H2pS2q, are critical spaces for 3D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that the energy balance above only gives control of the 9H1pS2q norm, hence the Peskin problem is supercritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The formulation of the problem, both in jump and Boundary Integral forms, is derived in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Once the formulation is provided, the main objective of the paper is to show that the problem is well-posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' More specifically, we will first show the existence and uniqueness of strong solutions with initial data in little H¨older spaces, h1,γpS2q, γ P p0, 1q, defined as the completion of the set of smooth functions in C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 (Strong solution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P Cpr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,γpS2qqXC1pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' CγpS2qq, γ P p0, 1q, and |Xptq|˚ ą 0 for t P r0, T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, X is a strong solution to the 3D Peskin problem with initial data Xp0q “ X0 if it satisfies equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) for t P p0, T s and Xptq Ñ X0 in C1,γpS2q as t Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The choice of little H¨older spaces will be needed to obtain the convergence to the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 7 we will prove the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the 3D Peskin problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) with initial data satisfying X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, there exists some time T ą 0 such that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) has a unique strong solution X, X P Cpr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' h1,γpS2qq X C1pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' hγpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is instructive to briefly recall the idea of the proof for the 2D linear Peskin problem [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For X a non-degenerate, closed simple plane curve, the boundary integral formulation in 2D is given by BtXpθ, tq “ ´ ż S1 BηGpXpθq ´ XpηqqBηXpηqdη, where G is the Stokeslet in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It turns out that one can perform a small-scale decomposition [30,37] to write it as follows BtX “ 1 4ΛX ` RpXq, ΛX “ HBθX, with RpXq a lower order operator compared to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, it is natural to construct the solution as a fixed point of the equation written in Duhamel form: Xptq “ etΛX0 ` ż t 0 ept´τqΛRpXpτqqdτ We notice two important facts: the semigroup is explicit, both in space and Fourier variables, and the equation is semilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Even for nonlinear elastic law, the lead- ing term has a kernel not depending on the curve itself, ´ 1 4HpT p|∇S2X|q∇S2Xq, making it possible to use the Λ-like structure via energy methods [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Here, we first consider the strategy adapted to nonlinear equations in [35] (see also [47] for 2D Peskin).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us write equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) as follows BX Bt “ FpXq, X|t“0 “ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, at least formally, linearization around the initial data would give (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) Xptq “ eLpX0qtX0 ` ż t 0 eLpX0qpt´τqEpXpτqqdτ, with LpX0q “ BXFpX0qX the Gateaux derivative of F at X0 and EpXq “ FpXq ´ LpX0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, while EpXq is not expected to be smoother than LpX0q, it should be small for short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, one first need to make sense of the above expression (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) by showing that LpX0q generates an analytic semigroup, which amounts to proving that the operator is sectorial in adequate spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is the core of the abstract Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1, whose proof encompasses a fixed point argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The application of this theorem to our problem soon becomes highly involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is done in Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the equation is not semilinear, the process will require to further decompose the operator LpX0q and then freeze the coefficients at a given point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The decomposition must be done maintaining a derivative structure for the kernel that allows extra cancellations, required to control the singular integral operators that appear, and so that we can invert the frozen-coefficient operator (the study of this part is done separately in Section 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Schematically, we decompose the kernel in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) as follows B Bηi ` GpXpθq´Xpηqq ˘ « ´ B Bxj G p∇Xpηqpθ ´ ηqq BXj Bηi pηq ` RpXqpθq « 1 8π BXpηq Bηi ¨ p∇Xpηqpθ´ηqq |∇Xpηqpθ´ηq|3 ` ¨ ¨ ¨ ` RpXqpθq, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) where one expects RpXq to be lower order and the dots represent additional terms of high order coming from the second term in Gppxq (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) (we note that in 2D these additional high-order terms cancel each other).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' These leading kernels are not of convolution type and cannot be written as a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For this purpose, one could be tempted to use ∇Xpθq in the approximation instead of ∇Xpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, higher derivatives of X would appear later in the proof and the argument would not close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, to take advantage of the derivative structure, we will be forced to estimate together the leading and remainder terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In a second step, we approximate ∇Xpηq in the leading kernels above by its value at a given point (see Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 for more details), which requires the introduction of a partition of unity for the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Due to the geometry of the problem, we need to work with charts, and due to the nonlocal character of the equation a second localization procedure will be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A fine implementation of these localization procedures will be crucial to avoid transition maps that would otherwise overcomplicate the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the fully nonlinear Peskin problem, we must linearize and freeze the coeffi- cient of the elastic force as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we show that the frozen-coefficient linear operator in the general force case is given by pLAY qkpθq “ ´ ż R2 B Bηi pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 7 where A is a constant matrix and TFpAq a tensor (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, in the general case, the multiplier for the frozen-coefficient linear operator becomes: LApξq “ I ` vpξq b vpξq 4detpBq |B´1ξ| ˆT p∥A∥F q ∥A∥F ˆ |ξ|2 I´ Aξ b Aξ ∥A∥F ˙ ` dT dλ p∥A∥F qAξ b Aξ ∥A∥F ˙ , where || ¨ ||F denotes the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is not difficult to see that, if T ą 0 and dT {dλ ě 0, then the above is coercive in |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, in contrast to the 2D case, dT {dλ “ 0 is allowed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, if T satisfies T ą 0 and pdT {dλq{T ą ´1, then the problem is expected to be locally well-posed if the initial condition is sufficiently close to the uniform sphere (pg´1g is close to a multiple of the identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is an interesting difference between 2D and 3D Peskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use this operator (in conjunction with the localization procedures) to show that the full operator LpX0q “ BXFpX0qX is sectorial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The approximation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) is done on the equation written in coordinates partly to obtain a linear leading operator given by a Fourier multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we notice that the regularity obtained in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 for the strong solu- tions is not enough to satisfy equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) in a classical sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Obtaining higher regularity for the solutions is also important since this further regularity is needed for the equivalence between different formulations to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The abstract theory for nonlinear equations in [35] does not yield gain of smoothness for the solution, and in fact this important point is left open in the 2D results in [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nevertheless, we are able to prove that initial data in little H¨older spaces become smooth for positive times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X be the solution to the Peskin problem with initial data X0 P h1,γpS2q constructed in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for any α P p0, 1q, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C3,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, for any 3 ď n P N and α P p0, 1q, assuming that T P Cn,α, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,βpS2qq, for any β ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the solutions constructed in the previous theorem and Duhamel formula (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) to perform a bootstrapping argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We build on the properties of the semigroup e´tLA (see Section 6 and Appendix B) to first gain regularity in mixed- type spaces Lpp0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq and then transfer this higher regularity in space to show regularity in time as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A key point is that, while the kernels are not of convolution type, we find that it is possible to move derivatives in θ to derivatives in η at the expense of new terms of the same order, but not higher (see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As explained above, we must work with the equation localized around a given point and later deal with the corresponding commutators and combine the estimates (see Section 8 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, the bootstrapping argument cannot be done on (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) directly, because the right-hand side contains terms of highest regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We combine this process with a regularization argument (see (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14)), where the use of little H¨older spaces becomes crucial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Related results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first analytical results for the 2D Peskin problem ap- peared recently in [34,37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In [34], energy arguments are used to prove local well- posedness for H 5 2 initial data and also exponential convergence to steady states for sufficiently close to equilibrium initial data is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The authors in [37] lowered the required initial regularity to barely subcritical spaces, h1,γ, γ P p0, 1q, showed instant smoothing, and provided a blow up criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 8 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN After these works, many improvements for the 2D Peskin problem have appeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The work [25] deals with the setting in which the enclosed fluid is different to the exterior one, and shows asymptotic stability for small data in Wiener algebra critical spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The result [8] shows the local well-posedness and smoothing for general data in the critical Besov space B 3 2 2,1, including the case of nonlinear elastic law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The sharpest result in terms of regularity appeared in [9], where the semilinear 2D Peskin problem is shown to be well-posed in B1 8,8, and thus with possibly non-Lipschitz curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In relation to the Peskin problem, the article [51] introduces a regularization of the problem inspired by the immersed boundary method and studies its conver- gence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Filaments that resist both bending and stretching are considered in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we mention two works that introduce simplified models of the 2D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The work [23] considers a model for the normal component and shows the existence of global solutions for Lipschitz data near the equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Very re- cently, [52] derives a PDE to model the tangential effects of the Peskin problem in the case of an infinitely long and straight string and obtains global solutions with initial data in the energy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, the author presents many connections of the model with well-known one-dimensional PDEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From a mathematical point of view, there are remarkable similarities between the 2D Peskin problem and the so-called Muskat problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, both problems have the same leading linear operator, they can be written in Boundary Integral form [13,30], they have the same scaling and satisfy an energy balance [12, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The Muskat problem, which describes the movement of the interface between incompressible fluids in a porous medium, has been intensively studied in the last two decades [1,2,4,7,12,13,18,36,39], and some of the techniques developed there have been successfully extended in the last years to lower the required regularity for the well-posedness of the 2D Peskin problem [8,9,23,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, while there are also results for the 3D Muskat problem [3,5,10,14,24], in all these results the interface is a surface given by graph, hence the geometry does not play a major role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Even in 2D, in the recent non-graph setting [22] that considers a bubble of fluid surrounded by another in a porous medium, a change of parametrization becomes crucial, which is not allowed in the Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We finally mention some results with more complex elastic interactions [6,11,16, 32,38,42,53,54], mostly dedicated to more qualitative results and weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Part of the interest generated by the Peskin problem is due to its relative simplic- ity, which makes it possible to initiate the analytical study of the rich variety of behaviors in FSI problems, including longtime dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The rest of the paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 2, we ob- tain the expression for the elastic law and show the Boundary Integral formulation for the 3D Peskin problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 contains the notation used along the paper as well as some definitions and standard results concerning the stereographic pro- jection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, in Section 4, we introduce the operators that will be used later in the paper, we decompose the equation and compute the multiplier of the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Section 5 is dedicated to study the operators previously defined, to show the needed commutators estimates, and to prove Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' These lemmas will be repeatedly used in the proof of the main theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In Section 6, we show that the frozen-coefficient operator generates an analytic semigroup (for which we need the WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 9 multiplier results contained in Appendix A, with further properties studied in Ap- pendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, Sections 7 and Section 8 contain the proofs of the main results: Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Formulation and Boundary Integral Reduction The formulation of the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) is closed once that the expression for F el is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To specify the elastic force F el in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6), we consider the elastic energy of the interface Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider an elastic energy EpXq of the form: EpXq “ ż S2 E ˆBX Bθ , θ ˙ dµS2, where µS2 is the standard measure on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From this, we may com- pute the elastic force by taking the variational derivative as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X “ pX1, X2, X3qT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Define the following metric tensors pg and g on S2 and Γ respec- tively, whose i, j components are given by: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) pgij “ Bx X Bθi ¨ Bx X Bθj , gij “ BX Bθi ¨ BX Bθj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We write the energy density as follows: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) E “ AEpsij, θq, sij “ BXi Bθj , i “ 1, ¨ ¨ ¨ 3, j “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Y “ pY1, Y2, Y3qT be a perturbation of the configuration that is compactly supported on the open set U on which the coordinate system θ is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have: d dτ EpX ` τY q ˇˇˇˇ τ“0 “ ż U BAE Bsij BYi Bθj a detpgdθ1dθ2 “ ´ ż U B Bθj ˆBAE Bsij a detpg ˙ Yidθ1dθ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) where the summation convention is in effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We set: Fel,i “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθj ˆBAE Bsij a detpg ˙ , where Fel,i are the components of the elastic force F el of equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' With this prescription of the elastic force, the solutions satisfy the following energy relation: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) dE dt “ ´ ˜ż Ω 2µ |∇Su|2 dx ` ż R3zΩ 2 |∇Su|2 ¸ dx, ∇Su “ 1 2 ` ∇u ` p∇uqT˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will now impose symmetry conditions to determine the explicit form of AE and hence E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let θ be a (local) orthogonal coordinate system on S2 so that the two coordinate tangent vectors are orthogonal: Bx X Bθ1 ¨ Bx X Bθ2 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus have an orthonormal frame on (a neighborhood of) S2 given by the two vectors: pei “ Bx X Bθi ����� Bx X Bθi ����� ´1 , i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN The deformation map X maps the above unit orthogonal vectors to the following two vectors: ei “ BX Bθi ����� Bx X Bθi ����� ´1 , i “ 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the matrix 3 ˆ 2 matrix B “ pe1, e2q whose column vectors are given by ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may say that the energy density E is a function of B and θ: E “ AEpB, θq, where we have continued to use the notation AE as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By homogeneity of the unit sphere, we impose that AE does not have an explicit dependence on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Furthermore, the value of AE should not depend on the choice of orthonormal frame pei or the coordinate system in which X resides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) AEpBq “ AEpR3BR2q for all R3 P SOp3q and R2 P SOp2q where SOp2q and SOp3q are the group on rotation matrices in 2 and 3 dimensions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let: H “ ˆ e1 ¨ e1 e1 ¨ e2 e1 ¨ e2 e2 ¨ e2 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The invariance condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) implies that AE can only be a function of the trace and determinants of H, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) E “ AEpλ, γq, λ “ a trpHq, γ “ a detpHq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In terms of the metric tensors g and pg, we can write λ and γ as: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) λ2 “ trppg´1gq, γ2 “ detppg´1gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The above expressions for λ and γ are valid even when θ is not an orthogonal coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may substitute (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) to obtain: Fel,k “ Fλ,k ` Fγ,k, Fλ,k “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθi ˆ 1 λ BAE Bλ a detpg pgij BXk Bθj ˙ , Fγ,k “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθi ˆBAE Bγ a detg gij BXk Bθj ˙ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) where we use the standard notation aij to denote the inverse tensor pa´1qij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that the expressions Fλ,k and Fγ,k are similar but differ crucially in whether pg or g features inside the force expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is most clearly seen in the following simple cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If we let AE “ λ2{2, we have: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) Fel,k “ Fλ,k “ γ∆S2Xk, ∆S2Xk “ 1 a detpg B Bθi ˆa detpg pgij BXk Bθj ˙ , where ∆S2 is the Laplace-Beltrami operator on the unit sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If we let AE “ γ, we have: Fel,k “ Fγ,k “ ∆ΓXk “ 1 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='detg B Bθi ˆa detg gij BXk Bθj ˙ “ ´2κΓnk, where ∆Γ is the Laplace-Beltrami operator of the closed elastic surface Γ, κΓ is the mean curvature of Γ and nk is the k-th component of the outward normal vector n of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This is just the well-known statement on the variation of surface area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We see from the above expressions that the Fλ,k expresses an elastic force that depends WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 11 strongly on the stretching of the spherical reference configuration whereas Fγ,k is a surface tension force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The prescription of interfacial elastic energy density as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) or (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) has its origins the classical work of [19], and may be called the membrane neo-Hookean model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Specific forms for this energy have been used extensively in the modeling and simulation of fluid-structure interaction problems [20,28,31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We now rewrite our evolution equation in a form suitable for our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence- forth we focus on the case when AE is only a function of λ, and the viscosity µ of the interior fluid is equal to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us rewrite the equations of motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let G be the Stokeslet tensor in R3: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) Gi,jpxq “ 1 8π ˜ δi,j |x| ` xixj |x|3 ¸ , x “ px1, x2, x3q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pFel,k “ 1 γ Fel,k “ 1 a detpg B Bθi ˆ λ´1T pλq a detpg pgij BXk Bθj ˙ “ ∇S2 ¨ ` λ´1T pλq∇S2Xk ˘ , where T pλq “ BAE{Bλ (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pF el “ pFel,1, Fel,2, Fel,3qT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that we can write λ in terms of X, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) λppxq2 “ |∇S2Xppxq|2, which can be seen from their definitions λ2 “ trppg´1gq “ pgijgji, |∇S2X|2 “ ∇S2Xk ¨ ∇S2Xk “ BXk Bxi BXk Bxl pgijpglmpgjm “ gilpgil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When µ “ 1, we may write the evolution of X as BX Bt ppxq “ ż S2 GpXppxq ´ XppyqqpF elppyqdµS2ppyq “ ż S2GpXppxq ´ Xppyqq∇S2 ¨ ´ T p|∇S2Xppyq|q ∇S2Xppyq |∇S2Xppyq| ¯ dµS2ppyq, and integrating by parts, we obtain (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) BX Bt ppxq“´p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż S2∇S2GpXppxq´Xppyqq¨T p|∇S2Xppyq|q ∇S2Xppyq |∇S2Xppyq|dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the following, we will suppress the principal value notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Introducing a smooth partition of unity tρnu, subordinate to a finite atlas of the sphere tUnu, we may write our problem as follows BX Bt ppxq“´ ÿ n ż S2∇S2GpXppxq´Xppyqq¨ T p|∇S2Xppyq|q |∇S2Xppyq| ∇S2` ρnppyqXppyq ˘ dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' This can be rewritten using the local charts: BX Bt ppxq “ ´ ÿ n ż Un pgijpηq B Bηi GpXppxq´Xnpηqq ˆ pgjmpηqT p a pgqrgrqpηqq a pgqrgrqpηq pgpmpηq B Bηp ` ρnpηqXnpηq ˘a detpgpηqdη1dη2, 12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where Xnpηq is the coordinate map on the n-th coordinate chart and ρnpηq “ ρnpx Xpηqq (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 for details of notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may take an isothermal coordinate system (the stereographic projection gives such a system, for example) on each chart Un, which yields: BX Bt ppxq “ ´ ÿ n ż Un B Bηi GpXppxq´XnpηqqT pλnpηqq λnpηq B Bηi ` ρnpηqXnpηq ˘ dη1dη2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) where we denote (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) λnpηq “ a trppg´1pηqgpηqq “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2}∇Xnpηq}F }∇x Xnpηq}´1 F , and }A}F :“ a trpAT Aq is the Frobenius norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Preliminaries In this section we introduce the notations that will be used in the rest of the paper and summarize some standard results about stereographic projection charts for the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Einstein notation over repeated indices will be of constant use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given vectors v, w and matrices A, B, C with the same size, we denote |v| :“ ∥v∥ “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='vivi, ∥A∥ :“ sup |v|ą0 |Av| |v| “ sup |v|“1 |Av| , A : B :“tr ` ATB ˘ “ AijBij, |A| :“ ∥A∥F :“ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A : A, v b w :“vwT , pA b Bqijkl :“AijBkl, ppA b Bq Cqij :“AijBklCkl “ pB : Cq Aij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote C‚,j to the vector given by the jth column of C, and Cj,‚ to the one given by the jth row.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote µS2 the standard measure on the unit sphere, and for simplicity we will write dpy instead of dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will write high partial derivatives in Rk by multi-index α, where multi-index α is a sequence of k nonegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α “ pα1, α2, ¨ ¨ ¨ , αkq P Nk 0, where N0 “ t0u Ş N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 (Multi-index).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given α, β P Nk 0, we have the following arithmetic about the multi-index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (i) |α| “α1 ` α2 ` ¨ ¨ ¨ ` αk α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' “α1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='α2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' αk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α ` β “ pα1 ` β1, α2 ` β2, ¨ ¨ ¨ , αk ` βkq WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 13 (ii) We set α ď β, which is αi ď βi for all i “ 1, 2, ¨ ¨ ¨ , k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have α ´ β “ pα1 ´ β1, α2 ´ β2, ¨ ¨ ¨ , αk ´ βkq ˆ α β ˙ “ α!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pα ´ βq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='β!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' “ α1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pα1 ´ β1q!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='β1!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ¨ ¨ ¨ αk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pαk ´ βkq!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='βk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' High partial derivatives can be written as Bα x f pxq :“ Bα1 Bxα1 1 Bα2 Bxα2 2 ¨ ¨ ¨ Bαk Bxαk k f pxq (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) where α :“ pα1, α2, ¨ ¨ ¨ , αkq and |α| “ α1 ` ¨ ¨ ¨ ` αk is the total number of deriva- tives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use the following set of non-singular matrices (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) DAσ1,σ2 :“ tA : @ξ ‰ 0, σ2 |ξ| ď |Aξ| ď σ1 |ξ|u where σ1 ě σ2 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Euclidean balls of radius R centered at px P Rn will be denoted by Bpx,R, and for balls centered at the origin we will also denote BpRq :“ B0,R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote Xppx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' tq : S2 ÞÑ Γptq the deformation map that describes the evolving membrane, and we will omit the dependence on time for simplicity of notation, Xppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will consider a finite atlas tUn, x Xnu of the sphere with 0 P Un, such that the coordinate functions x Xnpθq : Un Ă R2 Ťt8u ÞÑ S2 satisfy (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) Bx Xnpθq Bθ1 ¨ Bx Xnpθq Bθ2 “ 0, ����� Bx Xnpθq Bθ1 ����� “ ����� Bx Xnpθq Bθ2 ����� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, we will choose the standard stereographic coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We set tρnu to be a smooth partition of unity subordinate to the coordinate patches tUnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For convenience in the definition of H¨older continuity, we take our system tUn, x Xn, ρnu satisfying the following properties with some R, δ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 (System tUn, x Xn, ρnu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given R ą 2δ ą 0, we set our isothermal coordinate charts tUnu with the coordinate functions tx Xnpθqu and the partition tρnu to have the following properties: i) Set pxn “ x Xn p0q, then S2 Ă Ť n Bpxn,R, and there exists 0 ă Rn ă 8 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t Bpxn,4R X S2 Ă x Xn pB0,Rnq Ă x XnpUnq ii) @px P S2, Dn s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' x Xn pθq “ px for some θ P B0,Rn, and pBpx,2δ X S2q Ă x Xn pB0,Rnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' iii) 0 ď ρn ď 1, ρn P C8pS2q, supp pρnq Ă Bpxn,2R X S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' iv) @px P S2, ř n ρn ppxq “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If |x Xn pθq´ x Xn pηq | ě C |θ ´ η| on Un, then Un is totally bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given f ppxq : S2 Ñ R, we will denote fn pθq : Un Ă R2 Ñ R with fnpθq :“ fpx Xnpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Analogously, we will denote Xnpθq “ Xpx Xnpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If x Xn pθq “ x Xm pηq, then Xn pθq “ Xm pηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 14 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 (H¨older semi-norm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' �f ppxq�Cγ δ pS2q :“ sup 0ă|px´py|ăδ |f ppxq´f ppyq| |px ´ py|γ “ sup n sup 0ă|x Xnpθq´x Xnpηq|ăδ |fn pθq ´ fn pηq| |x Xn pθq´x Xn pηq |γ , �f ppxq�CγpS2q :“ sup 0ă|px´py| |f ppxq ´ f ppyq| |px ´ py|γ “ sup x Xnpθq‰x Xmpηq |fn pθq ´ fm pηq| |x Xn pθq ´ x Xm pηq |γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 (Arc-chord condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |f|˚ :“ inf px‰py |f ppxq ´ f ppyq| |px ´ py| “ inf x Xnpθq‰x Xmpηq |fn pθq ´ fm pηq| |x Xn pθq ´ x Xm pηq | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 (Locally Arc-chord condition in the Charts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Vn Ă Un, |f|˝,n :“ inf θ‰η,θ,ηPVn |fnpθq ´ fnpηq| |θ ´ η| , |f|˝ :“ inf n |f|˝,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8 (Lp norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ∥f ppxq∥p LppS2q :“ ÿ n ż Un ρn pθq |fn pθq|p a det pgndθ “ ÿ n ���pρnq 1 p fn ��� p LppUnq ď ÿ n ∥fn pθq∥p LppUnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Standard Stereographic Projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will see the properties of the standard stereographic projection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' the projection point is p0, 0, 1q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the other projection points, because S2 is centrosymmetric, we just need to rotate the coordinates of S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, most properties among the projection charts are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9 (Standard Stereographic Projection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We set x X : R2 Ťt8u Ñ S2 with x X pθq “ ˜ 2θ1 1 ` |θ|2 , 2θ2 1 ` |θ|2 , ´1 ` |θ|2 1 ` |θ|2 ¸ , x X p8q :“ lim |θ|Ñ8 x X pθq “ p0, 0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, θ ppxq “ ˆ px1 1 ´ px3 , px2 1 ´ px3 ˙ , θ p0, 0, 1q “ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We call the parameterization x X the standard stereographic projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will denote VR the coordinate balls in R2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) VR :“ B ´ R ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4 ´ R2 ¯ Ă R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The standard stereographic projection has the following prop- erties: WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 15 i) Bx Xpθq Bθ1 ¨ Bx Xpθq Bθ2 “ 0, ����� Bx Xpθq Bθ1 ����� “ ����� Bx Xpθq Bθ2 ����� “ 2 1 ` |θ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ii) For all R ą 0 and all VR (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), x XpVRq “ Bp0,0,´1q,R X S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' iii) For θ, η P R2, |x X pθq ´ x X pηq | ď 2 |θ ´ η| , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) and if θ, η P VR with R ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, |x X pθq ´ x X pηq | ě 2 π |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For iii), set pξ “ θ´η |θ´η|, then |x X pθq´x X pηq | “ ˇˇˇ ż |θ´η| 0 B Bs x Xpη`spξqds ˇˇˇ ď ż |θ´η| 0 |∇x Xpη`spξq ¨ pξ|ds “ ż |θ´η| 0 2 1`|η ` spξ|2 ds ď 2 |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set distppx, py;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q as the length of shortest curve connecting px and py on S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When θ, η P VR with R ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, the shortest curve ℓ for distpx X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q is on Bp0,0,´1q,R X S2 and 2 π ď R ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4 ´ R2 2 cos´1 2´R2 2 ď |x X pθq ´ x X pηq | distpx X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q ď 1, where the above function of R is a decreasing function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, since x X is isothermal and x X ´1 pℓq is in VR, distpx X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2q “ ż ℓ dl ppxq “ ż x X ´1pℓq 2 1 ` |θ|2 dl pθq ě4 ´ R2 2 ż x X ´1pℓq dl pθq ě 4 ´ R2 2 |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore |x X pθq ´ x X pηq | ě 2 π dist ´ x X pθq , x X pηq ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2¯ ě 2 π |θ ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The Quantitative Relationships between S2 and R2 on the Standard Stereographic Projection Chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a function f ppxq on S2, we may define f pθq :“ fpx X pθqq on R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' x X pθq is isothermal, but the chart is neither isometric nor area-preserving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, some quantities of f between S2 and R2 are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have to check their quantitative relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, let see the H¨older continuous seminorm Cγ and the arc-chord condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given f on S2, it holds that �f�CγpR2q ď 2γ�f�CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 16 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' �f�CγpR2q “ sup θ‰η ´|x X pθq ´ x X pηq | |θ ´ η| ¯γ |fpx X pθqq ´ fpx X pηqq| |x X pθq ´ x X pηq |γ ď 2γ�f�CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Notice that |f|˝ for R2 is zero, and the other sided inequality between Cγ pUq and Cγ pUq cannot hold with some constant C ą 0, thus we only can find local inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set BR “ Bp0,0,´1q,R X S2, VR as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), and ρ smooth on S2 and supported in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given R ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, it holds that �ρf�CγpS2q ď `π 2 ˘γ�ρf�CγpR2q, |f|˚ ďπ 2 |f|˝ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For �ρf�CγpS2q, when px “ x X pθq , py “ x X pηq P BR, θ, η P VR, so |ρf ppxq ´ ρf ppyq| |px ´ py|γ “ ´ |θ ´ η| |x X pθq ´ x X pηq | ¯γ |ρf pθq ´ ρf pηq| |θ ´ η|γ ď `π 2 ˘γ�ρf�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, when px “ x X pθq , py “ x X pηq P Bc R, |ρfppxq´ρfp pyq| |px´ py|γ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, when px “ x X pθq P BR, py “ x X pηq P Bc R, θ P VR, η P V c R, set pz “ x X pξq P BBR s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |px ´ pz| “ dist ppx, BBRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ρ is smooth on S2 and supported in BR, ρ ppzq “ 0 “ ρ ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, |θ ´ ξ| ď π 2 |px ´ pz| ď π 2 |px ´ py| , so |ρf ppxq ´ ρf ppyq| |px ´ py|γ “ |ρf ppxq ´ ρf ppzq| |px ´ py|γ “ ´ |θ ´ ξ| |x X pθq ´ x X pηq | ¯γ |ρf pθq ´ ρf pξq| |θ ´ ξ|γ ď `π 2 ˘γ�ρf�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, for |f|˝, |f|˝ “ inf θ‰η,θ,ηPV |x X pθq ´ x X pηq | |θ ´ η| |fpx X pθqq ´ fpx X pηqq| |x X pθq ´ x X pηq | ě 2 π |f|˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Next, let us discuss the relationship between C1,γ ` S2˘ and C1,γ ` R2˘ on the standard stereographic projection chart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the standard stereographic projection chart px “ x X pθq, the surface gradient of f, ∇S2f ppxq, is ∇S2f ppxq “ ÿ i,j pgi,j B Bθi f pθq B Bθj x X pθq “ ´1 ` |θ|2 2 ¯2 ÿ i B Bθi f pθq B Bθi x X pθq , where pgij denotes the inverse tensor of pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, 2 1 ` |θ|2 ���∇S2fpx Xpθqq ��� “ |∇f pθq| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We may use the above expressions to obtain the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 17 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' There exist R0 ą 0 and C ě 21`γ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all R ă R0 ∥f∥C1,γpR2q ďC ∥f∥C1,γpS2q , ∥ρf∥C1,γpS2q ď ∥ρf∥C1,γpR2q , |f|˚ ď |f|˝ , where ρ is smooth on S2 and supported in BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Stereographic Projection Charts x Xn Covering S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We consider a finite cover of the sphere consisting of balls of radius R ă R0 and center pxn, tBpxn,RXS2u, and a smooth partition of unity subordinated to it, tρnu, such that ρnppxq “ 0 if |px´ pxn| ě 2R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For convenience, we set px0 “ p0, 0, ´1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then we take stereographic projection charts x Xn covering S2 (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Stereographic projection charts, x Xnpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14 (Stereographic Projection Charts covering S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' x Xn are standard stereographic projection charts with x Xn : R2 Ñ S2 x Xnpθq “ Θnx Xpθq + , where Θn is the rotation matrix with pxn “ Θnpx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In each chart x Xn, given R ă R0 4 and R0, C from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13, since ρ are supported in Bn, ∥f∥C1,γpR2q ď C ∥f∥C1,γpS2q , ∥ρnf∥C1,γpS2q ď ∥ρnf∥C1,γpR2q , |f|˚ ď |f|˝,n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 18 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN As a consequence, |f|˚ ď |f|˝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nonlinear decomposition In this section we extract the leading structure of the equation and compute its symbol, introducing the notation for the operators that will appear along the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nonlinear decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will usually consider separately the two terms involved in the Stokeslet kernel (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) Gk,lpxq “ G1 k,lpxq ` G2 k,lpxq, G1 k,lpxq “ 1 8π δk,l |x| , G2 k,lpxq “ 1 8π xkxl |x|3 , and thus we will write our equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) BX Bt ppxq “ FpXqppxq “ F 1pXqppxq ` F 2pXqppxq, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) FpXqppxq “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy, F jpXqppxq “ ´ ż S2 ∇S2GjpXppxq ´ Xppyqq ¨ T p|∇S2Xppyq|q∇S2Xppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and we introduced the notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) T p|∇S2X|q “ T p|∇S2X|q |∇S2X| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Above, we use the shorter notation dpy “ dµS2ppyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We define the following associate linear operators, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) pNpXqZqkppxq “ ´ ż S2 ∇S2GklpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy, pN jpXqZqkppxq “ ´ ż S2 ∇S2Gj klpXppxq ´ Xppyqq ¨ Zl,‚ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we compute the kernels: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) B Bxi G1 k,lpxq “ ´1 8π xi |x|3 δk,l, B Bxi G2 k,lpxq “ 1 8π δk,ixl ` xkδi,l |x|3 ´ 3 8π xkxlxi |x|5 , and by the chain rule, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) qj k,lppx, pyq : “ ∇S2Gj k,lpXppxq ´ Xppyqq “ ´ B Bxi Gj k,lpXppxq ´ Xppyqq∇S2Xippyq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' so that we write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) pN jpXqZqkppxq “ ´ ż S2 qj k,lppx, pyq ¨ Zl,‚ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 19 The explicit expression for qj k,l is given by q1 k,lppx, pyq “ 1 8π ∆ pyXjppxq∇S2Xjppyq |∆ pyXppxq|3 δk,l |px ´ py|2 , and q2 k,lppx, pyq “ ´ 1 8π ∆ pyXlppxq∇S2Xkppyq ` ∆ pyXkppxq∇S2Xlppyq |∆ pyXppxq|3 1 |px ´ py|2 ` 3 8π ∆ pyXkppxq∆ pyXlppxq∆ pyXjppxq∇S2Xjppyq |∆ pyXppxq|5 1 |px ´ py|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using the standard stereographic projection (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and the notation Xpθq “ Xpx Xpθqq, the equation for each component of Xpθq becomes BXk Bt pθq “ pFpXqqkpθq “ ´ ż R2 B Bηi Gk,lpXpθq´XpηqqT pλpηqqBXl Bηi pηqdη1dη2, where λpηq is given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) and we denote accordingly F jpXqpθq, N jpXqZpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If we use the stereographic projection centered at pxn, then we denote N jpXqZnpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we take the derivative in Gk,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) to obtain B Bηi Gk,lpXpθq´Xpηqq “ qi,k,lpθ, ηq “ q1 i,k,lpθ, ηq ` q2 i,k,lpθ, ηq, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) q1 i,k,lpθ, ηq “ B Bηi G1 k,lpXpθq´Xpηqq “ 1 8πδk,l δηXpθq ¨ BX Bηi pηq |δηXpθq|3 , q2 i,k,lpθ, ηq “ B Bηi G2 k,lpXpθq´Xpηqq “ ´ 1 8π BXk Bηi pηqδηXlpθq ` δηXkpθq BXl Bηi |δηXpθq|3 ` 3 8π δηXkpθqδηXlpθq |δηXpθq|5 δηXpθq ¨ BXpηq Bηi , so that we can write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) pN jpXqZqkpθq “ ´ ż R2 qj m,k,lpθ, ηq B p Xi Bηm pηqZl,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We notice that the kernels in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) are given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) qj i,k,lpθ, ηq “ ´ B Bxm Gj k,lpXpθq ´ XpηqqBXm Bηi pηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We introduce the following notation for finite differences, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) δηgpθq “ gpθq ´ gpηq, ∆ηgpθq “ δηgpθq |θ ´ η|, and we extract the expected leading terms by replacing δηXpθq « ∇Xpηqpθ ´ ηq, p∇Xqp,qpηq “ BXn,p Bηq pηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Hence, we define the associate kernels (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) mi,k,lpθ, ηq “ ´ B Bxj Gk,l p∇Xpηqpθ ´ ηqq BXj Bηi pηq “ m1 i,k,lpθ, ηq ` m2 i,k,lpθ, ηq, and define the linear operators Mp∇Xqz as follows: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) pMp∇XqZqkpθq “ ´ ż R2 mm,k,lpθ, ηq B p Xi Bηm pηqZl,ipηqdη1dη2 “ pM1p∇XqZqkpθq ` pM2p∇XqZqkpθq, with pMjp∇XqZqkpθq “ ´ ż R2 mj m,k,lpθ, ηq B p Xi Bηm pηqZl,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We compute the explicit expression of these kernels mj i,k,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), m1 i,k,lpθ, ηq “ 1 8π BXpηq Bηi ¨ p∇Xpηqpθ´ηqq |∇Xpηqpθ´ηq|3 , m2 i,k,lpθ, ηq “ ´ 1 8π BXkpηq Bηi p∇Xpηqpθ´ηqql ` BXlpηq Bηi p∇Xpηqpθ´ηqqk |∇Xpηqpθ´ηq|3 ` 3 8π p∇Xpηqpθ´ηqqkp∇Xpηqpθ´ηqql |∇Xpηqpθ´ηq|5 p∇Xpηqpθ´ηqq ¨ BXpηq Bηi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use the notation ∇Xpηqpθ ´ ηq “ pθ ´ ηq ¨ ∇Xpηq, pz “ z |z|, and we define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) EηXpθq :“ p{ θ ´ ηq ¨ ∇Xpηq ´ ∆ηXpθq, for which we have that, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) |EηpXpθqq| |θ ´ η|γ ď �∇X�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we can write (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) NpXqZpθq “ Mp∇XqZpθq ` RpXqZpθq, where the remainder term RpXqZpθq “ 2ÿ j“1 RjpXqZpθq is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) pRjpXqZqqkpθq “ pN jpXqZqkpθq ´ pMjp∇XqpZqqkpθq “ ż R2 Kj m,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq B p Xi Bηm pηqZl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ipηqdη1dη2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 21 with kernels K1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ ´q1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq ` m1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ 1 8π δk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l |θ ´ η|2 BXpηq Bηi ¨ ˆ EηXpθq |∆ηXpθq|3 ´ pp{ θ ´ ηq ¨ ∇Xpηqq ´ 1 |∆ηXpθq|3 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|3 ¯˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and K2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ ´q2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq ` m2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='l pθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq ` K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ 1 8π 1 |θ ´ η|2 ˆ ´ BXkpηq Bηi |∆ηXpθq|3 EηXlpθq ` BXkpηq Bηi pp{ θ ´ ηq ¨ ∇Xpηqql ´ 1 |∆ηXpθq|3 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|3 ¯ ´ BXlpηq Bηi |∆ηXpθq|3 EηXkpθq ` BXlpηq Bηi pp{ θ ´ ηq ¨ ∇Xpηqqk ´ 1 |∆ηXpθq|3 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|3 ¯˙ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' K2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lpθ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ηq “ 1 8π 3 |θ ´ η|2 ˆ∆ηXkpθq∆ηXlpθq |∆ηXpθq|5 BXpηq Bηi ¨ EηXpθq ` BXpηq Bηi ¨ pp{ θ ´ ηq ¨ ∇Xpηqq |∆ηXpθq|5 ∆ηXkpθqEηXlpθq ` BXpηq Bηi ¨ pp{ θ ´ ηq ¨ ∇Xpηqq |∆ηXpθq|5 pp{ θ ´ ηq ¨ ∇XpηqqlEηXkpθq ´ BXpηq Bηi ¨ pp{ θ ´ ηq ¨ ∇Xpηqq |∆ηXpθq|5 pp{ θ ´ ηq ¨ ∇Xpηqqlpp{ θ ´ ηq ¨ ∇Xpηqqk ˆ ´ 1 |∆ηXpθq|5 ´ 1 |p{ θ ´ ηq ¨ ∇Xpηq|5 ¯˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that for all positive odd integers k, it holds that 1 |u|k ´ 1 |v|k “ pv ´ uq ¨ pu ` vq |u|k ` |v|k řk i“1 |u|2pi´1q|v|2pk´iq |u|k|v|k (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) and ���� 1 |u|k ´ 1 |v|k ���� “ ||v| ´ |u|| řk i“1 |u|i´1 |v|k´i |u|k |v|k ď |v ´ u| řk i“1 |u|i´1 |v|k´i |u|k |v|k (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) In particular, formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) with u “ ∆ηXpθq and v “ p{ θ ´ ηq ¨ ∇Xpηq, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15)-(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16), makes it clear that there is an extra cancellation in the kernels of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18), 22 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN In summary, our equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) is given by BX Bt pθq “ FpXqpθq “ NpXqpT p|∇S2X|q∇S2Xqpθq “ Mp∇XqpT p|∇S2X|q∇S2Xqpθq ` RpXqpT p|∇S2X|q∇S2Xqpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Symbol of the leading term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As a preliminary step towards studying the leading term M (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), let us consider its frozen-coefficient counterpart, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', re- placing ∇Xpηq by a constant matrix A and letting pg “ I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We start with the case T “ Id, that is, T ” 1: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) pLL AY qkpθq “ p ˜ MpAq∇Y qkpθq, where we define ˜ MpAq “ ˜ M1pAq ` ˜ M2pAq and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) p ˜ MjpAqZqkpθq “ ´ ż R2 B Bηi pGj k,lpA pθ ´ ηqqqZl,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Fθ be the 2D Fourier transform in θ and ξ “ pξ1, ξ2qT: Fθwpξq “ ż R2 wpθq expp´iθ ¨ ξqdθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We now compute the Fourier transform of the function GA: GApθq “ GpAθq “ 1 8π ˜ I |Aθ| ` Aθ b Aθ |Aθ|3 ¸ , where I3 is the 3 ˆ 3 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given that θ P R2 and A is a 3 ˆ 2 matrix, it is convenient to rewrite GA as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, note that: |Aθ|2 “ Aθ ¨ Aθ “ θ ¨ ` ATAθ ˘ “ |Bθ|2 , B “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ATA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that B is a 2 ˆ 2 symmetric positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using this B, we have: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) GApθq “ 1 8π ˜ I |Bθ| ` Q ˜ Bθ b Bθ |Bθ|3 ¸ QT ¸ , Q “ AB´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We note that Q is an isometry in the sense that QTQ “ I2 where I2 is the 2 ˆ 2 identity matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We are now ready to compute the Fourier transform of GA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, note that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) Fθ ˆ 1 |θ| ˙ “ 2π |ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, a simple change of variable yields: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) Fθ ˆ 1 |Bθ| ˙ “ 2π detpBq |B´1ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, note that: Fθ ˜ θiθj |θ|3 ¸ “ Fθ ˆ θiθj∆θ ˆ 1 |θ| ˙˙ “ B2 BξiBξj ˆ |ξ|2 Fθ ˆ 1 |θ| ˙˙ “ 2π B2 BξiBξj |ξ| “ 2π ˜ δij |ξ| ´ ξiξj |ξ|3 ¸ , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 23 where ∆θ is the Laplacian in R2, δij is the Kronecker delta and we used (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) in the third equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In matrix notation, the above can be written as: Fθ ˜ θ b θ |θ|3 ¸ “ 2π ˜ I2 |ξ| ´ ξ b ξ |ξ|3 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Again, by changing variables, we see that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) Fθ ˜ Bθ b Bθ |Bθ|3 ¸ “ 2π detpBq ˜ I2 |B´1ξ| ´ B´1ξ b B´1ξ |B´1ξ|3 ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23), we obtain: FθGA “ 1 4detpBq ˜ I ` QQT |B´1ξ| ´ QB´1ξ b QB´1ξ |B´1ξ|3 ¸ “ 1 4 a detpATAq ˜ I ` ApATAq´1AT pξ ¨ pATAq´1ξq1{2 ´ ApATAq´1ξ b ApATAq´1ξ pξ ¨ pATAq´1ξq3{2 ¸ This implies that the Fourier symbol of LL A is given by: LL AY “ ´F´1 ξ LL ApξqFθY , LL Apξq “ |ξ|2 pFθGAq pξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To better understand the properties of Fourier multiplier LL Apξq, we first note that: QQT ´ QB´1ξ b QB´1ξ |B´1ξ|2 “ Q ˜ I2 ´ B´1ξ b B´1ξ |B´1ξ|2 ¸ QT “ vpξq b vpξq, vpξq “ QRπ{2 B´1ξ |B´1ξ|, Rπ{2 “ ˆ 0 ´1 1 0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) Note that vpξq P R3 is a unit vector, and hence, the above matrix 3 ˆ 3 matrix is an orthogonal projection on to the subspace spanned by vpξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We see that: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) LL Apξq “ |ξ|2 4detpBq |B´1ξ| pI ` vpξq b vpξqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is now immediate that LL Apξq is a symmetric positive definite matrix for each ξ ‰ 0 with eigenvalues: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) λ “ µ 4 |ξ|2 and µ 2 |ξ|2 , µ “ 1 detpBq |B´1ξ|, where the eigenspace for µ{2 is spanned by vpξq and the two-dimensional eigenspace of µ{4 is spanned by the orthogonal complement of vpξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We also have: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) µ 4 |ξ|2 |w|2 ď w ¨ Lpξqw ď µ 2 |ξ|2 |w|2 for any w P R3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the case of general T , the frozen coefficient linear operator can be obtained by a further linearization of the force function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the expression: T pλτq λτ BpXl ` τYlq Bθi , λτ “ λpX ` τY q 24 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where λ is here viewed as a function of X through its dependence on g (see (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, d dτ ˆT pλτq λτ BpXl ` τYlq Bθi ˙ˇˇˇˇ τ“0 “ ˆ 1 λ dT dλ ´ T λ2 ˙ dλτ dτ ˇˇˇˇ τ“0 BXl Bθi ` T λ BYl Bθi “ ˆ 1 λ dT dλ ´ T λ2 ˙ 1 λppg´1qm,n BXq Bθm BYq Bθn BXl Bθi ` T λ BYl Bθi .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, the frozen coefficient approximation amounts to taking pg “ I2, BXl{Bθi “ Al,i and λ “ ∥A∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, d dτ ˆT pλτq λτ BpXl ` τYlq Bθi ˙ˇˇˇˇ τ“0 « pTF pAqqi,l,m,q BYq Bθm , with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) TFpAq “ T p∥A∥F q ∥A∥F I2 b I2 ´ ˆT p∥A∥F q ∥A∥F ´ dT dλ p∥A∥F q ˙ A b A ∥A∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, the frozen-coefficient linear operator in the general force case is given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) pLAY qkpθq “ ´ ż R2 B Bηi pGk,lpA pθ ´ ηqqqpTF pAq∇Y ql,ipηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us now take the Fourier transform of the divergence of the above: Fp∇ ¨ pTF pAq∇Y qqpξq “ ´MApξqFY pξq, where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) MApξq “ T p∥A∥F q ∥A∥F ˜ |ξ|2 I ´ Aξ b Aξ ∥A∥2 F ¸ ` dT dλ p∥A∥F qAξ b Aξ ∥A∥2 F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that, if we set T “ Id, then TF “ Id and the above reduces to Mpξq “ |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, in the general case, the multiplier in LApξq of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) becomes: LApξq “ pFθGAq pξqMApξq “ I ` vpξq b vpξq 4detpBq |B´1ξ| ˆT p∥A∥F ∥A∥F ˆ |ξ|2 I ´ Aξ b Aξ ∥A∥F ˙ ` dT dλ p∥A∥F qAξ b Aξ ∥A∥F ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) It is not difficult to see that, if T ą 0 and dT {dλ ě 0, then the above is coercive in |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Calculus estimates In this section we include some estimates of the operators that will be frequently used in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 25 Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1pS2q, such that |X|˚ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the kernels Gj k,lpxq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and qj k,lppx, pyq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) satisfy the following bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) ���Bα x Gj k,lpxq ��� ď C |x|1`|α| , ���∆y ´ Bα x Gj k,lpxq ¯��� ď C M 1`|α| m3`2|α| , |qj k,lppx, pyq| ď C |∇S2Xppyq| |∆ pyXppxq|2 1 |px ´ py|2 ď C }∇S2X}C0pS2q |X|2˚ 1 |px ´ py|2 , where |α| defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1), M “ max p|x| , |y|q, and m “ min p|x| , |y|q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the sake of completeness, we include a version of the divergence theorem that will be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that, following standard convention, we will not explicitly write the principal values elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A and a compact set D Ă R2 containing 0, then p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż Dc ∇GpAηqdη : “ lim LÑ8 ż DcXBpLq ∇GpAηqdη “ ´ ż BD GpAηqn pηq dl pηq , where B pLq Ă R2 is the ball centered at 0 of radius L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż R2 ∇GpAηqdη :“ lim LÑ8,εÑ0 ż BpLqzBpεq ∇GpAηqdη “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since D is compact and contains 0, D Ă B pLq when L is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by integration by parts ż DcXBpLq ∇GpAηqdη “ ´ ż BD GpAηqn pηq dl pηq ` ż BBpLq GpAηqn pηq dl pηq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since GpAηq is even, the boundary term vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ż Dc ∇GpAηqdη “ lim LÑ8 ż DcXBpLq ∇GpAηqdη “ ´ ż BD GpAηqn pηq dl pηq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, set D “ B pεq, ż R2 ∇GpAηqdη “ lim LÑ8,εÑ0 ż BpLqXBpεqc ∇GpAηqdη “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let A be a matrix in the set DAσ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the linear operator ˜ MpAq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) maps CγpR2q X L2pR2q to CγpR2q X L2pR2q for any any γ P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, } ˜ MjpAqZ}CγpR2q ď C σ2 ´ 1 ` ´σ1 σ2 ¯2¯ }Z}CγpR2qXL2pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' And given A1, A2 P DAσ1,σ2 } ˜ MjpA1qZ ´ ˜ MjpA2qZ}CγpR2q ď C σ2 2 ´ 1 ` ´σ1 σ2 ¯5¯ }Z}CγpR2qXL2pR2q ∥A1 ´ A2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 26 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Taking into account Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we have p ˜ MjpAqZqkpθq “ ´ ż R2 B Bηm Gj k,lpApθ ´ ηqq ` Zl,mpηq ´ Cl,m ˘ dη1dη2, where Cl,m is an arbitrary constant, that we will take to be zero or Zl,mpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the estimate for |p ˜ MjpAqZqkpθq| follows by splitting the integral in two terms, p ˜ MjpAqZqkpθq “ I1pθq ` I2pθq, with I1pθq “ ´ ż |θ´η|ď1 B Bηm Gj k,lpApθ ´ ηqq ` Zl,mpηq ´ Zl,mpθq ˘ dη1dη2, I2pθq “ ´ ż |θ´η|ě1 B Bηm Gj k,lpApθ ´ ηqqZl,mpηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since B Bηm Gj k,lpApθ ´ ηqq “ ´ BGj k,l Bxi pApθ ´ ηqqAi,m, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) the kernel bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and the fact that A P DAσ1,σ2 provide that |I1pθq| ď C σ1 σ2 2 �Z�CγpR2q, |I2pθq| ď C σ1 σ2 2 }Z}L2pR2q, hence |p ˜ MjpAqZqkpθq| ď C σ1 σ2 2 }Z}CγpR2qXL2pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed with the seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let h P R2, |h| ď 1, and perform the following splitting p ˜ MjpAqZqkpθq ´ p ˜ MjpAqZqkpθ ` hq “ J1 ` J2 ` J3 ` J4, where J1 “ ´ ż |θ´η|ď2|h| B Bηm Gj k,lpApθ ´ ηqqδηZl,mpθqdη, J2 “ ż |θ´η|ď2|h| B Bηm Gj k,lpApθ ` h ´ ηqqδηZl,mpθ ` hqdη, J3 “ δθZl,mpθ ` hq ż |θ´η|ą2|h| B Bηm Gj k,lpθ ´ ηqdη, J4 “ ż |θ´η|ą2|h| ´ B Bηm Gj k,l pApθ`h´ηqq´ B Bηm Gj k,lpApθ´ηqq ¯ δηZl,mpθ`hqdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Absolutely, |J1| ` |J2| ď C σ1 σ2 2 �Z�CγpR2q |h|γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, J3 “ pZl,mpθ ` hq ´ Zl,mpθqq ż |θ´η|“2|h| Gj k,l pApθ ´ ηqq nmpηqdlpηq, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 27 and so |J3| ď C�Z�CγpR2q |h|γ 1 σ2 ż |θ´η|“2|h| 1 |θ ´ η|dl pηq ď C 1 σ2 �Z�CγpR2q |h|γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since B Bθp B Bηm Gj k,lpApθ ´ ηqq “ ´ B Bxq B Bxi Gj k,lpApθ ´ ηqqAi,mAq,p, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) it follows that |J4| “ ˇˇˇ ż |θ´η|ą2|h| ż 1 0 hp B Bθp B Bηm Gj k,l pApθ ` sh ´ ηqq δηpZl,mpθ ` hqdsdη ˇˇˇ ď C σ2 1 σ3 2 �Z�CγpR2q |h| ż |θ´η|ą2|h| ż 1 0 |θ ` h ´ η|γ |θ ` sh ´ η|3 dsdη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the domain where |θ ´ η| ą 2 |h|, it holds that for s P r0, 1s, 1 2 |θ ` h ´ η| ď |θ ` sh ´ η| ď 3 2 |θ ` h ´ η| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, |J4| ď C σ2 1 σ3 2 �Z�CγpR2q |h|γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we obtain ��� ˜ MjpAqZ ��� CγpRq ď C σ2 ´ 1 ` σ2 1 σ2 2 ¯ }Z}CγpR2qXL2pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the L2 norm, since ξmFθ ” Gj k,l pAθq ı pξq is bounded by σ1 and σ2, we have ���p ˜ MjpAqZqk ��� L2pRq “ ���ξmFθ ” Gj k,l pAθq ı pξq FrZl,ms pξq ��� L2pRq ď C pσ1, σ2q ∥FrZs∥L2pRq “ C pσ1, σ2q ∥Z∥L2pRq (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) Next, through (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), we have ���Gj k,lpA1pθ ´ ηqq´Gj k,lpA2pθ ´ ηqq ��� ď C σ1 |pA1´A2q pθ ´ ηq| σ3 2 |θ ´ η|3 ď C σ1 σ3 2 ∥pA1´A2q∥ |θ ´ η| , ˇˇˇ B Bηm Gj k,lpA1pθ´ηqq´ B Bηm Gj k,lpA2pθ´ηqq ˇˇˇ ď ˇˇˇ BGj k,l Bxi pA1pθ´ηqq´ BGj k,l Bxi pA2pθ´ηqq ˇˇˇ|A1,i,m| ` ˇˇˇ BGj k,l Bxi pA2pθ´ηqq pA1,i,m´A2,i,mq ˇˇˇ ď C ´ 1 σ2 2 ` σ3 1 σ5 2 ¯∥pA1 ´ A2q∥ |θ ´ η|2 , 28 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN and ˇˇˇ B Bθp B Bηm Gj k,lpA1pθ´ηqq´ B Bθp B Bηm Gj k,lpA2pθ´ηqq ˇˇˇ ď ˇˇˇ B Bxq B Bxi Gj k,lpA1pθ´ηqq´ B Bxq B Bxi Gj k,lpA2pθ´ηqq ˇˇˇ|A1,i,mA1,q,p| ` ˇˇˇ B Bxq B Bxi Gj k,lpA2pθ´ηqq pA1,i,mA1,q,p´A2,i,mA2,q,pq ˇˇˇ ď C ´σ1 σ3 2 ` σ5 1 σ7 2 ¯∥pA1´A2q∥ |θ ´ η|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, we obtain } ˜ MjpA1qZ´ ˜ MjpA2qZ}CγpR2q ď C σ2 2 ´ 1` ´σ1 σ2 ¯5¯ }Z}CγpR2qXL2pR2q ∥A1´A2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ As an immediate consequence of the previous lemma with ˜Zl,m “ Bx Xi Bηm pηqZl,ipηq, we obtain the following lemma for MpAq: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let A be a matrix in the set DAσ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the linear operators MjpAq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) map CγpS2q to CγpR2q for any any γ P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, }MjpAqZ}CγpR2q ď C σ2 ´ 1 ` ´σ1 σ2 ¯2¯ sup l,m } Bx X Bηm pηq ¨ Zl,‚pηq}CγpR2qXL1pR2q ď Cpσ1, σ2q}Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In most cases, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 will be used with Z compactly sup- ported and given by a multiple of a gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Notice that in that case, for Z “ λ∇S2X, λ : R2 ÞÑ R, Bx X Bηm pηq ¨ Zl,‚pηq “ λpηq BXl Bηm pηq, and therefore }MjpAqpλ∇S2Xq}CγpR2q ď Cpσ1, σ2q}λ∇X}CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1pS2q such that |X|˚ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the linear operators N jpXq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) map CγpS2q to CγpS2q for any γ P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) }N jpXqZ}CγpS2q ď C |X|˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We first notice that we can introduce an arbitrary constant (matrix), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) NpXqZppxq “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq ¨ pZppyq ´ Cqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will usually take C “ 0 or C “ Zppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the kernels (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8), we write the equation for each component pN jpXqZqkppxq “ ´ ż S2 qj k,lppx, pyq ¨ pZl,‚ppyq ´ Cl,‚qdpy, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 29 where Cl,‚ “ 0 or Cl,‚ “ Zl,‚ppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We first perform the estimate for |NpXqZppxq|, px P S2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) |NpXqZppxq| ď 2ÿ j“1 |N jpXqZppxq|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) for the kernel, we have (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) |NpXqZppxq| ď C }∇S2X}C0pS2q |X|2˚ ż S2 |Zl,‚ppxq ´ Zl,‚ppyq| |px ´ py|2 dpy ď C }∇S2X}C0pS2q |X|2˚ }Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to estimate the H¨older seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let px, pxh P S2, and denote h “ |px ´ pxh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We write pN jpXqZqkppxq´pN jpXqZqkppxhq “ ż S2qj k,lppx, pyq ¨ pZl,‚ppxq´Zl,‚ppyqqdpy ´ ż S2qj k,lppxh, pyq¨pZl,‚ppxhq´Zl,‚ppyqqdpy, and perform the following splitting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) pN jpXqZqkppxq ´ pN jpXqZqkppxhq “ I1 ` I2 ` I3 ` I4, where I1 “ ż t|px´ py|ď2huXS2 qj k,lppx, pyq ¨ pZl,‚ppxq ´ Zl,‚ppyqqdpy, I2 “ ´ ż tpx´ py|ď2huXS2 qj k,lppxh, pyq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy, I3 “ pZl,‚ppxq ´ Zl,‚ppxhqq ¨ ż t|px´ py|ě2huXS2 qj k,lppx, pyqdpy, I4 “ ż t|px´ py|ě2huXS2pqj k,lppx, pyq ´ qj k,lppxh, pyqq ¨ pZl,‚ppxhq ´ Zl,‚ppyqqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first two terms are estimated directly (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) |I1| ` |I2| ď C }∇S2X}C0pS2q |X|2˚ }Z}CγpS2qhγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the third, we use that the kernel is a derivative to integrate by parts and obtain that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) |I3| “ |pZl,‚ppxq ´ Zl,‚ppxhqq ¨ ż t|px´ py|“2huXS2 GjpXppxq ´ XppyqqnppyqdlS2ppyq| ď C }Z}CγpS2q |X|˚ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we use the mean-value theorem on the kernel to estimate I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set ℓpsq the shortest path function from pxh to px respect to arc-length s variable, and L “ 30 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN dist ` pxh, px;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' S2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have Gj k,lpXppxq´Xppyqq ´ Gj k,lpXppxhq´Xppyqq “ ż L 0 B BsGj k,lpXpℓ psqq´Xppyqqds “ ż L 0 ∇S2Gj k,lpXpℓ psqq´Xppyqq ¨ B Bsℓ psq ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, for qj k,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), |qj k,lppx, pyq ´ qj k,lppxh, pyq| “ ˇˇˇ∇S2 ż L 0 ∇S2Gk,lpXpℓ psqq´Xppyqq ¨ B Bsℓ psq ds ˇˇˇ “ ˇˇˇ∇S2 ż L 0 B Bxi Gk,lpXpℓ psqq´Xppyqq ˆ ∇S2Xi pℓ psqq ¨ B Bsℓ psq ˙ ds ˇˇˇ “ ˇˇˇ ż L 0 B Bxj B Bxi Gk,lpXpℓ psqq´Xppyqq ˆ ∇S2Xi pℓ psqq¨ B Bsℓ psq ˙ ∇S2Xj ppyq ds ˇˇˇ, and recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) we obtain the bound |qj k,lppx, pyq ´ qj k,lppxh, pyq| ďC ∥∇S2X∥2 C0pS2q |X|3 ˚ ż L 0 1 |ℓ psq ´ py|3 ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have that |I4| ď C }∇S2X}2 C0pS2q}Z}CγpS2q |X|3 ˚ ż t|px´ py|ě2huXS2 |pxh´ py|γ ż L 0 ds |ℓ psq´ py|3 dpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We notice that since ℓpsq is the shortest path function from pxh to px on S2, it holds that |ℓ psq ´ px| ď |px ´ pxh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, |ℓ psq ´ py| ě |px ´ py| ´ |ℓ psq ´ px| ě |px ´ py| ´ |pxh ´ px| ě 1 2 |px ´ py| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In addition, |pxh ´ py| ď |px ´ py| ` |pxh ´ px| ď 3 2 |px ´ py| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since L ď Ch, we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) |I4| ď C }∇S2X}2 C0pS2q}Z}CγpS2q |X|3 ˚ h ż tpx´ py|ě2huXS2 dpy |px ´ py|3´γ ď C }∇S2X}2 C0pS2q}Z}CγpS2q |X|3 ˚ hγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Joining the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) back in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) rNpXqZsCγpS2q ď C |X|˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }Z}CγpS2q, and, with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), the same bound holds for the H¨older norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 31 We will need to localize the operators above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For that purpose, let us define the cutoff function pρn, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) pρnppxq “ # 1 if |px ´ pxn| ď 3R, 0 if |px ´ pxn| ě 4R, and recall the partition of unity tρnu based on the points pxn (see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1pS2q such that |X|˚ ą 0, NpXq the linear operator defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), and pρn the cutoff function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for Z P C0pS2q compactly supported on Bpxn,2R X S2, it holds that, }p1 ´ pρnqN jpXqZ}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Ippxq “ p1 ´ pρnppxqqN jpXqZppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since 1 ´ pρppxq “ 0 when px P Bpxn,3R, let px P S2zBpxn,3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, recalling the condition on the support of Z, Ippxq“ppρnppxq´1q ż B pxn,2RXS2 ∇S2GpXppxq´Xppyqq¨Zppyqdpy, and using the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) for the kernel, |Ippxq| ď C }∇S2X}C0pS2q |X|2˚ }Z}C0pS2q ż B pxn,2RXS2 |px´ py|´2dpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since we have that |px ´ py| ě R, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) |Ippxq| ď Cp|X|˚, }∇S2X}C0pS2qq}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To estimate the H¨older seminorm, consider two points px, pxh P S2, h “ |px ´ pxh|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Due to the cut-off function pρn, the only non-trivial case is px, pxh P S2zBpxn,3R: |Ippxhq ´ Ippxq| “ ppρnppxq ´ pρnppxhqq ż B pxn,2RXS2qk,lppx, pyq ¨ Zl,‚ppyqdpy ` p1´pρnppxhqq ż B pxn,2RXS2 ` qk,lppx, pyq´qk,lppxh, pyq ˘ ¨ Zl,‚ppyqdpy “ J1 ` J2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term is bounded as (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15), |J1| ď C}pρn}C1pS2q }∇S2X}C0pS2q |X|2˚ }Z}C0pS2qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the expression of qk,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), we can check that, since |px´py| ě R, |pxh´py| ě R, |qk,lppx, pyq ´ qk,lppxh, pyq| ď C }∇S2X}2 C0pS2q |X|3˚R3 h, hence |J2| ď C R }∇S2X}2 C0pS2q |X|3˚ }Z}C0pS2qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) }I}C1pS2q ď CpR, |X|˚, }∇S2X}C0pS2q}C1q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 32 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN The previous lemma holds analogously for the operators ˜ MpAq: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let A P DAσ1,σ2, ˜ MjpAq the linear operator defined by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22), and pρn the cutoff function (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for Z P C0pR2q compactly supported on V2R, it holds that, }p1 ´ pρnq ˜ MjpAqZ}C1pR2q ď CpR, σ1, σ2q}Z}C0pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pxn P S2, X P C1pS2q, |X|˚ ą 0, and ρn P C8pBpxn,2R X S2q, R ă 1{10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the commutator rρn,NpXqsZppxq “ ´ ż S2∇S2GpXppxq´Xppyqq¨ Zppyqpρnppyq´ρnppxqqdpy, satisfies that, for any γ P p0, 1q, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) ∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the kernel bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1), |rρn, NpXqsZppxq| ď C }∇S2ρn}C0pS2q}Z}C0pS2q}∇S2X}C0pS2q |X|2˚ ż S2 1 |px ´ py|dpy ď Cp|X˚, }∇S2X|C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we study the H¨older seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We take two points px, pxh, denote h “ |px ´ pxh|, and perform a splitting analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using the kernel notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsZppxq´rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsZppxhq“ ż S2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq´ρnppxqq¨Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy ´ ż S2qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppxh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq´ρnppxhqq¨Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy “ I1 n ` I2 n ` I3 n ` I4 n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' where I1 n “ ż t|px´ py|ď2huXS2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq ´ ρnppxqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy I2 n “ ´ ż t|px´ py|ď2huXS2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppxh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppyq ´ ρnppxhqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy I3 n “ ż t|px´ py|ě2huXS2 qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq pρnppxhq ´ ρnppxqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy and I4 n “ ż t|px´ py|ě2huXS2 pqk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyq´qk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='lppxh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pyqq pρnppyq´ρnppxhqq ¨ Zl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='‚ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for I1 n and I2 n, |I1 n| ` |I2 n| ď C ∥∇S2ρn∥C0pS2q ∥∇S2X∥C0pS2q |X|2 ˚ }Z}C0pS2qh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 33 Next, for I3 n, integration by parts gives that ��I3 n �� ď C |ρnppxhq´ρnppxq| ż t|px´ py|ě2huXS2 }Z}C0pS2q ∥∇S2X∥C0pS2q |X|2 ˚ |px ´ py|2 dpy ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2qh log h´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, in I4 n, the use of the mean-value theorem provides that ��I4 n �� ď C ∥∇S2ρn∥C0pS2q }Z}C0pS2q ∥∇S2X∥2 C0pS2q |X|3 ˚ ˆ ż t|px´ py|ď2huXS2 ż L 0 |pxh ´ py| |ℓ psq ´ py|3 dsdpy ď Cp|X˚, ∥∇S2X∥C0pS2qq ∥∇S2ρn∥C0pS2q ∥Z∥C0pS2q h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, ∥rρn, NpXqsZ∥CγpS2q ď Cp|X˚, }∇S2X}C0pS2qq}∇S2ρn}C0pS2q}Z}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let pxn P S2 and x Xn : R2 Y t8u Ñ S2 the stereographic projection centered at pxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X P C1,γpS2q, |X|˚ ą 0, and pρn given by (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let the linear operators NpXq and MpAq be defined by in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), with A “ ∇Xnp0q, where we are using the notation Xnpθq “ Xpx Xnpθqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Denote Inpθq “ pρnpθqrMpAq ´ NpXqsZnpθq, Then, for Z P CγpS2q compactly supported on Bpxn,2R X S2, the following estimates hold: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) }In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` p1 ` }∇S2X}CγpB pxn,5RXS2qq}Z}C0pS2q ` εpRq}Z}CγpS2q ˘ , and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) }In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` }Z}C0pS2q`}∇S2X}C γ 2 pB pxn,5RXS2q}Z}C γ 2 pS2q ` εpRq}Z}CγpS2q ˘ , with εpRq Ñ 0 as R Ñ 0 given by the modulus of continuity of ∇S2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, we write (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) In “ I1 n ` I2 n “ 2ÿ j“1 pρnpθqrMjpAq ´ N jpXqsZnpθq “ 2ÿ j“1 pρnpθqrMjpAq ´ Mjp∇XnqsZnpθq ´ RjpXnqZnpθq, and focus on the term I1 n, as the other I2 n will follow similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We then write I1 n “ ´pρnpθq ż R2 P 1 m,k,lpθ, ηq B p Xi Bηm pηqZn,lipηqdη1dη2, 34 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where we used the notation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) P 1 m,k,lpθ, ηq “δk,l 8π B Bηm ´ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ¯ “ ´ 1 8π pBpθ´ηqqm |Apθ´ηq|3 δk,l ` q1 m,k,lpθ, ηq “ ´ 1 8π pBpθ´ηqqm |Apθ´ηq|3 δk,l ` m1 m,k,lpθ, ηq ´ K1 m,k,lpθ, ηq “ ´ 1 8π ˆpBpθ´ηqqm |Apθ´ηq|3 ´ pBnpηqpθ´ηqqm |Anpηqpθ´ηq|3 ˙ δk,l ´ K1 m,k,lpθ, ηq :“P1 m,k,lpθ, ηq ´ K1 m,k,lpθ, ηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Above, we are denoting B “ AT A, A “ ∇Xnp0q, Anpηq “ ∇Xnpηq, and K1 m,k,l was given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We denote ˜Zl,mpηq “ B p Xi Bηm pηqZn,lipηq, and note that } ˜Z}CγpR2q ď C}Z}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We start with a bound for |I1 n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We split it as follows (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) I1 n “ O1 ` O2, with O1 “ pρnpθq ż V5R P 1 mklpθ, ηq ` ˜Zl,mpηq ´ ˜Zl,mpθq ˘ dη1dη2 O2 “ pρnpθq ˜Zl,mpθq ż V5R P 1 m,k,lpθ, ηqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we split O1 further |O1| ď O1,1 ` O1,2, where O1,1 “ } ˜Z}CγpV5Rq ż V5R |P1 m,k,lpθ, ηq||θ ´ η|γdη1dη2, O1,2 “ pρnpθq ż V5R |K1 m,k,lpθ, ηq||δη ˜Zl,mpθq|dη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For θ P V4R, η P V5R, we have the following bound for P1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) |P1 m,k,l| ď 1 8π |pBnpηq ´ Bqpθ ´ ηq |Apθ ´ ηq|3 | ` 1 8π |Bnpηqpθ ´ ηq|| 1 |Apθ ´ ηq|3 ´ 1 |Anpηqpθ ´ ηq|3 | ď C }Bnp¨q ´ B}C0pV5Rq |X|3˝,n|θ ´ η|2 ` C }Anp¨q ´ A}C0pV5Rq}∇Xn}7 C0pV5Rq |X|9˝,n|θ ´ η|2 , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 35 where (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) has been used for the last term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the first term O1,1 is high-order but with small coefficients, O1,1 ď C ´}Bnp¨q ´ B}C0pV5Rq |X|3˝,n ` }Anp¨q ´ A}C0pV5Rq}∇Xn}7 C0pV5Rq |X|9˝,n ¯ Rγ ˆ } ˜Z}CγpV5Rq, since we have that }Anp¨q ´ A}C0pV5Rq ď εpRqCp}∇X}C0pV5Rqq, with εpRq Ñ 0 as R Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The kernel bound, with θ P V4R, η P V5R, |K1 m,k,lpθ, ηq| ď C }∇Xn}C0pV5Rq |X|3˝,n r∇XnsCγpV5Rq 1 |θ ´ η|2´γ , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) gives that O1,2 ď C }∇Xn}C0pV5Rq}∇Xn}CγpV5Rq |X|3˝,n Rγ} ˜Z}C0pV5Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' But one also has the bound |K1 m,k,lpθ, ηq| ď C }∇Xn}C0pV5Rq}∇Xn}C γ 2 pV5Rq |X|3˝,n 1 |θ ´ η|2´ γ 2 , which gives that O1,2 ď C }∇Xn}C0pV5Rq}∇Xn}C γ 2 |X|3˝,n Rγ} ˜Z}C γ 2 pV5Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Joining the two bounds, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) |O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` } ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q ` εpRq} ˜Z}CγpV5Rq ˘ , and |O1| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` }∇S2X}C γ 2 pB pxn,5RXS2q} ˜Z}C γ 2 pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To estimate O2, we use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) to integrate by parts, |O2| ď C|pρnpθq|| ˜Zl,mpθq| ż BV5R ˇˇˇ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ˇˇˇdlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we note that since θ P V4R, we have that for η P BV5R, |θ ´ η| ě R 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We compute the difference (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| “ 1 |θ ´ η| ´ 1 |Ap{ θ ´ ηq| ´ 1 |∆ηXnpθq| ¯ “ 1 |θ´η| ` ∆ηXnpθq´Ap{ θ ´ ηq ˘ ¨ ` ∆ηXnpθq`Apz θ´ηq ˘ |Apz θ´ηq||∆ηXnpθq| ` |Apz θ´ηq|`|∆ηXnpθq| ˘ , 36 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN and ∆ηXnpθq´Ap{ θ ´ ηq “ ∆ηXnpθq´∇Xnpηqp{ θ ´ ηq`p∇Xnpηq ´ Aqp{ θ ´ ηq “ ´EηXnpθq ` pAnpηq ´ Aqp{ θ ´ ηq, where EηXnpθq is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, |O2| ď C} ˜Z}C0pV5Rq }∇Xn}C0pV5Rq |X|3˝,n ż BV5R |EηXnpθq|`}Anp¨q´A}C0pV5Rq |θ´η| dlpηq, hence |O2| ď Cp|X|˚, }∇S2X}C0pS2qq} ˜Z}C0pV5RqpR γ 2 }∇S2X}C γ 2 pB pxn,5RXS2q ` εpRqq, and |O2| ď Cp|X|˚, }∇S2X}C0pS2qqRγ}∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) back in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) |I1 n| ď Cp|X|˚, }∇S2X}C0pS2qqRγ` } ˜Z}C0pV5Rq}∇S2X}CγpB pxn,5RXS2q ` εpRq} ˜Z}CγpV5Rq ˘ , and also |I1 n| ď Cp|X|˚, }∇S2X}C0pS2qq ` R γ 2 }∇S2X}C γ 2 pB pxn,5RXS2q} ˜Z}C γ 2 pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to estimate the H¨older seminorm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Take θ, θ ` h P V4R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the splitting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22), and start with the estimate for O1: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) |O1pθ ` hq ´ O1pθq| “ |Q1 ` Q2 ` Q3 ` Q4 ` Q5|, where, using the notation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12), Q1 “ ´pρnpθ ` hq ż t|θ´η|ď2|h|uXV5R P 1 m,k,lpθ ` h, ηqδη ˜Zl,mpθ ` hqdη1dη2, Q2 “ ´pρnpθ ` hq ż t|θ´η|ď2|h|uXV5R P 1 m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2, Q3 “ pρnpθ ` hqδθ ˜Zl,mpθ ` hq ż t|θ´η|ě2|h|uXV5R P 1 m,k,lpθ, ηqdη1dη2, Q4 “ pρnpθ ` hq ż t|θ´η|ě2|h|uXV5R pP 1 m,k,lpθ, ηq ´ P 1 m,k,lpθ ` h, ηqqδη ˜Zl,mpθ ` hqdη1dη2, Q5 “ ppρnpθq ´ pρnpθ ` hqq ż V5R P 1 m,k,lpθ, ηqδη ˜Zl,mpθqdη1dη2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and the bounds for P1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and K1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24), we obtain |Q1| ` |Q2| ď C ´}Bnp¨q ´ B}C0pV5Rq |X|3˝,n ` }Anp¨q ´ A}C0pV5Rq}∇Xn}7 C0pV5Rq |X|9˝,n ¯ ˆ ż t|θ´η|ď2|h|uXV5R r ˜ZsCγpV5Rq |θ ´ η|2´γ dη1dη2 ` C }∇Xn}C0pV5Rq |X|3˝,n ż t|θ´η|ď2|h|uXV5R r∇XnsCγpV5Rq} ˜Z}C0pV5Rq |θ ´ η|2´γ dη1dη2, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 37 thus (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) |Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is clear that we could also obtain the estimate |Q1|`|Q2| ď Cp|X|˚, }∇S2X}C0pS2qq|h|γ` }∇S2X}C γ 2 pB pxn,5RXS2q} ˜Z}C γ 2 pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We see that the difference between those two type of estimate comes only from the kernel K, where we can distribute half a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The same idea propagates along the lines below, hence we only show the first estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, integration by parts in Q3 gives |Q3| ď C|pρnpθ ` hq||δθ ˜Zpθ ` hq| ˆ ż t|θ´η|“2|h|uYBV5R ˇˇˇ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ˇˇˇdlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26), |h| ď 8R and that θ P V4R, we obtain that ż t|θ´η|“2|h|uYBV5R ˇˇˇ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ˇˇˇdlpηq ď C }∇X}C0pV5Rq |X|3˝,n ż t|θ´η|“2|h|uYBV5R |EηXnpθq| |θ´η| ` }Anp¨q´A}C0pV5Rq |θ´η| dlpηq ď C }∇X}C0pV5Rq |X|3˝,n ´ }∇X}C0pV5Rqpεp|h|q ` εpRqq ` }∇X}C0pV5RqεpRq ¯ ď C }∇X}2 C0pV5Rq |X|3˝,n εpRq, hence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) |Q3| ď εpRqCp|X|˚, }∇S2X}C0pS2qq} ˜Z}CγpV5Rq|h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term Q4 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) is estimated by applying the mean-value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As in the previous terms, we need to consider separately the two kernels in P 1 m,k,l (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We take a derivative on P1 m,k,l, B Bθj P1 m,k,lpθ, ηq “ δk,l 8π ´ ´ Bm,j |Apθ ´ ηq|3 ` pBnpηqqm,j |Anpηqpθ ´ ηq|3 ¯ ` δk,l 8π ´ 3pBpθ ´ ηqqmpBpθ ´ ηqqj |Apθ ´ ηq|5 ´ 3pBnpηqpθ ´ ηqqmpBnpηqpθ ´ ηqqj |Anpηqpθ ´ ηq|5 ¯ , and thus | B Bθj P1 m,k,lpθ, ηq| ď Cp}∇Xn}C0pV4Rq, |X|˝,nq εpRq |θ ´ η|3 , 38 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN while the derivative of K1 m,k,l is computed and bounded below B Bθj K1,1 m,k,lpθ, ηq “ ´δk,l BXnpηq Bηm ¨ ˆ ¨ ˝3 ∆ηXnpθq ¨ BXnpθq Bθj |∆ηXnpθq|5 EηXnpθq |θ ´ η|3 ` δη BXnpθq Bθj |∆ηXnpθq|3|θ ´ η|3 ˛ ‚, | B Bθj K1,1pθ, ηq| ď }∇Xn}C0pV4Rq |X|3˝,n r∇XnsCγpV4Rq |θ ´ η|3´γ ˆ 1`3}∇Xn}C0pV4Rq |X|˝,n ˙ ď Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpB pxn,2RXS2q |θ ´ η|3´γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) |Q4| ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, the estimate for Q5 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) follows from the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) |Q5| ď Cp}∇Xn}C0pV5Rq, |X|˝,nq}pρn}CγpV5Rq|h|γ ˆ ´ εpRq} ˜Z}CγpV5Rq`}∇Xn}CγpV5Rq} ˜Z}C0pV5Rq ¯ ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We combine the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) to conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) |O1pθ ` hq ´ O1pθq| ď Cp|X|˚, }∇S2X}C0pS2qq ` }∇S2X}CγpB pxn,5RXS2q} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We continue with the H¨older seminorm for O2 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Integrating by parts O2pθ ` hq ´ O2pθq “ δk,l 8π pρnpθ ` hq ˜Zl,mpθ ` hq ˆ ż BV5R ´ 1 |Apθ ` h ´ ηq| ´ 1 |Xpθ ` hq ´ Xnpηq| ¯ nmpηqdlpηq ´ δk,l 8π pρnpθq ˜Zl,mpθq ż BV5R ´ 1 |Apθ´ηq| ´ 1 |Xnpθq´Xnpηq| ¯ nmpηqdlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) |O2pθ ` hq ´ O2pθq| “ |Q6 ` Q7|, with Q6 “ δk,l 8π ´ pρnpθ ` hq ˜Zl,mpθ ` hq ´ pρnpθq ˜Zl,mpθq ¯ ˆ ż BV5R | 1 |Apθ ` h ´ ηq| ´ 1 |Xpθ ` hq ´ Xnpηq||nmpηqdlpηq, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 39 Q7 “ δk,l 8π pρnpθq ˜Zl,mpθq ˆ ˆ ż BV5R ´ 1 |Apθ ` h ´ ηq| ´ 1 |Xnpθ ` hq ´ Xnpηq| ¯ nmpηqdlpηq ´ ż BV5R ´ 1 |Apθ ´ ηq| ´ 1 |Xnpθq ´ Xnpηq| ¯ nmpηqdlpηq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) and that θ P V4R, we obtain |Q6| ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq |X|3˝,n ˆ ż BV5R |EηXnpθq|`}Anp¨q´A}C0pV5Rq |θ´η| dlpηq ď C} ˜Z}CγpV5Rq|h|γ }∇Xn}C0pV5Rq |X|3˝,n }∇Xn}C0pV5RqεpRq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we estimate Q7, |Q7| ď C} ˜Z}C0pV5Rq ż BV5R | 1 |Apθ ` h ´ ηq| ´ 1 |Apθ ´ ηq||dlpηq ` C} ˜Z}C0pV5Rq ż BV5R | 1 |Xnpθ ` hq ´ Xnpηq| ´ 1 |Xnpθq ´ Xnpηq||dlpηq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Performing the differences we obtain that |Q7| ď Cp|X|˝,n, }∇Xn}C0pV5Rqqp|h| ` |h|1´γRγq R } ˜Z}C0pV5Rq|h|γ ď Cp|X|˝,n, }∇Xn}C0pV5Rqq} ˜Z}C0pV5Rq|h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, going back to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34), we conclude that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='35) |O2pθ`hq´O2pθq|ďCp|X|˚, }∇S2X}C0pS2qq ` } ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ, and together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) back into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) we obtain the H¨older norm estimate for I1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the kernel in I2 n (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) satisfy the same estimates, we conclude that }In}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` p1 ` }∇S2X}CγpB pxn,5RXS2qq} ˜Z}C0pV5Rq ` εpRq} ˜Z}CγpV5Rq ˘ |h|γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Frozen-coefficient Semigroup We will need later in the proof (see Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) to deal with the following kernels, with 0 ď α ď 1: GαpAθq “ G1pAθq ` αG2pAθq, 40 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where G1, G2 are given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we have that pFθGα,Aq pξq “ ` FθG1pAθq ˘ pξq ` α ` FθG2pAθq ˘ pξq, ` FθG1pAθq ˘ pξq “ 1 4 detpBq |Uξ|, ` FθG2pAθq ˘ pξq “ 1 4 detpBq |Uξ| ˆ P ´ Uξ |Uξ| b Uξ |Uξ| ˙ , where λ “ ∥A∥F , B “ ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ATA, P “ ApATAq´1AT, U “ ApATAq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let us consider the operator defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In preparation for this, we consider the operator with ∇X given by a constant matrix and with pg “ I2, as defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32): Lα,AY :“ ˜ Mα pAq pTFpAq∇Y q “ ´ ˜ M1pAq ` α ˜ M2pAq ¯ pTF pAq∇Y q where TFpAq is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The parameter α is useful in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will prove Lα,A is a sectorial operator first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' That is to say, we have to estimate pz ` Lα,Aq´1 where z is in a set with some ω P R, 0 ă δ ă π 2 in the complex plane: Sω,δ “ tz P C : |argpz ´ ωq| ď π ´ δu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ˜ MjpAq is a singular integral operator, it is difficult to compute its inverse operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' However, ˜ MjpAq is a convolution with kernel Gj pAθq, so we may use its Fourier multiplier to obtain pz ` Lα,Aq´1 Y “F´1ppz ´ Lα,Apξqq´1 FY pξqqpθq where Lα,A is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) (in this section, we will write LA instead of Lα,A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we will use the Fourier multiplier to estimate the original operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In harmonic analysis, the Fourier multiplier theorem in Lp norms is well-known [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will use a Fourier multiplier theorem in semi-norms �¨�CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If T is a Fourier multiplier operator with multiplier m pξq P Cs pRnzt0uq X L8 pRnq , s ą n 2 , and ��Bα ξ m pξq �� ď Cα |ξ|´|α| for all |α| ď s, then for all u P Cγ pRnq X L2 pRnq where 0 ă γ ă 1, �T u�CγpRnq ď Cγ,s,nDm�u�CγpRnq, where Dm “ max|α|ďs Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 may be split into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first part is the well-known equivalence between the homogeneous Besov norm ∥¨∥ 9Bγ 8,8pRnq and H¨older seminorm �¨�CγpRnq [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The second part is proving the Fourier mul- tiplier theorem in homogeneous Besov norms ∥¨∥ 9Bγ 8,8pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Although these results are classical, we include the proof of the version we need in Appendix A for the covenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will compute pz ´ Lα,Apξqq´1 in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, for the norm ∥¨∥C0pR2q, we may expand the result in [47, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1] in R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 41 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 ( [47, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If �u�CγpR2q ă 8 for some γ P p0, 1q and Fu pξq “ 0 in a neighborhood of ξ “ 0, then ∥u∥C0pR2q ď C�u�CγpR2q, where C depends on the neighborhood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we may choose a suitable cutting function ϕ pξq with ϕ pξq “ 1 in a neighborhood of ξ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, the rest of the work for ���pz ` Lα,Aq´1 Y ��� C0pR2q is only ���F´1ppz ´ Lα,Apξqq´1 ϕ pξq FY pξqqpθq ��� C0pR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Fundamental estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We need some elementary estimates on operators LA and LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' To achieve it, we first compute the estimate of TFpAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given matrice A1, A2 in DAσ1,σ2, we have σ2 ď ∥A1∥F , ∥A2∥F ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have the following estimates for TF pAq: |TF pA1qijkl| ďzp0q M , |TFpA1qijkl ´ TF pA2qijkl| ďC ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ∥A1 ´ A2∥ , where zp0q M “ max σ2ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 |f1 pλq| ` |f2 pλq| , zp1q M “ max σ2ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 ���� df1 dλ ���� ` ���� df2 dλ ���� , f1 pλq “ T λ , f2 pλq “ T λ ´ dT dλ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' More specific, given Z P Cγ ` R2˘ Ş L2 ` R2˘ with the size of A1, ∥TFpA1qZ∥CγpR2q Ş L2pR2q ďCzp0q M ∥Z∥CγpR2q Ş L2pR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) ∥pTF pA1q ´ TFpA2qq Z∥CγpR2q Ş L2pR2q ďC ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ∥A1 ´ A2∥ ∥Z∥CγpR2q Ş L2pR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set λi “ ∥Ai∥F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since TF pA1qijkl “ f1 pλ1q δikδjl ´ f2 pλ1q pA1qij pA1qkl λ2 1 , It is obvious to obtain the result of TF pAqijkl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, TF pA1qijkl ´ TF pA2qijkl “ pf1 pλ1q ´ f1 pλ2qq δikδjl ´ pf2 pλ1q ´ f2 pλ2qq pA1qij pA1qkl λ2 1 ´ f2 pλ2q ˆpA1qij pA1qkl λ2 1 ´ pA2qij pA2qkl λ2 2 ˙ Since |λ1 ´ λ2| ď ∥A1 ´ A2∥F ď C ∥A1 ´ A2∥ , we obtain |fi pλ1q ´ fi pλ2q| ď zp1q M |λ1 ´ λ2| ď Czp1q M ∥A1 ´ A2∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 42 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN pA1qij pA1qkl λ2 1 ´ pA2qij pA2qkl λ2 2 “ pA1 ´ A2qij pA1qkl ` pA2qij pA1 ´ A2qkl λ2 1 ` pA2qij pA2qkl pλ2 ` λ1q pλ2 ´ λ1q λ2 1λ2 2 , so ���� pA1qij pA1qkl λ2 1 ´ pA2qij pA2qkl λ2 2 ���� ď C σ1 σ2 2 ∥A1 ´ A2∥ Therefore, |TF pA1qijkl ´ TF pA2qijkl| ď C ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ∥A1 ´ A2∥ Since TF pA1q, TF pA2q are linear operators, we may obtain the estimates in Cγ ` R2˘ X L2 ` R2˘ of TFpA1qZ and pTF pA1q ´ TFpA2qq Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Then, because we have estimated ˜ Mα in Thoerem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, we can obtain the bounds of Lα,A and its difference Lα,A1 ´ Lα,A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A1, A2 P DAσ1,σ2, then for all Y P C1,γ ` R2˘ compactly supported, Lα,AY P Cγ ` R2˘ X L2 ` R2˘ , and ∥Lα,AY ∥CγpRq ď zp0q M σ2 ´ 1 ` ´σ1 σ2 ¯2¯ ∥∇Y ∥CγpR2q Ş L2pR2q , ∥Lα,A1Y ´ Lα,A2Y ∥CγpRq ď C σ2 ´zp0q M σ2 ´ 1 ` σ1 σ2 ¯ ` zp1q M ¯´ 1 ` ´σ1 σ2 ¯5¯ ∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Y P C1,γ ` R2˘ compactly supported, ∇Y is also in L2 ` R2˘ , so pTF pAq∇Y q P Cγ ` R2˘ X L2 ` R2˘ by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, Lα,AY P Cγ ` R2˘ X L2 ` R2˘ and ∥Lα,AY ∥CγpRq ď C σ2 ´ 1 ` ´σ1 σ2 ¯2¯ ∥TF pAq∇Y ∥CγpR2q Ş L2pR2q ďzp0q M σ2 ´ 1 ` ´σ1 σ2 ¯2¯ ∥∇Y ∥CγpR2q Ş L2pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, since Lα,A1Y ´ Lα,A2Y “ ´ ˜ Mα pA1q ´ ˜ Mα pA2q ¯ pTF pA1q∇Y q ´ ˜ Mα pA2q ppTF pA1q ´ TF pA2qq ∇Y q , by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, ∥Lα,A1Y ´ Lα,A2Y ∥CγpRq ď C zp0q M σ2 2 ´ 1 ` ´σ1 σ2 ¯5¯ ∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ ` C σ2 ˆ zp0q M σ1 σ2 2 ` zp1q M ˙ ´ 1 ` ´σ1 σ2 ¯2¯ ∥∇Y ∥CγpR2q Ş L2pR2q ∥A1 ´ A2∥ □ WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 43 Next, we compute some elementary estimates on the symbol LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We denote Gα,A pθq :“ G1 pAθq ` αG2 pAθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The pairs of eigenvalues and respective eigenvec- tors of pFθGα,Aq pξq are ´ p1 ` αq µ 4 , v pξq ¯ , ´µ 4 , vK pξq ¯ , ´µ 4 , v0 ¯ , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) and the pairs of MApξq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) are ˜ T λ ˜ |ξ|2 ´ |Aξ|2 λ2 ¸ ` dT dλ |Aξ|2 λ2 , Aξ ¸ , ˆT λ |ξ|2 , URπ{2ξ ˙ , ˆT λ |ξ|2 , v0 ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) Since pFθGα,Aq pξq and MApξq are symmetric positive definite (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ), LA pξq is diagonalizable and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have some estimates of LA and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given A P DAσ1,σ2, LA and its derivatives satisfy (i) σ2 σ2 1 |ξ|´1 ď µ ď σ1 σ2 2 |ξ|´1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) zm |ξ|2 ď ∥MApξq∥ ď zp0q M |ξ|2 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) σ2zm 4σ2 1 |ξ| ď ∥LApξq∥ ď σ1zp0q M 2σ2 2 |ξ| , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) where zm “ minσ1ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 ´ T λ , ` T λ ` dT dλ ˘ σ2 2 λ2 ¯ , zp0q M “ maxσ1ďλď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 ` T λ ` dT dλ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (ii) ���� BLA Bξk ���� ď C1 σ2 1 σ3 2 zp0q M , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) ���� B2LA BξkBξl ���� ď C1 σ3 1 σ4 2 zp0q M |ξ|´1 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) where C1 is a constant that does not depend on α or A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (iii) B Bξj ∆ξLApξq, p∆ξq2 LApξq and B Bξj p∆ξq2 LApξq may be written as B Bξj ∆ξLApξq “ 1 |ξ|2 Φp3q A,jpˆξq, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) p∆ξq2 LApξq “ 1 |ξ|3 Φp4q A pˆξq, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) B Bξj p∆ξq2 LApξq “ 1 |ξ|4 Φp5q A,jpˆξq, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) where Φp3q A,j, Φp4q A , Φp5q A,j are bounded on ���ˆξ ��� “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (i) We first note the following inequalities: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) σ2 ď ∥B∥ ď σ1, 1 σ1 ď ��B´1�� “ ∥U∥ ď 1 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) σ2 2 ď detpBq “ ∥B∥ ��B´1��´1 ď σ2 1, 44 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where we used the fact that B is a 2 ˆ 2 symmetric positive definite matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), we immediately have: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) |Uξ| “ ��B´1ξ �� ě 1 σ1 |ξ| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, σ2 σ2 1 |ξ|´1 ď µ “ 1 detpBq |B´1ξ| ď σ1 σ2 2 |ξ|´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, through (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), one of the eigenvalues of MA is bounded by ˆT λ ` dT dλ ˙ σ2 2 λ2 |ξ|2 ď T λ ˜ |ξ|2 ´ |Aξ|2 λ2 ¸ ` dT dλ |Aξ|2 λ2 ď ˆT λ ` dT dλ ˙ |ξ|2 , so we may obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since LA is diagonalizable and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with LA “ FθGα,AMA, the eigenvalues of LA are between µ 4 zm |ξ|2 and µ 2 zp0q M |ξ|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, we get the bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (ii) We now turn to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Note that: B Bξk ˆ 1 |Uξ| ˙ “ ´ ` U TUξ ˘ k |Uξ|3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) B Bξk B Bξl ˆ 1 |Uξ| ˙ “ ´ ` U TU ˘ k,l |Uξ|3 ` 3 ` U TUξ ˘ k ` U TUξ ˘ l |Uξ|5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) Likewise, we have: B Bξk ˆpUξqj |Uξ| ˙ “ Ujk |Uξ| ´ pUξqj ` U TUξ ˘ k |Uξ|3 , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) B Bξk B Bξl ˆpUξqj |Uξ| ˙ “ ´Ujk ` U TUξ ˘ l |Uξ|3 ´ Ujl ` U TUξ ˘ k |Uξ|3 ´ pUξqj ` U TU ˘ k,l |Uξ|3 ` 3pUξqj ` U TUξ ˘ k ` U TUξ ˘ l |Uξ|5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) The above relations, together with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), show that: ���� B Bξk ˆ 1 |Uξ| ˙���� ď σ2 1 σ2 1 |ξ|2 , ���� B Bξk B Bξl ˆ 1 |Uξ| ˙���� ď 4σ3 1 σ2 2 1 |ξ|3 , ���� B Bξk ˆpUξqj |Uξ| ˙���� ď 2σ1 σ2 1 |ξ|, ���� B Bξk B Bξl ˆpUξqj |Uξ| ˙���� ď 6σ2 1 σ2 2 1 |ξ|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we obtain ���� BFθGα,A Bξk ���� ď C σ2 1 σ3 2 |ξ|´2 , ���� B2FθGα,A BξkBξl ���� ď C σ3 1 σ4 2 |ξ|´3 , Next, set A “ rA1A2s, since ���� B Bξk Aξ ���� “ |Ak| ď λ, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 45 we have ���� B Bξk pAξ b Aξq ���� ď Cσ1λ |ξ| , ���� B Bξk B Bξl pAξ b Aξq ���� ď Cλ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���� B Bξk MA ���� ď C ˆT λ ` dT dλ ˙ |ξ| , ���� B Bξk B Bξl MA ���� ď C ˆT λ ` dT dλ ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The desired bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) now follows easily by combining the above estimates and 1 ď σ1 σ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (iii) By lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1, we may obtain B Bξj ∆ξLApξq “ ÿ i“1,2 B Bξjii LApξq “ ÿ i“1,2 1 |ξ|2 Pjii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since σ2 ď ���Uˆξ ��� ď σ1 and Pjii ´ ˆξ1, ˆξ2 ¯ is a matrix of polynomials on the domain ���ˆξ ��� “ 1, Pjiipˆξ1,ˆξ2q |Uˆξ| 9 is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we have Φp3q A,jpˆξq “ ÿ i“1,2 Pjii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 9 , B Bξj ∆ξLApξq “ 1 |ξ|2 Φp3q A,jpˆξq, p∆ξq2 LApξq “ 1 |ξ|3 Φp4q A pˆξq, B Bξj p∆ξq2 LApξq “ 1 |ξ|4 Φp5q A,jpˆξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Similarly, Φp4q A pˆξq “ ÿ i,k“1,2 Pkkii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 11 , Φp5q A,jpˆξq “ ÿ i,k“1,2 Pjkkii ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 13 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 46 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Some estimates for z ` Lα,A and pz ` Lα,Aq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since LA pξq is p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and diagonalizable, P ´1 pξq LA pξq P pξq “ D pξq where D is a positive diagonal matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, P ´1 pξq pz ` LA pξqq P pξq “ z ` D pξq , P ´1 pξq pz ` LA pξqq´1 P pξq “ pz ` D pξqq´1 , and the eigenvalues of pz ` LA pξqq´1 are on a curve # 1 z ` a ˇˇˇ a P ”σ1zp0q M 2σ2 2 |ξ| , σ2zm 4σ2 1 |ξ| ı+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let λ ą 0 and z P Sω,δ with ω ą 0, β :“ π´argpz ` λ ´ ωq ą δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we obtain the following inequality |z ` λ|2 “λ2 ` |z|2 ´ 2 |z| λ cos β ěλ2 ` |z|2 ´ 2 |z| λ cos δ “ cos δ pλ ´ |z|q2 ` p1 ´ cos δq ´ λ2 ` |z|2¯ ě p1 ´ cos δq ´ λ2 ` |z|2¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, we estimate ���Bα ξ pz ` LA pξqq´1��� with |α| ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Sω,δ with ω ą 0 and A P DAσ1,σ2, we have the following estimates for all z P Sω,δ, 1 |z| ` σ1zp0q M 2σ2 2 |ξ| ď ���pz ` LA pξqq´1��� ď 2 b p1 ´ cos δq ` p σ2zm 4σ2 1 |ξ|q2 ` |z|2˘, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) ���� B Bξk pz ` LA pξqq´1 ���� ď C1 σ2 1 σ3 2 zp0q M 4 p1 ´ cos δq ` p σ2zm 4σ2 1 |ξ|q2 ` |z|2 ˘, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and ���� B2 BξlBξk pz ` LA pξqq´1 ���� ď C2 1 σ4 1 σ6 2 zp0q M 2 16 ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 ` C1 σ3 1 σ4 2 zp0q M 4 p1 ´ cos δq |ξ| ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all |α| ď 2, ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |z| |ξ|´|α| , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |z|2 |ξ|1´|α| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 47 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the eigenvalues of pz ` LA pξqq´1 are between ´ z ` σ1zp0q M 2σ2 2 |ξ| ¯´1 and ´ z ` σ2zm 4σ2 1 |ξ| ¯´1 , it follows that 1 |z| ` σ1zp0q M 2σ2 2 |ξ| ď ���pz ` LA pξqq´1��� ď 2 d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, B Bξk pz ` LA pξqq´1 “ ´ pz ` LA pξqq´1 B Bξk LA pξq pz ` LA pξqq´1 , so by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8), ���� B Bξk pz ` LA pξqq´1 ���� ď C1 σ2 1 σ3 2 zp0q M 4 p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, B2 BξlBξk pz ` LA pξqq´1 “ pz ` LA pξqq´1 B Bξl LA pξq pz ` LA pξqq´1 B Bξk LA pξq pz ` LA pξqq´1 ` pz ` LA pξqq´1 B Bξl LA pξq pz ` LA pξqq´1 B Bξk LA pξq pz ` LA pξqq´1 ´ pz ` LA pξqq´1 B2 BξlBξk LA pξq pz ` LA pξqq´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���� B2 BξlBξk pz ` LA pξqq´1 ���� ď C2 1 σ4 1 σ6 2 zp0q M 2 16 ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 ` C1 σ3 1 σ4 2 zp0q M 4 p1 ´ cos δq |ξ| ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From the inequalities 1 dˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ď 1 |z| and 1 dˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ď 1 σ2zm 4σ2 1 |ξ|, we obtain ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2 |z| |ξ|´|α| , ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2 |z|2 |ξ|1´|α| for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Next, let us consider ���Bα ξ |ξ| pz ` LA pξqq´1��� with |α| ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 48 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Sω,δ with ω ą 0 and A P DAσ1,σ2, the following estimates hold for all z P Sω,δ ���|ξ| pz ` LA pξqq´1��� ď 2 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) ���� B Bξk ´ |ξ| pz ` LA pξqq´1¯���� ď 2 d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ2 1 σ3 2 zp0q M 4 |ξ| p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) ���� B2 BξlBξk ´ |ξ| pz ` LA pξqq´1¯���� ď 4 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ3 1 σ4 2 zp0q M 12 p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C2 1 σ4 1 σ6 2 zp0q M 2 16 |ξ| ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) Moreover, there exists a constant Cδ,σ1,σ2,T depending on δ, σ1, σ2 and T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all |α| ď 2, ���Bα ξ |ξ| pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20), it is easy to obtain ���|ξ| pz ` LA pξqq´1��� ď 2 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, B Bξk ´ |ξ| pz ` LA pξqq´1¯ “B |ξ| Bξk pz ` LA pξqq´1 ` |ξ| B Bξk pz ` LA pξqq´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 49 Therefore, with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21), ���� B Bξk ´ |ξ| pz ` LA pξqq´1¯���� ď 2 d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ2 1 σ3 2 zp0q M 4 |ξ| p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, B2 BξlBξk ´ |ξ| pz ` LA pξqq´1¯ “ B2 BξlBξk |ξ| pz ` LA pξqq´1 ` B Bξk |ξ| B Bξl pz ` LA pξqq´1 ` B Bξl |ξ| B Bξk pz ` LA pξqq´1 ` |ξ| B2 BξlBξk pz ` LA pξqq´1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, ���� B2 BξlBξk ´ |ξ| pz ` LA pξqq´1¯���� ď 4 |ξ| d p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C1 σ3 1 σ4 2 zp0q M 12 p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙ ` C2 1 σ4 1 σ6 2 zp0q M 2 16 |ξ| ˆ p1 ´ cos δq ˆ´ σ2zm 4σ2 1 |ξ| ¯2 ` |z|2 ˙˙ 3 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' With 1 dˆˆ σ2zm 4σ2 1 |ξ| ˙2 `|z|2 ˙ ď 1 σ2zm 4σ2 1 |ξ|, we obtain ���Bα ξ pz ` LA pξqq´1��� ď Cδ,σ1,σ2,T |ξ|´|α| for all |α| ď 2, where the constant Cδ,σ1,σ2,T only depends on δ, σ1, σ2 and T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Now, we may prove Lα,A is a sectorial operator and obtain the estimate of pz ´ Lα,Aq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A satisfying the condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) and a constant K ą 0 , there exists Sω,δ with ω, δ ą 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all z P Sω,δ ���pz ´ Lα,Aq´1 Y ��� CγpR2q ď Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q , and ���pz ´ Lα,Aq´1 Y ��� C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q , for all Y P CγpR2q X LppR2q with 1 ď p ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 50 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since Lα,AY pθq “ ´F´1LApξqFY , the Fourier multiplier of pz ´ Lα,Aq´1 is pz ` LAq´1 and of B Bθi pz ´ Lα,Aq´1 is ξi pz ` LAq´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' With (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28), we obtain there exists Cω,δ,σ1,σ2,T s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' �pz ´ Lα,Aq´1 Y �CγpR2q ďCω,δ,σ1,σ2,T |z| �Y �CγpR2q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) �pz ´ Lα,Aq´1 Y �C1,γpR2q ďCω,δ,σ1,σ2,T �Y �CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) Next, for ���pz ´ Lα,Aq´1 Y ��� C0pR2q, set ϕpξq to be a smooth and radial cutting func- tion with a compact support in B p1q and ϕpξq “ 1 in a neighborhood of ξ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, ���pz ´ Lα,Aq´1 Y ��� C0pR2q “ ���F´1 pz ` LAq´1 pξqFY ��� C0pR2q ď ���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY ��� C0pR2q ` ���F´1 pz ` LAq´1 pξqϕpξqFY ��� C0pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the first term, since 1´ϕpξq “ 0 in a neighborhood of ξ “ 0 and |1 ´ ϕpξq| ď 1, by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, we obtain ���F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY ��� C0pR2q ďC�F´1 pz ` LAq´1 pξq p1 ´ ϕpξqq FY �CγpR2q ď Cω,δ,σ1,σ2,T |z| �Y �CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the second term, define the kernel K0 pθq :“ F´1 pz ` LAq´1 pξqϕpξq, so ���F´1 pz ` LAq´1 pξqϕpξqFY ��� C0pR2q “ ∥K0 ˚ Y ∥C0pR2q ď ∥K0∥L1pR2q ∥Y ∥C0pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we estimate ∥K0∥L1, by Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2, we have ∥K0 pθq∥ ďCω,δ,σ1,σ2,T |z| 1 1 ` |θ|3 , so ∥K0∥L1 ď Cω,δ,σ1,σ2,T |z| ż R2 1 1 ` |θ|3 dθ ď Cω,δ,σ1,σ2,T |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���F´1 pz ` LAq´1 pξqϕpξqFY ��� C0pR2q ď Cω,δ,σ1,σ2,T |z| ∥Y ∥C0pR2q , so ���pz ´ Lα,Aq´1 Y ��� CγpR2q ď Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 51 Similarly, for ��� B Bθi pz ´ Lα,Aq´1 Y ��� C0pR2q, we may use the above technique with the kernel K1,i K1,i pθq :“F´1ξi pz ` LAq´1 pξqϕpξq, ∥K1,j pθq∥ ďCω,δ,σ1,σ2,T 1 1 ` |θ|4 to obtain ���� B Bθi pz ´ Lα,Aq´1 Y ���� C0pR2q ď Cω,δ,σ1,σ2,T ∥Y ∥C0pR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, ���pz ´ Lα,Aq´1 Y ��� C1,γpR2q ď Cω,δ,σ1,σ2,T ∥Y ∥CγpR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a matrix A in DAσ1,σ2, there exists Sω,δ with ω, δ ą 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all z P Sω,δ ∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q , and ∥pz ´ Lα,Aq Y ∥CγpR2q ě Cω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q , for all compactly supported Y P C1,γ ` R2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given Y P C1,γ with a compact support, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5, W “ pz ´ Lα,Aq Y P Cγ ` R2˘ X L2 ` R2˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Through Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10, ∥W ∥CγpR2q ě Cp1q ω,δ,σ1,σ2,T |z| ���pz ´ Lα,Aq´1 W ��� CγpR2q “ Cp1q ω,δ,σ1,σ2,T |z| ∥Y ∥CγpR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) and ∥W ∥CγpR2q ě Cp2q ω,δ,σ1,σ2,T �pz ´ Lα,Aq´1 W �C1,γpR2q “ Cp2q ω,δ,σ1,σ2,T ∥Y ∥C1,γpR2q (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) □ In each chart x Xn pθq and Y n pθq, A is ∇Xn p0q and ρnY n is supported in V4R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lα,A will be used in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since Apξ “ lim hÑ0 Xnphpξq ´ Xnp0q h , it is clear that |X|˚ ď |X|˝,n ď lim inf ξÑ0 |Xn pξq ´ Xn p0q| |ξ| ď ���Apξ ��� , C ∥X∥C1,γpS2q ě ∥Xn∥C1pV4Rq ě lim sup ξÑ0 |Xn pξq ´ Xn p0q| |ξ| ě ���Apξ ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we may set σ1 “ C ∥X∥C1,γpS2q , σ2 “ |X|˚ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 52 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Next, set A0 “ ∇x Xn p0q, so A0 “ » – 2 0 0 2 0 0 fi fl , and L0,A0Y pθq “ ´ ż R2 B Bηk 1 4π |θ|W kdη, L0,A0 pξq “ |ξ| 8 I Set Lpσq A “ p1 ´ σq L0,A0 ` σL0,A, which will be used in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 , then Lpσq A Y pθq “ ´ 1 8π ż R2 B Bηk ˆ2 p1 ´ σq |θ| ` σ |Aθ| ˙ W kdη, Lpσq A pξq “|ξ| 4 ˆ1 ´ σ 2 ` σ |ξ| det pBq |Uξ| ˙ We just have to adapt the above theorems and their proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we obtain Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given X P C1,γpS2q, and our Stereoghraphic projection charts and the partition functions tx Xn, ρnu with the radius R, set σ1 “ C ∥X∥C1,γpS2q, σ2 “ |X|˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' There exists Sω,δ with ω, δ ą 0 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' in each chart x Xn, for Y P C1,γpS2q, we have the following inequalities: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) ∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp1q ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q , ∥pz ´ Lα,Aq ρnY n∥CγpR2q ě Cp2q ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q , ��� ´ z ´ Lpσq A ¯ ρnY n ��� CγpR2q ě Cp3q ω,δ,σ1,σ2,T |z| ∥ρnY n∥CγpR2q , ��� ´ z ´ Lpσq A ¯ ρnY n ��� CγpR2q ě Cp4q ω,δ,σ1,σ2,T ∥ρnY n∥C1,γpR2q , where A “ ∇Xn p0q, and σ, α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Some estimates for etLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have two ways of representing the semigroup e´tLA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' One is by the Dunford integral etLA “ 1 2πi ż ω`γr,η etz pz ` LAq´1 dz, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) where r ą 0, δ ă η ă π 2 and the curve γr,η “ tz P C : |argz| “ π ´ η, |z| ě ru X tz P C : |argz| ď π ´ η, |z| “ ru.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The other is through Fourier transform, etLAf pθq :“ F´1 ” e´tLApξqFrfs pξq ı pθq “ ż R2 KA pt, θ ´ ηq f pηq dη, where KA pt, θq “ F´1 “ e´tLApξq‰ pθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given 0 ď t0 ď t ď T and 0 ď β´α ă 1 2, we have the following estimates: }ept´t0qLAfpt0q}CβpR2q ď C pt ´ t0qβ´α }fpt0q}CαpR2q, }ept´t0qLAfpt0q}L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CβpR2qq ď C}fpt0q}CαpR2q, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 53 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34), Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 and 0 ď t ´ t0 ď T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' we have ���ept´t0qLAfpt0q ��� CαpR2q “ ����� 1 2πi ż ω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='η ept´t0qz pz ` LAq´1 fpt0qdz ����� CαpR2q ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ż ω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='η ���ept´t0qz��� 1 |z|d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ż pt´t0qpω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ηq |ez| 1 |z|d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and ���ept´t0qLAfpt0q ��� C1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='αpR2q ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥fpt0q∥CαpR2q ż ω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='η ���ept´t0qz��� d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T t ´ t0 ∥fpt0q∥CαpR2q ż pt´t0qpω`γr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ηq |ez| d |z| ď Cω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T t ´ t0 ∥fpt0q∥CαpR2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by interpolation theorem, we obtain ���ept´t0qLAfpt0q ��� CβpR2q ď Cω,δ,σ1,σ2,T ,T,β´α pt ´ t0qβ´α ∥fpt0q∥CαpR2q , so }ept´t0qLAfpt0q}L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CβpR2qq ď Cω,δ,σ1,σ2,T ,T,β´α}fpt0q}CαpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given 0 ď t0 ď t ď T and 0 ď α ă 1, we have the following estimates: ���� ż t t0 ept´sqLAfpsqds ���� C1,αpR2q ď C sup t0ďsďt }fpsq}CαpR2q, ���� ż t t0 ept´sqLAfpsqds ���� L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,αpR2qq ď C}f}L2pt0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CαpR2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, define u as upt, t0, θq :“ urfs pθq“ ż t t0 ept´sqLAfps, θqds “ ż t t0 ż R2KApt´s, θ´ηqfps, ηqdηds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since tLA pξq “ LA ptξq, KApt ´ s, x ´ yq “ 1 pt ´ sq2 KA ˆ 1, θ ´ η t ´ s ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 54 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN By (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='48) in Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, |upt, t0, θq| “ ����� ż t t0 ż R2 1 pt ´ sq2 KA ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds ����� ď ż t t0 ż R2 1 pt ´ sq2 C 1 ` ��� θ´η t´s ��� 3 |fps, ηq| dηds ďC ∥f∥C0 ż t t0 ż R2 1 1 ` |θ ´ η|3 dηds ď C ∥f∥C0 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we assume f ps, θq “ 0 when s ă t0, so Bu Bθi “ ż t t0 ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds “ ż t ´8 ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Through (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='49), ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdη “ ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ pfps, ηq ´ f ps, θqq dη, and ����� ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ pfps, ηq ´ f ps, θqq dη ����� ď 1 pt ´ sq3 ż R2 |fps, ηq ´ f ps, θq| 1 ` ��� θ´η t´s ��� 4 dη ď C�f ps, ¨q�Cα 1 pt ´ sq3 ż R2 |θ ´ η|α 1 ` ��� θ´η t´s ��� 4 dη “ C�f ps, ¨q�Cα 1 pt ´ sq1´α ż R2 |θ ´ η|α 1 ` |θ ´ η|4 dη “ C�f ps, ¨q�Cα 1 pt ´ sq1´α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5, for all m ď t, we obtain ����� ż t m ż R2 1 pt ´ sq3 BKA Bθi ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds ����� ďC ż t m �f ps, ¨q�Cα 1 pt ´ sq1´α ds ď C pt ´ mqα Ml r�f ps, ¨q�Cαs ptq , so set m “ 0, ���� Bu Bθi pt, t0, θq ���� ď CT αMl r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 55 For � Bu Bθi �Cα, we first compute B2 BθiBθj ż M ´8 ż R2KApt ´ s, θ ´ ηqfps, ηqdηds “ ż M 8 ż R2 1 pt ´ sq4 B2KA BθiBj ˆ 1, θ ´ η t ´ s ˙ fps, ηqdηds, where M ă t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='50) and Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5, we have ����� ż R2 1 pt´sq4 B2KA BθiBj ´ 1, θ ´ η t´s ¯ fps, ηqdη ����� ď C�f ps, ¨q�Cα 1 pt´sq2´α ż R2 |θ´η|α 1`|θ´η|4 dη ď C�f ps, ¨q�Cα 1 pt ´ sq2´α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and ����� B2 BθiBθj ż M ´8 ż R2 KApt ´ s, θ ´ ηqfps, ηqdηds ����� ď C pt ´ Mqα´1 Ml r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When |θ ´ η| “ 1, we set a cutting function φpsq s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' φpsq “ 1 on r´8, ´2s and φpsq “ 0 on r´1, 8s, and define φptqpsq “ φps ´ tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We obtain ���� Bu Bθi pt, t0, θq ´ Bu Bηi pt, t0, ηq ���� “ ���� Bu Bθi rfspθq ´ Bu Bηi rfspηq ���� ď ���� Bu Bθi rφptqfspθq ´ Bu Bηi rφptqfspηq ����` ���� Bu Bθi rp1´φptqqfspθq ����` ���� Bu Bηi rp1´φptqqfspηq ���� ď Cpt´Mqα´1Ml “ �φptqfps, ¨q�Cα‰ ptq` Cpt´mqαMl “ �p1´φptqqfps, ¨q�Cα‰ ptq, where M “ t ´ 1 and m “ t ´ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, since 0 ď φ ď 1 and �f ps, ¨q�Cα ě 0, ���� Bu Bθi pt, t0, θq ´ Bu Bηi pt, t0, ηq ���� ď CMl r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' When ρ “ |θ ´ η| ‰ 1, we define uρ pt, t0, θq “ 1 ρu pρt, ρt0, ρθq , f ρ pt, θq “ f pρt, ρθq and ¯θ “ θ ρ , ¯η “ η ρ , ¯t “ t ρ, ¯t0 “ t0 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have Buρ Bt pt, θq “LAuρ pt, θq ` f ρ pt, θq , Buρ Bθi pt, θq “ Bu Bθi pρt, ρθq , so ˇˇˇ Bu Bθi pt, t0, θq´ Bu Bηi pt, t0, ηq ˇˇˇ“ ˇˇˇBuρ B¯θi p¯t, ¯t0, ¯θq´ Buρ B¯ηi p¯t, ¯t0, ¯ηq ˇˇˇďCMlr�f ρ p¯s, ¨q�Cαsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since �f ρ�Cα p¯sq “ ρα�f�Cα pρ¯sq, Ml “ �f ρ p¯s, ¨q�Cα‰ p¯tq “ sup ¯rą0 1 ¯r ż ¯t ¯t´¯r �f ρ�Cα p¯sq d¯s “ ρα sup ¯rą0 1 ¯r ż ¯t ¯t´¯r �f�Cα pρ¯sq d¯s “ρα sup rą0 1 r ż t t´r �f�Cα psq ds “ ραMl r�f ps, ¨q�Cαs ptq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 56 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Hence, ���� Bu Bθi pt, t0, θq ´ Bu Bηi pt, t0, ηq ���� ď CMl r�f ps, ¨q�Cαs ptq |θ ´ η|α , and by the Hardy-Littlewood maximal function theorem, for all 1 ď p ď 8 ���∥u∥CγpR2q ��� Lppt0,T q ď ���C ∥f∥C0pR2q ` CMl “ �f ps, ¨q�CαpR2q ‰ ptq ��� Lppt0,T q ďC ���∥f∥C0pR2q ��� Lppt0,T q ` C ��Ml “ �f ps, ¨q�CαpR2q ‰ ptq �� Lppt0,T q ďC ���∥f∥C0pR2q ��� Lppt0,T q ` C ���f pt, ¨q�CαpR2q �� Lppt0,T q ďC ���∥f∥CγpR2q ��� Lppt0,T q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Local well-posedness We write the Peskin problem as an evolution equation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) BX Bt “ FpXq, t ą 0, Xp0q “ X0, where FpXq is given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will make use of Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 in [35]: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let E1 Ă E0 Ă E be Banach spaces and let 0 ă σ ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given T ą 0, open set O1 Ă E1 and a function F : r0, T s ˆ O1 ÞÑ E0, pt, uq ÞÑ Fpt, uq such that F and Fu are continuous in r0, T s ˆ O1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If for every p¯t, ¯uq P r0, T s ˆ O1 we have Fup¯t, ¯uq : E1 ÞÑ E0 is the part of a sectorial operator S : DpSq Ă E ÞÑ E with DSpσq » E0 and DSpσ ` 1q » E1, then for every ¯t P r0, ts and ¯u P O1 there are δ ą 0, r ą 0 such that if t0 P r0, T q, |t0 ´ ¯t| ď δ, and }u0 ´ ¯u} ď r then the problem v1ptq “ Fpt, vptqq, t0 ď t ď t0 ` δ, vpt0q “ u0, has a unique solution v P Cprt0, t0 ` δs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' E1q X C1prt0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t0 ` δs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' E0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, our main result is the following Theorem: Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the 3D Peskin problem (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) with initial data satisfying X0 P h1,γpS2q, |X0|˚ ą 0, and T P C3 such that T ą 0, dT {dλ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, there exists some time T ą 0 such that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) has a unique solution X, X P Cpr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' h1,γpS2qq X C1pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' hγpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Om “ tY P h1,γpS2q : |Y |˚ ě m ą 0u, E1 “ h1,γpS2q, E0 “ hγpS2q, and E “ hαpS2q, with 0 ă α ă γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Define the operator S as the linearization of F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) around X0: SpX0qY :“ BXFpX0qY “ d dεFpX0 ` εY q|ε“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since X0 P Om is arbitrary, we can study the Gˆateaux derivative of F at any X P Om, which is given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) SpXqY “ S1pXqY ` S2pXqY , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 57 with S1pXqY “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq¨ ˆ d dε ´ T p|∇S2pXppyq ` εY ppyqq|qp∇S2pX ` εY qqppyq ¯ |ε“0dpy, S2pXqY “ ´ ż S2∇S2 d dε ´ GpXppxq´Xppyq`εpY ppxq ´ Y ppyqqq ¯ˇˇˇ ε“0¨ ˆ T p|∇S2Xppyq|q∇S2Xppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It remains to check that the hypothesis of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1 are satisfied, which follow from Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If m ą 0 and γ P p0, 1q, T P C2, then F (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) is a continuous map from Om Ă h1,γpS2q to hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given that FpXq “ NpXqpT p|∇S2X|q∇S2Xq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), we apply Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 to obtain that }FpXq}CγpS2q ď C 1 |X|˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }T p|∇S2Xq|q∇S2Xq}CγpS2q, hence recalling the expression for T (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), the bound above yields that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) }FpXq}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We have thus proved that F maps C1,γpS2q to CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We need to show that it also maps h1,γpS2q to hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Having the estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), it suffices to show that if X P h1,γpS2q, then FpXq P hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since hk,γpS2q is the completion of Ck,γpS2q in any Ck,αpS2q with 0 ă γ ă α ă 1, k ě 0, let X P h1,γpS2q, and tXmum a sequence Xm P C1,αpS2q, α ą γ, such that Xm Ñ X in C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is clear that the previous estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) also holds replacing γ by α, thus FpXmq P Cα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We conclude that FpXq P hγpS2q by showing that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) }FpXmq ´ FpXq}CγpS2q ď C}Xm ´ X}C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The estimate will follow from the previous ones by writing FpXmq ´ FpXq as follows: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) FpXmqppxq ´ FpXqppxq “ ∆1ppxq ` ∆2ppxq, with ∆1ppxq “ ´ ż S2∇S2GpXmppxq´Xmppyqq¨ ˆ ` T p∇S2Xmppyqq∇S2Xmppyq´T p∇S2Xppyqq∇S2Xppyq ˘ dpy, ∆2ppxq “ ż S2∇S2 ´ GpXmppxq´Xmppyqq´GpXppxq´Xppyqq ¯ ¨ ˆ T p∇S2Xppyqq∇S2Xppyqdpy, 58 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where both terms have kernels given by a derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term has thus already been treated, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) }∆1}CγpS2q ď Cp}∇S2Xm}C0pS2q, |Xm|˚q}T p∇S2Xmq∇S2Xm´T p∇S2Xq∇S2X}CγpS2q ďCp}∇S2Xm}C0pS2q, |Xm|˚, }∇S2X}C0pS2q, |X|˚, }T }C2q ´ }∇S2pXm´Xq}CγpS2q ` p}∇S2X}CγpS2q ` }∇S2Xm}CγpS2qq}∇S2pXm ´ Xq}C0pS2q ¯ , while the second one can be estimated in a similar manner by noticing that one can always extract Xm ´ X from the difference of kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider for example the kernel q1 k,l (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), q1 k,lppx, pyq “ ´ 1 8π δ pyXippxq |δ pyXppxq|3 ∇S2Xippyqδk,l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can write 1 8π δ pyXm i ppxq |δ pyXmppxq|3 ∇S2Xm i ppyqδk,l ´ 1 8π δ pyXippxq |δ pyXppxq|3 ∇S2Xippyqδk,l “ 1 8π δ pypXm i ´ Xiqppxq∇S2Xm i ppyq |δ pyXmppxq|3 ` 1 8π δ pyXippxq∇S2pXm i ´ Xiqppyq |δ pyXmppxq|3 ` 1 8π δ pyXippxq∇S2Xippyq ´ 1 |δ pyXmppxq|3 ´ 1 |δ pyXppxq|3 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, it holds that }∆2}CγpS2q ď Cp}∇S2X}C0pS2q, |X|˚, }∇S2Xm}C0pS2q, |Xm|˚, }T }C1q ˆ }∇S2X}CγpS2q}∇S2pXm ´ Xq}C0pS2q, which together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) proves (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' If m ą 0 and γ P p0, 1q, T P C3, then the Gˆateaux derivative of F at any X P Om Ă h1,γpS2q (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) is continuous and maps h1,γpS2q to hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term S1pXqY in the Gˆateaux derivative of F (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) is given in terms of the operator NpXq (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) S1pXqY ppxq “ NpXqpTSp∇S2Xq∇S2Y qppxq, with TS given by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) TSp∇S2Xq“ T p|∇S2X|q |∇S2X| ` ´ T 1p|∇S2X|q´ T p|∇S2X|q |∇S2X| ¯∇S2X b ∇S2X |∇S2X|2 , and, in index notation, pTSp∇S2Xq∇S2Y ql,i “ T p|∇S2X|q |∇S2X| p∇S2Y ql,i ` ´ T 1p|∇S2X|q´ T p|∇S2X|q |∇S2X| ¯p∇S2Xql,ip∇S2Xqq,m |∇S2X|2 p∇S2Y qq,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 59 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 then gives that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) }S1pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2Y }CγpSq ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq ` Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pSq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed with S2pXqY , S2pXqY “ S2,1pXqY ` S2,2pXqY , pS2,jpXqY qkppxq “ ż S2 Qj k,lppx, pyq ¨ pT p∇S2Xq∇S2Xlppyq ´ Clqdpy, where we define the kernels Qj k,lppx, pyq “ ∇S2 d dε ´ GjpXppxq ´ Xppyq ` εpY ppxq ´ Y ppyqqq ¯ˇˇˇ ε“0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Taking the derivatives we see that Qj k,lppx, pyq “ ∇S2 ´ B Bxi GjpXppxq ´ XppyqqpYippxq ´ Yippyqq ¯ “ ´ B Bxl B Bxi GjpXppxq ´ Xppyqq∇S2XlppyqpYippxq ´ Yippyqq ´ B Bxi GjpXppxq ´ Xppyqq∇S2Yippyq, hence we have the following bound, similarly as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) |Qj k,lppx, pyq| ď C ´ |∇S2Y ppyq| |∆ pyXppxq|2 ` |∇S2Xppyq||∆ pyY ppxq| |∆ pyXppxq|3 ¯ ď C }∇S2Y }C0pS2q |X|2˚ ´ 1 ` }∇S2X}C0pS2q |X|˚ ¯ 1 |px ´ py|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, |S2,jpXqY ppxq|ďC }T p∇S2Xq∇S2X}CγpS2q |X|2˚ ´ 1` }∇S2X}C0pS2q |X|˚ ¯ }∇S2Y }C0pS2q, hence |S2,jpXqY ppxq| ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given that the kernels Qj k,l are also a derivative, the estimate of the H¨older semi- norm follows the same steps as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, performing the splitting as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), we find that rS2,jpXqY sCγpS2q ď C }T p∇S2Xq∇S2X}CγpS2q |X|2˚ ´ 1 ` ´}∇S2X}C0pS2q |X˚| ¯2¯ }∇S2Y }C0pS2q, thus (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) }S2pXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2X}CγpS2q}∇S2Y }C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 60 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Together with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9), this shows that SpXq maps C1,γpS2q to CγpS2q, }SpXqY }CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2X}CγpS2q}∇S2Y }C0pS2q ` Cp|X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2Y }CγpSq ď Cp|X|˚ , }∇S2X}CγpS2q, }T }C2q ∥∇S2Y ∥CγpS2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) We are left to show that SpXq also maps h1,γpS2q to hγpS2q and that it is continuous with respect to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We follow the lines below (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It suffices to show that if Y P h1,γpS2q, then SpXqY P hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let Y P h1,γpS2q, and tY mum a sequence Y m P C1,αpS2q, α ą γ, such that Y m Ñ Y in C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since SpXqY m P CαpS2q, we conclude that SpXqY P hγpS2q by showing that }SpXqY m ´ SpXqY }CγpS2q ď C}Y m ´ Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' But since we are dealing with a linear operator, the estimate is trivially satisfied from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' That the Gˆateaux derivative is continuous in X follows along the lines below (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, ` SpX1q ´ SpX2q ˘ Y “ pS1pX1q ´ S1pX2qqY ` pS2pX1q ´ S2pX2qqY , and we decompose each Sj as in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, it is not hard to see that the following bound holds }pSpX2q ´ SpX1qqY }CγpS2q ď Cp}∇S2X1}C0pS2q, |X1|˚, }∇S2X2}C0pS2q, |X2|˚, }T }C3q ˆ ´ }∇S2Y }CγpS2q}∇S2pX1 ´ X2q}CγpS2q ` }∇S2Y }C0pS2q}∇S2pX1 ´ X2q}C0pS2qp}∇S2X1}CγpS2q ` }∇S2X2}CγpS2qq ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Consider the linear operator SpXq : C1,γpS2q Ñ CγpS2q defined in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) with X P C1,γpS2q, T P C2, T ą 0, dT {dλ ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, there exists a sector such that for all z in the sector }z ´ SpXqY }CγpS2q ě Cp}Y }C1,γpS2q ` |z|}Y }CγpS2qq, where the constant C depends only on the sector, γ, the norms }X}C1,γpS2q and }T }C2, and the arc-chord condition |X|˚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) }pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ }S2pXqY }CγpS2q, and using (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) we obtain that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) }pz ´ SpXqqY }CγpS2q ě }pz ´ S1pXqqY }CγpS2q ´ C ∥∇S2Y ∥C0pS2q .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the notation (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we can write S1pXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) as (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) S1pXqY ppxq “ ´ ż S2 ∇S2GpXppxq ´ Xppyqq ¨ pTSp∇S2Xppyqq∇S2Y ppyq ´ Cqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 61 Next, we introduce the partition of unity ρn (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) to write S1pXqY ppxq “ ÿ n S1pXq pρnY q ppxq “´ ÿ n ż S2∇S2GpXppxq´Xppyqq¨pTSp∇S2Xppyqq∇S2pρnY qppyq´Cnqdpy, where now we will choose Cn “ 0 or Cn “ TSp∇S2Xppxqq∇S2pρnY qppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will extensively use that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) supp ρn Ă Bpxn,2R X S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We notice that ρnppxqzY ppxq ´ S1pXqpρnY qppxq “ ρnppxq ´ zY ppxq ´ S1pXqY ppxq ¯ ` ´ ρnppxqS1pXqY ´ S1pXqpρnY qppxq ¯ , hence 2}ρn}CγpS2q}pz ´ S1pXqqY }CγpS2q ě }ρn ´ zY ´ S1pXqY ¯ }CγpS2q ě }zρnY ´ S1pXqpρnY q}CγpS2q ´ }ρnS1pXqY ´ S1pXqpρnY q}CγpS2q, and summing in n we obtain (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) }pz ´ S1pXqqY }CγpS2q ě C ÿ n p}I1 n}CγpS2q ´ }I2 n}CγpS2qq, where (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) I1 n “ zρnY ´ S1pXqpρnY q, I2 n “ ρnS1pXqY ´ S1pXqpρnY q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling that S1pXq is given in terms of NpXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), we split I2 n further: I2 n “ rρn, NpXqspTSp∇S2Xq∇S2Y q ` NpXq ` Tsp∇S2XqY b ∇S2ρn ˘ , and by Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9, }I2 n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2qq ´ }∇S2ρn}C0pS2q}TSp∇S2Xq∇S2Y }C0pS2q ` }Tsp∇S2XqY b ∇S2ρn}CγpS2q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, }I2 n}CγpS2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q}∇S2ρn}CγpS2q ˆ p}∇S2X}CγpS2q}Y }CγpS2q`}∇S2Y }C0pS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to deal with the term I1 n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We introduce the cutoff (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) so that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) }I1 n}CγpS2q ě }zρnY ´pρnS1pXqpρnY q}CγpS2q´ }p1 ´ pρnqS1pXqpρnY q}CγpS2q “ }I1,1 n }CγpS2q ´ }I1,2 n }CγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The last term will be smoother because the integral is not singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, recalling again the expression of S1pXq in terms of NpXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7), we use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 to obtain }I1,2 n }CγpS2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 62 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Although the constant in the bound above (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) becomes large for R small, it will suffice since it is lower order in terms of regularity for Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we proceed to estimate I1,1 n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can decompose further by intro- ducing the frozen-coefficient linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We denote by x Xn the stereographic projection centered at pxn, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' pxn “ x Xnp0q, and Xnpθq “ Xpx Xnpθqq (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17), we have (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) }I1,1 n }CγpS2q ě C}zρnY n ´ pρnNpXqpTSp∇S2Xq∇S2pρnY qqn}CγpR2q, and I1,1 n pθq “ zρnpθqY npθq ´ pρnpθqNpXqpTSp∇S2Xq∇S2pρnY qqnpθq “ J3 ` J4 ` J5 ` J6, with J3 “ zρnpθqY npθq ´ LApρnY nqpθq, J4 “ pρnpθqrMpAq ´ NpXqspTSp∇S2Xq∇S2pρnY qqnpθq “ pρnpθqrMpAq ´ Mp∇Xnq ´ RnpXnqspTSp∇S2Xq∇S2pρnY qqnpθq, J5 “ pρnpθq ´ LApρnY nqpθq ´ MpAqpTSp∇S2Xqq∇S2pρnY q ˘ npθq ¯ , J6 “ p1 ´ pρnpθqqLApρnY nqpθq, where we denote A the constant matrix A “ ∇Xnp0q, Mp∇Xnq is defined in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), RpXnq in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18), LA in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32), pxn “ x Xnp0q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The bound for J6 follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8 together with Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12, }J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2qq}TFpAq∇pρnY nq}C0pR2q, thus, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) }J6}C1pR2q ď CpR, |X|˚, }∇S2X}C0pS2q, }T }C1q}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 with Z “ TSp∇S2Xq∇S2pρnY q provides the following bound for J4: }J4}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2qq ` p1 ` }∇S2X}CγpS2qq}TSp∇S2Xq∇S2pρnY q}C0pS2q ` εpRq}TSp∇S2Xq∇S2pρnY q}CγpS2q ˘ , where εpRq Ñ 0 as R Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, }J4}CγpR2q ď Cp|X|˚, }∇S2X}C0pS2q, }T }C2q ´ εpRq}∇S2pρnY q}CγpS2q ` p1 ` }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed with J5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Recalling the expression for TS (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8), we have pTSp∇S2Xq∇S2pρnY qqn,lipηq “ pTSp∇S2Xq∇S2pρnY q ˝ x Xnql,ipηq “ pdetppgpηqqq´ 1 2 B Bηr pρnYn,qqpηqB p Xm Bηr pηq ´T pλnpηqq λnpηq δlqδim ` ` T 1pλnpηqq ´ T pλnpηq λnpηq ˘ BXn,l Bηj pηq Bx Xi Bηj pηq BXn,q Bηp pηq Bx Xm Bηp pηq pλnpηqq2detppgpηqq ¯ , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 63 with λnpηq given in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Substituting into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) pMpAqpTSp∇S2Xq∇S2pρnY qqnqkpθq “ ´ ż R2 mm,k,lpθ, ηq B p Xi Bηm pηqpTSp∇S2Xpηqq∇S2pρnY nqpηqql,idη1dη2 “ ´ ż R2 mi,k,lpθ, ηqp ˜TSqipqlpηq B Bηp pρnYn,qqpηqdη1dη2 “ ˜ MpAqp ˜TS∇pρnY nqqpθq, where we denote (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) p ˜TSp∇Xqqipqlpηq“ T pλnpηqq λnpηq δpiδql` ` T 1pλnpηqq´ T pλnpηq λnpηqq ˘ BXn,l Bηi pηq BXl,q Bηp pηq pλnpηqq2a detppgpηqq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus we can write J5 “ pρnpθq ˜ MpAqppTF pAq ´ ˜TSp∇Xqq∇pρnY nqqpθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3 with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12 gives that }J5}CγpR2q ďCp|X˚, }∇S2X}C0pS2qq}pTF pAq ´ ˜TSp∇Xqq∇pρnY nq}CγpR2q, and since TF pAq “ ˜TSp∇Xp0qqp0q, we obtain }J5}CγpR2q ď Cp|X˚, }∇S2X}C0pS2q, }T }C2q ´ εpRq}∇S2pρnY q}CγpS2q ` }∇S2X}CγpS2q}∇S2pρnY q}C0pS2q ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we continue from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19), }I1,1 n }CγpS2q ě }J3 n}CγpR2q ´ }J4 n}CγpR2q ´ }J5 n}CγpR2q ´ }J6 n}CγpR2q, so inserting back the bounds for J4 n, J5 n, and J6 n, we have that }I1,1 n }CγpS2q ě }J3 n}CγpR2q ´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q ´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q ´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we use the frozen-coefficient estimate in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13 for J3 n (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We first interpolate the inequalities in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11 to control the lower-order terms, J3 n “ pz ´ LAqpρnY nqpθq, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) }J3 n}CγpR2q ě C|z|}ρnY n}CγpR2q ` C}ρnY n}C1,γpR2q ` C|z|1´σ}ρnY n}Cγ`σpR2q, where σ P r0, 1s is chosen so that 1 ă γ ` σ ă 1 ` γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we have }I1,1 n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q ´ Cp|X|˚, }∇S2X}C0pS2qqεpRq}∇S2pρnY q}CγpS2q ´ Cp|X|˚, }∇S2X}C0pS2qq}∇S2X}CγpS2q}∇S2pρnY q}C0pS2q ´ CpR, |X|˚, }∇S2X}C0pS2qq}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 64 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN and taking R small enough, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) }I1,1 n }CγpR2q ě C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q ´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2pρnY q}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we go back to (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) and substitute the above bound together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24), }pz ´ S1pXqqY }CγpS2q ě ÿ n ´ C|z|}ρnY }CγpS2q ` C}ρnY }C1,γpS2q ` C|z|1´σ}ρnY }Cγ`σpS2q ¯ ´ CpR, |X|˚, }∇S2X}CγpS2qq}∇S2Y }C0pS2q ´ Cp|X|˚, }∇S2X}CγpS2qq}∇S2ρn}CγpS2qp}Y }CγpS2q ` }∇S2Y }C0pS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Plugging this inequality in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13), and then using the triangle inequality and the fact that Cα ãÑ Cβ for α ě β, we obtain }pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q ` C|z|1´σ}Y }Cγ`σpS2q ´ CpR, |X|˚, }∇S2X}CγpS2qq}Y }Cγ`σpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, by moving the sector if necessary to make |z| big, we conclude the result }pz ´ SpXqqY }CγpS2q ě C|z|}Y }CγpS2q ` C}Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The Gˆateaux derivative of F at any X P Om, SpXq (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), generates an analytic semigroup on the space h0,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We need to prove that the operator SpXq is sectorial, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', that there exists a sector such that for any z in the sector }pz ´ SpXqq´1Y }hγpS2q ď C |z|}Y }hγpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the norm on little H¨older spaces hγpS2q is the same as in the usual H¨older spaces CγpS2q, from the previous Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 we are left to prove that the operator pz ´ SpXqq is invertible from hγpS2q to h1,γpS2q for any z in the sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Similarly as we did in Section 6, define the following family of operators SαpXq, α P r0, 1s, SαpXqY ppxq “ ´α ż S2∇S2 ´ G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq ¯ ¨pTSp∇S2Xq∇S2Y ppyqqdpy ´p1 ´ αq ż S2∇S2 ´ G1pXppxq´Xppyqq`αG2pXppxq´Xppyqq ¯ ¨ ∇S2Y ppyqdpy ` αS2pXqY ppxq, with Gαpxq “ 1 8π pG1pxq ` αG2pxqq , x “ px1, x2, x3q, pG1qi,jpxq “ δij |x|, pG2qi,jpxq “ xixj |x|3 , WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 65 and TS given in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, SpXq “ S1pXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Propositions 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5 hold analogously for SαpXq, as all the remainder estimates were always done inde- pendently for each part of the kernel G and Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13 already included the parameter α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, for all α P r0, 1s, it holds that 1 C }Y }C1,γpS2q ě }pz ´ SαpXqqY }CγpS2q ě C}Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by the method of continuity, it suffices to show that the inverse of pz´S0pXqq exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Additionally, define a new family of operators S0,σpXq, σ P r0, 1s, as follows S0,σpXqY ppxq “ ´ ż S2∇S2 ´ p1´σqG1ppx´ pyq`σG1pXppxq´Xppyqq ¯ ¨ ∇S2Y ppyqdpy, so that S0,1pXq “ S0pXq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, taking into account (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33), it is clear that the following bound holds for all σ P r0, 1s, 1 C }Y }C1,γpS2q ě }pz ´ S0,σpXqqY }CγpS2q ě C}Y }C1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, by the method of continuity again we just need to show that pz ´ S0,0pXqq is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since the range is closed, it suffices to show that it is also dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The operator S0,0pXq is linear and explicit, so we can compute its eigenspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since S0,0pXqY “ 1 8π ż S2 1 |px ´ py|∆S2Y ppyqdpy, we only have to check a component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From [21], a single layer potential u pxq of g ppyq with u pxq “ 1 4π ż S2 1 |x ´ py|g ppyq dpy can be transformed into a harmonic problem with ∆u “ 0 in R3zS2 �u� “ 0, �∇u ¨ n� “ g on S2 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) If we denote the standard spherical coordinate system pr, θ, ϕq, where r is the radial coordinate, θ is the polar angle, and ϕ is the azimuthal angle, then for the harmonic equation on R3zS2, by separation of variables [15], we obtain some solutions uℓm pr, θ, ϕq with l ě 0 and |m| ď ℓ : uℓm pr, θ, ϕq “ " ArℓYℓm, |r| ă 1 Br´pℓ`1qYℓm, |r| ą 1 where Yl,m pθ, ϕq is the usual spherical harmonic function of degree l and order m, which satisfies the following equation: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) ∆S2Yℓm “ 1 sin θ B Bθ ˆ sin θBYℓm Bθ ˙ ` 1 sin2 θ B2Yℓm Bϕ2 “ ´ℓpℓ ` 1qYℓm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By plugging uℓm into (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25), we obtain uℓm “ 1 2ℓ ` 1Yℓm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, combining (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26), S0,0pXqYℓ,m “ ´ ℓpℓ ` 1q 2p2ℓ ` 1qYℓ,m, ℓ ě 0, |m| ď ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 66 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Finally, since finite linear combinations of Yl,m are dense in C8pS2q, we conclude the existence of the inverse pz ´ S0,0pXqq´1 : hγpS2q Ñ h1,γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Higher Regularity Following the notation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1, we recall that BX Bt ppxq “ NpXqpT p∇S2Xqqppxq, where we denote, with T given in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), T p∇S2Xq “ T p|∇S2X|q∇S2X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We localize using the partition tρnu (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), and linearize T at Xppxnq, B BtpρnXqppxq “ ρnppxqNpXqpTSp∇S2Xppxnqq∇S2Xqqppxq ` ρnppxqNpXq ´ T p∇S2Xq´TSp∇S2Xppxnqq∇S2X ¯ ppxq, where we recall that TSp∇S2Xq∇S2Y “ d dsT p∇S2pX `sY qq|s“0 was given in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we introduce the commutators, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) B BtpρnXqppxq “ NpXqpTSp∇S2Xppxnqq∇S2pρnXqqppxq ` NpXq ´ ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘¯ ppxq ` rρn, NpXqspTSp∇S2Xppxnqq∇S2Xqppxq ´ NpXqpTSp∇S2XppxnqqX∇S2ρnqppxq ` rρn, NpXqs ´ T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ¯ ppxq, and we move to stereographic coordinates to introduce the frozen-coefficient (at t “ 0, px “ pxn) operator and the cutoff pρn (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) B BtpρnXnqpθq “ LA0pρnXnqpθq ` 7ÿ j“1 f jpXqpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with f 1pXqpθq “ NpXq ` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 2pXqpθq “ pρnpθq “ NpXq ´ MpAq ‰ pTSp∇S2Xppxnqq∇S2pρnXqqnpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 3pXqpθq “ pρnpθq ` MpAqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq´LApρnXnqpθq ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 4pXqpθq “ p1 ´ pρnpθqqNpXqpTSp∇S2Xppxnqq∇S2pρnXqqnpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 5pXqpθq “ ´p1 ´ pρnpθqqLApρnXnqpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 6pXqpθq “ rLA ´ LA0spρnXnqpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and f 7pXqpθq “ ´NpXqpTSp∇S2XppxnqqX∇S2ρnqpx Xnpθqq ` rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsT p∇S2Xqpx Xnpθqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' where A0 “ ∇X0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='np0q and A “ ∇Xnp0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 67 Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Let X be the solution to the Peskin problem with initial data X0 P h1,γpS2q constructed in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, for any α P p0, 1q, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C3,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, for any 3 ď n P N and α P p0, 1q, assuming that T P Cn,α, it holds that X P C1pp0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,βpS2qq, for any β ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The main difficulty is to show the smoothing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, assume we have the higher regularity information X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2 states that BtX P C0pr0, T s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' CγpS2qq, and using the equation together with X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn`1,αpS2qq, it is straightforward to see that BtX P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, to get the continuity in time for the higher regularity, it suffices to inter- polate taking into account the higher regularity bounds and the continuity in the lower norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We proceed to show the smoothing in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will consider the following mollified version of the system (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) B BtpρnXδ nqpθq “ LA0pρnXδ nqpθq ` 7ÿ j“1 Jδf jpXδqpθq, with mollified initial data Xδ 0,npθq “ JδX0,npθq, where Jδ is the standard mollifier by convolution with a Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Our main goal is to obtain uniform in δ bounds for Xδ in L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, by construction, Xδ is smooth, and it is not hard to show that the limit of tXδu in L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,γpS2qq is given by the solution X in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, by interpolation and using the uniform bounds, we would conclude that X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,βpS2qq for any β ă α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus proceed to obtain the uniform bounds first, and show the convergence Xδ Ñ X at the end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We use the semigroup etLA0 to write ρnXδ nptq in Duhamel form: (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) ρnXδ nptq “ ept´t0qLA0 pρnXδ npt0qq ` 7ÿ j“1 ż t t0 ept´τqLA0Jδf jpXδqpτqdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In the following, we will repeatedly use the estimates in Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For simplicity of notation, we will drop the index δ and the mollifier Jδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Improving regularity to C1,αpS2q: We proceed to obtain bounds in Cα, α P p0, 1q, for the terms f j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We will be denoting C “ Cp|X|˚, }X}C1pS2q, }T }C2q, CpRq “ Cp|X|˚, }X}C1pS2q, }T }C2, Rq in the bounds that follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 gives that }f1}CαpR2q ďC}ρnpT p∇S2Xq´TSp∇S2Xppxnqq∇S2Xq}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We note that I :“ T p∇S2Xppx1qq´TSp∇S2Xppxnqq∇S2Xppx1q ´ T p∇S2Xppx2qq`TSp∇S2Xppxnqq∇S2Xppx2q “ T p∇S2Xppx1qq´Tp∇S2Xppx2qq´TSp∇S2Xppxnqqp∇S2Xppx1q´∇S2Xppx2qq, thus |I| “ | ż 1 0 ` TSps∇S2Xppx1q ` p1 ´ sq∇S2Xppx2qq ´ TSp∇S2Xppxnqq ˘ ds ˆ p∇S2Xppx1q´∇S2Xppx2qq| ď C maxt|∇S2Xppx1q´∇S2Xppxnq|, |∇S2Xppx2q´∇S2Xppxnqu ˆ |∇S2Xppx1q´∇S2Xppx2q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 68 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Hence, thanks to the presence of ρn, we obtain (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) }f 1}CαpR2q ďCεpRq}ρn}CαpS2q}∇S2X}CαpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) }f 2}CαpS2q ď C ´ εpRq}∇S2pρnXq}CαpS2q ` }∇S2X}C α 2 pB5RppxnqXS2q}∇S2pρnXq}C α 2 pS2q ¯ , while f 3 is identically zero (see (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 provides the estimate for f 4, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) }f 4}CαpS2q ď CpRq}∇S2pρnXq}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) }f 7}CαpS2q ď C}∇S2ρn}CαpS2q, and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) }f 5}CαpS2q ď C}∇S2pρnXq}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, by writing rLA ´ LA0spρnXnqpθq “ r ˜ MpAq ´ ˜ MpA0qspTF pAq∇pρnXnqqpθq ` ˜ MpA0qppTF pAq ´ TF pA0qq∇pρnXnqqpθq, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 yields that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) }f 6}CαpS2q ď C}A ´ A0}}∇S2pρnXq}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We thus see from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) and Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15 that we can bootstrap to get that X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,αpS2qq for all α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, consider the case γ ă 1 2 and take α such that γ ă α ` 1 2 ď 2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, with 0 ă ǫ ď γ ´ α arbitrarily small, and substituting the bounds for f j, we obtain }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď C}ρnp0qX0}C1`α`ǫpS2qq ` C 7ÿ j“1 }f j}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 1 2 `αpR2qq ď C}ρnp0qX0}C1`γpS2q ` CpR, T q ` }X}2 L4p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1` 1 4 ` α 2 pS2qq ` CεpR, T q ` }X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq ` }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2q ˘ , and so (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď C}ρnp0qX0}C1`γpS2q ` CpR, T q ` CεpR, T q}X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, we can write }X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq ď } ÿ mPMn ρmX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpB5RppxnqXS2qq, where the cardinal number |Mn| can be picked independent of R and n, since the radius of the support of ρn and B5Rppxnq X S2 are comparable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, adding in n in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) we obtain ÿ n }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď CpR, T q ` CεpR, T q|Mn| ÿ n }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 69 hence we conclude that }X}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď ÿ n }ρnX}L2p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C 3 2 `αpS2qq ď CpR, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In particular, choosing α “ 2γ ´ 1 2, this uniform bound allows us to conclude that X P L2p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1`2γpS2qq, and thus Xptq P C1`2γpS2q for a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t P p0, T q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, pick t0 P p0, T q arbitrarily close to 0 and such that Xpt0q P C1`2γpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is clear that we can repeat the process to find t1 ą t0 such that Xpt1q P C1,αpS2q for any α P p0, 1q (the case γ ą 1 2 follows in one step).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Starting at t1, we find that }ρnXptq}C1,αpS2q ď }ρnXpt1q}C1,αpS2q ` C sup t1ďτďt 7ÿ m“1 }fm}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can thus take the supremum in t P pt1, T q and use the previous estimates on f m to conclude that X P L8pt1, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,αpS2qq for any t1 ą 0 and any α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Higher regularity: To study further smoothing, we first show that we can move derivatives in px to derivatives in py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, denoting ∆X “ Xppxq ´ Xppyq, ∇S2NpXqY ppxq “ “ ´ ż S2 ∇S2,px∇S2, pyGp∆Xq ¨ ∆∇S2Y dpy “ ´ ż S2 ´ ´ ∇S2, py∇S2, pyGp∆Xq`∇S2, py ` ∇S2,pxGp∆Xq`∇S2, pyGp∆Xq ˘¯ ¨∆∇S2Y dpy, so further integration by parts gives that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) ∇S2NpXqY ppxq “ NpXq∇S2Y ppxq ´ ż S2 ∇S2, py ` ∇S2,pxGp∆Xq`∇S2, pyGp∆Xq ˘ ¨ ∆∇S2Y ppyqdpy “ NpXq∇S2Y ppxq ´ ż S2 ∇S2, py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ∆∇S2Y ppyqdpy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we take a derivative in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) to get B Bt∇S2pρnXqppxq “ NpXq ` ∇S2 ` TSp∇S2Xppxnqq∇S2pρnXq ˘˘ ppxq ` NpXq ´ ∇S2` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘˘¯ ppxq ´ ż S2 ∇S2, py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ ppyqdpy ´ ż S2 ∇S2, py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘ ppyqdpy ` ∇S2rρn, NpXqsT p∇S2Xqppxq ´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqppxqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 70 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN We introduce the frozen-coefficient operator and the cutoff pρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' B Bt∇S2pρnXnqpθq “ LA1p∇S2pρnXq ˘ npθq ` 8ÿ j“1 f jpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 1pθq “ NpXq ´ ∇S2` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘˘¯ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 2pθq “ pρnpθqrNpXq ´ MpAqs `` TSp∇S2Xppxnqq∇S2∇S2pρnXq ˘˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 3pθq“ pρnpθq ´ MpAqpTSp∇S2Xppxnqq∇S2∇S2pρnXqqnpθq´LAp∇S2pρnXqqnpθq ¯ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 4pθq “ p1´pρnpθqqNpXq ` TSp∇S2Xppxnqq∇S2∇S2pρnXq ˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 5pθq “ p1´pρnpθqqLAp∇S2pρnXqqnpθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 6pθq “ rLA ´ LA1s ` ∇S2pρnXq ˘ npθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 7pθq “ ∇S2rρn,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' NpXqsT p∇S2Xqpx Xnpθqq ´ ∇S2NpXqpTSp∇S2XppxnqqX∇S2ρnqpx Xnpθqq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and f 8pθq “ f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1pθq ` f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2pθq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1pθq “ ´ ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi GpXpx Xpθq ´ Xppyqq ` ∇S2Xipx Xpθqq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ ppyqdpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2pθq “ ´ ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi GpXpx Xpθqq ´ Xppyqq ` ∇S2Xipx Xpθqq ´ ∇S2Xippyq ˘¯ ¨ ˆ ∆ ` ρn ` T p∇S2Xq ´ TSp∇S2Xppxnqq∇S2X ˘ ppyqdpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and A1 “ ∇Xnp0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A “ ∇Xnp0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' tq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, we proceed as we previously did in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) ∇S2pρnXnqptq “ ept´t0qLA1p∇S2pρnXnqpt0qq ` 8ÿ j“1 ż t t0 ept´τqLA1f jpτqdτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, to bootstrap and get C2,α regularity we need to use Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14- 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15 and obtain Cα estimates for the forced terms above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10 gives that }f 2}CαpS2q ď C ´ εpRq}∇2 S2pρnXq}CαpS2q ` }∇S2X}CαpS2q}∇2 S2pρnXq}C0pS2q ` }∇2 S2pρnXq}C0pS2q ¯ , while f 3 ” 0, and Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8 provide that }f 4}CαpS2q ` }f 5}CαpS2q ď CpRq}∇2 S2X}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As done before in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), we have that }f 6}CαpS2q ď C}A ´ A1}}∇2 S2pρnXq}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We interpolate the C2pS2q norm followed by Young’s inequality to get a small coefficient for the higher regularity part: }∇2 S2pρnXq}C0pS2q ď Cpεq}∇S2pρnXq}C1´αpS2q ` ε}∇2 S2pρnXq}CαpS2q, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 71 so that 6ÿ j“2 }f j}CαpS2q ď CεpR, ∆tq}∇2 S2pρnXq}CαpS2q`CpRq}∇S2pρnXq}C1´αpS2q, where from now on the constants C and CpR, ∆tq, ∆t “ t ´ t1, also depend on the controlled norm }X}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,maxtα,1´αupS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, }f 1}CαpS2q ď Cε}∇S2X}CαpS2q ` C}ρn ` TSp∇S2Xq∇2 S2X ´ TSp∇S2ppxnqq∇2 S2X ˘ }CαpS2q ď C ` C}∇S2X}CαpS2q}∇2 S2X}C0pB pxn,2RXS2q ` CεpRq}∇2 S2X}CαpB pxn,2RXS2q, so by interpolation again }f 1}CαpS2q ď CpRq ` CεpRq}∇2 S2X}CαpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term f 8 is lower order, and thus we can control it using interpolation once more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' In fact, taking the derivative in the kernel, we have for f 8,1 f 8,1ppxq “ ż S2 B Bxj B Bxi Gp∆Xq∇S2Xjppyq∆∇S2Xi ¨ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ dpy ´ ż S2 B Bxi Gp∆Xq∇2 S2Xippyq ¨ ∆ ` TSp∇S2Xppxnqq∇S2pρnXq ˘ dpy, and therefore, proceeding as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9, we obtain }f 8,1}CαpS2q ď C ` C}∇2 S2X}C0pB pxn,2RXS2q ď CpRq ` CεpRq}∇2 S2X}CαpB2RppxqXS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The estimate for f 8,2 follows in the same manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we estimate the commu- tator terms, f 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Using (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' we write f 7ppxq “ f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1ppxq ` f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2ppxq ` f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3ppxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1ppxq “ ´rNpXq∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ρnsT p∇S2Xqppxq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2ppxq “ ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ∆ ` ρnT p∇S2Xqppyq ˘ dpy ´ ρnppxq ż S2 ∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi Gp∆Xq ` ∇S2Xippxq ´ ∇S2Xippyq ˘¯ ¨ ∆ ` T p∇S2Xqppyq ˘ dpy ` ż S2∇S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' py ´ B Bxi Gp∆Xq ` ∇S2Xippxq´∇S2Xippyq ˘¯ ¨∆ ` TSp∇S2XppxnqqX∇S2ρnppyq ˘ dpy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' and f 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3ppxq “ ´NpXq ` TSp∇S2Xppxnqq∇S2pX∇S2ρnq ˘ ppxq ` ∇S2ρnNpXqpT p∇S2Xqqppxq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term f 7,3 is lower order and it only requires C1,αpS2q regularity for X, while the estimate for f 7,2 follows taking the derivative of the kernel, as done for f 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We get that }f 7,2}CαpS2q ď C ` C}∇2 S2X}C0pB pxn,2RXS2q ` C}∇2 S2X}C0pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By interpolation, }f 7,2}CαpS2q ` }f 7,3}CαpS2q ď Cp˜εq ` C˜ε}∇2 S2X}CαpS2q, 72 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN with ˜ε ą 0 to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The term f 7,1 is written as follows: f 7,1ppxq “ ´ ż S2 ∇S2, pyGpXppxq´Xppyqq ¨ ` ρnppxq∇S2T p∇S2Xppyqq ´ ∇S2 ` ρnppyqT p∇S2Xppyqq ˘˘ dpy “ rρn, NpXqs∇S2T p∇S2Xqppxq ` ż S2 ∇S2, pyGpXppxq´Xppyqq ¨ ∇S2ρnppyqT p∇S2Xppyqqdpy, hence, by Lemmas 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6, we have that }f 7,1}CαpS2q ď C}∇S2ρn}C1,αpS2q ` C}∇S2ρn}C0pS2q}∇2 S2X}C0pS2q, and, by interpolation, }f 7,1}CαpS2q ď Cp˜εq ` C˜ε}∇2 S2X}CαpS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, we have that for any t1 ą 0 and α P p0, 1 2q, }∇S2pρnXqptq}L2pt1,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,αpS2qq ď C}ρnpt1qXpt1q}C 3 2 `α´ǫ ` 7ÿ m“1 }f mpτq}L2pt1,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CαpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, introducing the estimates above for f j and summing in n, we take the partition so that εpRq is small enough and then choose ˜ε small enough (depending on the partition ρn), to obtain that X P L2pt1, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C2,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, we can take t2 ą t1 so that Xpt2q P C2,αpS2q and use (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) to conclude that X P L8pt2, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C2,αpS2qq for any t2 ą 0, α P p0, 1 2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, starting with the upgraded regularity and repeating the same steps with no changes, we conclude that X P L8pt2, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C2,αpS2qq for any t2 ą 0, α P p0, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It is not difficult to show by induction that an analogous formula to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) holds for higher derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, by repeating the steps above one can continue the bootstrapping argument, concluding that for any n P N, X P L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Cn,αpS2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Xδ Ñ X in L8p0, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' C1,γpS2qq: We write the difference ∆δX :“ Xδ´X as follows: ρn∆δXnptq “ etLA0 ` ρn∆δX0,n ˘ ` 7ÿ j“1 ż t 0 ept´τqLA0 ´ pJδ ´ 1qf jpXδqpτq ` f jpXδqpτq ´ f jpXqpτq ¯ dτ, with f jpXq given in (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) }ρn∆δXn}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q ` C 7ÿ j“1 }pJδ ´ 1qf jpXδq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ` C sup tPr0,T s 7ÿ j“1 } ż t 0 ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since X0 P h1,γpS2q and f jpXδq P hγpS2q, the first two terms converge to zero as δ Ñ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the third term, we need to show that it can be absorbed by the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' As in the previous arguments, we will show that for the quasilinear terms we WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 73 can find a small coefficient, while for the lower order ones we will take advantage of the extra regularity via (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) to get a small coefficient for T small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' From previous estimates we immediately get that for j “ 4, 5, 7 and 0 ă ǫ ă 1´γ, }f jpXδq ´ f jpXq}Cγ`ǫpS2q ď C}∆δX}C1pS2q, hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) gives that sup tPr0,T s ÿ j“4,5,7 } ż t 0 ept´τqLA0pf jpXδq ´ f jpXqqpτq}CγpR2q ď C T ǫ}∆δX}C1pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Also, for f 6, }f 6pXδq ´ f 6pXq}CγpS2q ď C}A ´ A0}}∇S2pρn∆δXq}CγpS2q ` C}∆δX}C1pS2q}ρnXδ}C1,γpS2q, so that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) give } ż t 0 ept´τqLA0pf 6pXδq ´ f 6pXqqpτq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ď CεpT q}ρn∆δX}C1,γpS2q ` CT ε}∆δX}C1pS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, we proceed with the term f 1: pf 1pXδq ´ f 1pXqqpθq “ I1 ` I2, with I1pθq “ pNpXδq ´ NpXqq ` ρn ` T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ˘˘ npθq I2pθq “ NpXq ´ ρn ´ T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ ´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X ¯¯ npθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term is estimated easily from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 by noticing that one can always extract ∆δX “ Xδ ´X from the difference of the kernels (similarly as done in the proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3), (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) }I1}CγpR2q ďC}∇S2∆δX}C0pS2q}ρnpT p∇S2Xδq´TSp∇S2Xδppxnqq∇S2Xδq}CγpS2q ď CεpRq}∇S2∆δX}C0pS2q}∇S2X}CγpB pxn,2RXS2q, where we have used (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) in the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 gives that (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) }I2}CγpR2q “ C}ρn ´ T p∇S2Xδq ´ TSp∇S2Xδppxnqq∇S2Xδ ´ T p∇S2Xq ` TSp∇S2Xppxnqq∇S2X ¯ }CγpR2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Denote JpY qppxq “ T p∇S2Y ppxqq ´ TSp∇S2Y ppxnqq∇S2Y ppxq, so that we can write JpXδppx1qq ´ JpXδppx2qq “ p∇S2Xδppx1q ´ ∇S2Xδppx2qq ˆ ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 74 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Then, JpXδppx1qq ´ JpXδppx2qq ´ JpXppx1qq ` JpXppx2qq “ J1 ` J2, with J1 “ p∇S2∆δXppx1q ´ ∇S2∆δXppx2qq ˆ ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1, and J2 “ p∇S2Xppx1q ´ ∇S2Xppx2qq ˆ ´ ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1 ´ ż 1 0 ` TSps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ´ TSp∇S2Xppxnqq ˘ ds1 ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It follows that |J1| ď C maxt|∇S2Xδppx1q´∇S2Xδppxnq|, |∇S2Xδppx2q´∇S2Xδppxnqu ˆ |∇S2∆δXppx1q´∇S2∆δXppx2q|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We apply the mean-value theorem again in J2: ż 1 0 ` TSps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ´ TSp∇S2Xδppxnqq ˘ ds1 “ ż 1 0 ż 1 0 DTS ` s2ps1∇S2Xδppx1q`p1´s1q∇S2Xδppx2qq`p1´s2q∇S2Xδppxnq ˘ ds1ds2 ˆ ` s1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2q ´ ∇S2Xδppxnq ˘ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' hence adding and subtracting we obtain |J2| ď |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1| ` |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' with |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1| ď |∇S2Xppx1q ´ ∇S2Xppx2q| ˆ maxt|∇S2Xδppx1q´∇S2Xδppxnq|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |∇S2Xδppx2q´∇S2Xδppxnq|u ˆ ˇˇˇDTSps2ps1∇S2Xδppx1q ` p1 ´ s1q∇S2Xδppx2qq ` p1 ´ s2q∇S2Xδppxnqq ´ DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq ˇˇˇ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |J2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2| ď |∇S2Xppx1q ´ ∇S2Xppx2q| ˆ |DTSps2ps1∇S2Xppx1q ` p1 ´ s1q∇S2Xppx2qq ` p1 ´ s2q∇S2Xppxnqq| ˆ maxt|∇S2∆δXppx1q´∇S2∆δXppxnq|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' |∇S2∆δXppx2q´∇S2∆δXppxnqu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Going back to (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16), we thus conclude that }I2}CγpR2q ď CεpRq}∇S2∆δX}CγpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Together with the bound for I1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15), we obtain the following estimate for f 1: } ż t 0 ept´τqLA0pf 1pXδq ´ f 1pXqqpτq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ď C T ǫ}∆δX}C1pS2q ` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q ` CεpRq}∆δX}C1,γpB pxn,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 75 The estimate for f 2 follows in the same way than those for f 1 and f 6, from which we conclude that }ρn∆δXn}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,γpR2qq ď C}ρn∆δX0,n}C1,γpR2q ` C 7ÿ j“1 }pJδ ´ 1qf jpXδq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq ` CT ε}∆δX}C1pS2q ` CεpRq}∆δX}C1pS2q}X}C1,γpB pxn,2RXS2q ` CpεpT q ` εpRqq}∆δX}C1,γpB px,2RXS2q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Taking R and T small enough, the last term is absorbed by the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, adding in n, we conclude that, for T and R small enough, the desired estimate holds }∆δX}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='C1,γpS2qq ďC}∆δX0}C1,γpS2q`C 7ÿ j“1 }pJδ´1qfjpXδq}L8p0,T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='CγpR2qq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Besov Spaces and Fourier Multiplier Theorems In this section, we will proof Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, define Besov Spaces Bγ p,q by a dyadic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Set a function ψ pξq P C8 pRnq s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ψ pξq “ " 1 |ξ| ď 1 0 |ξ| ě 2 and define φ pξq :“ ψ pξq ´ ψ p2ξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, φ pξq P C8 pRnq and φ pξq “ 0, |ξ| ď 1 2, |ξ| ě 2, 8 ÿ j“´8 φ ` 2´jξ ˘ “ 1, |ξ| ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, the homogeneous dyadic blocks 9∆j are defined by 9∆jf pθq :“ F´1 ` φ ` 2´jξ ˘ Ff pξq ˘ pθq “ Kj ˚ f (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) where K pθq :“ F´1 pφ pξqq pθq and Kj pθq :“ F´1 ` φ ` 2´jξ ˘˘ pθq “ 2jnK ` 2jθ ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Now, given γ a real number and p, q ě 1, we may define homogeneous Besov spaces 9Bγ p,q pRnq with its seminorm ∥¨∥ 9Bγ p,qpRnq by ∥f∥ 9Bγ p,qpRnq :“ ˜ 8 ÿ j“´8 ˆ 2jγ ��� 9∆jf ��� LppRnq ˙q¸ 1 q , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) ∥f∥ 9Bγ p,8pRnq :“ sup jPZ ˆ 2jγ ��� 9∆jf ��� LppRnq ˙ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) According to [27, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2] and [48, Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2], we know for all 0 ă γ ă 1, ∥¨∥ 9Bγ p,qpRnq and �¨�CγpRnq are equivalent, so we only need to prove the Fourier multiplier theorem on 9Bγ p,q pRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The proof is from [48, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given T a Fourier multiplier operator with multiplier m pξq P Cs pRnzt0uq X L8 pRnq, for s ą n 2 and for all |α| ď s, such that ��Bα ξ m pξq �� ď Cα |ξ|´|α| , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) 76 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN we first define a related kernel with λ ą 0 by Kλ pθq :“ F´1 pφ pξq m pλξqq pθq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given m pξq satisfying (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4), then Kλ pθq is bounded by ż Rn |Kλ pθq| dθ ď Cs,n,φDm, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) where Dm “ max|α|ďs Cα Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since there exist Cs s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all θ P Rn ´ 1 ` |θ|2¯s ď Cs ÿ |α|ďs |θα|2 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) we obtain ż Rn |Kλ pθq|2 ´ 1 ` |θ|2¯s dθ ďCs ÿ |α|ďs ż Rn |θαKλ pθq|2 dθ “Cn,s ÿ |α|ďs ż Rn ��Bα ξ pφ pξq m pλξqq ��2 dξ “Cn,s ÿ |α|ďs ż Rn ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq λ|α´β| ´ Bα´β ξ m ¯ pλξq ����� 2 dξ ďCn,sD2 m ÿ |α|ďs ż Rn ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq λ|α´β| |λξ|´|α´β| ����� 2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) supp pφq Ă ␣ ξ| 1 2 ď |ξ| ď 2 ( , so ż Rn ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq λ|α´β| |λξ|´|α´β| ����� 2 dξ “ ż 1 2 ď|ξ|ď2 ����� ÿ βďα ˆ α β ˙ ´ Bβ ξ φ ¯ pξq |ξ|´|α´β| ����� 2 dξ ď Cφ,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) Thus, ż Rn |Kλ pθq|2 ´ 1 ` |θ|2¯s dθ ď Cn,sD2 m ÿ |α|ďs Cφ,α ď Cs,n,φD2 m, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) and by Holder inequality, ż Rn |Kλ pθq| dθ ď ˆż Rn |Kλ pθq|2 ´ 1 ` |θ|2¯s dθ ˙ 1 2 ˆż Rn ´ 1 ` |θ|2¯´s dθ ˙ 1 2 ď a Cs,n,φDm ˆż Rn ´ 1 ` |θ|2¯´s dθ ˙ 1 2 ď Cs,n,φDm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) □ Next, we may use Kλ pθq and homogeneous Besov semi norm ∥¨∥ 9Bγ p,qpRnq to prove the Fourier multiplier theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 77 Proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, set T j mf pθq :“ 9∆jTmf pθq “ F´1 ` φ ` 2´jξ ˘ m pξq Ff pξq ˘ pθq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) Since F´1 ` φ ` 2´jξ ˘ m pξq ˘ pθq “ 2njK2j ` 2jθ ˘ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) ��T j mf pθq �� “ ��F´1 ` φ ` 2´jξ ˘ m pξq ˘ ˚ f pθq �� ď ��F´1 ` φ ` 2´jξ ˘ m pξq ˘ pθq �� L1pRnq ∥f∥L8pRnq ď ��2njK2j ` 2jθ ˘�� L1pRnq ∥f∥L8pRnq ď ∥K2j pθq∥L1pRnq ∥f∥L8pRnq ďCs,n,φDm ∥f∥L8pRnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) Next, supp ` φ ` 2´jξ ˘˘ Ă ␣ ξ ˇˇ2j´1 ď |ξ| ď 2j`1 ( , so for all j ` 1 ď k ´ 1 or j ´ 1 ě k ` 1 φ ` 2´jξ ˘ φ ` 2´kξ ˘ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) Therefore, φ ` 2´jξ ˘ Ff pξq “ φ ` 2´jξ ˘ ´ F ´ 9∆j´1f ¯ pξq ` F ´ 9∆jf ¯ pξq ` F ´ 9∆j`1f ¯ pξq ¯ , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) so we obtain T j mf pθq “ T j m ´ 9∆j´1f ¯ pθq ` T j m ´ 9∆jf ¯ pθq ` T j m ´ 9∆j`1f ¯ pθq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) Finally, ∥Tmf∥ 9Bγ 8,8pRnq “ sup jPZ 2jγ ��T j mf �� L8pRnq ď sup jPZ 2jγ ���T j m ´ 9∆j´1f ¯��� L8pRnq ` sup jPZ 2jγ ���T j m ´ 9∆jf ¯��� L8pRnq ` sup jPZ 2jγ ���T j m ´ 9∆j`1f ¯��� L8pRnq ďCs,nDm ˆ sup jPZ 2jγ ��� 9∆j´1f ��� L8pRnq ` sup jPZ 2jγ ��� 9∆jf ��� L8pRnq ` sup jPZ 2jγ ��� 9∆j`1f ��� L8pRnq ˙ “Cs,nDm ` 2γ ` 1 ` 2´γ˘ ∥f∥ 9Bγ 8,8pRnq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) Since ∥¨∥ 9Bγ p,qpRnq and �¨�CγpRnq are equivalent, �Tmu�CγpRnq ď Cγ,s,nDm�u�CγpRnq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Estimates for the semigroup e´tLApξq Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For all β “ β1β2 ¨ ¨ ¨ βk, there exists a matrix Pβ ´ ˆξ1, ˆξ2 ¯ of polyno- mials with degree deg pPβq ď 3 |β| ` 4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' BβLA pξq “ 1 |ξ||β|´1 Pβ ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 2|β|`3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) 78 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN More specifically, Pβ ´ ˆξ1, ˆξ2 ¯ can be written as pPβqi1i2 ´ ˆξ1, ˆξ2 ¯ “ ÿ j1,j2ě0,j1`j2ď3|β|`4 cpβ,i1,i2q j1,j2 ˆ A, U, P, 1 detpBq, T , dT dλ , 1 λ2 ˙ ˆξj1 1 ˆξj2 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) where cpβ,i1,i2q j1,j2 ˆ A, U, P, 1 detpBq, T , dT dλ , 1 λ2 ˙ “cpβ,i1,i2q j1,j2 ˆ A11, ¨ ¨ ¨ , A32, U11, ¨ ¨ ¨ , U32, P11, ¨ ¨ ¨ , P33, 1 detpBq, T λ , dT dλ , 1 λ2 ˙ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3) is a polynomial function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Moreover, ���Pβ ´ ˆξ1, ˆξ2 ¯��� C1pDAσ1,σ2q and ���Uˆξ ��� C1pDAσ1,σ2q are uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' there exists Cpβq σ1,σ2,T and Cpβq σ1,σ2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' for all ˆξ P S1, ���Pβ ´ ˆξ1, ˆξ2 ¯��� C1pDAσ1,σ2q ď Cpβq σ1,σ2,T , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4) ���Uˆξ ��� C1pDAσ1,σ2q ď Cpβq σ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since pFθGα,Aq pξq “ 1 |ξ| pFθGα,Aq pˆξq (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6) “ 1 |ξ| pI ` αPq ���Uˆξ ��� 2 ´ αUˆξ b Uˆξ 4 detpBq ���Uˆξ ��� 3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='7) and Zpξq “ |ξ|2 Zpˆξq where Zpˆξq is a matrix of polynomials with degree 2, LA pξq “ |ξ| P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='8) where P0 “ pI ` αPq ���Uˆξ ��� 2 ´ αUˆξ b Uˆξ 4 detpBq ˜ T λ ˜ I ´ Aˆξ b Aˆξ λ2 ¸ ` dT dλ Aˆξ b Aˆξ λ2 ¸ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='9) where the degree of P0 is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Obviously, P0 can be written as (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 79 When |β| “ 1, BLA Bξi pξq “ ξi |ξ| P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 ` |ξ| ÿ j“1,2 B Bˆξj P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 B Bξi ξj |ξ| “ ξi |ξ| P0 ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 3 ` |ξ| ÿ j“1,2 BP0 Bˆξj ���Uˆξ ��� 2 ´ 3P0 ´ U T Uˆξ ¯ j ���Uˆξ ��� 5 δij ´ ˆξi ˆξj |ξ| “ Pi ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 5 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10) where Pi ´ ˆξ1, ˆξ2 ¯ “ ˆξiP0 ���Uˆξ ��� 2 ` ÿ j“1,2 ˜ BP0 Bˆξj ���Uˆξ ��� 2 ´ 3P0 ´ U T Uˆξ ¯ j ¸ ´ δij ´ ˆξi ˆξj ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='11) The degrees of all terms are at most 1 ` 4 ` 2 “ 3 ` 2 ` 2 “ 4 ` 1 ` 2 “ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ���Uˆξ ��� 2 “ 2ÿ j1,j2“1 3ÿ k“1 Ukj1Ukj2 ˆξj1 ˆξj2, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12) ´ U T Uˆξ ¯ j “ „ř3 k“1 UkjUk1 ˆξj ř3 k“1 UkjUk2 ˆξj \uf6be , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13) and BP0 Bˆξj can be written as the form of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), Pi ´ ˆξ1, ˆξ2 ¯ can be written as (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Thus, the case |β| “ 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Suppose |β| ď k´1 holds, for ��¯β �� “ k, we may rewrite ¯β as ββk where |β| “ k´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, B ¯βLA pξq “ BβkBβLA pξq “ B Bξβk ¨ ˚ ˝ 1 |ξ||β|´1 Pβ ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� 2|β|`3 ˛ ‹‚ “ ´ p|β| ´ 1q 1 |ξ||β| ˆξβk Pβ ���Uˆξ ��� 2|β|`3 ` 1 |ξ||β|´1 ÿ j“1,2 BPβ Bˆξj ���Uˆξ ��� 2 ´ p2 |β| ` 3q Pβ ´ U T Uˆξ ¯ j ���Uˆξ ��� 2|β|`5 δβkj ´ ˆξβk ˆξj |ξ| “ 1 |ξ|| ¯β| P¯β ´ ˆξ1, ˆξ2 ¯ ���Uˆξ ��� ¯β`3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14) 80 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where Pˆβ “ ´ p|β| ´ 1q ˆξiPβ ���Uˆξ ��� 2 ` ÿ j“1,2 ˜ BPβ Bˆξj ���Uˆξ ��� 2 ´ ` 2 ��¯β �� ` 1 ˘ Pβ ´ U T Uˆξ ¯ j ¸ ´ δβkj ´ ˆξβk ˆξj ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='15) The degrees of all terms are at most 1 ` p3 |β| ` 4q ` 2 “ p3 |β| ` 3q ` 2 ` 2 “ p3 |β| ` 4q ` 1 ` 2 “ 3 ��¯β �� ` 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Again, BPβ Bˆξj is still able to written as the form of (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2), so the case ��¯β �� “ k holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By Induction, for all β, the formulas (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1) and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, in Pβ, since for all square matrix M, ∥M∥ ď ř |Mij|, we just need to estimate each element pPβqi1i2, ���pPβqi1i2 ´ ˆξ1, ˆξ2 ¯��� ď ÿ ����cpβ,i1,i2q j1,j2 ˆ A, U, P, 1 detpBq, T λ , dT dλ , 1 λ2 ˙���� (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='16) and cpβ,i1,i2q j1,j2 ´ A, U, P, 1 detpBq, T λ , dT dλ , 1 λ2 ¯ is a form of a polynomial, so we may only check each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given A P DA, ��� ` AT A ˘´1��� , 1 detpBq ď 1 σ2 2 and σ1 ď λ ď ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1, so all variables are bounded by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ���� BAT A BAij ���� ď 2λ ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='17) ����� B ` AT A ˘´1 BAij ����� “ ���� ` AT A ˘´1 BAT A BAij ` AT A ˘´1 ���� ď 2 ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2σ1 σ4 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='18) so on DA, U, P are C1 functions and their derivatives are bounded by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Absolutely, ���Uˆξ ��� C1pDAσ1,σ2q ď Cpβq σ1,σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='19) Since ����� B det ` AT A ˘ BAij ����� ď λ2 ď 8σ2 1, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) ���� B BAij 1 detpBq ���� “ ����� 1 2 detpBq3 B det ` AT A ˘ BAij ����� ď 8σ2 1 σ8 2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21) and ���� Bλ BAij ���� “ ���� Aij λ ���� ď 1 (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='22) on DA, 1 detpBq, T λ , dT dλ , 1 λ2 are also C1 functions and their derivatives are bounded by σ1 and σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we obtain ���Pβ ´ ˆξ1, ˆξ2 ¯��� C1pDAσ1,σ2q ď Cpβq σ1,σ2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='23) □ WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 81 Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given A P DAσ1,σ2 and ϕ pξq “ ϕ p|ξ|q, a cutting and decreasing respect |ξ| and supported in B p1q, and set K0 pθq :“ F´1 ” pz ` LAq´1 pξqϕpξq ı , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) K1,j pθq :“ F´1 ” ξj pz ` LAq´1 pξqϕpξq ı .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='25) Then, for all z P Sω,δ, we have the following estimates ∥K0 pθq∥ ďCω,δ,σ1,σ2,T |z| 1 1 ` |θ|3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='26) ∥K1,j pθq∥ ďCω,δ,σ1,σ2,T 1 1 ` |θ|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='27) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For convenience, we define Hpξq :“ pz ` LAq´1 pξq First, since p1 ` |θ|3q ż R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ “ ż R2p1 ´ i θ |θ| ¨ iθ|θ|2qeiθ¨ξ pz ` LAq´1 pξqϕpξqdξ “ ż R2 „ p1 ´ i θ |θ| ¨ ∇ξ|∇ξ|2qeiθ¨ξ \uf6be pz ` LAq´1 pξqϕpξqdξ “ ż R2 eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ |θ| ¨ ∇ξ∆ξ ” pz ` LAq´1 pξqϕpξq ı dξ “ ż Bp1q eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ |θ| ¨ ∇ξ∆ξ ” pz ` LAq´1 pξqϕpξq ı dξ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='28) By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20), ����� ż Bp1q eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ ����� ď Cδ,σ1,σ2,T ∥ϕ∥C0 |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='29) Next, we compute B Bξj ∆ξ ” pz ` LAq´1 ϕ ı “ B Bξj ∆ξ pz ` LAq´1 ϕ ` ∆ξ pz ` LAq´1 B Bξj ϕ ` 2 B Bξj ∇ξ pz ` LAq´1 ¨ ∇ξϕ `2∇ξ pz ` LAq´1 ¨ B Bξj ∇ξϕ ` B Bξj pz ` LAq´1 ∆ξϕ ` pz ` LAq´1 B Bξj ∆ξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='30) Since ϕ is smooth, we may estimate all of the terms except the first by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Obviously, for the last term, �����i θj |θ| ż Bp1q eiθ¨ξ pz ` LAq´1 ˆ B Bξj ∆ξϕ ˙ dξ ����� ď Cδ,σ1,σ2,T ∥ϕ∥C3 |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='31) 82 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Then, for all |α| “ 1, 2, |α| ` |β| “ 3 �����i θj |θ| ż Bp1q eiθ¨ξBα ξ pz ` LAq´1 pξqBβ ξ ϕpξqdξ ����� ď ∥ϕ∥C2 ż Bp1q ���Bα ξ pz ` LAq´1 pξq ��� dξ ďCδ,σ1,σ2,T |z|2 ∥ϕ∥C2 ż Bp1q |ξ|1´|α| dξ ď Cδ,σ1,σ2,T |z|2 ∥ϕ∥C2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='32) Now, for the first term, B Bξj ∆ξH “2 ÿ k“1,2 “ ´ pEjkk ` Ekjk ` Ekkjq ` ` Ekpjkq ` Epjkqk ˘‰ ` ` Ejpkkq ` Epkkqj ˘ ´ H B Bξj ∆ξLAH, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33) where Eijk “H BLA Bξi H BLA Bξj H BLA Bξk H, Ekpjkq “H BLA Bξk H B2LA BξjBξk H, Ejpkkq “H BLA Bξj H∆ξLAH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ��� BLA Bξj ��� ≲ 1 and ��� B2LA BξjBξk ��� ≲ |ξ|´1, ∥Eijk∥ ≲ 1 and ��Ekpjkq �� , ��Ejpkkq �� ≲ |ξ|´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, we only have to check pz ` LAq´1 B Bξj ∆ξLA pz ` LAq´1 ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' LA is an even function, so the term is an odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By LA pξq “ |ξ| LA ´ ˆξ ¯ and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10), �����i θj |θ| ż Bp1q eiθ¨ξ pz ` LA pξqq´1 B Bξj ∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ ����� “ �����´ θj |θ| ż Bp1q sin pθ ¨ ξq pz ` LA pξqq´1 B Bξj ∆ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ ����� ď ������ ż Bp1q sin ´ θ ¨ ˆξ |ξ| ¯ ´ z ` |ξ| LA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ |ξ|2 ´ z ` |ξ| LA ´ ˆξ ¯¯´1 ϕ p|ξ|q dξ ������ “ ������ ż S1 ż 1 0 ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq sin ´ θ ¨ ˆξr ¯ r drdˆξ ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='34) WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 83 By Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, we obtain for all ˆξ P S1 ������ ż 1 0 ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq sin ´ θ ¨ ˆξr ¯ r dr ������ ď2 ���� ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq ���� C1pr0,1sq ď4 ���Φp3q A,j ´ ˆξ ¯��� ∥ϕ prq∥C1pr0,1sq ���� ´ z ` rLA ´ ˆξ ¯¯´1���� 2 C1pr0,1sq ďCδ,σ1,σ2,T ∥ϕ prq∥C1pr0,1sq ˜ 1 |z| ` 1 |z|2 ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='35) Therefore, ������ ż S1 ż 1 0 ´ z ` rLA ´ ˆξ ¯¯´1 Φp3q A,j ´ ˆξ ¯ ´ z ` rLA ´ ˆξ ¯¯´1 ϕ prq sin ´ θ ¨ ˆξr ¯ r drdˆξ ������ ďCδ,σ1,σ2,T ∥ϕ prq∥C1 ˜ 1 |z| ` 1 |z|2 ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='36) Since 1 |z| ď Cω,δ if z P Sω,δ, ����� ż Bp1q eiθ¨ξ pz ` LAq´1 pξqϕpξq ` ieiθ¨ξ θ |θ| ¨ ∇ξ∆ξ ” pz ` LAq´1 pξqϕpξq ı dξ ����� ď ∥ϕ prq∥C3 4ÿ k“1 Cpkq δ,σ1,σ2,T 1 |z|k ďCω,δ,σ1,σ2,T ∥ϕ∥C3 |z| , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='37) and ���� ż R2 eiθ¨ξ pz ` LAq´1 pξqϕpξqdξ ���� ďCω,δ,σ1,σ2,T ∥ϕ∥C3 |z| 1 1 ` |θ|3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='38) Next, for K1,j, we use the same technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ` |θ|4qK1,j pθq becomes p1 ` |θ|4q ż R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ “ ż R2 “ p1 ` |∇ξ|4qeiθ¨ξ‰ ξj pz ` LAq´1 pξqϕpξqdξ “ ż Bp1q eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2 ξ ” ξj pz ` LAq´1 pξqϕpξq ı dξ, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='39) By (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20), ����� ż Bp1q eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ ����� ď Cδ,σ1,σ2,T ∥ϕ∥C0 |z| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='40) 84 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Next, ∆2 ξ ” ξj pz ` LAq´1 pξqϕpξq ı “ 4 B Bξj ∆ξ ” pz ` LAq´1 ϕ ı ` ξj∆2 ξ ” pz ` LAq´1 pξqϕpξq ı (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='41) We have estimated the first term, so let us compute the second, ∆2 ξ ” pz ` LAq´1 ϕ ı “ ∆2 ξ pz ` LAq´1 ϕ ` 4∇ξ∆ξ pz ` LAq´1 ¨ ∇ξϕ ` 4 ” ∇2 ξ pz ` LAq´1 : ∇2 ξϕ ı ` 2∆ξ pz ` LAq´1 ∆ξϕ ` 4∇ξ pz ` LAq´1 ¨ ∇ξ∆ξϕ ` pz ` LAq´1 ∆2 ξϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='42) For the last four terms, we may estimate them by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='20) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='24) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the second term, since Bα ξ LA ≲ |ξ|1´|α|, by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='33), we may obtain ����� ż Bp1q eiθ¨ξξj∇ξ∆ξ pz ` LAq´1 pξq ¨ ∇ξϕ pξq dξ ����� ď ∥ϕ∥C1 4ÿ k“1 Cpkq δ,σ1,σ2,T 1 |z|k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='43) In the first term, we have ∆2 ξH “ ÿ j,k“1,2 r 8 pEjjkk ` Ejkjk ` Ejkkjq ´8 ` Ejkpjkq ` Ejpjkqk ` Epjkqjk ˘ ` 4H B2LA BξjBξk H B2LA BξjBξk H \uf6be ` ÿ j“1,2 “ ´4 ` Ejjpkkq ` Ejpkkqj ` Epkkqjj ˘ ` 4 ` Ejpjkkq ` Epjkkqj ˘‰ ` 2H∆ξLAH∆ξLAH ´ H∆2 ξLAH, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='44) where Ejjkk “H BLA Bξj H BLA Bξj H BLA Bξk H BLA Bξk H, Ejkpjkq “H BLA Bξj H BLA Bξk H B2LA BξjBξk H, Ejjpkkq “H BLA Bξj H BLA Bξj H∆ξLAH, Ejpjkkq “H BLA Bξk H B∆ξLA Bξj H, and we only have to compute the ´ pz ` LAq´1 ∆2 ξLA pz ` LAq´1 term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since LA is an even function, ´ξj pz ` LAq´1 ∆2 ξLA pz ` LAq´1 pξq ϕpξq is odd, again, by WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 85 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6 and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='�����´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Bp1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='eiθ¨ξξj pz ` LAq´1 ∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ξLA pz ` LAq´1 pξq ϕpξqdξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Bp1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ξj sin pθ ¨ ξq pz ` LA pξqq´1 ∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ξLA pξq pz ` LA pξqq´1 ϕ pξq dξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ď ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Bp1q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='θ ¨ ˆξ |ξ| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` |ξ| LA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ξjΦp4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='|ξ|3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` |ξ| LA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ϕ p|ξ|q dξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='“ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` rLA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Φp4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='z ` rLA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯¯´1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ϕ prq ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='sin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='θ ¨ ˆξr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='r ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='drdˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ď4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ż ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='S1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='���ˆξj ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='���Φp4q ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='´ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='ˆξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='¯��� ∥ϕ prq∥C1pr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1sq ���� ´ z ` rLA ´ ˆξ ¯¯´1���� 2 C1pr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1sq dˆξ ďCδ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='T ∥ϕ prq∥C1pr0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1sq ˜ 1 |z| ` 1 |z|2 ¸2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='45) Hence, ����� ż Bp1q eiθ¨ξξj pz ` LAq´1 pξqϕpξq ` eiθ¨ξ∆2 ξ ” ξj pz ` LAq´1 pξqϕpξq ı dξ ����� ď ∥ϕ prq∥C4 5ÿ k“1 Cpkq δ,σ1,σ2,T 1 |z|k ďCω,δ,σ1,σ2,T ∥ϕ∥C4 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='46) and ���� ż R2 eiθ¨ξξj pz ` LAq´1 pξqϕpξqdξ ���� ďCω,δ,σ1,σ2,T ∥ϕ∥C4 1 1 ` |θ|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='47) □ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given kpxq “ F´1re´LApξqs, then we have the following estimates ∥kpxq∥ ď C 1 1 ` |x|3 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='48) ���� B Bxi kpxq ���� ď C 1 1 ` |x|4 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='49) ���� B Bxi B Bxj kpxq ���� ď C 1 1 ` |x|5 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='50) 86 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' First, p1 ` |x|3q ż R2 eix¨ξe´LApξqdξ “ ż R2p1 ´ i x |x| ¨ ix|x|2qeix¨ξe´LApξqdξ “ ż R2p1 ´ i x |x| ¨ ∇ξ|∇ξ|2qeix¨ξe´LApξqdξ “ ż R2 eix¨ξe´LApξq ` i x |x| ¨ ∇ξ∆ξe´LApξqdξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since e´LApξq ≲ e´|ξ|, we obtain (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='51) p1 ` |x|3q ∥kpxq∥ ≲ ´ 1 ` ����� ż B1p0q eix¨ξ x |x| ¨ ∇ξ∆ξe´LApξqdξ ����� ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' since B Bξi e´LApξq “ ´ ż 1 0 e´p1´tqLApξq B Bξi LApξqe´tLApξqdt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Bjkk ξ e´LApξq “ ´ ż 1 0 e´p1´t1qLApξqBjkk ξ LApξqe´t1LApξqdt1 ` H21 pj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' kkq ` H21 pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jkq ` H21 pjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' kq ` H22 pkk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jq ` H22 pjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' kq ` H22 pk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jkq ´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ´ H3 pp1 ´ t1q p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2 p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ´ H3 pp1 ´ t1q p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t3q ´ H3 pp1 ´ t1q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2q ´ H3 pp1 ´ t1q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2q ´ H3 pp1 ´ t1q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1 p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2 p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1t2t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' where H21 pα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' βq “ ż 1 0 ż 1 0 e´p1´t1qp1´t2qLApξqBα ξ LApξqe´p1´t1qt2LApξqBβ ξ LApξqe´t1LApξqdt1dt2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' H22 pα,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' βq “ ż 1 0 ż 1 0 e´t1LApξqBα ξ LApξqe´p1´t1qt2LApξqBβ ξ LApξqe´p1´t1qp1´t2qLApξqdt1dt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' H3 ps1, α, s2, β, s3, γ, s4q “ ż 1 0 ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´t2LApξqBβ ξ LApξqe´s3LApξqBγ ξ LApξqe´s4LApξqdt1dt2dt3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since LApξq is even and homogeneous of degree one, we have that the third deriva- tives of LApξq are odd and homogeneous of degree minus two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The other terms are less singular and thus the corresponding integrals in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='51) are bounded directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' That is to say, ���Bα ξ LApξq ��� ≲ |ξ|1´|ξ| and ��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, so ∥H21∥ , ∥H22∥ ≲ |ξ|´1 e´C|ξ|,∥H3∥ ≲ e´C|ξ|, and all of them are integrable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 87 Therefore, we only have to estimate the first, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż B1p0q ż 1 0 e´p1´t1qLApξqeix¨ξ x |x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We can write e´p1´t1qLApξq B Bξj ∆ξLApξqe´t1LApξq “ 1 |ξ|2 e´p1´t1qLApξqΦp3q A,jpˆξqe´t1LApξq, where ˆξ “ ξ{|ξ| and Φp3q A,jpˆξq is even and bounded from below and above, thanks to the arc-chord condition and the C1 regularity (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2) and Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' We then have that ż B1p0q ż 1 0 e´p1´t1qLApξqeix¨ξ x |x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ “ ÿ j“1,2 ixj |x| ż S1 ż 1 0 ż 1 0 sin px ¨ ˆξ rq r e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξqdrdt1dˆξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' By lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4, we obtain for all x P R2, ˆξ P S1 and t1 P r0, 1s ����� ż 1 0 sin px ¨ ˆξ rq r e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξqdr ����� ď2 ���e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξq��� C1pr0,1s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='rq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='52) LApˆξq is positive definite and diagonalizable and Φp3q A,jpˆξq is boundned, so for all ˆξ P S1 and t1 P r0, 1s, ���e´p1´t1qrLApˆξqΦp3q A,jpˆξqe´t1rLApˆξq��� C1pr0,1s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='rq ď C ´ 1 ` ���LApˆξq ��� ¯ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='53) Therefore, since LApˆξq is bounded on S1, ż B1p0q ż 1 0 e´p1´t1qLApξqeix¨ξ x |x| ¨ ∇ξ∆ξLApξqe´t1LApξqdt1dξ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='54) is bounded, and we may conclude that ∥kpxq∥ ≲ p1 ` |x|3q´1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, since B Bxi kpxq “ ż R2 iξie´LApξqeix¨ξdξ, we have ���� B Bxi kpxq ���� “ ż R2 |ξi| ���e´LApξq��� dξ ď C, where C only depends on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Then, |x|4 B Bxi kpxq “ ż R2 p∆ξq2 ´ iξie´LApξq¯ eix¨ξdξ, 88 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN where p∆ξq2 ` ξie´LApξq˘ “ 4 B Bξi ∆ξe´LApξq ` ξi p∆ξq2 e´LApξq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' The first term is the i component in ∇ξ∆ξe´LApξq, so we may claim the term is integratable by the previous techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' For the second term,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Bjjkk ξ LApξq “ ´ ż 1 0 e´p1´t1qLApξqBjjkk ξ LApξqe´t1LApξqdt1 ` H2 pp1 ´ t1q p1 ´ t2q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jjk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ` ¨ ¨ ¨ ´ H3 pp1 ´ t1q p1 ´ t2q p1 ´ t3q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' jj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ´ ¨ ¨ ¨ ` H4 pp1 ´ t1q p1 ´ t2q p1 ´ t3q p1 ´ t4q ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q p1 ´ t3q t4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q p1 ´ t2q t3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' p1 ´ t1q t2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' t1q ` ¨ ¨ ¨ where H2 ps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s3q “ ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´s2LApξqBβ ξ LApξqe´s3LApξqdt1dt2 H3 ps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s4q “ ż 1 0 ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´s2LApξqBβ ξ LApξqe´s3LApξqBγ ξ LApξqe´s4LApξq dt1dt2dt3 H4 ps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' β,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' γ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' s5q “ ż 1 0 ż 1 0 ż 1 0 ż 1 0 e´s1LApξqBα ξ LApξqe´s2LApξqBβ ξ LApξqe´s3LApξqBγ ξ LApξqe´s4LApξq Bδ ξLApξqe´s5LApξqdt1dt2dt3dt4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' There are 14 H2-type terms, 36 H3-type terms, 24 H4-type terms in Bjjkk ξ LA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ���Bα ξ LApξq ��� ≲ |ξ|1´|ξ| and ��e´sLApξq�� ≲ e´sC|ξ| for some C ą 0, ∥ξiH2∥ ≲ |ξ|´1 e´C|ξ|,∥ξiH3∥ ≲ e´C|ξ| and ∥ξiH4∥ ≲ |ξ| e´C|ξ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hence, we only have to check ż R2 ż 1 0 e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since ş1 0 e´p1´t1qLApξqξi p∆ξq2 LApξqe´t1LApξqdt1 is odd, we may use the same tech- nique in kpxq term to obtain |x|4 ���� B Bxi kpxq ���� ď C, so ���� B Bxi kpxq ���� ≲ 1 1 ` |x|4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Finally, since B Bxi kpxq “ ż R2 ξie´LApξqeix¨ξdξ, WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 89 we have ���� B Bxi kpxq ���� “ ż R2 |ξi| ���e´LApξq��� dξ ď C, where C only depends on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given a vector function M prq in C1 pr0, 1sq, we have the following inequality: for all A ě 0, ���� ż 1 0 M prq sin pArq r dr ���� ď 2 ∥Mp0q∥ ` ���� dM prq dr ���� C0pr0,1sq ď 2 ∥M∥C1pr0,1sq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='55) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' ż 1 0 M prq sin pArq r dr “ ż 1 0 M p0q sin pArq r dr ` ż 1 0 M prq ´ M p0q r sin pArqdr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='56) For the first term, ���� ż 1 0 M p0q sin pArq r dr ���� “ ����M p0q ż 1 0 sin pArq r dr ���� “ ∥Mp0q∥ ����� ż A 0 sin prq r dr ����� ď ∥Mp0q∥ ż π 0 sin prq r dr « 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='852 ∥Mp0q∥ ď 2 ∥Mp0q∥ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='57) For the second term, since ���� M prq ´ M p0q r ���� “ ��şr 0 dM ds psq ds �� r ď ���� dM prq dr ���� C0pr0,1sq , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='58) ���� ż 1 0 M prq ´ M p0q r sin pArqdr ���� ď ż 1 0 ���� M prq ´ M p0q r ���� |sin pArq| dr ď ���� dM prq dr ���� C0pr0,1sq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='59) Therefore, we obtain the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Given fptq ě 0 is a locally integrable function on R, we have the following estimates: (i) If α ą ´1, for all m ă t, ���� ż t m pt ´ sqα fds ���� ď C pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='60) (ii) If α ă ´1, for all M ă t, ����� ż M ´8 pt ´ sqα fds ����� ď C pt ´ Mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='61) 90 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (i) By integration by part theorem, ż t m pt ´ sqα fds “ ´ ż t m pt ´ sqα ˆ B Bs ż t s f prq dr ˙ ds “ pt ´ sqα ż t s f prq dr ˇˇˇˇ m s“t ´ α ż t m pt ´ sqα ˆ 1 t ´ s ż t s f prq dr ˙ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Since lim sup sÑt´ ����pt ´ sqα ż t s f prq dr ���� “ lim sup sÑt´ ����pt ´ sqα`1 1 t ´ s ż t s f prq dr ���� ď lim sup sÑt´ pt ´ sqα`1 Mlrfsptq “ 0, �����pt ´ sqα ż t s f prq dr ˇˇˇˇ m s“t ����� ď pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Next, ���� ż t m pt ´ sqα ˆ 1 t ´ s ż t s f prq dr ˙ ds ���� ďMlrfsptq ż t m pt ´ sqα ds “ 1 α ` 1 pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ���� ż t m pt ´ sqα fds ���� ď C pt ´ mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (ii) The proof is basically the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' It will be ����� ż M ´8 pt ´ sqα fds ����� “ �����pt ´ sqα ż t s f prq dr ˇˇˇˇ M s“´8 ´ α ż M ´8 pt ´ sqα ˆ 1 t ´ s ż t s f prq dr ˙ ds ����� ď pt ´ Mqα`1 Mlrfsptq ` lim sup sÑ´8 pt ´ sqα`1 Mlrfsptq ` Mlrfsptq ż M ´8 pt ´ sqα ds “ pt ´ Mqα`1 Mlrfsptq ´ 1 α ` 1 pt ´ Mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' since α ` 1 ă 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Therefore, ����� ż M ´8 pt ´ sqα fds ����� ď C pt ´ Mqα`1 Mlrfsptq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' □ References [1] Thomas Alazard and Omar Lazar, Paralinearization of the Muskat equation and appli- cation to the Cauchy problem, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 237 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 545–583, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-020-01514-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [2] Thomas Alazard and Quoc-Hung Nguyen, Endpoint sobolev theory for the Muskat equation, Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 91 [3] , Quasilinearization of the 3D Muskat equation, and applications to the critical Cauchy problem, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 399 (2022), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 108278, 52, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='aim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='108278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [4] David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ambrose, Well-posedness of two-phase Hele-Shaw flow without surface tension, European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 15 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5, 597–607, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/S0956792504005662.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [5] , Well-posedness of two-phase Darcy flow in 3D, Quart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 65 (2007), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 189–203, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1090/S0033-569X-07-01055-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [6] David M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ambrose and Michael Siegel, Well-posedness of two-dimensional hydroe- lastic waves, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Edinburgh Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A 147 (2017), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 529–570, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/S0308210516000238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [7] Stephen Cameron, Global well-posedness for the two-dimensional Muskat problem with slope less than 1, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE 12 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 4, 997–1022, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2140/apde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [8] Stephen Cameron and Robert M Strain, Critical local well-posedness for the fully nonlinear Peskin problem, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='00692 (2021), Accepted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [9] Ke Chen and Quoc-Hung Nguyen, The Peskin problem with 9B1 8,8 initial data, arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='13854 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [10] Ke Chen, Quoc-Hung Nguyen, and Yiran Xu, The Muskat problem with C1 data, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 375 (2021), 3039–3060.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Arthur Cheng and Steve Shkoller, The interaction of the 3D Navier-Stokes equations with a moving nonlinear Koiter elastic shell, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 42 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 1094– 1155, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1137/080741628.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [12] Peter Constantin, Diego C´ordoba, Francisco Gancedo, and Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Strain, On the global existence for the Muskat problem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (JEMS) 15 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 201–227, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4171/JEMS/360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [13] Antonio C´ordoba, Diego C´ordoba, and Francisco Gancedo, Interface evolution: the Hele-Shaw and Muskat problems, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' (2) 173 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 477–542, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4007/annals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [14] Antonio C´ordoba, Diego C´ordoba, and Francisco Gancedo, Porous media: the Muskat prob- lem in three dimensions, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE 6 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 447–497, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2140/apde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='447.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Courant and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hilbert, Methods of mathematical physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' I, Interscience Publishers, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', New York, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [16] Daniel Coutand and Steve Shkoller, The interaction between quasilinear elastodynamics and the Navier-Stokes equations, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 179 (2006), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 303–352, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-005-0385-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [17] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Denisova, Problem of the motion of two viscous incompressible fluids separated by a closed free interface, Acta Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 37 (1994), 31–40, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/BF00995127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [18] Joachim Escher and Bogdan-Vasile Matioc, On the parabolicity of the Muskat problem: well- posedness, fingering, and stability results, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anwend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 30 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 193–218, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='4171/ZAA/1431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [19] Evan A Evans and Richard Skalak, Mechanics and thermodynamics of biomembranes, CRC Press, 1980, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1201/9781351074339.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [20] Thomas G Fai, Boyce E Griffith, Yoichiro Mori, and Charles S Peskin, Immersed boundary method for variable viscosity and variable density problems using fast constant-coefficient linear solvers i: Numerical method and results, SIAM Journal on Scientific Computing 35 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5, B1132–B1161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [21] Gerald B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Folland, Introduction to partial differential equations, second ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Princeton Uni- versity Press, Princeton, NJ, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Gancedo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Garc´ıa-Ju´arez, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Patel, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Strain, Global regularity for gravity unstable muskat bubbles, Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', arXiv:1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='02318 (2021), To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [23] Francisco Gancedo, Rafael Granero-Belinch´on, and Stefano Scrobogna, Global existence in the lipschitz class for the N-peskin problem, Indiana Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='01787 (2021), To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [24] Francisco Gancedo and Omar Lazar, Global well-posedness for the three dimensional Muskat problem in the critical Sobolev space, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 246 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 141–207, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-022-01808-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [25] Eduardo Garc´ıa-Ju´arez, Yoichiro Mori, and Robert M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Strain, The Peskin Problem with Viscosity Contrast, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE, arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='03360 (2020), To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 92 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' GARC´IA-JU´AREZ, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' KUO, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' MORI, AND R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' STRAIN [26] Loukas Grafakos, Classical fourier analysis, third ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Graduate Texts in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 249, Springer, 2014, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/978-1-4939-1194-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [27] , Modern fourier analysis, third ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=', Graduate Texts in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 250, Springer, 2014, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/978-0-387-09434-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [28] Boyce E Griffith and Neelesh A Patankar, Immersed methods for fluid–structure interaction, Annual review of fluid mechanics 52 (2020), 421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [29] Susanna V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Haziot and Benoˆıt Pausader, Note on the dissipation for the general muskat problem, Preprint, arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='04395 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [30] Thomas Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Hou, John S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lowengrub, and Michael J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Shelley, Removing the stiffness from interfacial flows with surface tension, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 114 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 312–338, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1006/jcph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1994.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [31] William Ko and John M Stockie, Parametric resonance in spherical immersed elastic shells, SIAM Journal on Applied Mathematics 76 (2016), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 58–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [32] Daniel Lengeler and Michael Ruˇziˇcka, Weak solutions for an incompressible Newtonian fluid interacting with a Koiter type shell, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 211 (2014), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 205–255, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-013-0686-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [33] Hui Li, Stability of the Stokes immersed boundary problem with bending and stretching energy, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 281 (2021), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 9, Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 109204, 65, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='jfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='109204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [34] Fang-Hua Lin and Jiajun Tong, Solvability of the Stokes immersed boundary problem in two dimensions, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 72 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 159–226, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1002/cpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21764.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [35] Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems, Mod- ern Birkh¨auser Classics, Birkh¨auser/Springer Basel AG, Basel, 1995, [2013 reprint of the 1995 original] [MR1329547].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [36] Bogdan-Vasile Matioc, The Muskat problem in two dimensions: equivalence of formu- lations, well-posedness, and regularity results, Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' PDE 12 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 2, 281–332, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2140/apde.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [37] Yoichiro Mori, Analise Rodenberg, and Daniel Spirn, Well-posedness and global behavior of the Peskin problem of an immersed elastic filament in Stokes flow, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 72 (2019), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 5, 887–980, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1002/cpa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='21802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [38] Boris Muha and Suncica Cani´c, Existence of a weak solution to a nonlinear fluid- structure interaction problem modeling the flow of an incompressible, viscous fluid in a cylinder with deformable walls, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 207 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 919–968, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-012-0585-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [39] Huy Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nguyen and Benoˆıt Pausader, A paradifferential approach for well-posedness of the Muskat problem, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 237 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 35–100, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-020-01494-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [40] Charles S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Peskin, Numerical analysis of blood flow in the heart, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 25 (1977), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 220–252, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1016/0021-9991(77)90100-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [41] , The immersed boundary method, Acta Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 11 (2002), 479–517, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/S0962492902000077.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [42] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Plotnikov and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Toland, Modelling nonlinear hydroelastic waves, Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Lond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 369 (2011), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1947, 2942–2956, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1098/rsta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='0104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [43] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pozrikidis, Boundary integral and singularity methods for linearized viscous flow, Cam- bridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 1992, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1017/CBO9780511624124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [44] , Modeling and simulation of capsules and biological cells, Chapman & Hall/CRC Mathematical Biology and Medicine Series, Chapman & Hall/CRC, Boca Raton, FL, 2003, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1201/9780203503959.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [45] Jan Pr¨uss and Gieri Simonett, Moving interfaces and quasilinear parabolic evolution equations, Monographs in Mathematics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 105, Birkh¨auser/Springer, [Cham], 2016, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/978-3-319-27698-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [46] Thomas Richter, Fluid-structure interactions: models, analysis and finite elements, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 118, Springer, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [47] Analise Rodenberg, 2D Peskin Problems of an Immersed Elastic Filament in Stokes Flow, ProQuest LLC, Ann Arbor, MI, 2018, Thesis (Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=')–University of Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [48] Yoshihiro Shibata, Theory of the lebesgue integral, Uchida Rokakuho, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' WELL-POSEDNESS OF THE 3D PESKIN PROBLEM 93 [49] Senjo Shimizu, Maximal regularity and its application to free boundary problems for the Navier-Stokes equations [translation of mr2656036], Sugaku Expositions 25 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 1, 105–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [50] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Solonnikov, Solvability of the problem of the motion of a viscous incompressible fluid that is bounded by a free surface, Izv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Nauk SSSR Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 41 (1977), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 6, 1388–1424, 1448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [51] Jiajun Tong, Regularized Stokes immersed boundary problems in two dimensions: Well- posedness, singular limit, and error estimates, Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Pure Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 74(2):366–449 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [52] , Global solutions to the tangential Peskin problem in 2-D, Preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='14723 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [53] Sunˇcica ˇCani´c, Marija Gali´c, and Boris Muha, Analysis of a 3D nonlinear, moving boundary problem describing fluid-mesh-shell interaction, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 373 (2020), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 9, 6621–6681, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1090/tran/8125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' [54] Wei Wang, Pingwen Zhang, and Zhifei Zhang, Well-posedness of hydrodynamics on the moving elastic surface, Arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 206 (2012), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content=' 3, 953–995, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} +page_content='1007/s00205-012-0548-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9NFLT4oBgHgl3EQfty_-/content/2301.12153v1.pdf'} diff --git a/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf b/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..c27080af14cf3d1c024d44c9ecb46dae8d8a9198 --- /dev/null +++ b/ANE4T4oBgHgl3EQf4w6U/content/2301.05316v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6ff1ff8731882df85ec979ac1a59abd977322026c8354c969295e02163c2d68f +size 742490 diff --git a/ANE4T4oBgHgl3EQf4w6U/vector_store/index.faiss b/ANE4T4oBgHgl3EQf4w6U/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b76bdac35520d9811f72ac407131bac72f702e11 --- /dev/null +++ b/ANE4T4oBgHgl3EQf4w6U/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:14ccc38c1b8d82ede940a3455e31a06c7670070a2822fa76a588397b3a38891a +size 2162733 diff --git a/ANE4T4oBgHgl3EQf4w6U/vector_store/index.pkl b/ANE4T4oBgHgl3EQf4w6U/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..758fb53a36b542eb719ec74d5bc57436e492ca84 --- /dev/null +++ b/ANE4T4oBgHgl3EQf4w6U/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c2f2dedb859c33fbab51e4a3ce8e8d55a123af2f47b6ba9518324d4742314a5e +size 80119 diff --git a/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf b/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..7c4a65b0ebf0be57d2ddf0aaeebde4f007cdf259 --- /dev/null +++ b/BtE0T4oBgHgl3EQfQAAH/content/2301.02185v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c18c696c66da1c392216386e6a4f273c9241c9333314159339a4b27fac8f0448 +size 4790017 diff --git a/BtE0T4oBgHgl3EQfQAAH/vector_store/index.faiss b/BtE0T4oBgHgl3EQfQAAH/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..3fff7c86c5b694dcc493666fc59dd1fc9bc0e131 --- /dev/null +++ b/BtE0T4oBgHgl3EQfQAAH/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c7eda24ed050c634c71b03e000df684072a84adc23d97552d53cdb2feaf7227f +size 3014701 diff --git a/BtE0T4oBgHgl3EQfQAAH/vector_store/index.pkl b/BtE0T4oBgHgl3EQfQAAH/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ceecd3ee1f303e886541b911475925f89c047017 --- /dev/null +++ b/BtE0T4oBgHgl3EQfQAAH/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5c5800104e3fc010345389af276a39a5df559cd18a71bf6f6446299001e831a1 +size 117129 diff --git a/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf b/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f52cbee575311fe8788eeac82dc93e420100ea77 --- /dev/null +++ b/ENE0T4oBgHgl3EQfQgDK/content/2301.02195v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7d64d32be44a11d5fd48ecdb8989d9ffc8b8845b6682633a45ab852546dd68a0 +size 331712 diff --git a/ENE0T4oBgHgl3EQfQgDK/vector_store/index.faiss b/ENE0T4oBgHgl3EQfQgDK/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..c2a296bb8eb4116fd83180555bb2b71f745b6d01 --- /dev/null +++ b/ENE0T4oBgHgl3EQfQgDK/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:094ab4794dbb183a14cc936f97a6625cc256ab9bd0271e4865d85feb3ac23143 +size 2162733 diff --git a/ENE0T4oBgHgl3EQfQgDK/vector_store/index.pkl b/ENE0T4oBgHgl3EQfQgDK/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3d10858a1bc22232cf97bdd90d0c496e9c72cbf4 --- /dev/null +++ b/ENE0T4oBgHgl3EQfQgDK/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:52dbfcfdc65a76e22254f0b7bc2133517ec92752d7433013b1c78ce1e80fa29e +size 84358 diff --git a/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/2301.03343v1.pdf.txt b/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/2301.03343v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..04159b336f14090dc960f2bcb0750d298753b704 --- /dev/null +++ b/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/2301.03343v1.pdf.txt @@ -0,0 +1,1037 @@ +High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock +William R. Milner,1, ∗ Lingfeng Yan,1 Ross B. Hutson,1 Christian Sanner,1 and Jun Ye1, † +1JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA +We report on the observation of a high-density, band insulating state in a three-dimensional optical lattice +clock. Filled with a nuclear-spin polarized degenerate Fermi gas of 87Sr, the 3D lattice has one atom per site in +the ground motional state, thus guarding against frequency shifts due to contact interactions. At this high density +where the average distance between atoms is comparable to the probe wavelength, standard imaging techniques +suffer from large systematic errors. To spatially probe frequency shifts in the clock and measure thermodynamic +properties of this system, accurate imaging techniques at high optical depths are required. Using a combination +of highly saturated fluorescence and absorption imaging, we confirm the density distribution in our 3D optical +lattice in agreement with a single spin band insulating state. Combining our clock platform with this high filling +fraction opens the door to studying new classes of long-lived, many-body states arising from dipolar interactions. +Optical +lattice +clocks +integrate +quantum +many-body +physics and precision metrology to achieve state-of-the-art +measurement precision [1–5]. To advance clock performance, +one wishes to probe as many atoms as feasible for the longest +possible coherence time. Improvements in both precision and +accuracy of optical lattice clocks, with increased atom num- +bers, have been enabled by the development of high-fidelity, +microscopic imaging of the atomic cloud to spatially resolve +clock shifts [6, 7]. The combination of high density and long +coherence time will allow characterization of novel systematic +effects such as that arising from dipolar interactions between +atoms on neighboring lattice sites [8–11]. Lattice thermom- +etry [12] and studies of novel physics such as SU(N) mag- +netism [13, 14] will also benefit from accurate imaging at high +density where these phenomena emerge. +To optimize atom number while minimizing interaction- +related dephasing, a clock platform based on a 3D lattice +geometry and Fermi-degenerate matter has been developed +[7, 15]. Following nuclear spin polarization [16, 17], the Pauli +exclusion principle mandates there is at most one atom per lat- +tice site in the ground motional state. To ensure this ground +state motional occupation during lattice loading we operate +with kBT < kBTF < ℏωbg, where T, TF , ℏωbg refers to the +atomic temperature, Fermi temperature and lattice bandgap +respectively [18]. At the highest density affordable with one +fermion per lattice site, this system realizes an insulating state +of matter where tunnelling is suppressed [15, 19]. Combining +this high-density system with spin-orbit coupling generated +from clock addressing will enable exploring cluster state gen- +eration and tunable spin models [20, 21]. +Differential frequency shifts across the optical lattice en- +coding potential systematic effects can be spatially resolved +by combining in situ imaging and narrow-line clock spec- +troscopy [6]. To extract these frequency shifts, two subse- +quent images of the ground and excited state density distribu- +tions are required. Thus for our clock platform, accurate in +situ imaging at high density is imperative. In our lattice where +the average distance between atoms (407 nm) is comparable +to the probe wavelength (461 nm), imaging with a weak, reso- +nant probe is strongly perturbed. Both collective effects medi- +ated by dipolar interactions [22] and systematic defects such +as lensing of the probe beam [23, 24] introduce errors to the +reconstructed density distribution at high density. +To mitigate these systematic effects, different techniques +can be used to reduce the absorption cross section and make +the cloud ”optically thin”. These techniques can be broadly +divided into two categories: dispersive imaging at large de- +tuning from resonance [25–27] and saturated imaging at high +intensity [28–31]. For dispersive imaging extracting informa- +tion about the atomic density often requires spatially filtering +the scattered and unscattered light in the Fourier plane of the +imaging system, demanding precise fabrication and alignment +of custom optics. Additionally, careful studies of dispersive +imaging show that residual systematic effects at finite detun- +ing are non-negligible and can be addressed using differential +measurement schemes at opposite detuning [32]. To address +these imaging errors in this work, we use both highly saturated +fluorescence and absorption imaging. +In this Letter, we report on the observation of a band in- +sulating state in our 3D optical lattice clock. Using highly +saturated imaging to mitigate imaging errors, with a satura- +tion parameter far greater than the optical depth, we accu- +rately confirm the density distribution in our 3D optical lattice +in good agreement with thermodynamic calculation. We ex- +tend previous work using high intensity fluorescence imaging +[28], confirming the accuracy of this imaging technique in a +new high density regime with a degenerate Fermi gas of 87Sr +[16, 33]. With atomic densities as high as 6×1014 atoms/cm3, +we observe systematic agreement with atom counts obtained +via time-of-flight absorption imaging and identify the range +where the extracted atomic density distribution is not blurred +by our imaging pulse. +Our high intensity imaging scheme is outlined in Fig. 1. +The combination of atomic level structure and relatively large +mass of 87Sr is particularly well suited for our imaging tech- +nique, providing a cycling transition with a large scattering +rate while avoiding significant motional effects from the imag- +ing pulse. +The transition from 1S0 to 1P1 with linewidth +Γ = 2π × 30.5 MHz provides a large photon scattering rate +with minimal depumping to dark states during the imaging +time [34]. During a 1 µs pulse at full saturation about 100 +photons per atom are scattered and the atoms accelerate at +a = +ℏkΓ +2m where k is the imaging light wavenumber and m +is the atomic mass. The net momentum transfer amounts to +arXiv:2301.03343v1 [physics.atom-ph] 9 Jan 2023 + +2 +FIG. 1. Schematic of our clock platform. Vertical and horizontal +imaging systems with numerical apertures of 0.2 and 0.1 respectively +provide measurements of the 2D density distribution ˜n. Accounting +for the lattice spacing a = 407 nm, ˜na2 is determined from highly +saturated absorption imaging. To mitigate imaging errors, the atoms +are highly saturated and each scatters photons with a maximum rate +of Γ/2. Measurements from our high resolution imaging system in- +tegrated along gravity are presented in panel (a), where the density +distribution is extracted for thermodynamic modeling. Images from +the horizontal imaging system in panel (b) are just used to determine +our atom cloud aspect ratio for our inverse Abel transform. +a Doppler shift of kaτ = 2.8 MHz which is much less than +the transition linewidth Γ/2π. +Finally, the linear displace- +ment for a 1 µs pulse at full saturation is just aτ 2 +2 += 0.6 µm. +This linear displacement and corresponding Doppler shift can +be largely cancelled in fluorescence imaging by retroreflect- +ing the incident beam. The spread in transverse position due +to random momentum transfer from spontaneous emission +is +ℏk +6mt3/2� +Γ/2 < 0.1 µm over a 1 µs pulse duration and +small compared to our 1.3 µm imaging resolution [35]. Using +highly saturated absorption imaging, we measure the column +density distribution ˜n in our optical lattice in Fig. 1(a). Ac- +counting for the lattice spacing a = 407 nm corresponding +to the 87Sr magic wavelength at 813 nm, the scaled column +density ˜na2 is plotted. +Saturated absorption and fluorescence imaging are benefi- +cial in comparison to standard imaging techniques in a num- +ber of ways. In this highly saturated regime the scattering rate +is largely immune to beam intensity, frequency, and pointing +fluctuations. Given the saturation intensity Isat = 40 mW/cm2 +for the imaging transition, a Gaussian probe beam with 20 +mW of optical power and a 100 µm waist corresponds to a +peak intensity of I ∼ 3000 Isat, within the typical constraints +of a standard imaging laser system. Given that the probe beam +is attenuated through the atom cloud, a saturation parameter +I/Isat much greater than the optical depth is required to fully +saturate the imaging transition. We note parallels between flu- +orescence and absorption imaging at high saturation. In both +cases, the extracted atom number is determined by a single +variable. For fluorescence imaging, this corresponds to the +number of collected photons per atom and for saturated ab- +sorption imaging the number of missing photons per atom in +the probe beam. Thus, both fluorescence and saturated ab- +sorption imaging can be calibrated via a single absolute atom +number measurement. For images in our 3D lattice, we de- +termine our atom number via clock excitation fraction fluctu- +ations arising from quantum projection noise (QPN) [36, 37]. +For fluorescence imaging, only a single image of collected +fluorescence in an arbitrary direction is required, minimizing +fringing and simplifying image processing substantially. Flu- +orescence imaging also avoids limited dynamic range issues +suffered from high intensity absorption imaging. Strategies +such as multiple measurements at varying intensity to deter- +mine the atomic density in different regions of the cloud may +be taken to confront this issue [30, 31]. The primary disad- +vantage of fluorescence imaging in comparison to absorption +imaging is that the signal-to-noise is generally worse [37]. To +optimize signal-to-noise ratio (SNR) in fluorescence imaging, +the photon collection efficiency and therefore the numerical +aperture (NA) of the imaging system, must be maximized. In +our experiment, the vertical and horizontal imaging systems +have numerical apertures of 0.2 and 0.1, corresponding to col- +lection efficiencies of approximately 1 and 0.2 percent. Al- +ternatively, if spatial resolution is not required then the pulse +duration can be extended enhancing the number of detected +photons. +To motivate the development of our high intensity imaging +technique, systematic errors associated with standard in situ +imaging techniques at high density are presented in Fig. 2. +Absorption imaging at I ∼ Isat and high intensity fluorescence +imaging are presented side-by-side for comparison. To study +these systematic errors at high density, we prepare a sample +with optical depth > 200 by producing a degenerate Fermi +gas with 10 nuclear spin components, ≈ 2 × 105 atoms and a +T/TF of approximately 0.1 in a crossed dipole trap. The errors +associated with low intensity absorption imaging can be seen +twofold. First, the reconstructed optical depth from absorp- +tion detection in the upper left panel is far too low, two orders +of magnitude less than the expected value of ∼ 200. This +erroneously low optical depth is attributed to effects such as +enhanced forward emission and lensing of probe light [23]. +Secondly, the reconstructed optical depth in the upper right +panel increases after a 500 µs time-of-flight expansion con- +clusively demonstrating the density dependence of these ob- +served systematic errors. +In comparison, saturated fluorescence imaging yields a far +larger reconstructed optical depth and diffuses following ex- +pansion as expected. We compare this reconstructed 2D den- +sity distribution with the expected distribution corresponding +to a Fermi gas. Using independently measured experimental +values, we calculate this distribution with no free parameters +[38]. The total atom number and reduced temperature T/TF +are determined from time-of-flight absorption imaging at low +density with an optical density ∼ 1. The trapping frequencies +are extracted from parametric confinement modulation. Using + +皖A·23 +FIG. 2. A comparison of high intensity fluorescence and standard ab- +sorption imaging (I ∼ Isat) at optical depths exceeding 200 in our +highly degenerate Fermi gas is shown. In situ absorption imaging at +low intensity yields strikingly erroneous measurements at high den- +sity. The calculated 2D Fermi gas distribution according to our ex- +perimental parameters is shared for comparison in qualitative agree- +ment. +these parameters, we calculate both an in situ and 500 µs time- +of-flight Fermi gas profile for comparison with our measure- +ments. We observe qualitative agreement between measure- +ment and calculation at these extremely high optical depths. +Intrigued by the measurements presented in Fig. 2, we un- +dertake a quantitative study on the fidelity of our saturated +imaging technique. We present a calibration method for flu- +orescence detection, using the total number of collected fluo- +rescence photons for comparison with an accurate atom num- +ber reference. Absorption imaging at low density following +time-of-flight expansion serves as an appropriate calibration. +Following expansion for 7 ms, the optical depth is ∼ 1 and +systematic imaging errors can be safely ignored. To inde- +pendently calibrate the atom number in our 10 spin Fermi +gas, we prepare a thermal sample and use measured density +fluctuations to determine the effective absorption cross sec- +tion [39–41]. In Fig. 3(a) we ensure this calibration shows +systematic agreement with atom numbers between approxi- +mately 1 × 105 and 4 × 105, varied by increasing our final +evaporation trap depth. For the 3 µs pulse duration used, the +fitted calibration is in reasonable agreement with calculation +using the measured quantum efficiency and imaging system +numerical aperture [37]. To ensure that the imaging transition +is fully saturated, the laser intensity at 1 µs pulse duration is +increased until the collected photon number plateaus, as seen +in the figure inset. +To perform accurate spatially resolved measurements, we +must also determine the blurring induced by our imaging +pulse. +Just as collective effects introduce errors to the re- +FIG. 3. (a) Calibration method for in situ fluorescence detection us- +ing atom counts from time-of-flight absorption imaging. Collected +photon counts from both the vertical and horizontal imaging systems +are plotted, with solid and dashed lines representing fits to the hori- +zontal and vertical measurements respectively. Inset: Collected pho- +ton count with vertical imaging system as a function of I/Isat at 1 +µs pulse duration. (b) Peak column density as a function of fluo- +rescence pulse duration. Measurements are normalized by 1.9×1011 +atoms/cm2, the column density at the shortest pulse duration of 500 +ns. Images at 500 ns and 2 µs in inset are plotted for comparison. +The error bars denote the standard error of the mean. +constructed density distribution, any systematic changes to ˜n +introduced by our imaging pulse must be determined. To cal- +ibrate this blurring in Fig. 3(b), we extend the fluorescence +pulse duration and examine the peak column density as atoms +diffuse. The inset shows averaged images from 500 ns and +2 µs pulse durations. We note that we observe no atom loss +or molecular formation over the full 2 µs range, confirmed by +the detected photon count increasing linearly with pulse dura- +tion. To minimize blurring, we carefully retroreflect our probe +beam by optimizing the backcoupled light through the probe +optical fiber. At pulse durations up to 1 µs, we confirm that +the peak column density decreases by < 5%. [37]. +Motivated by the calibration reported in Fig. 3, we directly +determine the 3D density distribution in a deep optical lattice +via saturated in situ absorption imaging. We form a cubic lat- +tice with trap depths of approximately 60, 70, and 50 Er in +three orthogonal directions, where Er is the lattice photon re- +coil energy ≈ h × 3.5 kHz. Following forced evaporation +with 10 nuclear spin states we spin polarize using a focused +beam detuned from the 3P1 intercombination line to form a +state-dependent potential, removing nearly all but the mF = - +9/2 atoms [16, 17]. Clock spectroscopy confirms ≈ 90% spin +purity. An additional step of spin purification is applied by + +4 +FIG. 4. (a) The three-dimensional density distribution and the corre- +sponding lattice filling fraction are determined from in situ absorp- +tion image in Fig. 1(a) and the use of an inverse Abel transformation. +(b) A linecut along z = 0 provides the data points in circle. Errorbars +are both the statistical uncertainty of the Abel transformation and +atom number uncertainty added in quadrature. We start with a pre- +diction based on HTSE calculation, using independently measured +values for the temperature, atom number, and harmonic confinement. +The best fit to the data results in a 10% reduction of the measured as- +pect ratio ωy/ωx and 5% reduction of the measured T/TF . The red +line captures this fit, with temperature uncertainty in the shaded band. +The blue dashed line is a fit to Gaussian in qualitative disagreement +with na3. +coherently driving the mF = -9/2 atoms into the excited clock +state and removing any residual spins with a resonant imaging +pulse. Absorption imaging directly provides us with the col- +umn density distribution ˜n, integrated through the vertical axis +along gravity as depicted in Fig. 1(a). Based on our Fig. 3(b) +analysis, we choose a pulse duration of 1 µs to minimize blur- +ring and a saturation intensity of 54(4), substantially larger +than peak optical density of ∼ 15. To spatially probe the band +insulator plateau we use an imaging magnification of 38.8 to +achieve an effective pixel size of 412 nm, roughly equal to the +lattice constant a = 407 nm. We note that our effective pixel +size is smaller than our optical resolution of 1.3 µm, thus our +imaging system is optically oversampled. To extract the 3D +density distribution, we use an inverse Abel transform [42]. +Given our vertical imaging is not along an axis of cylindri- +cal symmetry, n must be appropriately scaled by the aspect +ratio of the spatial density distribution [37]. The aspect ra- +tio is independently calibrated using the absorption imaging +measurement in Fig. 1(b). +At this high magnification, the SNR in fluorescence imag- +ing for a 1 µs pulse duration is limited by a combination of +read noise and photon shot noise. We found that even after +extensive averaging the extracted 3D density distribution us- +ing an inverse Abel transform was sensitive to small fluctua- +tions in ˜n. Thus, saturated absorption imaging with a superior +SNR provides a more robust technique to characterize the 3D +density distribution. This extracted 3D density distribution is +plotted in Fig. 4(a). +To judge the fidelity of our measured 3D density distri- +bution, we compare the line cut at z = 0 with calculation +in Fig. 4(b). To estimate the density distribution, we use a +High Temperature Series Expansion (HTSE) calculation in +the atomic limit [12, 14, 37, 43]. The ingredients of this cal- +culation include values for the atomic temperature, harmonic +confinement, and total atom number. Given the density dis- +tribution only depends on the ratio of the respective harmonic +confinements, the measured aspect ratios from Fig. 1 are used +for our HTSE calculation. The total atom number N is de- +termined from quantum projection noise measurements [37]. +To estimate the temperature including heating during lattice +loading, we measure the reduced temperature T/TF in time- +of-flight after a round-trip from the lattice back to the dipole +trap and determine an entropy-per-particle increase of 0.25(6) +kB. Inferring an entropy increase of 0.13(3) kB in a single +lattice loading sequence, we estimate a T/TF of 0.165(7). +Although we did not perform a cross-dimensional thermaliza- +tion measurement to directly verify thermal equilibrium, the +uncertainty in our temperature is included in the shaded band +of the HTSE calculation in Fig. 4(b) [44, 45]. We note that the +extended plateau region is larger than our 1.3 µm imaging res- +olution. To further quantify the imaging fidelity, we compare +na3 to a Gaussian fit in clear disagreement with data. +In conclusion, we report on the observation of a spin- +polarized, band insulating state in our 3D optical lattice clock. +This has been enabled by characterizing saturated in situ +imaging techniques to accurately determine our density dis- +tribution. Broadly, the saturated imaging techniques in this +work will be applicable for studies of SU(N) magnetism and +thermodynamics in the Mott-insulating regime [46, 47]. With +the high filling fraction demonstrated in this work, many-body +states arising from dipolar interactions can be generated be- +tween atoms on neighboring lattice sites [8, 9]. +Acknowledgement. We thank D. Kedar for maintain- +ing the ultrastable clock laser used in this work and A. Aep- +pli, K. Kim, J. M. Robinson, M. Miklos, and Y. M. Tso for +useful discussions. We thank K. Kim, N. D. Oppong, and +L. Sonderhouse for careful reading of the manuscript and for +providing insightful comments. Funding for this work is pro- +vided by NSF QLCI OMA-2016244, DOE Center of Quan- +tum System Accelerator, DARPA, AFOSR, V. Bush Fellow- +ship, NIST, and NSF Phys-1734006. + +5 +∗ william.milner@colorado.edu +† ye@jila.colorado.edu +[1] T. Bothwell, C. J. Kennedy, A. Aeppli, D. Kedar, J. M. Robin- +son, E. Oelker, A. Staron, and J. Ye, Nature 602, 420 (2022). +[2] W. F. McGrew, X. Zhang, R. J. Fasano, S. A. Sch¨affer, K. Beloy, +D. Nicolodi, R. C. Brown, N. Hinkley, G. Milani, M. Schioppo, +et al., Nature 564, 87 (2018). +[3] A. Aeppli, A. Chu, T. Bothwell, C. J. Kennedy, D. Kedar, P. He, +A. M. Rey, and J. Ye, Science Advances 8, 42 (2022). +[4] B. J. Bloom, T. L. Nicholson, J. R. Williams, S. L. Campbell, +M. Bishof, X. Zhang, W. Zhang, S. L. Bromley, +and J. Ye, +Nature 506, 71 (2014). +[5] T. L. Nicholson, S. L. Campbell, R. B. Hutson, G. E. Marti, +B. J. Bloom, R. L. McNally, W. Zhang, M. D. Barrett, M. S. +Safronova, G. F. Strouse, et al., Nat. Comm. 6, 1 (2015). +[6] G. E. Marti, R. B. Hutson, A. Goban, S. L. Campbell, N. Poli, +and J. Ye, Phys. Rev. Lett. 120, 103201 (2018). +[7] S. L. Campbell, R. B. Hutson, G. E. Marti, A. Goban, N. D. +Oppong, R. L. McNally, L. Sonderhouse, J. M. Robinson, +W. Zhang, B. J. Bloom, et al., Science 358, 90 (2017). +[8] D. E. Chang, J. Ye, and M. D. Lukin, Phys. Rev. A 69, 023810 +(2004). +[9] A. Cidrim, A. PineiroOrioli, C. Sanner, R. B. Hutson, J. Ye, +R. Bachelard, and A. M. Rey, Physical Review Letters 127, +013401 (2021). +[10] S. Kr¨amer, L. Ostermann, and H. Ritsch, EPL (Europhysics +Letters) 114, 14003 (2016). +[11] R. B. Hutson, et al. Collective dipolar interactions in a near +unity filled three-dimensional optical lattice clock, in prepara- +tion (2023). +[12] C. Hofrichter, L. Riegger, F. Scazza, M. H¨ofer, D. R. Fernandes, +I. Bloch, and S. F¨olling, Phys. Rev. X 6, 021030 (2016). +[13] A. V. Gorshkov, M. Hermele, V. Gurarie, C. Xu, P. S. Julienne, +J. Ye, P. Zoller, E. Demler, M. D. Lukin, and A. Rey, Nat. Phys. +6, 289 (2010). +[14] S. Taie, R. Yamazaki, S. Sugawa, and Y. Takahashi, Nature +Physics 8, 825 (2012). +[15] R. B. Hutson, A. Goban, G. E. Marti, L. Sonderhouse, C. San- +ner, and J. Ye, Phys. Rev. Lett. 123, 123401 (2019). +[16] L. Sonderhouse, +C. Sanner, +R. B. Hutson, +A. Goban, +T. Bilitewski, L. Yan, W. R. Milner, A. M. Rey, +and J. Ye, +Nat. Phys. 16, 1216 (2020). +[17] S. Stellmer, R. Grimm, +and F. Schreck, Phys. Rev. A 84, +043611 (2011). +[18] S. Will, PhD Thesis (Springer Science & Business Media, +2012). +[19] U. Schneider, L. Hackermuller, S. Will, T. Best, I. Bloch, T. A. +Costi, R. Helmes, D. Rasch, and A. Rosch, Science 322, 1520 +(2008). +[20] M. Mamaev, R. Blatt, J. Ye, and A. M. Rey, Phys. Rev. Lett. +122, 160402 (2019). +[21] M. Mamaev, I. Kimchi, R. M. Nandkishore, and A. M. Rey, +Phys. Rev. Research 3, 013178 (2021). +[22] F. Andreoli, M. J. Gullans, A. A. High, A. Browaeys, and D. E. +Chang, Phys. Rev. X 11, 011026 (2021). +[23] W. Ketterle, D. S. Durfee, and D. Stamper-Kurn, arXiv (1999). +[24] S. P. Rath, T. Yefsah, K. J. G¨unter, M. Cheneau, R. Desbuquois, +M. Holzmann, W. Krauth, and J. Dalibard, Phys. Rev. A 82, +013609 (2010). +[25] S. Kadlecek, J. Sebby, R. Newell, and T. Walker, Opt. Lett. 26, +137 (2001). +[26] C. C. Bradley, C. A. Sackett, and R. G. Hulet, Phys. Rev. Lett. +78, 985 (1997). +[27] M. R. Andrews, M. O. Mewes, N. J. Van Druten, D. S. Durfee, +D. M. Kurn, and W. Ketterle, Science 273, 84 (1996). +[28] M. T. DePue, S. L. Winoto, D. Han, and D. S. Weiss, Opt. +Comm. 180, 73 (2000). +[29] G. Reinaudi, T. Lahaye, Z. Wang, and D. Gu´ery-Odelin, Opt. +Lett. 32, 3143 (2007). +[30] T. Yefsah, R. Desbuquois, L. Chomaz, K. J. G¨unter, and J. Dal- +ibard, Phys. Rev. Lett. 107, 130401 (2011). +[31] L. Chomaz, L. Corman, T. Yefsah, R. Desbuquois, and J. Dal- +ibard, New Journal of Physics 14, 055001 (2012). +[32] Y.-R. Lee, M.-S. Heo, J.-H. Choi, T. T. Wang, C. A. Chris- +tensen, T. M. Rvachov, +and W. Ketterle, Phys. Rev. A 85, +063615 (2012). +[33] C. Sanner, L. Sonderhouse, R. B. Hutson, L. Yan, W. R. Milner, +and J. Ye, Science 374, 979 (2021). +[34] D. S. Barker, B. J. Reschovsky, N. C. Pisenti, and G. K. Camp- +bell, Phys. Rev. A 92, 043418 (2015). +[35] M. A. Joffe, W. Ketterle, A. Martin, and D. E. Pritchard, JOSA +B 10, 2257 (1993). +[36] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. +Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. +Rev. A 47, 3554 (1993). +[37] Supplemental materials with description of imaging signal-to- +noise comparison, QPN atom number calibration, density dif- +fusion, and HTSE calculations. +[38] W. Ketterle and M. Zwierlein, arXiv (1999). +[39] C. Sanner, E. J. Su, A. Keshet, R. Gommers, Y.-i. Shin, +W. Huang, +and W. Ketterle, Phys. Rev. Lett. 105, 040402 +(2010). +[40] W. G. Tobias, K. Matsuda, G. Valtolina, L. De Marco, J.-R. Li, +and J. Ye, Phys. Rev. Lett. 124, 033401 (2020). +[41] T. M¨uller, +B. Zimmermann, +J. Meineke, +J.-P. Brantut, +T. Esslinger, +and H. Moritz, Phys. Rev. Lett. 105, 040401 +(2010). +[42] V. Dribinski, A. Ossadtchi, V. A. Mandelshtam, and H. Reisler, +Rev. Sci. Inst. 73, 2634 (2002). +[43] K. R. A. Hazzard, V. Gurarie, M. Hermele, and A. M. Rey, +Phys. Rev. A 85, 041604(R) (2012). +[44] C. R. Monroe, E. A. Cornell, C. A. Sackett, C. J. Myatt, and +C. E. Wieman, Phys. Rev. Lett. 70, 414 (1993). +[45] G. Valtolina, K. Matsuda, W. G. Tobias, J.-R. Li, L. De Marco, +and J. Ye, Nature 588, 239 (2020). +[46] A. Goban, R. B. Hutson, G. Marti, S. Campbell, M. Perlin, +P. Julienne, J. D’incao, A. Rey, +and J. Ye, Nature 563, 369 +(2018). +[47] X. Zhang, M. Bishof, S. L. Bromley, C. V. Kraus, M. S. +Safronova, P. Zoller, A. M. Rey, and J. Ye, Science 345, 1467 +(2014). + +Supplemental material to +High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock +Density diffusion +Here we provide supplemental analysis to the data pre- +sented in Fig. 3(b). In panel A of Fig. S1, we plot the in- +tegrated counts along the x axis of each image. We see an +asymmetry emerge along the direction of the probe beam as +the pulse duration is extended. This asymmetry suggests that +the observed density diffusion may arise from inhomogeni- +ety between the incident and retroreflected beams. While the +power is certainly mismatched, this could also be due to ei- +ther imperfect spatial alignment or mode mismatch given the +divergence of the probe beam. +We also plot the total counts in each image as a function of +pulse duration in panel B. The linear character of the counts +over the full pulse duration range suggests that we do not ob- +serve appreciable atom loss or pumping to dark states. The +counts at each pulse duration are normalized to the counts at +500 ns. The inset shows the Gaussian RMS width of the cloud +as a function of pulse duration. +Signal-to-noise comparison +In the main text of the paper we refer to both saturated ab- +sorption and fluorescence imaging. We provide a quantita- +tive comparison of the signal-to-noise ratio (SNR) between +the two techniques here. We express our signal-to-noise for a +detection pixel in terms of the normalized variance V(N)/N, +where N denotes the number of atoms within the respective +detection region. For fluorescence imaging the SNR is simply +determined by the shot noise associated with the number of +detected photons. To calculate the total atom number, we first +convert the fluorescence counts detected on our camera to the +number of collected photons. Then, using the collection effi- +ciency of our imaging system and scattering rate of our atomic +transition we determine the conversion of detected photons +per atom. On our CCD camera, we measure na counts in a +given pixel. Using the quantum efficiency q of the imaging +system, and the camera conversion gain g in units of counts +per photo electron, we infer na +qg photons. At full saturation, +the atomic scattering rate is Γ +2 and the number of photons scat- +tered per atom is Psc = Γ +2 × τ, where τ is the pulse duration. +Finally, we denote the collection efficiency as Y , determined +by the numerical aperture of our imaging system and by ra- +diation pattern anisotropies. Combining terms, the total atom +number is N = +na +gqY Psc . Using error propagation, we deter- +mine the variance VF l(N). +VF l(N) = +� ∂N +∂na +�2 +V(na) = +� +1 +gqY Psc +�2 +gna +(1) +FIG. S1. Panel (a) shows the integrated counts from the images in +Fig. 3(b) of the main text along the x axis as a function of pulse du- +ration. The total counts at each pulse duration is plotted in panel (b), +normalized by the counts at 500 ns. Given the detected photon count +increases linearly with pulse duration, we observe minimal atom loss +or molecular formation over the full 2 µs range. The inset shows the +Gaussian RMS width of the cloud as a function of pulse duration. +Here, we have used the fact that the distribution of gener- +ated photo electrons ne is binomial. Thus, V(na) = V(g × +ne) = g2V(ne) = g2ne = gna. Combining terms: +VF l(N)/N = +1 +qY Psc +(2) +The SNR associated with absorption imaging is more com- +plicated given the formula for the atom number in Eq. 3 has +both logarithmic and linear terms and involves two images na +and nb with and without atoms present. Here, A and σ0 refer +to the effective pixel size accounting for the imaging system +magnification and effective atomic absorption cross section, +respectively. Similar to fluorescence imaging, an appropriate +error propagation of the na and nb terms determines Eq. 4 and +Eq. 5. We summarize the formulas here and point a reader to +reference [1] for a full derivation. +arXiv:2301.03343v1 [physics.atom-ph] 9 Jan 2023 + +2 +N = A +σ0 +log( nb +na +) + +2 +Γτgq (nb − na) +(3) +VAbs(N) = g ˜A2( 1 +na ++ 1 +nb +) + g ˜B2(na + nb) + 4g ˜A ˜B (4) +˜A = A +σ0 +, ˜B = +2 +qgτΓ +(5) +We compare the different techniques in Fig. S2 using the +experimentally relevant parameters for our imaging system. +In both cases, a 1 µs resonant pulse is used with a numerical +aperture of 0.2 and a quantum efficiency of 85%. For the fluo- +rescence SNR in blue, the transition is assumed to be fully sat- +urated and scatters photons with a rate of Γ/2. For the I/Isat += ∼ 55 we use for our inverse Abel measurements, the SNR +in absorption imaging is superior to fluorescence imaging in +regions where the column density is higher than 2 atoms/a2. +Particularly given our peak density of ˜na2 = ∼ 20 in Fig. 1(a), +absorption imaging provides a better SNR in the regions of +high density where we extract our peak filling fraction. At a +critical OD of 0.17, fluorescence detection under our experi- +mental parameters provides a superior SNR at all imaging in- +tensities. We note these calculations neglect technical noise, +in particular camera readout noise, which can be accounted +for by offsetting V(na) accordingly. This contribution will +disproportionately reduce the SNR of fluorescence imaging, +as the fluorescence counts are substantially lower than the ab- +sorption counts. +To probe fine spatial details in our atomic cloud, an imag- +ing resolution smaller than the length scale of these spatial +features is required. To achieve this condition, a sufficiently +large numerical aperture imaging system must be utilized and +aberrations must be minimized. In this case, the imaging res- +olution is fundamentally limited by diffraction. We verified +the diffraction-limited performance of our NA = 0.2 objec- +tive lens by propagating a point source at 461 nm through +a test setup (including all imaging path optics and vacuum +viewports) and measuring the point-spread function. +While absorption and fluorescence imaging rely on the +same light scattering process (they only collect different parts +of the scattered EM field [2]), the signal amplitudes for these +two methods scale differently with the NA. When collect- +ing fluorescence, the solid angle coverage of the imaging sys- +tem proportionally affects the signal down to the lowest spa- +tial frequencies. This is not the case for absorption imaging, +where the amplitude of spatial frequency components below +the NA-dependent bandwidth is constant as the NA is further +increased (assuming the lens fully covers the probe beam). In +other words, for fluorescence imaging, most of the signal light +gets collected in the outer ring fraction of the lens aperture, +which renders it particularly susceptible to lens imperfections. +0 +20 +40 +60 +80 +100 +I/Isat +0.0 +0.5 +1.0 +1.5 +2.0 +Var(N)/N +Absorption beam intensity +1 s pulse duration, NA = 0.2 +Saturated fluorescence +OD = 0.17, 0.28 atoms/a2 +OD = 1.2, 2 atoms/a2 +OD = 3.1, 5 atoms/a2 +OD = 6.1, 10 atoms/a2 +OD = 12.2, 20 atoms/a2 +FIG. S2. +SNR comparison between absorption and fluorescence +imaging. The relevant imaging parameters from the main figures of +the paper are used for this calculation. For absorption imaging the +atom count variance scales inversely proportional with intensity in +the non-saturated limit I ≪ Isat, and proportional with intensity in +the high saturation limit. The variance is for both imaging methods +proportional to 1/τ. In the fully saturated regime (and assuming no +technical noise) the normalized variance for fluorescence imaging is +independent of atomic column density. To avoid imaging defects at +the high densities used in clock operation, an I/Isat > 50 was used +in all imaging measurements. The black dashed line indicates the +intensity used for our inverse Abel measurements. +HTSE calculation +To accurately model the density distribution in our 3D lat- +tice, we use a High Temperature Series Expansion (HTSE) +calculation in the atomic limit. The general Hamiltonian for +SU(N) symmetric fermions in a 3D lattice in the atomic limit +takes the following form: +HAL = U +2 +� +i,σ̸=σ′ +ˆni,σˆni,σ′ + +� +i,σ +Viˆni,σ +(6) +On a lattice site i, there are just two competing energy +scales: an interaction energy U between particles and a po- +sition dependent energy offset Vi according to the harmonic +confinement. By using the local density approximation µ = +V (x, y, z)−µ0, where V (x, y, z) = 1 +2m(ω2 +xx2+ω2 +yy2+ω2 +zz2) +and µ0 corresponds to the peak chemical potential in the lat- +tice. For the spin-polarized system in this work, U = 0 and the +calculations are substantially simplified. +Ultimately, we want to express the density distribution +n(µ, T, r) in terms of the chemical potential, atomic tempera- +ture, and position in the lattice. On a lattice site i, we express +the Grand partition function Z and Grand potential Ω : +Z(µ, T, r) = +N=1 +� +σ=0 +�N +σ +� +e−βµσ +(7) + +3 +Ω = −kBTln(Z) +From here, we determine the entropy and occupancy per +lattice site i: +s(µ, T, r) = −∂Ω +∂T = kB ln(Z) + ∆s +∆s = kB +Z βµe−βµ +(8) +n(µ, T, r) = −∂Ω +∂µ = +1 +Z(µ, T)e−βµ +(9) +We accurately determine the total atom number Nlat from +in situ absorption imaging and total entropy Slat via time-of- +flight fitting to a non-interacting Fermi-Dirac profile. Simi- +larly, we express the entropy s and occupation n on a given +lattice site using Eq. 8 and Eq. 9 expressed in terms of T +and µ. We then determine global fitting parameters T and µ0 +to ensure the integrated entropy and occupancy over all lat- +tice sites equals our experimentally measured values. After +determining µ0 and T to realize the equality in Eq. 9, we cal- +culate n(µ, T, r). A linecut of n(µ, T, r) at z = 0 is plotted in +Fig. 4(b). +Inverse Abel transform +We outline our reconstruction procedure here using mea- +surements of the atomic cloud aspect ratios and an inverse +Abel transform: First, we use saturated absorption images +along a vertical axis aligned with z and a horizontal axis +aligned with x corresponding to Fig. 1(a) and Fig. 1(b) to +determine the aspect ratios ωx/ωy and ωx/ωz respectively. +Next, we perform an inverse Abel transform on the Fig. 1(a) +image to reconstruct an initial three-dimensional density dis- +tribution. Given there is no axis of cylindrical symmetry in +our system geometry, the reconstructed density from the in- +verse Abel transform must be appropriately re-scaled. +Treating our system as an ellipsoid with radii rx, ry, rz +and N atoms the density is nlat = N/Vlat where Vlat = +4 +3πrxryrz. +We extract the inverse Abel transform for the +Fig. 1(a) image along the x axis, given the largest Band in- +sulator plateau will occur along the axis with the weakest har- +monic confinement. The density distribution from this pro- +cedure assumes a volume of VAbel = +4 +3πrxrxry. Thus we +scale the extracted density by nAbel/nlat = rz +rx = ωz/ωx us- +ing the measured aspect ratio from Fig. 1(b). Given excess +noise around the origin, the x = 0 point is interpolated with +the neighboring point in Fig. 4(a). This reconstruction proce- +dure was cross-checked with simulated density distributions +to ensure its fidelity. The three-point Abel transform method +was used for this work, which has been independently studied +to verify its fidelity [3]. +QPN calculation +To calibrate our atom number, we analyze quantum projec- +tion fluctuations using the narrow-linewidth clock transition +between the 1S0 and 3P0 states in 87Sr. Using a clock laser +stabilized to our 8 mHz linewidth silicon reference cavity, ro- +tation noise due to laser instability can be neglected in these +measurements [4]. Additionally, fluctuations in total counts +are < 2% and not a limiting systematic for determining the +atom number calibration. Referenced in many texts [5], by +preparing atoms in a superposition of 1S0 to 3P0 the variance +V of the measured excitation fraction is related to the mean +atom number ¯N and mean excitation ¯pe by: +VQP N = ¯pe(1 − ¯pe) +¯N +(10) +To determine this variance, we do many subsequent mea- +surements of pe under identical operating conditions. For a +measurement i to determine pi +e, two fluorescence counts ˜Ci +g +and ˜Ci +e are read off a region of interest of our camera includ- +ing our atoms. These counts are subtracted by two averaged +dark frames ¯Bg and ¯Be to yield Ci +g = ˜Ci +g− ¯Bg, Ci +e = ˜Ci +e− ¯Be. +We would like to determine the coefficient a that satisfies +N i +e = aCi +e/τ, N i +g = aCi +g/τ. We can immediately see that +the excitation fraction has no dependence on this coefficient: +pi +e = +�aCi +e +�aCie + �aCig +(11) +However, the total atom number N i = a(Ci +e + Ci +g)/τ = +aCi +t/τ does. Rewriting Eq. 10, we see a measurement of the +variance VQP N, the mean excitation ¯pe, and the mean total +counts ¯Ct can determine a. +VQP N = ¯pe(1 − ¯pe) +a ¯Ct/τ +(12) +The coefficient a can be interpreted as the ”atoms per count +per pulse duration”. In principle, with knowledge of the quan- +tum efficiency, gain, scattering rate, numerical aperture, and +radiation pattern one could calculate this value. Practically, +assumptions about the radiation pattern based on the quanti- +zation axis and probe light polarization make this calculation +more difficult. In practice, it is much more straightforward to +directly measure a than to individually measure each of these +values with high accuracy. +The observed variance of the excitation fraction Vpe has +contributions from quantum projection noise (QPN), photon +shot noise (PSN), and camera readout noise (RN): +Vpe = VQP N + VP SN + VRN +(13) +Here g is the detector gain in units of counts per electron. + +4 +VP SN = ¯pe(1 − ¯pe) +¯Ct +× g +(14) +VRN = R2 +¯Ct +2 (2¯p2 +e − 2¯pe + 1) +(15) +VP SN can be understood intuitively considering the ratio +VQP N/VP SN. The number of signal electrons (equivalently +the number of collected photons multiplied by the camera +quantum efficiency) per atom determines the relative scaling +of VQP N and VP SN. +VQP N +VP SN += +1 +g × a +(16) +105 +3 × 104 +4 × 104 +6 × 104 +Ct (counts) +10 +5 +Var(pe) +FIG. S3. Readout noise calibration. A π pulse on our optical clock +transition is used so pe ≈ 1 and Vpe = +R2 +¯ +Ct2 + C. We use 4 pulse +durations between 5 and 20 µs to vary Ct. We fit R = 100.2 ± 24.6 +and C = 2.73 × 10−6 ± 1.02 × 10−6. +To determine a we need to accurately calibrate VRN and +VP SN. We see at pe = 1, VP SN, VQP N = 0. Thus, measur- +ing Vpe at pe = 1 will independently determine VRN. +We wish to fit R and ensure it is consistent with the cameras +specified readout noise. To extract this value, we use 4 pulse +durations between 5 and 20 µs to vary Ct. This is illustrated +in Fig. S3. In practice, we fit +Vpe = R2 +¯Ct +2 + C +(17) +We fit R = 100.2 ± 24.6 and C = 2.73 × 10−6 ± 1.02 × +10−6. For our circular ROI there are X = 889 pixels in the +masked radius. For the calibrated gain g = 1.59 counts/e- and +readout noise r = 2.4 e- respectively , Rcalc = √Xgr = +94.7 in agreement with R = 100.2 ± 24.6. We note that the +gain and readout noise of the camera are close to specifica- +tion. Dark counts over our 30 ms exposure are < .1 e- and +considered negligible. +Next, we wish to determine aQP N. To do so, we perform a +second measurement at pe = 0.5. The variance of this dataset +contains contributions from VQP N, VP SN, and VRN. Us- +ing the measured R value, we subtract the VRN contribution. +Next, we fit the data in Fig. S4 to: +Vpe = 0.5(1 − 0.5) +a ¯Ct/τ ++ 0.5(1 − 0.5) +¯Ct +× g +(18) +We fit aQP N = 1.72 ± 0.16. This is in reasonable agree- +ment with the calculated value of 1.43 assuming Γ/2 scatter- +ing into 4 π while also accounting for the measured quantum +efficiency. +105 +4 × 104 +6 × 104 +Ct (counts) +2 × 10 +5 +3 × 10 +5 +4 × 10 +5 +Var(pe) +FIG. S4. aQP N calibration. The atoms in our optical lattice are +placed in a superposition of the ground and clock states with a π/2 +pulse so pe ≈ 0.5 for these measurements and Vpe is fit to Eq. 18. +We determine aQP N = 1.72 ± 0.16. +READOUT NOISE +Here, we derive the readout noise term used in our variance +measurements. The expressions used are somewhat different +than other literature, given that we use averaged dark frames +¯Be and ¯Bg. Recall, pe = +Ce +Ce+Cg . To determine the readnoise +contribution to the excitation fraction, we perform standard +error propagation: +VRN = +� ∂pe +∂Ce +�2 +V(Ce) + +� ∂pe +∂Cg +�2 +V(Cg) +(19) +Here, +∂pe +∂Cg += +Ce +(Ce + Cg)2 = +pe +(Ce + Cg) +(20) + +5 +∂pe +∂Ce += +Cg +(Ce + Cg)2 = +1 − pe +(Ce + Cg) +(21) +To determine V(Ce) consider an X pixel region-of-interest +for which we extract Cg, Ce in two separate measurements. +Each pixel contains r read noise in electrons. The single pixel +read noise in units of counts is thus g × ri. The total noise in +this region of interest is summed in quadrature pixel-by-pixel +V(Cg), V(Ce) = � +X (ri × g)2 = Xr2g2 = R2. Plugging +terms in Eq. 19: +VRN = R2 +¯Ct +2 (2¯p2 +e − 2¯pe + 1) +(22) +Imaging system parameters for Fig. 3(a) +In Table 1 is a summary of the imaging parameters for the +measurements in Fig. 3(a). For Fig. 1 and Fig. 4, a 1 µs pulse +duration was used. In Fig. 3(b), we vary the pulse length be- +tween 500 ns and 2 µs. Atom number fluctuations in time- +of-flight absorption imaging for these measurements have a +standard deviation less than 2 %. +Table 1 +Vertical imaging system +Numerical aperture +0.23 +Pulse duration +3 µs +Total photons scattered per atom at full +saturation +287 +Collection efficiency +1.3 % +Camera quantum efficiency +0.85 +Imaging system quantum efficiency +0.65 +Calculated photon count per atom +2.06 +Measured photon count per atom +1.91(1) +Horizontal imaging system +Numerical aperture +0.10 +Pulse duration +3 µs +Total photons scattered per atom at full +saturation +287 +Collection efficiency +0.25 % +Camera quantum efficiency +0.78 +Imaging system quantum efficiency +0.72 +Calculated photon count per atom +0.402 +Measured photon count per atom +0.445(3) +[1] G. E. Marti, PhD Thesis (University of California, Berkeley, +2014). +[2] W. Ketterle, D. S. Durfee, and D. Stamper-Kurn, arXiv (1999). +[3] D. D. Hickstein, S. T. Gibson, R. Yurchak, D. D. Das, +and +M. Ryazanov, Review of Scientific Instruments 90, 065115 +(2019). +[4] D. Matei, T. Legero, S. H¨afner, C. Grebing, R. Weyrich, +W. Zhang, L. Sonderhouse, J. Robinson, J. Ye, F. Riehle, et al., +Phys. Rev. Lett. 118, 263202 (2017). +[5] W. M. Itano, J. C. Bergquist, J. J. Bollinger, J. M. Gilligan, D. J. +Heinzen, F. L. Moore, M. G. Raizen, and D. J. Wineland, Phys. +Rev. A 47, 3554 (1993). + diff --git a/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/load_file.txt b/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f638dc09a843019ff4e197f2aa4dee79a49284f2 --- /dev/null +++ b/EtE1T4oBgHgl3EQfqgU3/content/tmp_files/load_file.txt @@ -0,0 +1,971 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf,len=970 +page_content='High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock William R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Milner,1, ∗ Lingfeng Yan,1 Ross B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson,1 Christian Sanner,1 and Jun Ye1, † 1JILA, NIST and University of Colorado, 440 UCB, Boulder, Colorado 80309, USA We report on the observation of a high-density, band insulating state in a three-dimensional optical lattice clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Filled with a nuclear-spin polarized degenerate Fermi gas of 87Sr, the 3D lattice has one atom per site in the ground motional state, thus guarding against frequency shifts due to contact interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At this high density where the average distance between atoms is comparable to the probe wavelength, standard imaging techniques suffer from large systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To spatially probe frequency shifts in the clock and measure thermodynamic properties of this system, accurate imaging techniques at high optical depths are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using a combination of highly saturated fluorescence and absorption imaging, we confirm the density distribution in our 3D optical lattice in agreement with a single spin band insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining our clock platform with this high filling fraction opens the door to studying new classes of long-lived, many-body states arising from dipolar interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Optical lattice clocks integrate quantum many-body physics and precision metrology to achieve state-of-the-art measurement precision [1–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To advance clock performance, one wishes to probe as many atoms as feasible for the longest possible coherence time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Improvements in both precision and accuracy of optical lattice clocks, with increased atom num- bers, have been enabled by the development of high-fidelity, microscopic imaging of the atomic cloud to spatially resolve clock shifts [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The combination of high density and long coherence time will allow characterization of novel systematic effects such as that arising from dipolar interactions between atoms on neighboring lattice sites [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lattice thermom- etry [12] and studies of novel physics such as SU(N) mag- netism [13, 14] will also benefit from accurate imaging at high density where these phenomena emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To optimize atom number while minimizing interaction- related dephasing, a clock platform based on a 3D lattice geometry and Fermi-degenerate matter has been developed [7, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Following nuclear spin polarization [16, 17], the Pauli exclusion principle mandates there is at most one atom per lat- tice site in the ground motional state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To ensure this ground state motional occupation during lattice loading we operate with kBT < kBTF < ℏωbg, where T, TF , ℏωbg refers to the atomic temperature, Fermi temperature and lattice bandgap respectively [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At the highest density affordable with one fermion per lattice site, this system realizes an insulating state of matter where tunnelling is suppressed [15, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining this high-density system with spin-orbit coupling generated from clock addressing will enable exploring cluster state gen- eration and tunable spin models [20, 21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Differential frequency shifts across the optical lattice en- coding potential systematic effects can be spatially resolved by combining in situ imaging and narrow-line clock spec- troscopy [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To extract these frequency shifts, two subse- quent images of the ground and excited state density distribu- tions are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus for our clock platform, accurate in situ imaging at high density is imperative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In our lattice where the average distance between atoms (407 nm) is comparable to the probe wavelength (461 nm), imaging with a weak, reso- nant probe is strongly perturbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Both collective effects medi- ated by dipolar interactions [22] and systematic defects such as lensing of the probe beam [23, 24] introduce errors to the reconstructed density distribution at high density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To mitigate these systematic effects, different techniques can be used to reduce the absorption cross section and make the cloud ”optically thin”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' These techniques can be broadly divided into two categories: dispersive imaging at large de- tuning from resonance [25–27] and saturated imaging at high intensity [28–31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For dispersive imaging extracting informa- tion about the atomic density often requires spatially filtering the scattered and unscattered light in the Fourier plane of the imaging system, demanding precise fabrication and alignment of custom optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Additionally, careful studies of dispersive imaging show that residual systematic effects at finite detun- ing are non-negligible and can be addressed using differential measurement schemes at opposite detuning [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To address these imaging errors in this work, we use both highly saturated fluorescence and absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In this Letter, we report on the observation of a band in- sulating state in our 3D optical lattice clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using highly saturated imaging to mitigate imaging errors, with a satura- tion parameter far greater than the optical depth, we accu- rately confirm the density distribution in our 3D optical lattice in good agreement with thermodynamic calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We ex- tend previous work using high intensity fluorescence imaging [28], confirming the accuracy of this imaging technique in a new high density regime with a degenerate Fermi gas of 87Sr [16, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' With atomic densities as high as 6×1014 atoms/cm3, we observe systematic agreement with atom counts obtained via time-of-flight absorption imaging and identify the range where the extracted atomic density distribution is not blurred by our imaging pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Our high intensity imaging scheme is outlined in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The combination of atomic level structure and relatively large mass of 87Sr is particularly well suited for our imaging tech- nique, providing a cycling transition with a large scattering rate while avoiding significant motional effects from the imag- ing pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The transition from 1S0 to 1P1 with linewidth Γ = 2π × 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 MHz provides a large photon scattering rate with minimal depumping to dark states during the imaging time [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' During a 1 µs pulse at full saturation about 100 photons per atom are scattered and the atoms accelerate at a = ℏkΓ 2m where k is the imaging light wavenumber and m is the atomic mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The net momentum transfer amounts to arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='03343v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='atom-ph] 9 Jan 2023 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Schematic of our clock platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Vertical and horizontal imaging systems with numerical apertures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 respectively provide measurements of the 2D density distribution ˜n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Accounting for the lattice spacing a = 407 nm, ˜na2 is determined from highly saturated absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To mitigate imaging errors, the atoms are highly saturated and each scatters photons with a maximum rate of Γ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Measurements from our high resolution imaging system in- tegrated along gravity are presented in panel (a), where the density distribution is extracted for thermodynamic modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Images from the horizontal imaging system in panel (b) are just used to determine our atom cloud aspect ratio for our inverse Abel transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' a Doppler shift of kaτ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='8 MHz which is much less than the transition linewidth Γ/2π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Finally, the linear displace- ment for a 1 µs pulse at full saturation is just aτ 2 2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This linear displacement and corresponding Doppler shift can be largely cancelled in fluorescence imaging by retroreflect- ing the incident beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The spread in transverse position due to random momentum transfer from spontaneous emission is ℏk 6mt3/2� Γ/2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 µm over a 1 µs pulse duration and small compared to our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 µm imaging resolution [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using highly saturated absorption imaging, we measure the column density distribution ˜n in our optical lattice in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ac- counting for the lattice spacing a = 407 nm corresponding to the 87Sr magic wavelength at 813 nm, the scaled column density ˜na2 is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Saturated absorption and fluorescence imaging are benefi- cial in comparison to standard imaging techniques in a num- ber of ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In this highly saturated regime the scattering rate is largely immune to beam intensity, frequency, and pointing fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given the saturation intensity Isat = 40 mW/cm2 for the imaging transition, a Gaussian probe beam with 20 mW of optical power and a 100 µm waist corresponds to a peak intensity of I ∼ 3000 Isat, within the typical constraints of a standard imaging laser system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given that the probe beam is attenuated through the atom cloud, a saturation parameter I/Isat much greater than the optical depth is required to fully saturate the imaging transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note parallels between flu- orescence and absorption imaging at high saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In both cases, the extracted atom number is determined by a single variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For fluorescence imaging, this corresponds to the number of collected photons per atom and for saturated ab- sorption imaging the number of missing photons per atom in the probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, both fluorescence and saturated ab- sorption imaging can be calibrated via a single absolute atom number measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For images in our 3D lattice, we de- termine our atom number via clock excitation fraction fluctu- ations arising from quantum projection noise (QPN) [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For fluorescence imaging, only a single image of collected fluorescence in an arbitrary direction is required, minimizing fringing and simplifying image processing substantially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Flu- orescence imaging also avoids limited dynamic range issues suffered from high intensity absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Strategies such as multiple measurements at varying intensity to deter- mine the atomic density in different regions of the cloud may be taken to confront this issue [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The primary disad- vantage of fluorescence imaging in comparison to absorption imaging is that the signal-to-noise is generally worse [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To optimize signal-to-noise ratio (SNR) in fluorescence imaging, the photon collection efficiency and therefore the numerical aperture (NA) of the imaging system, must be maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In our experiment, the vertical and horizontal imaging systems have numerical apertures of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1, corresponding to col- lection efficiencies of approximately 1 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 percent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Al- ternatively, if spatial resolution is not required then the pulse duration can be extended enhancing the number of detected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To motivate the development of our high intensity imaging technique, systematic errors associated with standard in situ imaging techniques at high density are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Absorption imaging at I ∼ Isat and high intensity fluorescence imaging are presented side-by-side for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To study these systematic errors at high density, we prepare a sample with optical depth > 200 by producing a degenerate Fermi gas with 10 nuclear spin components, ≈ 2 × 105 atoms and a T/TF of approximately 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 in a crossed dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The errors associated with low intensity absorption imaging can be seen twofold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' First, the reconstructed optical depth from absorp- tion detection in the upper left panel is far too low, two orders of magnitude less than the expected value of ∼ 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This erroneously low optical depth is attributed to effects such as enhanced forward emission and lensing of probe light [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Secondly, the reconstructed optical depth in the upper right panel increases after a 500 µs time-of-flight expansion con- clusively demonstrating the density dependence of these ob- served systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In comparison, saturated fluorescence imaging yields a far larger reconstructed optical depth and diffuses following ex- pansion as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We compare this reconstructed 2D den- sity distribution with the expected distribution corresponding to a Fermi gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using independently measured experimental values, we calculate this distribution with no free parameters [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total atom number and reduced temperature T/TF are determined from time-of-flight absorption imaging at low density with an optical density ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The trapping frequencies are extracted from parametric confinement modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using 皖A·23 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A comparison of high intensity fluorescence and standard ab- sorption imaging (I ∼ Isat) at optical depths exceeding 200 in our highly degenerate Fermi gas is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In situ absorption imaging at low intensity yields strikingly erroneous measurements at high den- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The calculated 2D Fermi gas distribution according to our ex- perimental parameters is shared for comparison in qualitative agree- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' these parameters, we calculate both an in situ and 500 µs time- of-flight Fermi gas profile for comparison with our measure- ments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We observe qualitative agreement between measure- ment and calculation at these extremely high optical depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Intrigued by the measurements presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 2, we un- dertake a quantitative study on the fidelity of our saturated imaging technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We present a calibration method for flu- orescence detection, using the total number of collected fluo- rescence photons for comparison with an accurate atom num- ber reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Absorption imaging at low density following time-of-flight expansion serves as an appropriate calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Following expansion for 7 ms, the optical depth is ∼ 1 and systematic imaging errors can be safely ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To inde- pendently calibrate the atom number in our 10 spin Fermi gas, we prepare a thermal sample and use measured density fluctuations to determine the effective absorption cross sec- tion [39–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(a) we ensure this calibration shows systematic agreement with atom numbers between approxi- mately 1 × 105 and 4 × 105, varied by increasing our final evaporation trap depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the 3 µs pulse duration used, the fitted calibration is in reasonable agreement with calculation using the measured quantum efficiency and imaging system numerical aperture [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To ensure that the imaging transition is fully saturated, the laser intensity at 1 µs pulse duration is increased until the collected photon number plateaus, as seen in the figure inset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To perform accurate spatially resolved measurements, we must also determine the blurring induced by our imaging pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Just as collective effects introduce errors to the re- FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (a) Calibration method for in situ fluorescence detection us- ing atom counts from time-of-flight absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Collected photon counts from both the vertical and horizontal imaging systems are plotted, with solid and dashed lines representing fits to the hori- zontal and vertical measurements respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inset: Collected pho- ton count with vertical imaging system as a function of I/Isat at 1 µs pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (b) Peak column density as a function of fluo- rescence pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Measurements are normalized by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='9×1011 atoms/cm2, the column density at the shortest pulse duration of 500 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Images at 500 ns and 2 µs in inset are plotted for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The error bars denote the standard error of the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' constructed density distribution, any systematic changes to ˜n introduced by our imaging pulse must be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To cal- ibrate this blurring in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b), we extend the fluorescence pulse duration and examine the peak column density as atoms diffuse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The inset shows averaged images from 500 ns and 2 µs pulse durations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that we observe no atom loss or molecular formation over the full 2 µs range, confirmed by the detected photon count increasing linearly with pulse dura- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To minimize blurring, we carefully retroreflect our probe beam by optimizing the backcoupled light through the probe optical fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At pulse durations up to 1 µs, we confirm that the peak column density decreases by < 5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Motivated by the calibration reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3, we directly determine the 3D density distribution in a deep optical lattice via saturated in situ absorption imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We form a cubic lat- tice with trap depths of approximately 60, 70, and 50 Er in three orthogonal directions, where Er is the lattice photon re- coil energy ≈ h × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 kHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Following forced evaporation with 10 nuclear spin states we spin polarize using a focused beam detuned from the 3P1 intercombination line to form a state-dependent potential, removing nearly all but the mF = - 9/2 atoms [16, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Clock spectroscopy confirms ≈ 90% spin purity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' An additional step of spin purification is applied by 4 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (a) The three-dimensional density distribution and the corre- sponding lattice filling fraction are determined from in situ absorp- tion image in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) and the use of an inverse Abel transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' (b) A linecut along z = 0 provides the data points in circle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Errorbars are both the statistical uncertainty of the Abel transformation and atom number uncertainty added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We start with a pre- diction based on HTSE calculation, using independently measured values for the temperature, atom number, and harmonic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The best fit to the data results in a 10% reduction of the measured as- pect ratio ωy/ωx and 5% reduction of the measured T/TF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The red line captures this fit, with temperature uncertainty in the shaded band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The blue dashed line is a fit to Gaussian in qualitative disagreement with na3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' coherently driving the mF = -9/2 atoms into the excited clock state and removing any residual spins with a resonant imaging pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Absorption imaging directly provides us with the col- umn density distribution ˜n, integrated through the vertical axis along gravity as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Based on our Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b) analysis, we choose a pulse duration of 1 µs to minimize blur- ring and a saturation intensity of 54(4), substantially larger than peak optical density of ∼ 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To spatially probe the band insulator plateau we use an imaging magnification of 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='8 to achieve an effective pixel size of 412 nm, roughly equal to the lattice constant a = 407 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that our effective pixel size is smaller than our optical resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 µm, thus our imaging system is optically oversampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To extract the 3D density distribution, we use an inverse Abel transform [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given our vertical imaging is not along an axis of cylindri- cal symmetry, n must be appropriately scaled by the aspect ratio of the spatial density distribution [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The aspect ra- tio is independently calibrated using the absorption imaging measurement in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At this high magnification, the SNR in fluorescence imag- ing for a 1 µs pulse duration is limited by a combination of read noise and photon shot noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We found that even after extensive averaging the extracted 3D density distribution us- ing an inverse Abel transform was sensitive to small fluctua- tions in ˜n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, saturated absorption imaging with a superior SNR provides a more robust technique to characterize the 3D density distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This extracted 3D density distribution is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To judge the fidelity of our measured 3D density distri- bution, we compare the line cut at z = 0 with calculation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To estimate the density distribution, we use a High Temperature Series Expansion (HTSE) calculation in the atomic limit [12, 14, 37, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The ingredients of this cal- culation include values for the atomic temperature, harmonic confinement, and total atom number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given the density dis- tribution only depends on the ratio of the respective harmonic confinements, the measured aspect ratios from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1 are used for our HTSE calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total atom number N is de- termined from quantum projection noise measurements [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To estimate the temperature including heating during lattice loading, we measure the reduced temperature T/TF in time- of-flight after a round-trip from the lattice back to the dipole trap and determine an entropy-per-particle increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='25(6) kB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inferring an entropy increase of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='13(3) kB in a single lattice loading sequence, we estimate a T/TF of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='165(7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Although we did not perform a cross-dimensional thermaliza- tion measurement to directly verify thermal equilibrium, the uncertainty in our temperature is included in the shaded band of the HTSE calculation in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(b) [44, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that the extended plateau region is larger than our 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 µm imaging res- olution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To further quantify the imaging fidelity, we compare na3 to a Gaussian fit in clear disagreement with data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In conclusion, we report on the observation of a spin- polarized, band insulating state in our 3D optical lattice clock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This has been enabled by characterizing saturated in situ imaging techniques to accurately determine our density dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Broadly, the saturated imaging techniques in this work will be applicable for studies of SU(N) magnetism and thermodynamics in the Mott-insulating regime [46, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' With the high filling fraction demonstrated in this work, many-body states arising from dipolar interactions can be generated be- tween atoms on neighboring lattice sites [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Acknowledgement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We thank D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kedar for maintain- ing the ultrastable clock laser used in this work and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Aep- pli, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Robinson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Miklos, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Tso for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We thank K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kim, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Oppong, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sonderhouse for careful reading of the manuscript and for providing insightful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Funding for this work is pro- vided by NSF QLCI OMA-2016244, DOE Center of Quan- tum System Accelerator, DARPA, AFOSR, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bush Fellow- ship, NIST, and NSF Phys-1734006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 5 ∗ william.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='milner@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='edu † ye@jila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='edu [1] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bothwell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kennedy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Aeppli, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kedar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Robin- son, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Oelker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Staron, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Nature 602, 420 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' McGrew, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Fasano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sch¨affer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Beloy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Nicolodi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Brown, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hinkley, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Milani, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Schioppo, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=', Nature 564, 87 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Aeppli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Chu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bothwell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kennedy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kedar, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' He, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Science Advances 8, 42 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bloom, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Nicholson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Williams, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Campbell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bishof, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bromley, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Nature 506, 71 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [5] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Nicholson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Marti, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bloom, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' McNally, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Barrett, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Safronova, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Strouse, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=', Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 6, 1 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Marti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Goban, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Campbell, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Poli, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 120, 103201 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [7] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Campbell, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Marti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Goban, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Oppong, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' McNally, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sonderhouse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Robinson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bloom, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=', Science 358, 90 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Chang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lukin, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 69, 023810 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Cidrim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' PineiroOrioli, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sanner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bachelard, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, Physical Review Letters 127, 013401 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [10] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kr¨amer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ostermann, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ritsch, EPL (Europhysics Letters) 114, 14003 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [11] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Collective dipolar interactions in a near unity filled three-dimensional optical lattice clock, in prepara- tion (2023).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [12] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hofrichter, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Riegger, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Scazza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' H¨ofer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Fernandes, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bloch, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' F¨olling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' X 6, 021030 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [13] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gorshkov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hermele, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gurarie, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Xu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Julienne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zoller, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Demler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lukin, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 6, 289 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [14] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Taie, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yamazaki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sugawa, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Takahashi, Nature Physics 8, 825 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [15] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Goban, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Marti, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sonderhouse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' San- ner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 123, 123401 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [16] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sonderhouse, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sanner, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Goban, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bilitewski, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Milner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 16, 1216 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [17] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Stellmer, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Grimm, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Schreck, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 84, 043611 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Will, PhD Thesis (Springer Science & Business Media, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [19] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Schneider, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hackermuller, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Will, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Best, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bloch, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Costi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Helmes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rasch, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rosch, Science 322, 1520 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [20] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Mamaev, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Blatt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 122, 160402 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [21] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Mamaev, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kimchi, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Nandkishore, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Research 3, 013178 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [22] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Andreoli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gullans, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' High, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Browaeys, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Chang, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' X 11, 011026 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [23] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Durfee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Stamper-Kurn, arXiv (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rath, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yefsah, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G¨unter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Cheneau, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Desbuquois, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Holzmann, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Krauth, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Dalibard, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 82, 013609 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kadlecek, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sebby, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Newell, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Walker, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 26, 137 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bradley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sackett, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hulet, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 78, 985 (1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Andrews, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Mewes, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Van Druten, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Durfee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kurn, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle, Science 273, 84 (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' DePue, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Winoto, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Han, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Weiss, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 180, 73 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [29] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Reinaudi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lahaye, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Wang, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gu´ery-Odelin, Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 32, 3143 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [30] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yefsah, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Desbuquois, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Chomaz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G¨unter, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Dal- ibard, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 107, 130401 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [31] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Chomaz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Corman, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yefsah, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Desbuquois, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Dal- ibard, New Journal of Physics 14, 055001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [32] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Heo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Choi, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Chris- tensen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rvachov, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 85, 063615 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [33] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sanner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sonderhouse, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Milner, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Science 374, 979 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Barker, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Reschovsky, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Pisenti, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Camp- bell, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 92, 043418 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [35] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Joffe, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Martin, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Pritchard, JOSA B 10, 2257 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [36] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Itano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bergquist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bollinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gilligan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Heinzen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Moore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Raizen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Wineland, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 47, 3554 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [37] Supplemental materials with description of imaging signal-to- noise comparison, QPN atom number calibration, density dif- fusion, and HTSE calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [38] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zwierlein, arXiv (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [39] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sanner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Su, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Keshet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gommers, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Shin, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Huang, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 105, 040402 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [40] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Tobias, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Matsuda, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Valtolina, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' De Marco, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Li, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 124, 033401 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [41] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M¨uller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zimmermann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Meineke, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Brantut, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Esslinger, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Moritz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 105, 040401 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [42] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Dribinski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ossadtchi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Mandelshtam, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Reisler, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 73, 2634 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [43] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hazzard, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gurarie, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hermele, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 85, 041604(R) (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Monroe, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Cornell, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sackett, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Myatt, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Wieman, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 70, 414 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [45] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Valtolina, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Matsuda, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Tobias, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' De Marco, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Nature 588, 239 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [46] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Goban, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hutson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Marti, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Campbell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Perlin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Julienne, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D’incao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Nature 563, 369 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [47] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bishof, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bromley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Kraus, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Safronova, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zoller, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rey, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, Science 345, 1467 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Supplemental material to High-fidelity imaging of a band insulator in a three-dimensional optical lattice clock Density diffusion Here we provide supplemental analysis to the data pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In panel A of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S1, we plot the in- tegrated counts along the x axis of each image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We see an asymmetry emerge along the direction of the probe beam as the pulse duration is extended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This asymmetry suggests that the observed density diffusion may arise from inhomogeni- ety between the incident and retroreflected beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' While the power is certainly mismatched, this could also be due to ei- ther imperfect spatial alignment or mode mismatch given the divergence of the probe beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We also plot the total counts in each image as a function of pulse duration in panel B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The linear character of the counts over the full pulse duration range suggests that we do not ob- serve appreciable atom loss or pumping to dark states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The counts at each pulse duration are normalized to the counts at 500 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The inset shows the Gaussian RMS width of the cloud as a function of pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Signal-to-noise comparison In the main text of the paper we refer to both saturated ab- sorption and fluorescence imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We provide a quantita- tive comparison of the signal-to-noise ratio (SNR) between the two techniques here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We express our signal-to-noise for a detection pixel in terms of the normalized variance V(N)/N, where N denotes the number of atoms within the respective detection region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For fluorescence imaging the SNR is simply determined by the shot noise associated with the number of detected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To calculate the total atom number, we first convert the fluorescence counts detected on our camera to the number of collected photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Then, using the collection effi- ciency of our imaging system and scattering rate of our atomic transition we determine the conversion of detected photons per atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' On our CCD camera, we measure na counts in a given pixel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using the quantum efficiency q of the imaging system, and the camera conversion gain g in units of counts per photo electron, we infer na qg photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At full saturation, the atomic scattering rate is Γ 2 and the number of photons scat- tered per atom is Psc = Γ 2 × τ, where τ is the pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Finally, we denote the collection efficiency as Y , determined by the numerical aperture of our imaging system and by ra- diation pattern anisotropies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining terms, the total atom number is N = na gqY Psc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using error propagation, we deter- mine the variance VF l(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' VF l(N) = � ∂N ∂na �2 V(na) = � 1 gqY Psc �2 gna (1) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Panel (a) shows the integrated counts from the images in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b) of the main text along the x axis as a function of pulse du- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total counts at each pulse duration is plotted in panel (b), normalized by the counts at 500 ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given the detected photon count increases linearly with pulse duration, we observe minimal atom loss or molecular formation over the full 2 µs range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The inset shows the Gaussian RMS width of the cloud as a function of pulse duration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Here, we have used the fact that the distribution of gener- ated photo electrons ne is binomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, V(na) = V(g × ne) = g2V(ne) = g2ne = gna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Combining terms: VF l(N)/N = 1 qY Psc (2) The SNR associated with absorption imaging is more com- plicated given the formula for the atom number in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3 has both logarithmic and linear terms and involves two images na and nb with and without atoms present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Here, A and σ0 refer to the effective pixel size accounting for the imaging system magnification and effective atomic absorption cross section, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Similar to fluorescence imaging, an appropriate error propagation of the na and nb terms determines Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We summarize the formulas here and point a reader to reference [1] for a full derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='03343v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='atom-ph] 9 Jan 2023 2 N = A σ0 log( nb na ) + 2 Γτgq (nb − na) (3) VAbs(N) = g ˜A2( 1 na + 1 nb ) + g ˜B2(na + nb) + 4g ˜A ˜B (4) ˜A = A σ0 , ˜B = 2 qgτΓ (5) We compare the different techniques in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S2 using the experimentally relevant parameters for our imaging system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In both cases, a 1 µs resonant pulse is used with a numerical aperture of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 and a quantum efficiency of 85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the fluo- rescence SNR in blue, the transition is assumed to be fully sat- urated and scatters photons with a rate of Γ/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the I/Isat = ∼ 55 we use for our inverse Abel measurements, the SNR in absorption imaging is superior to fluorescence imaging in regions where the column density is higher than 2 atoms/a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Particularly given our peak density of ˜na2 = ∼ 20 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a), absorption imaging provides a better SNR in the regions of high density where we extract our peak filling fraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' At a critical OD of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='17, fluorescence detection under our experi- mental parameters provides a superior SNR at all imaging in- tensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note these calculations neglect technical noise, in particular camera readout noise, which can be accounted for by offsetting V(na) accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This contribution will disproportionately reduce the SNR of fluorescence imaging, as the fluorescence counts are substantially lower than the ab- sorption counts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To probe fine spatial details in our atomic cloud, an imag- ing resolution smaller than the length scale of these spatial features is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To achieve this condition, a sufficiently large numerical aperture imaging system must be utilized and aberrations must be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In this case, the imaging res- olution is fundamentally limited by diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We verified the diffraction-limited performance of our NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 objec- tive lens by propagating a point source at 461 nm through a test setup (including all imaging path optics and vacuum viewports) and measuring the point-spread function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' While absorption and fluorescence imaging rely on the same light scattering process (they only collect different parts of the scattered EM field [2]), the signal amplitudes for these two methods scale differently with the NA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' When collect- ing fluorescence, the solid angle coverage of the imaging sys- tem proportionally affects the signal down to the lowest spa- tial frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This is not the case for absorption imaging, where the amplitude of spatial frequency components below the NA-dependent bandwidth is constant as the NA is further increased (assuming the lens fully covers the probe beam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In other words, for fluorescence imaging, most of the signal light gets collected in the outer ring fraction of the lens aperture, which renders it particularly susceptible to lens imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 0 20 40 60 80 100 I/Isat 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='0 Var(N)/N Absorption beam intensity 1 s pulse duration, NA = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 Saturated fluorescence OD = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='17, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='28 atoms/a2 OD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2, 2 atoms/a2 OD = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1, 5 atoms/a2 OD = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1, 10 atoms/a2 OD = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2, 20 atoms/a2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' SNR comparison between absorption and fluorescence imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The relevant imaging parameters from the main figures of the paper are used for this calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For absorption imaging the atom count variance scales inversely proportional with intensity in the non-saturated limit I ≪ Isat, and proportional with intensity in the high saturation limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The variance is for both imaging methods proportional to 1/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In the fully saturated regime (and assuming no technical noise) the normalized variance for fluorescence imaging is independent of atomic column density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To avoid imaging defects at the high densities used in clock operation, an I/Isat > 50 was used in all imaging measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The black dashed line indicates the intensity used for our inverse Abel measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' HTSE calculation To accurately model the density distribution in our 3D lat- tice, we use a High Temperature Series Expansion (HTSE) calculation in the atomic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The general Hamiltonian for SU(N) symmetric fermions in a 3D lattice in the atomic limit takes the following form: HAL = U 2 � i,σ̸=σ′ ˆni,σˆni,σ′ + � i,σ Viˆni,σ (6) On a lattice site i, there are just two competing energy scales: an interaction energy U between particles and a po- sition dependent energy offset Vi according to the harmonic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' By using the local density approximation µ = V (x, y, z)−µ0, where V (x, y, z) = 1 2m(ω2 xx2+ω2 yy2+ω2 zz2) and µ0 corresponds to the peak chemical potential in the lat- tice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the spin-polarized system in this work, U = 0 and the calculations are substantially simplified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ultimately, we want to express the density distribution n(µ, T, r) in terms of the chemical potential, atomic tempera- ture, and position in the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' On a lattice site i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' we express the Grand partition function Z and Grand potential Ω : Z(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' r) = N=1 � σ=0 �N σ � e−βµσ (7) 3 Ω = −kBTln(Z) From here,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' we determine the entropy and occupancy per lattice site i: s(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' r) = −∂Ω ∂T = kB ln(Z) + ∆s ∆s = kB Z βµe−βµ (8) n(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' r) = −∂Ω ∂µ = 1 Z(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T)e−βµ (9) We accurately determine the total atom number Nlat from in situ absorption imaging and total entropy Slat via time-of- flight fitting to a non-interacting Fermi-Dirac profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Simi- larly, we express the entropy s and occupation n on a given lattice site using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 8 and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 9 expressed in terms of T and µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We then determine global fitting parameters T and µ0 to ensure the integrated entropy and occupancy over all lat- tice sites equals our experimentally measured values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' After determining µ0 and T to realize the equality in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 9, we cal- culate n(µ, T, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A linecut of n(µ, T, r) at z = 0 is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Inverse Abel transform We outline our reconstruction procedure here using mea- surements of the atomic cloud aspect ratios and an inverse Abel transform: First, we use saturated absorption images along a vertical axis aligned with z and a horizontal axis aligned with x corresponding to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(b) to determine the aspect ratios ωx/ωy and ωx/ωz respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Next, we perform an inverse Abel transform on the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) image to reconstruct an initial three-dimensional density dis- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given there is no axis of cylindrical symmetry in our system geometry, the reconstructed density from the in- verse Abel transform must be appropriately re-scaled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Treating our system as an ellipsoid with radii rx, ry, rz and N atoms the density is nlat = N/Vlat where Vlat = 4 3πrxryrz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We extract the inverse Abel transform for the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(a) image along the x axis, given the largest Band in- sulator plateau will occur along the axis with the weakest har- monic confinement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The density distribution from this pro- cedure assumes a volume of VAbel = 4 3πrxrxry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus we scale the extracted density by nAbel/nlat = rz rx = ωz/ωx us- ing the measured aspect ratio from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Given excess noise around the origin, the x = 0 point is interpolated with the neighboring point in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This reconstruction proce- dure was cross-checked with simulated density distributions to ensure its fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The three-point Abel transform method was used for this work, which has been independently studied to verify its fidelity [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' QPN calculation To calibrate our atom number, we analyze quantum projec- tion fluctuations using the narrow-linewidth clock transition between the 1S0 and 3P0 states in 87Sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Using a clock laser stabilized to our 8 mHz linewidth silicon reference cavity, ro- tation noise due to laser instability can be neglected in these measurements [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Additionally, fluctuations in total counts are < 2% and not a limiting systematic for determining the atom number calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Referenced in many texts [5], by preparing atoms in a superposition of 1S0 to 3P0 the variance V of the measured excitation fraction is related to the mean atom number ¯N and mean excitation ¯pe by: VQP N = ¯pe(1 − ¯pe) ¯N (10) To determine this variance, we do many subsequent mea- surements of pe under identical operating conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For a measurement i to determine pi e, two fluorescence counts ˜Ci g and ˜Ci e are read off a region of interest of our camera includ- ing our atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' These counts are subtracted by two averaged dark frames ¯Bg and ¯Be to yield Ci g = ˜Ci g− ¯Bg, Ci e = ˜Ci e− ¯Be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We would like to determine the coefficient a that satisfies N i e = aCi e/τ, N i g = aCi g/τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We can immediately see that the excitation fraction has no dependence on this coefficient: pi e = �aCi e �aCie + �aCig (11) However, the total atom number N i = a(Ci e + Ci g)/τ = aCi t/τ does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rewriting Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 10, we see a measurement of the variance VQP N, the mean excitation ¯pe, and the mean total counts ¯Ct can determine a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' VQP N = ¯pe(1 − ¯pe) a ¯Ct/τ (12) The coefficient a can be interpreted as the ”atoms per count per pulse duration”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In principle, with knowledge of the quan- tum efficiency, gain, scattering rate, numerical aperture, and radiation pattern one could calculate this value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Practically, assumptions about the radiation pattern based on the quanti- zation axis and probe light polarization make this calculation more difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In practice, it is much more straightforward to directly measure a than to individually measure each of these values with high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The observed variance of the excitation fraction Vpe has contributions from quantum projection noise (QPN), photon shot noise (PSN), and camera readout noise (RN): Vpe = VQP N + VP SN + VRN (13) Here g is the detector gain in units of counts per electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4 VP SN = ¯pe(1 − ¯pe) ¯Ct × g (14) VRN = R2 ¯Ct 2 (2¯p2 e − 2¯pe + 1) (15) VP SN can be understood intuitively considering the ratio VQP N/VP SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The number of signal electrons (equivalently the number of collected photons multiplied by the camera quantum efficiency) per atom determines the relative scaling of VQP N and VP SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' VQP N VP SN = 1 g × a (16) 105 3 × 104 4 × 104 6 × 104 Ct (counts) 10 5 Var(pe) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Readout noise calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A π pulse on our optical clock transition is used so pe ≈ 1 and Vpe = R2 ¯ Ct2 + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We use 4 pulse durations between 5 and 20 µs to vary Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We fit R = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6 and C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='73 × 10−6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='02 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To determine a we need to accurately calibrate VRN and VP SN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We see at pe = 1, VP SN, VQP N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Thus, measur- ing Vpe at pe = 1 will independently determine VRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We wish to fit R and ensure it is consistent with the cameras specified readout noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To extract this value, we use 4 pulse durations between 5 and 20 µs to vary Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In practice, we fit Vpe = R2 ¯Ct 2 + C (17) We fit R = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6 and C = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='73 × 10−6 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='02 × 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For our circular ROI there are X = 889 pixels in the masked radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For the calibrated gain g = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='59 counts/e- and readout noise r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='4 e- respectively , Rcalc = √Xgr = 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='7 in agreement with R = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='2 ± 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We note that the gain and readout noise of the camera are close to specifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Dark counts over our 30 ms exposure are < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='1 e- and considered negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Next, we wish to determine aQP N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To do so, we perform a second measurement at pe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The variance of this dataset contains contributions from VQP N, VP SN, and VRN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Us- ing the measured R value, we subtract the VRN contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Next, we fit the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S4 to: Vpe = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5) a ¯Ct/τ + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5(1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5) ¯Ct × g (18) We fit aQP N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' This is in reasonable agree- ment with the calculated value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='43 assuming Γ/2 scatter- ing into 4 π while also accounting for the measured quantum efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 105 4 × 104 6 × 104 Ct (counts) 2 × 10 5 3 × 10 5 4 × 10 5 Var(pe) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' aQP N calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The atoms in our optical lattice are placed in a superposition of the ground and clock states with a π/2 pulse so pe ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='5 for these measurements and Vpe is fit to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' We determine aQP N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' READOUT NOISE Here, we derive the readout noise term used in our variance measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The expressions used are somewhat different than other literature, given that we use averaged dark frames ¯Be and ¯Bg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Recall, pe = Ce Ce+Cg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' To determine the readnoise contribution to the excitation fraction, we perform standard error propagation: VRN = � ∂pe ∂Ce �2 V(Ce) + � ∂pe ∂Cg �2 V(Cg) (19) Here, ∂pe ∂Cg = Ce (Ce + Cg)2 = pe (Ce + Cg) (20) 5 ∂pe ∂Ce = Cg (Ce + Cg)2 = 1 − pe (Ce + Cg) (21) To determine V(Ce) consider an X pixel region-of-interest for which we extract Cg, Ce in two separate measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Each pixel contains r read noise in electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The single pixel read noise in units of counts is thus g × ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' The total noise in this region of interest is summed in quadrature pixel-by-pixel V(Cg), V(Ce) = � X (ri × g)2 = Xr2g2 = R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Plugging terms in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 19: VRN = R2 ¯Ct 2 (2¯p2 e − 2¯pe + 1) (22) Imaging system parameters for Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(a) In Table 1 is a summary of the imaging parameters for the measurements in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' For Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 4, a 1 µs pulse duration was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 3(b), we vary the pulse length be- tween 500 ns and 2 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Atom number fluctuations in time- of-flight absorption imaging for these measurements have a standard deviation less than 2 %.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Table 1 Vertical imaging system Numerical aperture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='23 Pulse duration 3 µs Total photons scattered per atom at full saturation 287 Collection efficiency 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='3 % Camera quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='85 Imaging system quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='65 Calculated photon count per atom 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='06 Measured photon count per atom 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='91(1) Horizontal imaging system Numerical aperture 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='10 Pulse duration 3 µs Total photons scattered per atom at full saturation 287 Collection efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='25 % Camera quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='78 Imaging system quantum efficiency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='72 Calculated photon count per atom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='402 Measured photon count per atom 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content='445(3) [1] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Marti, PhD Thesis (University of California, Berkeley, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [2] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ketterle, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Durfee, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Stamper-Kurn, arXiv (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [3] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Hickstein, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gibson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Yurchak, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Das, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ryazanov, Review of Scientific Instruments 90, 065115 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Matei, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Legero, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' H¨afner, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Grebing, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Weyrich, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Sonderhouse, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Robinson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Ye, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Riehle, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' 118, 263202 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Itano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bergquist, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Bollinger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Gilligan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Heinzen, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Moore, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Raizen, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Wineland, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} +page_content=' A 47, 3554 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE1T4oBgHgl3EQfqgU3/content/2301.03343v1.pdf'} diff --git a/GdAyT4oBgHgl3EQfSvfs/vector_store/index.faiss b/GdAyT4oBgHgl3EQfSvfs/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..240bf143ee87e5af1e39404638a040de9c84ab92 --- /dev/null +++ b/GdAyT4oBgHgl3EQfSvfs/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:378f06e83045775fe1e271169a47227108b8212cc57d27c8138ec4656a9aaa98 +size 3342381 diff --git a/GdE0T4oBgHgl3EQfRQBK/content/tmp_files/2301.02204v1.pdf.txt b/GdE0T4oBgHgl3EQfRQBK/content/tmp_files/2301.02204v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..461099dc9167e1d60c9fe49352d77684878fdf74 --- /dev/null +++ b/GdE0T4oBgHgl3EQfRQBK/content/tmp_files/2301.02204v1.pdf.txt @@ -0,0 +1,397 @@ +arXiv:2301.02204v1 [math.CO] 5 Jan 2023 +ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL +SEMILINEAR GROUPS +DOM VITO A. BRIONES +Abstract. Association schemes on triples (ASTs) are 3-dimensional analogues of classical +association schemes. +If a group acts two-transitively on a set, the orbits of the action +induced on the triple Cartesian product of that set yields an AST. By considering the +actions of semidirect products of the affine special linear group ASL(k, n) with subgroups of +the Galois group Gal(GF(n)), we obtain the sizes, third valencies, and intersection numbers +of the ASTs obtained from subgroups of the affine special semilinear group. +1. Introduction +A classical association is an algebraic-combinatorial object with certain symmetry prop- +erties. These properties suffice to afford classical association with desirable structural char- +acteristics and are pliant enough to allow classical association schemes to be applicable to +several areas of mathematics. For example, the adjacency algebra of a classical association +scheme is semisimple and, when the adjacency matrices define a distance-regular graph, the +structure constants of this algebra can be expressed in terms of certain families of orthogonal +polynomials. [4] +Mesner and Bhattacharya introduced the notion of association schemes on triples (or +ASTs), a ternary analogue for classical association schemes [5]. +An AST on a set Ω is +a partition of the triple Cartesian product Ω × Ω × Ω subject to regularity requirements +paralleling the symmetry conditions for classical association schemes. In ASTs, the resulting +adjacency hypermatrices produce a ternary algebra under a ternary product that extends +the usual matrix multiplication. +However, the structural properties of ASTs remain unclear, partly due to the ternary +adjacency algebra not being associative nor commutative. As first steps in the investigation +of ASTs, studies were conducted regarding analogues of identity and inverse elements [6], +enumerations of ASTs over the smallest number of vertices [1], possible sources of ASTs such +as those from group actions, two-graphs, designs, and other ASTs [5, 7, 3], as well as the +intersection numbers of known families of ASTs [5, 2, 3]. +In particular, the actions of two-transitive groups yield ASTs [5]. The orbits of these +actions are closely related to the parameters of the AST, providing their sizes, third valencies, +and intersection numbers [5, 2]. In fact, [5] provides the sizes, third valencies, and intersection +numbers of the ASTs obtained from the affine general linear group AGL(1, n) where n is a +prime power. This was extended in [2], wherein these parameters were obtained for the ASTs +(D.V.A. Briones, Corresponding author) Institute of Mathematics, University of the Philippines +Diliman, 1101 Quezon City, Philippines +E-mail address: dabriones@up.edu.ph. +Date: January 6, 2023. +Key words and phrases. algebraic combinatorics, ternary algebra, association scheme on triples +MSC Classification: 05E30. +1 + +2 +D.V.A. BRIONES +obtained from subgroups of the affine semilinear group AΓL(k, n) of the form AGL(k, n)⋊H), +where k ≥ 1 and H ≤ Gal(GF(q). Further work was done in [3], where these parameters +were obtained from ASTs obtained from the affine special linear group ASL(2, n). +We extend this last result by determining the sizes, third valencies, and intersection num- +bers of ASTs obtained from subgroups of the affine special semilinear group ASL(k, n) ⋊ +Gal(GF(n)) of the form ASL(k, n) ⋊ H, where k ≥ 2, n is a prime power, and H is a +subgroup of Gal(GF(n)). In particular, we show that the ASTs obtained from ASLH(k, n) +are the same as the ASTs obtained from AGLH(k, n) = AGL(k, n) ⋊ H for k ≥ 3. +2. Preliminaries +We define association schemes on triples, remark how ASTs arise from two-transitive +groups, and review the actions of the affine special linear and affine special semilinear groups. +2.1. Association schemes on triples. We define association schemes on triples and men- +tion how the parameters of an AST obtained from a two-transitive group are related to the +group action. +Definition 2.1. [5, 7] Let Ω be a finite set with at least 3 elements. An association scheme +on triples (AST) on Ω is a partition X = {Ri}m +i=0 of Ω × Ω × Ω with m ≥ 4 such that the +following hold. +(1) For each i ∈ {0, . . . , m}, there exists an integer n(3) +i +such that for each pair of distinct +x, y ∈ Ω, the number of z ∈ Ω with (x, y, z) ∈ Ri is n(3) +i . +(2) (Principal Regularity Condition.) For any i, j, k, l ∈ {0, . . . , m}, there exists a con- +stant pl +ijk such that for any (x, y, z) ∈ Rl, the number of w such that (w, y, z) ∈ Ri, +(x, w, z) ∈ Rj, and (x, y, w) ∈ Rk is pl +ijk. +(3) For any i ∈ {0, . . . , m} and any σ ∈ S3, there exists a j ∈ {0, . . . , m} such that +Rj = {(xσ(1), xσ(2), xσ(3)) : (x1, x2, x3) ∈ Ri}. +(4) The first four relations are R0 = {(x, x, x) : x ∈ Ω}, R1 = {(x, y, y) : x, y ∈ Ω, x ̸= y}, +R2 = {(y, x, y) : x, y ∈ Ω, x ̸= y}, and R3 = {(y, y, x) : x, y ∈ Ω, x ̸= y}. +The integer n(3) +i +is the third valency of Ri, and is the analogue of valency from classical +association schemes. By Conditions 1 and 3 of Definition 2.1 there are for each i the constants +n(1) +i += |{z ∈ Ω : (z, x, y) ∈ Ri}| and n(2) +i += |{z ∈ Ω : (x, z, y) ∈ Ri}| independent of any pair +of distinct x, y ∈ Ω. Similarly, n(1) +i +is the first valency of Ri and n(2) +i +is the second valency of +Ri. The trivial relations are R0, R1, R2 and R3 while the other relations are the nontrivial +relations. Further, the numbers pl +ijk are called the intersection numbers. +ASTs arise naturally from the actions of two-transitive groups [5], mirroring how Schurian +classical association schemes are induced by the actions of transitive groups [4]. Indeed, +when a two-transitive group G acts on a set Ω, the orbits of the induced action on Ω×Ω×Ω +is an AST [5]. Correspondences between the action and the parameters of the induced AST +are summarized in the following remark. +Remark 1 ([5, 2]). Let G be a group acting two-transitively on a set Ω and let X be the AST +induced by this action. For any pair of distinct elements a, b ∈ Ω, the orbits of the two-point +stabilizer Ga,b on Ω \ {a, b} are in bijection with the nontrivial relations of the AST. As a +consequence of this bijection, the sizes of these orbits are also the third valencies. + +ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS +3 +2.2. Affine special groups. Given a prime power n and k ≥ 1, the affine special linear +group ASL(k, n) is the semidirect product GF(n) ⋊ SL(k, n), where SL(k, n) is the group +of invertible linear transformations on the k-dimensional vector space V over GF(n) of +determinant 1. Explicitly, the affine special linear group is the following group of maps from +V to itself. +ASL(k, n) = {x �→ Ax + b : A ∈ SL(k, n), b ∈ V } . +Similarly, the affine special semilinear group ASL(k, n) ⋊ Gal(GF(n)) is the semidirect +product of the affine special linear group ASL(k, n) with the Galois group Gal(GF(n)). +Explicitly, the affine special semilinear group is the following group of maps from V to itself. +ASL(k, n) ⋊ Gal(GF(n)) = {x �→ Aφ(x) + b : A ∈ SL(k, n), b ∈ GF(n), φ ∈ Gal(GF(n))} . +3. ASTs from subgroups of the affine special semilinear group +In this section we generalize work done in [3] by obtaining the sizes, third valencies, and +intersection numbers of ASTs obtained from the actions of subgroups of the affine special +semilinear group of the form +ASLH(k, n) = ASL(k, n) ⋊ H, +where k ≥ 2, n = pα a power of a prime number p, and H a subgroup of Gal(GF(n)). +We obtain the sizes and third valencies of these ASTs by obtaining a two-point stabilizer of +ASLH(k, n) and then determining its orbits. Finally, we obtain the intersection numbers of +these ASTs through explicit orbit computations. +For ease of discussion, we fix the following notations. Let n = pα be a power of a prime p, +k ≥ 2, V be the k-dimensional vector space over GF(n), H be a subgroup of Gal(GF(n)), +and X be the AST obtained from ASLH(k, n). For a ∈ GF(n), let ⃗a = (a, 0, . . . , 0)T ∈ V . +Further, for (u, v, w) ∈ V × V × V , let [(u, v, w)] ∈ X denote the orbit of (u, v, w) under +ASLH(k, n). +We begin with the case where k = 2. To determine the size and third valencies of X, we +exploit the relationships between these parameters and the orbits of a two-point stabilizer +of ASLH(k, n). +Theorem 3.1. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) +and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector space +V over GF(n). The two-point stabilizer ASLH(2, n)⃗0,⃗1 has the following orbits on V \{⃗0,⃗1}. +(1) There are −2 + ω +α +� α +ω +β=1 qgcd ( α +ω ,β) orbits of the form +� ⃗ +φ(a) : φ ∈ H +� +, a ̸= 0, 1 +each of size degGF (q)(a). +(2) There are −1 + ω +α +� α +ω +β=1 qgcd ( α +ω ,β) orbits of the form +� +(c, φ(a))T : c ∈ GF(n), φ ∈ H +� +, a ̸= 0 +each of size n degGF (q)(a). +As a consequence of Theorem 3.1, we obtain the sizes and third valencies of the ASTs +obtained from ASLH(2, n). +Theorem 3.2. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) +and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector + +4 +D.V.A. BRIONES +space V over GF(n). Then X has −3 + 2 +� +ω +α +� α +ω +β=1 qgcd ( α +ω ,β)� +nontrivial relations. There are +−2 + ω +α +� α +ω +β=1 qgcd ( α +ω ,β) nontrivial relations of the form +Ra = {[(⃗0,⃗1,⃗a)]}, a ̸= 0, 1, +with corresponding third valency degGF (q)(a). The remaining −1+ ω +α +� α +ω +β=1 qgcd ( α +ω ,β) nontrivial +relations of X are of the form +aR = {[(⃗0,⃗1, (0, a)T)]}, a ̸= 0, +with corresponding third valency n degGF (q)(a). +Proof. The two-point stabilizer is +ASLH(2, n)⃗0,⃗1 = {(x, y)T �→ +� +1 +c +0 +1 +� +(φ(x), φ(y))T : c ∈ GF(n), φ ∈ H}. +Direct computation shows that the orbits of ASLH(2, n)⃗0,⃗1 have the following forms. +(1) The first type of orbit has the form +{(φ(a), 0)T : φ ∈ H}, +which consists of those vectors whose second coordinate is 0 and whose first coordinate +is a Galois conjugate of an element a ∈ GF(n) with a ̸= 0, 1. +(2) The remaining orbits are of the form +{(x, φ(a))T : x ∈ GF(n), φ ∈ H}, +which consists of those vectors whose second coordinate is a Galois conjugate of an +element a ∈ GF(n) with a ̸= 0. +The sizes of these orbits follow directly from the Fundamental Theorem of Galois Theory. +The number of orbits of each type are then obtained through the Fundamental Theorem of +Galois Theory and a straightforward application of Burnside’s Orbit Counting Theorem to +the action of Gal(GF(n)) on GF(n). +□ +For notational convenience, let Aa denote the adjacency hypermatrix corresponding to +the relation Ra whenever a ̸= 0, 1. +Similarly, let aA denote the adjacency hypermatrix +corresponding to the relation aR whenever a ̸= 0. Further, let T be a transversal of the +orbits of H on GF(n) \ {0}. The intersection numbers of the subalgebra generated by the +adjacency hypermatrices of the nontrivial relations of X are given in the next theorem. +Theorem 3.3. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) +and X be the AST obtained from the action of ASLH(2, n). The following equations hold +for any a, b, c ̸= 0, 1 and a, b, c ̸= 0. +(1) AaAbAc = � +ℓ∈T\{1} pℓAℓ, where +pℓ = |{φ(c) : φ ∈ H and (∃ψ, τ ∈ H) [(1 − φ(c))τ(a) + φ(c) = ℓ = φ(c)ψ(b)]}| . +(2) AaAb cA = Aa cA Ab = cA AaAb = 0. +(3) aA bA Ac = � +ℓ∈T pℓ ℓA, where +pℓ = +���� +� +φ(c) : φ ∈ H and (∃ψ, τ ∈ H) +� +τ(a) +1 − φ(c) = ℓ = ψ(b) +φ(c) +������ . + +ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS +5 +(4) aA Ac bA = � +ℓ∈T pℓ ℓA, where +pℓ = |{ψ(b) : ψ ∈ H and (∃φ, τ ∈ H) [ψ(b)φ(c) = ℓ = τ(a) + ψ(b)]}| . +(5) Ac aA bA = � +ℓ∈T pℓ ℓA, where +pℓ = +���� +� +ψ(b) : ψ ∈ H and (∃φ, τ ∈ H) +� +ψ(b)(1 − φ(c)) = ℓ = τ(a)(φ(c) − 1) +φ(c) +������ . +(6) aA bA cA = � +ℓ∈T\{1} pℓAℓ + � +∈T p A, where +pℓ = q +���� +� +φ(c) : (∃ψ, τ ∈ H) +�τ(a) + φ(c) +φ(c) += d = −ψ(b) +φ(c) +������ , +p = |{ψ(b) : (∃φ, τ ∈ H) [τ(a) + ψ(b) + φ(c) = ]}| . +Proof. We prove only the third statement, as the other statements are shown similarly. With +Ri = aR, Rj = bR, and Rk = Rc, we determine the Rℓ such that the intersection number +pℓ +ijk is nonzero. If Rℓ =d R for some d ̸= 0, considering the viable w as in the the Principal +Regularity Condition from Definition 2.1 necessitates that φ(c)ψ(b) = 0 for some φ, ψ ∈ H, +which is impossible. If Rℓ = Rd for some d ̸= 0, 1, the Principal Regularity Conditions says +that the number of viable w, pℓ +ijk, is the number of vectors of the form (φ(c), 0)T with φ ∈ H +such that there are ψ and τ in H that satisfy +τ(a) +1 − φ(c) = ℓ = ψ(b) +φ(c) +□ +The succeeding theorem gives the intersection numbers pl +ijk of the ASTs obtained from +ASLH(2, q) whenever exactly one of Ri, Rj, and Rk is trivial. Here I1, I2, and I3 denote the +respective adjacency hypermatrices of the trivial relations R1, R2, and R3 of X. The proof, +similar to that of the proof of Theorem 3.3, is omitted. +Theorem 3.4. Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) +and X be the AST obtained from the action of ASLH(2, q). The following equations hold for +any a, b ̸= 0, 1 and a, b ̸= 0. +(1) I1AaAb = pI1, where +p1 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = 1]}|. +(2) AaI2Ab = p2I2, where +p2 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = τ(a) + ψ(b)]}|. +(3) AaAbI3 = p3I3, where +p3 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) + ψ(b) = 1]}|. +(4) I1Aa aA = I1 aA Aa = AaI2 aA = aA I2Aa = Aa aA I3 = aA AaI3 = 0. +(5) I1 aA bA = p∗I1, aA I2 bA = p∗I2, aA bA I3 = p∗I3, where +p∗ = q |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) = −ψ(b)]}| . +Here we consider the AST obtained from ASLH(k, n) for k ≥ 3, n a prime power, and +H a subgroup of Gal(GF(n)). The following theorem tells us that the AST obtained from +ASLH(k, n) is the same as the AST obtained from the subgroup AGLH(k, n) = AGL(k, n)⋊ +H of the affine semilinear group AΓL(k, n) whenever k ≥ 3. In particular, the parameters +of these ASTs have already been obtained in [3]. + +6 +D.V.A. BRIONES +Theorem 3.5. Let n = pα be a power of a prime p, q = pω with ω|α, and H = GalGF (q)(GF(n)). +Then the AST obtained from the action of ASLH(k, n) is equal to the AST obtained from +the action of AGLH(k, n). +Proof. Notice that if a group G and a subgroup K of G both act two-transitively on a set, +the orbits of G are unions of orbits of K. In particular, if G and K have the same number +of orbits, then these orbits are exactly the same. Thus, to prove the theorem, it suffices to +show that the ASTs obtained from AGLH(k, n) and ASLH(k, n) have the same size. By +Remark 1, it suffices to show that the two-point stabilizer ASLH(k, n)⃗0,⃗1 has the same orbits +as AGLH(k, n)⃗0,⃗1 on GF(n) \ {⃗0,⃗1}. +Indeed, the two-point stabilizers above are given by +ASLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ SL(k, n), φ ∈ H}, +and +AGLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ GL(k, n), φ ∈ H}. +Direct computation shows that the orbits of ASLH(k, n)⃗0,⃗1 have the following forms. +(1) One type of orbit has the form +{(φ(a), 0, . . . , 0)T : φ ∈ H}, +which consists of those vectors whose first coordinate is a Galois conjugate of an +element a ∈ GF(n) with a ̸= 0, 1. The other coordinates are 0. +(2) The remaining orbit is +(GF(n))k \ Span(⃗1), +consisting of the vectors linearly independent from ⃗1. +These are also the orbits of AGLH(k, n)⃗0,⃗1, completing the proof. +□ +References +1. J.M.P. +Balmaceda +and +D.V.A. +Briones, +Association +schemes +on +triples +over +few +vertices, +Matimyas +Matematika +45 +(2022), +13–26, +http://mathsociety.ph/matimyas/images/vol45/BalmacedaMatimyas.pdf. +2. +, Families of association schemes on triples from two-transitive groups (preprint), arXiv (2022), +https://arxiv.org/abs/2107.07753. +3. +, +A +survey +on +association +schemes +on +triples +(preprint), +arXiv +(2022), +https://arxiv.org/abs/2206.10500. +4. E. Bannai and T. Ito, Algebraic combinatorics I. Association schemes, Mathematics lecture note series, +no. 58, Benjamin/Cummings Pub. Co, San Francisco, 1984. +5. D.M. +Mesner +and +P. +Bhattacharya, +Association +schemes +on +triples +and +a +ternary +algebra, +Journal +of +Combinatorial +Theory, +Series +A +55 +(1990), +no. +2, +204–234, +https://www.sciencedirect.com/science/article/pii/0097316590900688. +6. +, A ternary algebra arising from association schemes on triples, Journal of Algebra 164 (1994), +no. 3, 595–613, https://www.sciencedirect.com/science/article/pii/S0021869384710817. +7. C.E. Praeger and P. Bhattacharya, Circulant association schemes on triples, New Zealand Journal of +Mathematics 52 (2021), 153–165, https://nzjmath.org/index.php/NZJMATH/article/view/106. + diff --git a/GdE0T4oBgHgl3EQfRQBK/content/tmp_files/load_file.txt b/GdE0T4oBgHgl3EQfRQBK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a4f803756c4295e1c69fb71b5e2c1c44f591ca86 --- /dev/null +++ b/GdE0T4oBgHgl3EQfRQBK/content/tmp_files/load_file.txt @@ -0,0 +1,223 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf,len=222 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='02204v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='CO] 5 Jan 2023 ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS DOM VITO A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' BRIONES Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Association schemes on triples (ASTs) are 3-dimensional analogues of classical association schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' If a group acts two-transitively on a set, the orbits of the action induced on the triple Cartesian product of that set yields an AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' By considering the actions of semidirect products of the affine special linear group ASL(k, n) with subgroups of the Galois group Gal(GF(n)), we obtain the sizes, third valencies, and intersection numbers of the ASTs obtained from subgroups of the affine special semilinear group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Introduction A classical association is an algebraic-combinatorial object with certain symmetry prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' These properties suffice to afford classical association with desirable structural char- acteristics and are pliant enough to allow classical association schemes to be applicable to several areas of mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' For example, the adjacency algebra of a classical association scheme is semisimple and, when the adjacency matrices define a distance-regular graph, the structure constants of this algebra can be expressed in terms of certain families of orthogonal polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' [4] Mesner and Bhattacharya introduced the notion of association schemes on triples (or ASTs), a ternary analogue for classical association schemes [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' An AST on a set Ω is a partition of the triple Cartesian product Ω × Ω × Ω subject to regularity requirements paralleling the symmetry conditions for classical association schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' In ASTs, the resulting adjacency hypermatrices produce a ternary algebra under a ternary product that extends the usual matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' However, the structural properties of ASTs remain unclear, partly due to the ternary adjacency algebra not being associative nor commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' As first steps in the investigation of ASTs, studies were conducted regarding analogues of identity and inverse elements [6], enumerations of ASTs over the smallest number of vertices [1], possible sources of ASTs such as those from group actions, two-graphs, designs, and other ASTs [5, 7, 3], as well as the intersection numbers of known families of ASTs [5, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' In particular, the actions of two-transitive groups yield ASTs [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The orbits of these actions are closely related to the parameters of the AST, providing their sizes, third valencies, and intersection numbers [5, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' In fact, [5] provides the sizes, third valencies, and intersection numbers of the ASTs obtained from the affine general linear group AGL(1, n) where n is a prime power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' This was extended in [2], wherein these parameters were obtained for the ASTs (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Briones, Corresponding author) Institute of Mathematics, University of the Philippines Diliman, 1101 Quezon City, Philippines E-mail address: dabriones@up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' algebraic combinatorics, ternary algebra, association scheme on triples MSC Classification: 05E30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 1 2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' BRIONES obtained from subgroups of the affine semilinear group AΓL(k, n) of the form AGL(k, n)⋊H), where k ≥ 1 and H ≤ Gal(GF(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Further work was done in [3], where these parameters were obtained from ASTs obtained from the affine special linear group ASL(2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' We extend this last result by determining the sizes, third valencies, and intersection num- bers of ASTs obtained from subgroups of the affine special semilinear group ASL(k, n) ⋊ Gal(GF(n)) of the form ASL(k, n) ⋊ H, where k ≥ 2, n is a prime power, and H is a subgroup of Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' In particular, we show that the ASTs obtained from ASLH(k, n) are the same as the ASTs obtained from AGLH(k, n) = AGL(k, n) ⋊ H for k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Preliminaries We define association schemes on triples, remark how ASTs arise from two-transitive groups, and review the actions of the affine special linear and affine special semilinear groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Association schemes on triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' We define association schemes on triples and men- tion how the parameters of an AST obtained from a two-transitive group are related to the group action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' [5, 7] Let Ω be a finite set with at least 3 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' An association scheme on triples (AST) on Ω is a partition X = {Ri}m i=0 of Ω × Ω × Ω with m ≥ 4 such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (1) For each i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , m}, there exists an integer n(3) i such that for each pair of distinct x, y ∈ Ω, the number of z ∈ Ω with (x, y, z) ∈ Ri is n(3) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (2) (Principal Regularity Condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=') For any i, j, k, l ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , m}, there exists a con- stant pl ijk such that for any (x, y, z) ∈ Rl, the number of w such that (w, y, z) ∈ Ri, (x, w, z) ∈ Rj, and (x, y, w) ∈ Rk is pl ijk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (3) For any i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , m} and any σ ∈ S3, there exists a j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , m} such that Rj = {(xσ(1), xσ(2), xσ(3)) : (x1, x2, x3) ∈ Ri}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (4) The first four relations are R0 = {(x, x, x) : x ∈ Ω}, R1 = {(x, y, y) : x, y ∈ Ω, x ̸= y}, R2 = {(y, x, y) : x, y ∈ Ω, x ̸= y}, and R3 = {(y, y, x) : x, y ∈ Ω, x ̸= y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The integer n(3) i is the third valency of Ri, and is the analogue of valency from classical association schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' By Conditions 1 and 3 of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='1 there are for each i the constants n(1) i = |{z ∈ Ω : (z, x, y) ∈ Ri}| and n(2) i = |{z ∈ Ω : (x, z, y) ∈ Ri}| independent of any pair of distinct x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Similarly, n(1) i is the first valency of Ri and n(2) i is the second valency of Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The trivial relations are R0, R1, R2 and R3 while the other relations are the nontrivial relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Further, the numbers pl ijk are called the intersection numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' ASTs arise naturally from the actions of two-transitive groups [5], mirroring how Schurian classical association schemes are induced by the actions of transitive groups [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Indeed, when a two-transitive group G acts on a set Ω, the orbits of the induced action on Ω×Ω×Ω is an AST [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Correspondences between the action and the parameters of the induced AST are summarized in the following remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Remark 1 ([5, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let G be a group acting two-transitively on a set Ω and let X be the AST induced by this action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' For any pair of distinct elements a, b ∈ Ω, the orbits of the two-point stabilizer Ga,b on Ω \\ {a, b} are in bijection with the nontrivial relations of the AST.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' As a consequence of this bijection, the sizes of these orbits are also the third valencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Affine special groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Given a prime power n and k ≥ 1, the affine special linear group ASL(k, n) is the semidirect product GF(n) ⋊ SL(k, n), where SL(k, n) is the group of invertible linear transformations on the k-dimensional vector space V over GF(n) of determinant 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Explicitly, the affine special linear group is the following group of maps from V to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' ASL(k, n) = {x �→ Ax + b : A ∈ SL(k, n), b ∈ V } .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Similarly, the affine special semilinear group ASL(k, n) ⋊ Gal(GF(n)) is the semidirect product of the affine special linear group ASL(k, n) with the Galois group Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Explicitly, the affine special semilinear group is the following group of maps from V to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' ASL(k, n) ⋊ Gal(GF(n)) = {x �→ Aφ(x) + b : A ∈ SL(k, n), b ∈ GF(n), φ ∈ Gal(GF(n))} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' ASTs from subgroups of the affine special semilinear group In this section we generalize work done in [3] by obtaining the sizes, third valencies, and intersection numbers of ASTs obtained from the actions of subgroups of the affine special semilinear group of the form ASLH(k, n) = ASL(k, n) ⋊ H, where k ≥ 2, n = pα a power of a prime number p, and H a subgroup of Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' We obtain the sizes and third valencies of these ASTs by obtaining a two-point stabilizer of ASLH(k, n) and then determining its orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Finally, we obtain the intersection numbers of these ASTs through explicit orbit computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' For ease of discussion, we fix the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let n = pα be a power of a prime p, k ≥ 2, V be the k-dimensional vector space over GF(n), H be a subgroup of Gal(GF(n)), and X be the AST obtained from ASLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' For a ∈ GF(n), let ⃗a = (a, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , 0)T ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Further, for (u, v, w) ∈ V × V × V , let [(u, v, w)] ∈ X denote the orbit of (u, v, w) under ASLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' We begin with the case where k = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' To determine the size and third valencies of X, we exploit the relationships between these parameters and the orbits of a two-point stabilizer of ASLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector space V over GF(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The two-point stabilizer ASLH(2, n)⃗0,⃗1 has the following orbits on V \\{⃗0,⃗1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (1) There are −2 + ω α � α ω β=1 qgcd ( α ω ,β) orbits of the form � ⃗ φ(a) : φ ∈ H � , a ̸= 0, 1 each of size degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (2) There are −1 + ω α � α ω β=1 qgcd ( α ω ,β) orbits of the form � (c, φ(a))T : c ∈ GF(n), φ ∈ H � , a ̸= 0 each of size n degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' As a consequence of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='1, we obtain the sizes and third valencies of the ASTs obtained from ASLH(2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) and X be the AST obtained from the action of ASLH(2, n) on the 2-dimensional vector 4 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' BRIONES space V over GF(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Then X has −3 + 2 � ω α � α ω β=1 qgcd ( α ω ,β)� nontrivial relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' There are −2 + ω α � α ω β=1 qgcd ( α ω ,β) nontrivial relations of the form Ra = {[(⃗0,⃗1,⃗a)]}, a ̸= 0, 1, with corresponding third valency degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The remaining −1+ ω α � α ω β=1 qgcd ( α ω ,β) nontrivial relations of X are of the form aR = {[(⃗0,⃗1, (0, a)T)]}, a ̸= 0, with corresponding third valency n degGF (q)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The two-point stabilizer is ASLH(2, n)⃗0,⃗1 = {(x, y)T �→ � 1 c 0 1 � (φ(x), φ(y))T : c ∈ GF(n), φ ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Direct computation shows that the orbits of ASLH(2, n)⃗0,⃗1 have the following forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (1) The first type of orbit has the form {(φ(a), 0)T : φ ∈ H}, which consists of those vectors whose second coordinate is 0 and whose first coordinate is a Galois conjugate of an element a ∈ GF(n) with a ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (2) The remaining orbits are of the form {(x, φ(a))T : x ∈ GF(n), φ ∈ H}, which consists of those vectors whose second coordinate is a Galois conjugate of an element a ∈ GF(n) with a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The sizes of these orbits follow directly from the Fundamental Theorem of Galois Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The number of orbits of each type are then obtained through the Fundamental Theorem of Galois Theory and a straightforward application of Burnside’s Orbit Counting Theorem to the action of Gal(GF(n)) on GF(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' □ For notational convenience, let Aa denote the adjacency hypermatrix corresponding to the relation Ra whenever a ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Similarly, let aA denote the adjacency hypermatrix corresponding to the relation aR whenever a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Further, let T be a transversal of the orbits of H on GF(n) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The intersection numbers of the subalgebra generated by the adjacency hypermatrices of the nontrivial relations of X are given in the next theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) and X be the AST obtained from the action of ASLH(2, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The following equations hold for any a, b, c ̸= 0, 1 and a, b, c ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (1) AaAbAc = � ℓ∈T\\{1} pℓAℓ, where pℓ = |{φ(c) : φ ∈ H and (∃ψ, τ ∈ H) [(1 − φ(c))τ(a) + φ(c) = ℓ = φ(c)ψ(b)]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (2) AaAb cA = Aa cA Ab = cA AaAb = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (3) aA bA Ac = � ℓ∈T pℓ ℓA, where pℓ = ���� � φ(c) : φ ∈ H and (∃ψ, τ ∈ H) � τ(a) 1 − φ(c) = ℓ = ψ(b) φ(c) ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' ASSOCIATION SCHEMES ON TRIPLES FROM AFFINE SPECIAL SEMILINEAR GROUPS 5 (4) aA Ac bA = � ℓ∈T pℓ ℓA, where pℓ = |{ψ(b) : ψ ∈ H and (∃φ, τ ∈ H) [ψ(b)φ(c) = ℓ = τ(a) + ψ(b)]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (5) Ac aA bA = � ℓ∈T pℓ ℓA, where pℓ = ���� � ψ(b) : ψ ∈ H and (∃φ, τ ∈ H) � ψ(b)(1 − φ(c)) = ℓ = τ(a)(φ(c) − 1) φ(c) ������ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (6) aA bA cA = � ℓ∈T\\{1} pℓAℓ + � \uf6be∈T p\uf6be \uf6beA, where pℓ = q ���� � φ(c) : (∃ψ, τ ∈ H) �τ(a) + φ(c) φ(c) = d = −ψ(b) φ(c) ������ , p\uf6be = |{ψ(b) : (∃φ, τ ∈ H) [τ(a) + ψ(b) + φ(c) = \uf6be]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' We prove only the third statement, as the other statements are shown similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' With Ri = aR, Rj = bR, and Rk = Rc, we determine the Rℓ such that the intersection number pℓ ijk is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' If Rℓ =d R for some d ̸= 0, considering the viable w as in the the Principal Regularity Condition from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='1 necessitates that φ(c)ψ(b) = 0 for some φ, ψ ∈ H, which is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' If Rℓ = Rd for some d ̸= 0, 1, the Principal Regularity Conditions says that the number of viable w, pℓ ijk, is the number of vectors of the form (φ(c), 0)T with φ ∈ H such that there are ψ and τ in H that satisfy τ(a) 1 − φ(c) = ℓ = ψ(b) φ(c) □ The succeeding theorem gives the intersection numbers pl ijk of the ASTs obtained from ASLH(2, q) whenever exactly one of Ri, Rj, and Rk is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Here I1, I2, and I3 denote the respective adjacency hypermatrices of the trivial relations R1, R2, and R3 of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The proof, similar to that of the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='3, is omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, H = GalGF (q)(GF(n)) and X be the AST obtained from the action of ASLH(2, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The following equations hold for any a, b ̸= 0, 1 and a, b ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (1) I1AaAb = pI1, where p1 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = 1]}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (2) AaI2Ab = p2I2, where p2 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a)ψ(b) = τ(a) + ψ(b)]}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (3) AaAbI3 = p3I3, where p3 = |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) + ψ(b) = 1]}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (4) I1Aa aA = I1 aA Aa = AaI2 aA = aA I2Aa = Aa aA I3 = aA AaI3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (5) I1 aA bA = p∗I1, aA I2 bA = p∗I2, aA bA I3 = p∗I3, where p∗ = q |{ψ(b) : ψ ∈ H and (∃τ ∈ H)[τ(a) = −ψ(b)]}| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Here we consider the AST obtained from ASLH(k, n) for k ≥ 3, n a prime power, and H a subgroup of Gal(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The following theorem tells us that the AST obtained from ASLH(k, n) is the same as the AST obtained from the subgroup AGLH(k, n) = AGL(k, n)⋊ H of the affine semilinear group AΓL(k, n) whenever k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' In particular, the parameters of these ASTs have already been obtained in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 6 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' BRIONES Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Let n = pα be a power of a prime p, q = pω with ω|α, and H = GalGF (q)(GF(n)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Then the AST obtained from the action of ASLH(k, n) is equal to the AST obtained from the action of AGLH(k, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Notice that if a group G and a subgroup K of G both act two-transitively on a set, the orbits of G are unions of orbits of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' In particular, if G and K have the same number of orbits, then these orbits are exactly the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Thus, to prove the theorem, it suffices to show that the ASTs obtained from AGLH(k, n) and ASLH(k, n) have the same size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' By Remark 1, it suffices to show that the two-point stabilizer ASLH(k, n)⃗0,⃗1 has the same orbits as AGLH(k, n)⃗0,⃗1 on GF(n) \\ {⃗0,⃗1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Indeed, the two-point stabilizers above are given by ASLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ SL(k, n), φ ∈ H}, and AGLH(k, n)⃗0,⃗1 = {v �→ Aφ(v) : A ∈ GL(k, n), φ ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Direct computation shows that the orbits of ASLH(k, n)⃗0,⃗1 have the following forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (1) One type of orbit has the form {(φ(a), 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , 0)T : φ ∈ H}, which consists of those vectors whose first coordinate is a Galois conjugate of an element a ∈ GF(n) with a ̸= 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' The other coordinates are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' (2) The remaining orbit is (GF(n))k \\ Span(⃗1), consisting of the vectors linearly independent from ⃗1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' These are also the orbits of AGLH(k, n)⃗0,⃗1, completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' □ References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Balmaceda and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Briones, Association schemes on triples over few vertices, Matimyas Matematika 45 (2022), 13–26, http://mathsociety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='ph/matimyas/images/vol45/BalmacedaMatimyas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , Families of association schemes on triples from two-transitive groups (preprint), arXiv (2022), https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='org/abs/2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='07753.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , A survey on association schemes on triples (preprint), arXiv (2022), https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='org/abs/2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='10500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Bannai and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Ito, Algebraic combinatorics I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Association schemes, Mathematics lecture note series, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 58, Benjamin/Cummings Pub.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Co, San Francisco, 1984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Mesner and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Bhattacharya, Association schemes on triples and a ternary algebra, Journal of Combinatorial Theory, Series A 55 (1990), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 2, 204–234, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='com/science/article/pii/0097316590900688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' , A ternary algebra arising from association schemes on triples, Journal of Algebra 164 (1994), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 3, 595–613, https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='sciencedirect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='com/science/article/pii/S0021869384710817.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Praeger and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content=' Bhattacharya, Circulant association schemes on triples, New Zealand Journal of Mathematics 52 (2021), 153–165, https://nzjmath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='org/index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} +page_content='php/NZJMATH/article/view/106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GdE0T4oBgHgl3EQfRQBK/content/2301.02204v1.pdf'} diff --git a/GtAzT4oBgHgl3EQfHftK/content/tmp_files/2301.01045v1.pdf.txt b/GtAzT4oBgHgl3EQfHftK/content/tmp_files/2301.01045v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2d304541fa899b837610a7ded63cae535606ee68 --- /dev/null +++ b/GtAzT4oBgHgl3EQfHftK/content/tmp_files/2301.01045v1.pdf.txt @@ -0,0 +1,2577 @@ +Risk-Averse MDPs under Reward Ambiguity +Haolin Ruan +School of Data Science, City University of Hong Kong, Kowloon Tong, Hong Kong +haolin.ruan@my.cityu.edu.hk +Zhi Chen +Department of Management Sciences, College of Business, City University of Hong Kong, Kowloon Tong, Hong Kong +zhi.chen@cityu.edu.hk +Chin Pang Ho +School of Data Science, City University of Hong Kong, Kowloon Tong, Hong Kong +clint.ho@cityu.edu.hk +We propose a distributionally robust return-risk model for Markov decision processes (MDPs) under risk and +reward ambiguity. The proposed model optimizes the weighted average of mean and percentile performances, +and it covers the distributionally robust MDPs and the distributionally robust chance-constrained MDPs +(both under reward ambiguity) as special cases. By considering that the unknown reward distribution lies +in a Wasserstein ambiguity set, we derive the tractable reformulation for our model. In particular, we show +that that the return-risk model can also account for risk from uncertain transition kernel when one only +seeks deterministic policies, and that a distributionally robust MDP under the percentile criterion can be +reformulated as its nominal counterpart at an adjusted risk level. A scalable first-order algorithm is designed +to solve large-scale problems, and we demonstrate the advantages of our proposed model and algorithm +through numerical experiments. +1. +Introduction +Markov decision processes (MDPs) provide a powerful modeling framework for sequential decision- +making problems and reinforcement learning in stochastic dynamic environments (Puterman 2014). +Obtaining the model parameters of MDPs that perfectly reflect the environments, however, has +always been a challenge in practice, as these parameters are estimated from limited data that are +potentially contaminated (Mannor et al. 2007). Moreover, these parameters, such as transition +kernel and reward function, are often time-dependent or even uncertain, but they are approximated +as fixed values in an overly simplified setting (Mannor et al. 2016). Therefore, the output policies +of MDPs are often disappointing in practice. +Robust MDPs address the aforementioned issues of parameter ambiguity, by allowing the +unknown values of transition kernels and reward functions to lie in a given ambiguity set (Behza- +dian et al. 2021, Chen et al. 2019, Clement and Kroer 2021a, Delgado et al. 2016). Then, robust +MDPs seek for policies that maximize the worst-case expected return over all transition kernels +1 +arXiv:2301.01045v1 [cs.LG] 3 Jan 2023 + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +2 +and reward functions in the ambiguity sets. By specifying ambiguity sets that contain the unknown +transition kernels with high confidence, the optimal policies of robust MDPs are robust to param- +eter ambiguity (Iyengar 2005). +In this paper, we focus on the case where the reward function is ambiguous, which sometimes +is referred to as imprecise-reward MDPs (Alizadeh et al. 2015, Regan and Boutilier 2010, 2011a,b, +2012). This particular setting is also closely related to imitation learning, which trains an agent to +learn a certain behavior of an expert, while only some demonstrated trajectories of her is available +(Chen et al. 2020, Ho and Ermon 2016, Osa et al. 2018, Rashidinejad et al. 2021). When applying +inverse reinforcement learning approach to learn the reward function that completely represents +the expert’s preference (Brown et al. 2020, Choi and Kim 2012, Ng et al. 2000), the yielded policies, +which suffer from reward ambiguity, may perform poorly in practice. +To handle reward ambiguity, we utilize techniques from distributionally robust optimization +(DRO) (Derman and Mannor 2020) and distributionally robust chance-constrained program (Chen +et al. 2007, Postek et al. 2018), assuming that the true reward distribution resides in an ambiguity +set. This approach does not require the reward function to be precisely specified. Instead, only +the descriptions of common distribution information such as support, moments and shape in the +ambiguity set are needed, which are often much easier to be obtained/estimated (Hanasusanto +et al. 2015, 2017, Zymler et al. 2013). In this paper, we consider a Wasserstein ambiguity set for our +distributionally robust models as in Abdullah et al. (2019), Calafiore and Ghaoui (2006), Xie (2021). +Unlike phi-divergence ambiguity sets which may contain too extreme member distributions, the +closeness between points in the support set is incorporated in Wasserstein sets, thus their member +distributions may be more reasonable (Gao and Kleywegt 2022); on the other hand, Wasserstein +sets are often a better choice than moment-based ambiguity sets when the number of samples is +too small to obtain a reliable estimation on moments (Yang 2020). We choose Wasserstein sets +for these reasons, although other types of ambiguity sets such as nested ambiguity sets (Xu and +Mannor 2010, 2012) and the ambiguity sets based on Prohorov metric (Erdo˘gan and Iyengar 2006) +are also considered in literature. For our distributionally robust chance-constrained MDPs, we will +furthermore show its equivalence with the nominal counterparts with an adjusted risk level. To the +best of our knowledge, this is the first result in MDPs that establishes the mutual transformation +between distributional ambiguity and risk. +Our return-risk model (RR) is a risk-averse MDP model that not only takes into account reward +ambiguity, but also considers both the average and risk of the return. MDPs that minimize the risk +of the return instead of the expected cost are called risk-aware MDPs (also called risk-sensitive or +risk-averse MDPs) (Ahmadi et al. 2021, B¨aauerle and Rieder 2017, Carpin et al. 2016, Haskell and +Jain 2015, Huang and Haskell 2017). In risk-aware optimization, the objective function is taken as + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +3 +a risk measure, such as value-at-risk (VaR) (Delage and Mannor 2007, 2010, Gilbert et al. 2017), +conditional value-at-risk (CVaR) (B¨auerle and Ott 2011, Chow et al. 2017, Huang and Guo 2016) +and other spectral risk measures (B¨auerle and Glauner 2021), and variants of expected utility +(Bernard et al. 2022, Jaimungal et al. 2022, Pflug and Wozabal 2007). +Among these risk measures, VaR and CVaR are arguably the most popular ones and have +attracted the attention of many researchers (B¨auerle and Ott 2011, Chow et al. 2017, Delage and +Mannor 2007, 2010, Gilbert et al. 2017, Huang and Guo 2016). By using CVaR, one aims to give a +precise depiction of the extreme tail of the distribution (of the uncertain rewards), while VaR does +not reflect the extreme scenerios exceeding VaR. It is well-known that CVaR is a coherent risk +measure, which can be efficiently optimized by convex optimization tools (Chen and Xie 2021); in +contrast, VaR is a more challenging risk measure because it is not a coherent one. +One remarkable advantage of VaR is its stability of estimation (especially under fat-tailed reward +distribution (Sarykalin et al. 2008)), which is particularly important under data-driven settings +where the number of samples are limited and decision makers evaluate models based on their +out-of-sample performances (Bertsimas and Thiele 2006, van de Berg et al. 2022, Zheng et al. +2016). To demonstrate, we provide an example where we consider a one-step MDP with only 1 +state s and 2 actions a1 and a2 (Sutton and Barto 2018). In this one-step MDP, the decision +maker only makes one decision in each episode, and she aims to maximize her VaR/CVaR of +rewards for these episodes. We consider uncertain rewards ˜rs,a1 ∼ Pt-dist and ˜rs,a2 = ˜rs,a1 + ρ|s| +where Pt-dist is a Student’s t-distribution and we vary its degree of freedom δ ∈ {2,3,4}. We set +the shift ratios ρ = {0.05i}i∈[5], and for testing the estimation accuracy w.r.t. VaR (resp., CVaR) +(where we choose the risk threshold 10%), we set the shift quantity s as Pt-dist-VaR0.1[˜rs,a1] (resp., +Pt-dist-CVaR0.1[˜rs,a1]), where both risk measures can be efficiently calculated (see Appendix B for +more details). We evaluate the decision maker’s accuracy rate as the proportion of testing samples +where she has chosen the action with a higher VaR/CVaR of rewards (i.e., action a2); for each +pair of accuracy rate and shift ratio, following Yamai et al. (2002), 1000 random reward samples +for each state-action pair are available for the decision maker, and we test her accuracy rate based +on 10000 testing samples. +As illustrated in Figure 1, the accuracy rate increases with the shift ratio ρ. As δ decreases, F +becomes more fat-tailed, and the accuracy rate of VaR is remarkably higher than that of CVaR, +which indicates that the statistical inference on VaR would be more accurate than on CVaR. +Therefore, VaR may be a more preferable choice when only small sample sets are available. +Our return-risk model is motivated by the soft-robust criterion/model, which optimizes a convex +combination of the mean and a robust performance in the optimization literature (Ben-Tal et al. +2010). MDPs with soft-robustness are also popular in recent years, where decision makers aim to + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +4 +0.05 +0.10 +0.15 +0.20 +0.25 +Shift ratio +0.6 +0.7 +0.8 +0.9 +1.0 +Accuracy rate +VaR +CVaR +0.05 +0.10 +0.15 +0.20 +0.25 +Shift ratio +0.6 +0.7 +0.8 +0.9 +1.0 +VaR +CVaR +0.05 +0.10 +0.15 +0.20 +0.25 +Shift ratio +0.6 +0.7 +0.8 +0.9 +1.0 +VaR +CVaR +Figure 1 +The accuracy rates of the decision maker choosing the correct action (so that the VaR/CVaR of her +rewards is maximized): δ = 4 (left), δ = 3 (middle) and δ = 2 (right). +maximize a weighted average of the mean and percentile performances (Brown et al. 2020, Lobo +et al. 2020). Unlike these existing soft-robust MDPs, however, the proposed return-risk model is +fundamentally different in two aspects: first, these existing soft-robust models have no consideration +for reward ambiguity, while we utilize distributionally robustness to account for reward ambiguity, +by which we can hedge against the most adversarial realization of the distribution of rewards +(within the ambiguity set), thus our model is more robust to reward uncertainty (Chen et al. 2019, +Xu and Mannor 2010); second, we choose VaR as the risk measure which has a direct interpretation +to percentile performances, and, as illustrated above, tends to be more advantageous in data-driven +optimization. +Our work concentrates on model-based setting, where our proposed models are motivated by +the classical (dual formulation of) nominal MDPs (Puterman 2014) and the chance-constrained +MDPs (Delage and Mannor 2010). It is worth noting that, beyond model-based setting, there are +other inspiring and innovative researches on robust reinforcement learning, such as robust TDC +algorithms and robust Q-learning (Roy et al. 2017, Wang and Zou 2021), robust policy gradient +(Wang and Zou 2022), least squares policy iteration (Lagoudakis and Parr 2003) and sample +complexity analysis (Panaganti and Kalathil 2022). Note that, though model-free reinforcement +learning can be used to learn satisfactory policies for complex environment, the requirement of +large amounts of interaction (with environment) may render the learning process slow (Kaiser et al. +2019), while high sample efficiency is one strong advantage of model-based learning (Sutton and +Barto 2018). We also note that MDPs with transition kernel ambiguity is another active research +line where distributionally robustness is widely employed (Clement and Kroer 2021b, Shapiro 2016, +2021, Xu and Mannor 2012). +We may summarize our contributions as follows (and we also compare our contributions to those +of related works in Table 2 in Appendix I). + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +5 +(i) We show that the distributionally robust model of optimizing expected rewards can be +reformulated as a convex conic program, which is equivalent to the nominal MDP with a convex +regularization in the objective function. +(ii) For distributionally robust chance-constrained MDPs (DCC), we show that it can be refor- +mulated as nominal chance-constrained MDPs at adjusted risk levels. This observation bridges the +gap between risk and parameter ambiguity. +(iii) Combining the proposed models in (i) and (ii), we propose the return-risk MDP that +maximizes the weighted average of the expectation and VaR of reward (both under distributionally +robustness to reward uncertainty), which is flexible and can perform well under the criteria of mean +and percentile returns. +(iv) When only considering deterministic policies, we show that our return-risk model can also +account for risk from uncertain transition kernel, and we derive its equivalent reformulation as a +mixed-integer second-order cone program (MISOCP). +(v) To solve the proposed return-risk model, we design a first-order method that is more scalable +than the MOSEK solver, thus is faster with large-size problems. +(vi) In the simulation and empirical experiments, we adopt a data-driven setting, where the +decision maker aims at maximizing the expectation and VaR of the random reward. We compare +the performances of distributionally robust MDPs (DRMDPs), DCC, RR, robust MDPs (RMDPs) +(Delage and Mannor 2010) and BROIL (Brown et al. 2020), and results show that the third +one performs the best under both expectation and different VaR’s (with risk thresholds 5%, 10% +and 15%), which showcases its advantages and adjustability to the decision makers’ changeable +preferences between return and risk. +The remainder of this paper is organized as follows. We introduce the background in Section 2. +In Sections 3 and 4, we study DRMDPs as well as the DCC model, respectively, and we derive +their tractable reformulations. Combining these proposed models, we propose the RR model in +Section 5. The designed first-order algorithm for the RR model is detailed in Section 6. We compare +the performances of DRMDP, DCC, RR, RMDP and BROIL, and demonstrate the advantage of +our proposed algorithm in Section 7. Conclusion is drawn in Section 8. +2. +Background +We consider an infinite-horizon MDP with a finite state space S = {1,··· ,S} and a finite action +space A = {1,··· ,A}. Let P ∈ RS×A×S be the transition probability kernel such that ps,a,s′ is +denoted to be the transition probability of transiting to state s′ ∈ S when action a ∈ A is chosen +in state s ∈ S; thus, ps,a ∈ ∆S is the transition probability distribution for every (s,a) ∈ S × A. +Given the state-action pair (s,a), an agent will receive an expected reward rs,a ∈ R. To simplify +our notation, we denote the reward function as a vector r = {rs,a}(s,a)∈S×A. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +6 +We seek for the optimal stationary randomized policy π = {πs}s∈S with πs ∈ ∆A for all s ∈ S, +where an action a ∈ A will be taken in state s ∈ S with probability πs,a. A nominal MDP that +maximizes the expected reward can be formulated (Puterman 2014) as +ℓN = max +x∈X r⊤x, +(1) +where the feasible set X is given by X = +� +x ∈ RSA ++ +�� (E − γ · ¯P )x = p0 +� +. Here the coefficient +matrices E = diag(e⊤,··· ,e⊤) ∈ RS×SA with S all-ones vectors e ∈ RA and ¯P = (¯p1,··· , ¯pS)⊤ ∈ +RS×SA with ¯ps = {ps′,a,s}(s′,a)∈S×A for all s ∈ S. For each (s,a) ∈ S × A, we denote the sth sub- +vector of x as xs = {xi}i∈{(s−1)A+1,··· ,sA}; its ath component xs,a can be interpreted as the total +discounted probability one occupying state s and choosing action a when applying the policy +π⋆ +s,a = x⋆ +s,a/(� +a∈A x⋆ +s,a) ∀(s,a) ∈ S × A (Puterman 2014)1. We have a discount factor γ ∈ (0,1) and +the initial distribution p0 ∈ RS +++ of the initial states. Problem (1) is a linear program that can be +efficiently solved by simplex method and interior-point method (Nocedal and Wright 2006). One +can also compute the optimal policy efficiently by applying value iteration or policy iteration to +solve the associated Bellman equation of this problem (Bertsekas and Tsitsiklis 1995, Puterman +2014). +The nominal MDP (1) does not account for uncertainty in either rewards or transition kernel. +To account for reward uncertainty, Delage and Mannor (2010) assume that the random reward +vector ˜r follows a known Gaussian distribution P and propose a chance-constrained MDP model +as follows: +ℓCC(ε) = +� +� +� +� +� +� +� +max y +s.t. P[˜r⊤x ≥ y] ≥ 1 − ε +x ∈ X, y ∈ R. +(2) +In fact, the above chance-constrained model maximizes the VaR (at the risk level 1 − ε) of the +reward with respect to the distribution P. Since P is assumed Gaussian, by theorem 10.4.1 in +Pr´ekopa (2013), one can reformulate problem (2) as a second-order cone program as follows: +ℓCC(ε) = max +x∈X EP[˜r⊤x] − ∥F−1(1 − ε)Σ1/2x∥2, +where F−1(·) is the inverse of the cumulative density function of the Gaussian distribution P +and Σ is the covariance matrix of P. Second-order cone programs allow efficient solutions by +state-of-the-art commercial solvers such as CPLEX, Gurobi and MOSEK (see, e.g., Ben-Tal and +Nemirovski (2001)). Despite its tractability, the chance-constrained MDP (2) requires the precise +underlying reward distribution as input. Moreover, the above reformulation does not hold for +generic distribution P. +1 By Puterman (2014), any x ∈ X admits such interpretation, thus we can retrieve our policies of all the proposed +models in this paper in this way. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +7 +3. +Distributionally Robust MDPs +In many real-world situations, the true distribution of the uncertain reward is hard (if not impossi- +ble) to obtain. Instead, we may have some firm knowledge, such as moments and shape about it. As +one of the most efficacious treatments for such situations, the DRO approach models uncertainty +as a random variable governed by an unknown probability distribution residing in an ambiguity +set. Facing distributional ambiguity, a decision maker seeks for solutions that hedge against the +most adversarial distribution from within the ambiguity set. To be specific, in our context, we +assume that the true distribution of the uncertain reward resides in a Wasserstein ball of radius +θ ≥ 0 around some reference distribution ˆP: +F(θ) = {P ∈ P(RSA) | dW +� +P, ˆP +� +≤ θ}. +(3) +Here P(RSA) is the set of all probability distributions on RSA, and the Wasserstein distance +between two distributions P1 and P2, equipped with a general norm ∥ · ∥ in RSA, is given by +dW (P1,P2) = infP∈Q(P1,P2) EP[∥˜r1 − ˜r2∥], where Q(P1,P2) is the set of all joint distributions with +marginal distributions P1 and P2 that govern ˜r1 and ˜r2, respectively. +The random parameter in the nominal MDP (1) is the expectation of reward, which in practice, +is often estimated by the average of historical samples. However, when the sample size is small, +such a sample average is not close to the expectation but rather, is known to be optimistically +biased (see, e.g., Smith and Winkler (2006)). Hence, the nominal MDP (1) based on samples +may yield an unsatisfactory policy that does not perform well out-of-sample. For this reason, a +possible alternative is to maximize instead the worst-case expected reward as in the following +distributionally robust MDP: +ℓDRMDP(θ) = max +x∈X +inf +P∈F(θ)EP[˜r⊤x]. +(4) +The following proposition offers an equivalent conic program for (4). +Proposition 1. The distributionally robust MDP (4) can be reformulated a conic program +ℓDRMDP(θ) = max +x∈X EˆP[˜r⊤x] − θ · ∥x∥∗. +It is not hard to observe that the distributionally robust MDPs can be viewed as a convex reg- +ularization of the nominal MDP (4) under the reference distribution ˆP. In particular, the convex +regularizing term in the distributionally robust MDP is θ∥x∥∗, which is sized by the Wasserstein +radius θ. Interestingly, we have also found that an (distributionally) optimistic MDP can be refor- +mulated as a reverse conic program with a (concave) regularization term −θ∥x∥∗. We relegate this +result to Appendix D. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +8 +Figure 2 +Values of ε with respect to different θ’s: ε = 0.05 (left), ε = 0.1 (middle), and ε = 0.15 (right). +We remark that, problem (4) is indeed a special case of the robust optimization problem consid- +ered in Jaimungal et al. (2022), where we consider the expected utility framework. Compared to the +policy gradient methods provided in Jaimungal et al. (2022) where convergence is not established, +we have derived its equivalent reformulation as a tractable conic program which can be efficiently +solved by state-of-the-art commercial solvers such as Gurobi, Mosek and CPLEX, and can also +be seamlessly incorporated in the tractable reformulation of our proposed return-risk model in +Section 5. +4. +Distributionally Robust Chance-Constrained MDPs +In this section, we turn from optimizing the expectation of reward to its tailed performance, by +exploring chance-constrained MDPs. In particular, we still consider Wasserstein ambiguity sets (3) +to account for distributional ambiguity, meanwhile specifying the reference distribution ˆP and the +norm ∥ · ∥ in the definition of the Wasserstein distance. +For the former, we focus on an elliptical reference distribution ˆP = P(µ,Σ,g) +2 throughout this +section, whose probability density distribution is given by f(r) = k · g +� 1 +2(r − µ)⊤Σ−1(r − µ) +� +, +where k is a positive normalization scalar, µ is a mean vector, Σ is a positive definite matrix and g +is a generating function. We emphasize that this assumption on ˆP is mild as this is only the center +of the ambiguity set. In particular, our proposed distributionally robust chance-constrained MDPs +can account for all types of distributions (as long as they are inside the ambiguity set) and they are +not restricted to be all elliptical. As we shall see, such specifications lead to tractable reformulation +of our proposed models. Preliminaries on elliptical distributions are relegated to Appendix C. +For the latter, we adopt the Mahalanobis norm associated with the positive definite matrix Σ, +captured by ∥x∥Σ = +√ +x⊤Σ−1x. Note that the dual norm of a Mahalanobis norm ∥ · ∥Σ is another +Mahalanobis norm ∥ · ∥Σ−1 that is defined by the inverse matrix Σ−1. +2 Note that results in Section 3 hold for a general reference distribution. + +1e-03 +8e-04 +6e-04 +4e-04 +2e-04 +0e+00 +0.040 +0.045 +0.0501e-03 +8e-04 +6e-04 +4e-04 +2e-04 +0e+00 +0.085 +0.090 +0.095 +0.1001e-03 +8e-04 +6e-04 +4e-04 +2e-04 +0e+00 +0.130 +0.135 +0.140 +0.145 +0.150Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +9 +In a distributionally robust chance-constrained MDP, we hope that even in the worst-case, with +a high confidence the reward is no less than a lower bound, and we aim at maximizing such a lower +bound by solving +ℓDCC(θ,ε) = +� +� +� +� +� +� +� +� +� +max y +s.t. +inf +P∈F(θ)P[˜r⊤x ≥ y] ≥ 1 − ε +x ∈ X, y ∈ R. +(5) +Quite notably, the worst-case chance constraint in the pessimistic chance-constrained MDP (5) is +equivalent to a nominal chance constraint in (2) with a higher risky level. +Lemma 1. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an ellip- +tical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm +associated with the positive definite matrix Σ. The distributionally robust chance constraint +∀ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε +(6) +is satisfiable if and only if P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ε, where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ that can +be computed via bisection method which searches for the smallest η ≥ Φ−1(1 − ε) that satisfies +η(Φ(η) − (1 − ε)) − +� η2/2 +(Φ−1(1−ε)) +2/2 kg(z)dz ≥ θ. +Equipped with Lemma 1, it then turns out that the distributionally robust chance-constrained +MDP (5) is equivalent to a nominal chance-constrained MDP (2) at a higher risky level. Conse- +quently, the distributionally robust chance-constrained MDP (5) can be reformulated into a conic +program, or more precisely, a second-order cone program owing to our choice of the Mahalanobis +norm. +Proposition 2. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an +elliptical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis +norm associated with the positive definite matrix Σ. If the risk threshold satisfies ε < 0.5, then the +distributionally robust chance-constrained MDP (5) is equivalent to the second-order cone program +ℓDCC(θ,ε) = max +x∈X µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2, +where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ being the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) − +� η2/2 +(Φ−1(1−ε)) +2/2 kg(z)dz ≥ θ. +Similar to the distributionally robust MDPs in Section 3, the distributionally robust chance- +constrained MDPs also admit an optimistic counterpart, which is equivalent to the nominal chance- +constrained MDPs with a larger risk threshold. We relegate this result to Appendix E. +To conclude this section, we present in Figure 2 the relations between ε and ε. Indeed, for any +fixed ε, there is a one-to-one correspondence between the risk threshold ε and the Wasserstein +radius θ. Following from this fact, for the chance-constrained model in our numerical experiments +(Section 7), we only calibrate the risk threshold rather than the Wasserstein radius. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +10 +5. +Return-Risk MDP +For rational decision makers, two types of rewards are their chief concerns: the average and the +worst-case rewards. However, the risk-averse models often can not achieve decent average return +on which the model put no emphasis (Carpin et al. 2016, Delage and Mannor 2010, Jiang and +Powell 2018). To take both concerns into considerations, we leverage the established DRMDPs and +DCC model in Sections 3 and 4 as ingredients and propose the return-risk MDP that maximizes +the weighted average of the worst-case expectation and VaR of reward as follows: +ℓRR(α,θ,ε) = max +x∈X α inf +P∈F(θ)EP[˜r⊤x] + (1 − α) +inf +P∈F′(θ)P-VaRε[˜r⊤x]. +(7) +Here the Wasserstein ball F(θ) is assumed equipped with a general reference distribution and an +L2-norm in the definition of the Wasserstein distance, while an elliptical reference distribution +ˆP = P(µ,Σ,g) and a Mahalanobis norm associated with the positive definite matrix Σ are assumed for +F ′(θ). It is not hard to see that the return-risk MDP (7) takes the distributionally robust MDP (4) +and the distributionally robust chance-constrained MDP (5) in as special cases by varying ε, θ and +α ∈ {0,1}. Furthermore, by choosing a fractional α, the return-risk model enables one to tailor a +balance between risk and return. Proposition 3 below provides an equivalent second-order cone +program for the return-risk MDP (7) under these assumptions. +Proposition 3. Suppose in (7) the Wasserstein ball F(θ) (resp., F ′(θ)) is equipped with a +general distribution (resp., an elliptical reference distribution ˆP = P(µ,Σ,g)) and the norms in the +definitions of the Wasserstein distances of F(θ) and F ′(θ) are an L2-norm and the Mahalanobis +norm associated with Σ ≻ 0, respectively. Assume that the risk threshold satisfies ε < 0.5, then the +return-risk MDP (7) is equivalent to a second-order cone program +ℓRR(α,θ,ε) = max +x∈X µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2, +(8) +where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ being the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) − +� η2/2 +(Φ−1(1−ε)) +2/2 kg(z)dz ≥ θ, and it could be computed via bisection method. +5.1. +Risk-Awareness for Uncertain Transition Kernel +By adopting the static soft-robust framework in Lobo et al. (2020), one can indeed also account +for the uncertainty in transition kernel in our return-risk model. As in Lobo et al. (2020), suppose +we have finite samples of transition kernel { ˆP i}i∈[N] with weights w ∈ ∆N := {w ∈ RN ++ | e⊤w = 1} +that are generated by MCMC (see, e.g., Kruschke (2010)). Our proposed model is then as follows: +max +π∈(∆A)S ψ · EˆP[g(π, ˜P )] + (1 − ψ) · ˆP-CVaRι[g(π, ˜P )]. +(9) + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +11 +max (1 − ψ)(η − +1 +1 − ι +� +i∈[N] +yi) + ψ · +� +i∈[N] +(µ⊤xi − αθ · ∥xi∥2 − (1 − α)∥Φ−1(1 − ε)Σ1/2xi∥2) +s.t. yi − wiη ≥ αθ · ∥xi∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2 − µ⊤xi +∀i ∈ [N] +(E − γ · ¯P i)xi = wi · p0 +∀i ∈ [N] +xi ≤ +wi +1−γπ +∀i ∈ [N] +xi +s,a ≥ +wi +1 − γ (πs,a − 1) + +� +a′∈A +xi +s,a′ +∀(i,s,a) ∈ N × S × A +π ∈ (∆A)S ∩ {0,1}SA,η ∈ R,xi ∈ RSA ++ ,y ∈ RN ++ +∀i ∈ [N]. +Figure 3 +Reformulation of (9) as an MISOCP. +Here the objective function in (9) is again soft-robust against the uncertainty (in transition kernel), +with the weight ψ ∈ [0,1] as the controller for the robustness and ι ∈ [0,1] is the risk threshold (w.r.t. +the uncertain transition kernel). The weighted empirical distribution ˆP[ ˜P = ˆP i] = wi ∀i ∈ [N] and +the function +g(π,P ) = max µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2 +s.t. xs,a = πs,a · +� +a′∈A +xs,a′ +∀(s,a) ∈ S × A +(E − γ · ¯P )x = p0 +x ∈ RSA ++ +represents the optimal value of the return-risk model with the additional constraint that the optimal +policy should be the input π ∈ (∆A)S and with ¯P as the coefficient matrix corresponding to the +input transition kernel P . +Quite notably, when focusing on deterministic policies, one can reformulate (9) as an MISOCP. +Proposition 4. If π is restricted to be a deterministic policy (i.e., π ∈ (∆A)S ∩{0,1}SA), prob- +lem (9) has an equivalent MISOCP reformulation as in Figure 3. +We remark that, though deterministic policies seem to be restricted compared to the randomized +ones, they actually are more favored under some situations; for example, they may be a more +suitable choice in some medical domains where randomized policies are unworkable for practical +and philosophical reasons (Rosen et al. 2006). Also, randomized policies may be difficult to be +evaluated after they have been deployed and may have poor reproducibility (Lobo et al. 2020). +6. +First-Order Method +In this section, we introduce an efficient first-order algorithm to solve the equivalent formulation (8) +of our return-risk model. Our algorithm is based on an alternating direction linearized proximal + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +12 +method of multipliers (AD-LPMM) algorithm (Beck 2017, Shefi and Teboulle 2014), which is a +variant of the alternating direction method of multiplier (ADMM) algorithm and also has a con- +vergence rate of O(1/N) (here N is the number of iterations) proved by Beck (2017). The proposed +splitting allows efficient update of variables in AD-LPMM (where the solutions are analytical or +can be retrieved by an efficient bisection method). +For the primal update of the ADMM algorithm, one needs to solve minimization problems with +a quadratic term involved (in its objective function); in AD-LPMM, this quadratic term can be +linearized by adding a proximity term to the objective function, which could render the primal +update much easier. To implement our AD-LPMM algorithm, first we will introduce auxiliary +variables and rewrite (8) (as a minimization problem) as follows: +min αθ · ∥x∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2y∥2 − µ⊤z +s.t. (E − γ · ¯P )x = p0 +x = y +x = z +x ∈ RSA,y ∈ RSA,z ∈ RSA ++ , +(10) +where, in the spirit of AD-LPMM, we can split the decision variables into two groups and update +them separately. The augmented Lagrangian function of (10) is: +L(x,y,z;λ,ξ,η) += αθ · ∥x∥2 + (1 − α)Φ−1(1 − ε) · ∥Σ1/2y∥2 − µ⊤z + λ⊤((E − γ · ¯P )x − p0) + ξ⊤(x − y) ++η⊤(x − z) + c +2 · +�������� +(E − γ · ¯P )x − p0 +x − y +x − z +�������� +2 +2 +. +Based on our splitting method, we will update the two groups of variables (y,z) and x separately. +For the update of (y,z), we define two primal update operators +Py(x,ξ;c) = arg min +y +(1 − α)Φ−1(1 − ε) · ∥Σ1/2y∥2 − ξ⊤y + c +2 · ∥x − y∥2 +2 +and Pz(x,η;c) = arg min +z≥0 +−z⊤(µ + η) + c +2 · ∥x − z∥2 +2; while for the update of x (i.e., the second +group of variables), we define +Px(y,z,λ,ξ,η;c,ν, ˆx) = arg min +x +αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) ++ c +2 · +�������� +(E − γ · ¯P )x − p0 +x − y +x − z +�������� +2 +2 ++ 1 +2 · ℓ2 +Q(c,ν)(x − ˆx), + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +13 +Algorithm 1: AD-LPMM for Problem (10) +Input: Frobenius norm ν = ∥(E − γ · ¯P )⊤(E − γ · ¯P ) + 2 · I∥F, initial stepsize c0 > 0, +stepsize growth rate β > 0, desired precision δ, x0, y0, z0, λ0, ξ0, η0, k ← 0 +while +�������� +(E − γ · ¯P )xk − p0 +xk − yk +xk − zk +�������� +∞ +≥ δ do +// Primal update +step 1: yk+1 ← Py(xk,ξk;ck); +step 2: zk+1 ← Pz(xk,ηk;ck); +step 3: xk+1 ← Px(yk+1,zk+1,λk,ξk,ηk;ck,ν,xk); +// Dual update +step 4: λk+1 ← λk + ck · ((E − γ · ¯P )xk+1 − p0); +step 5: ξk+1 ← ξk + ck · (xk+1 − yk+1); +step 6: ηk+1 ← ηk + ck · (xk+1 − zk+1); +// Increase stepsize +step 7: ck+1 ← ck + βc0; +step 8: k ← k + 1; +end +Output: Solution xk +where Q(c,ν) = c · ((ν − 2) · I − (E − γ · ¯P )⊤(E − γ · ¯P )) and ℓQ(·) (equipped with a positive +semi-definite matrix Q) is a weighted vector norm such that ℓQ(x) = +� +x⊤Qx. As we shall see in +Section 6.3, the update of x is fast (where an analytical solution is available) with the proximity +term (1/2)·ℓ2 +Q(c,ν)(x− ˆx) added. Note that when Q(c,ν) ≡ 0, the update in AD-LPMM degenerates +to an ADMM’s one. +We now introduce our AD-LPMM in Algorithm 1. Basically, the most time-consuming computa- +tions lie in the primal update phase, where the updates are carried out by solving a minimization +problem with other variables fixed at values after their last updates. As shall be detailed soon, +owing to our variable splitting method, the primal updates are also quite fast, where analytical solu- +tions or solutions obtained by bisection are available. Here we choose a stepsize that is increasing +in every iteration (with a growth rate β > 0), which in practice accelerates the convergence. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +14 +6.1. +Subproblem in Step 1: Proximal Mapping and Projection +To solve Py(x,ξ;c), first we would utilize the technique of proximal mapping and establish the +following equivalences: +Py(x,ξ;c) = Prox (1−α)Φ−1(1−ε) +c +·∥·∥Σ(x + 1 +c · ξ) += x + 1 +c · ξ − (1−α)Φ−1(1−ε) +c +· ProjBℓΣ−1 (·) +� +1 +(1−α)Φ−1(1−ε) · (c · x + ξ) +� +, +(11) +where Proxf(·)(x) = arg minv f(v) + 1 +2 · ∥v − x∥2 +2 is the proximal mapping operator and +ProjBℓΣ(·)(x) = arg min +v:ℓΣ(v)≤1 +1 +2 · ∥v − x∥2 +2 +(12) +is the operator of projection on the unit ball BℓΣ(·) = {x ∈ RSA | ℓΣ(x) ≤ 1}. Here, the first equality +in (11) holds by the definition of the proximal mapping operator, and the second equality follows +from,e.g., example 6.4.7 in Beck (2017). Indeed, problem (12) allows an efficient solution obtained +by a bisection method to locate its optimal dual solution λ⋆ ≥ 0 (after which the optimal primal +solution can be retrieved immediately), where the upper bound of the bisection is provided in +Lemma 2 relegated to Appendix A.4. The time complexity of the solution process (11), as well as +the pseudocode for the bisection method, are provided in the following proposition. +Proposition 5. Problem Py(x,ξ;c) can be solved in time O(SAlog(1/δ′)), where δ′ is the +desired precision of the bisection method. +6.2. +Subproblem is Step 2: Componentwise Update +Problem Pz(x,η;c) can be decomposed into SA single-variable quadratic programming problems, +each allowing an analytical solution. We summarize the time complexity and details in the following +proposition. +Proposition 6. Problem Pz(x,η;c) can be solved in time O(SA). +6.3. +Subproblem in Step 3: Linearization and Proximal Mapping +Compared to the update in ADMM, in our AD-LPMM, a proximity term (1/2) · ℓ2 +Q(c,ν)(x − ˆx) +is added to the objective function of the update in step 3. By choosing Q(·,·) as mentioned in +Section 6, we can linearize all the quadratic terms in Px(y,z,λ,ξ,η;c,ν, ˆx), thus the solution can +be obtained analytically by the technique of proximal mapping (meanwhile assuring the positive +semi-definiteness of Q(ck,ν) in every iteration of Algorithm 1). This solution process, as well as its +time complexity, is provided in the following proposition. +Proposition 7. Problem Px(y,z,λ,ξ,η;c,ν, ˆx) can be solved in time O(SA). + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +15 +100 +200 +300 +400 +500 +Sample size +15 +14 +13 +VaR ( '=0.15) + +DRMDP +CC +RR +BROIL +RMDP +100 +200 +300 +400 +500 +Sample size +14 +12 +10 +Mean + +DRMDP +CC +RR +BROIL +RMDP +Figure 4 +Empirical study. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk +threshold ε′ = 15%) and mean of reward. The upper and lower edges of the shaded areas are respectively +the 95% and 5% percentiles of the 100 performances, while the solid lines are the medians. +7. +Numerical Experiments +In this section, we conduct two numerical experiments to compare the performances of +DRMDPs (4), CC (2)3, RR (7), RMDPs (Delage and Mannor 2010) and BROIL (Brown et al. +2020) (please see Appendices F and G for more details for the last two models). In both experi- +ments, we train our reward functions with different sample sizes (100,200,300,400,500). For each +sample size, performance of each model is evaluated for 100 times. The performance of each model +is evaluated by expectation and VaR with risk thresholds ε′ ∈ {5%,10%,15%}. Cross validations +are conducted for parameter selection (please see Appendix H.1 for details). +In Section 7.1, we conduct a simulation study where MDPs are generated randomly as in Regan +and Boutilier (2012); In Section 7.2, we study a machine replacement problem introduced in +Delage and Mannor (2010). As implied in our proofs, in this section, the Wasserstein ambiguity +set of DRMDPs (4) will be equipped with a general reference distribution and an L2-norm for the +Wasserstein distance; as for RR (7), we use a general reference distribution and an L2-norm in the +definition of the Wasserstein distance for the Wasserstein ambiguity set F(θ), while for F ′(θ), we +use an elliptical reference distribution ˆP = P(µ,Σ,g) and the Mahalanobis norm associated with the +positive definite matrix Σ for the Wasserstein distance. All optimization problems are solved by +MOSEK on a 2.3GHz processor with 32GB memory. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +16 +100 +200 +300 +400 +500 +Sample size +1600 +1650 +1700 +1750 +VaR ( '=0.15) + +DRMDP +CC +RR +BROIL +RMDP +100 +200 +300 +400 +500 +Sample size +1750 +1800 +1850 +Mean + +DRMDP +CC +RR +BROIL +RMDP +Figure 5 +Simulation. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold +ε′ = 15%) and mean of reward. The upper and lower edges of the shaded areas are respectively the 95% +and 5% percentiles of the 100 performances, while the solid lines are the medians. +7.1. +Simulation Study +In this experiment, we follow the experiment setup in Regan and Boutilier (2012) where the number +of reachable next-states and the transition kernel are randomly generated (both of which are known +to decision makers). More details of the experiment setting are relegated to Appendix H.2. +As illustrated in Figures 5 and 7 (where the latter for VaR with ε′ ∈ {5%,10%} is relegated +to Appendix H.4), when the decision maker aims to optimize her tailed performances, CC is a +preferable choice compared to DRMDPs; on the contrary, when pursuing optimizing the average +return, DRMDPs perform much better than CC. Observe that the RR model, which includes both +DRMDPs and the DCC model as special cases, remains as the best model under all criteria. In +particular, one can observe that, RR achieves higher percentile returns than BROIL (that is a +model without robustness), which demonstrates the benefits of distributionally robustness and the +advantage of the risk measure VaR for percentile performance optimization. As expected, RMDPs +end up yielding over-conservative policies; as a result, it performs poorly in most instances under +all criteria. +7.2. +Machine Replacement Problem +In this experiment, we follow the experiment setup in Delage and Mannor (2010) and consider +the case where a factory holds an extensive amount of machines, each of which is subject to the +same underlying MDP (more details of the experiment setting can be found in Appendix H.2). Our +setting is similar to Delage and Mannor (2010) except for the follows: we use a data-driven setting +3 As we demonstrated in Section 4, a DCC is equivalent to a nominal chance-constrained one with an adjusted risk +level, thus here we simply choose the latter as the benchmark. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +17 +as described above, and we evaluate our (policies of) models by looking at the various performance +measures as in Section 7.1. +We report the overall performances of the five models in Figures 4 and 8 (where the latter for +VaR with ε′ ∈ {5%,10%} is relegated to Appendix H.5). Similar to the previous experiment, RR +always performs better than or equal to the best model between CC and DRMDPs, and it provides +the best performance under all criteria, which again manifest the merit of taking both the expected +and worst-case performances into consideration and distributionally robustness. +7.3. +Computation Times of Different Algorithms +Table 1 +The average of the runtimes of the MOSEK solver and the AD-LPMM algorithm in seconds and the +relative gaps (%) to the optimal values computed by MOSEK. +S=A +Runtimes +Relative gaps +MOSEK AD-LPMM +40 +0.60 +2.79 +< 0.1 % +70 +5.58 +4.81 +< 0.1 % +100 +25.50 +19.98 +0.2 % +130 +93.54 +66.17 +< 0.1 % +160 +444.06 +168.34 +0.4 % +In this section, we compare the computation times of our AD-LPMM algorithm with the state- +of-the-art solver MOSEK. Table 1 reports the runtimes of the the AD-LPMM and MOSEK when +solving problem (8) at different problem sizes. Results indicate that, though our AD-LPMM is +slower than the MOSEK solver when problem size is small, it showcases its strong scalability and +become much faster than MOSEK with large-size problems (while always maintaining high solution +quality), where the advantage is more notable when the problem scales up. +8. +Conclusion +We consider risk-aware MDPs with ambiguous reward functions and propose the return-risk model, +which is versatile and can optimize any weighted combination of the average and quantile perfor- +mances of a policy. This model generalizes and combines the advantage of distributionally robust +MDPs and distributionally robust chance-constrained MDPs, thus is powerful in both average +and percentile performances optimization. In particular, risk from uncertain transition kernel can +also be captured by the return-risk model when output policies are deterministic. Tractable refor- +mulations are provided for all our proposed models, and we design an AD-LPMM algorithm for + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +18 +the return-risk model, which is well scalable and faster than the MOSEK solver with large-scale +problems. Experimental results showcase the versatility of the return-risk model as well as the +scalability of the algorithm. +In the future, we believe that it would be important to explore more efficient methods for +obtaining solution of RR, where function approximation and policy gradient (Sutton and Barto +2018) are possible choices to achieve this. +References +Abdullah, Mohammed Amin, Hang Ren, Haitham Bou Ammar, Vladimir Milenkovic, Rui Luo, Mingtian +Zhang, Jun Wang. 2019. Wasserstein robust reinforcement learning. arXiv preprint arXiv:1907.13196 +. +Ahmadi, Mohamadreza, Ugo Rosolia, Michel Ingham, Richard Murray, Aaron Ames. 2021. Constrained +risk-averse Markov decision processes. The 35th AAAI Conference on Artificial Intelligence (AAAI-21). +Alizadeh, Pegah, Yann Chevaleyre, Jean-Daniel Zucker. 2015. Approximate regret based elicitation in Markov +decision process. The 2015 IEEE RIVF International Conference on Computing & Communication +Technologies-Research, Innovation, and Vision for Future (RIVF). IEEE, 47–52. +B¨aauerle, Nicole, Ulrich Rieder. 2017. Partially observable risk-sensitive Markov decision processes. Mathe- +matics of Operations Research 42(4) 1180–1196. +B¨auerle, Nicole, Alexander Glauner. 2021. Minimizing spectral risk measures applied to Markov decision +processes. Mathematical Methods of Operations Research 94(1) 35–69. +B¨auerle, Nicole, Jonathan Ott. 2011. Markov decision processes with average-value-at-risk criteria. Mathe- +matical Methods of Operations Research 74(3) 361–379. +Beck, Amir. 2017. First-order methods in optimization. SIAM. +Behzadian, Bahram, Reazul Russel, Marek Petrik, Chin Pang Ho. 2021. Optimizing percentile criterion using +robust MDPs. International Conference on Artificial Intelligence and Statistics 1009–1017. +Ben-Tal, Aharon, Dimitris Bertsimas, David B Brown. 2010. A soft robust model for optimization under +ambiguity. Operations research 58(4-part-2) 1220–1234. +Ben-Tal, Aharon, Arkadi Nemirovski. 2001. Lectures on modern convex optimization: analysis, algorithms, +and engineering applications. SIAM. +Bernard, Carole, Silvana M Pesenti, Steven Vanduffel. 2022. Robust distortion risk measures. arXiv preprint +arXiv:2205.08850 . +Bertsekas, Dimitri, John Tsitsiklis. 1995. Neuro-dynamic programming: an overview. Proceedings of 1995 +34th IEEE Conference on Decision and Control, vol. 1. IEEE, 560–564. +Bertsimas, Dimitris, Aur´elie Thiele. 2006. Robust and data-driven optimization: modern decision making +under uncertainty. Models, methods, and applications for innovative decision making. INFORMS, 95– +122. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +19 +Blanchet, Jose, Karthyek Murthy. 2019. Quantifying distributional model risk via optimal transport. Math- +ematics of Operations Research 44(2) 565–600. +Brown, Daniel, Scott Niekum, Marek Petrik. 2020. Bayesian robust optimization for imitation learning. +Advances in Neural Information Processing Systems 33 2479–2491. +Calafiore, Carlo, L El Ghaoui. 2006. On distributionally robust chance-constrained linear programs. Journal +of Optimization Theory and Applications 130(1) 1–22. +Carpin, Stefano, Yin-Lam Chow, Marco Pavone. 2016. Risk aversion in finite Markov decision processes +using total cost criteria and average value at risk. 2016 IEEE International Conference on Robotics +and Automation (ICRA). IEEE, 335–342. +Chen, Xin, Melvyn Sim, Peng Sun. 2007. A robust optimization perspective on stochastic programming. +Operations Research 55(6) 1058–1071. +Chen, Xinyue, Zijian Zhou, Zheng Wang, Che Wang, Yanqiu Wu, Keith Ross. 2020. +Bail: Best-action +imitation learning for batch deep reinforcement learning. Advances in Neural Information Processing +Systems 33 18353–18363. +Chen, Zhi, Daniel Kuhn, Wolfram Wiesemann. 2018. Data-driven chance constrained programs over Wasser- +stein balls. arXiv preprint arXiv:1809.00210 . +Chen, Zhi, Melvyn Sim, Huan Xu. 2019. Distributionally robust optimization with infinitely constrained +ambiguity sets. Operations Research 67(5) 1328–1344. +Chen, Zhi, Weijun Xie. 2021. Sharing the value-at-risk under distributional ambiguity. Mathematical Finance +31(1) 531–559. +Choi, Jaedeug, Kee-Eung Kim. 2012. Nonparametric Bayesian inverse reinforcement learning for multiple +reward functions. Advances in Neural Information Processing Systems 25. +Chow, Yinlam, Mohammad Ghavamzadeh, Lucas Janson, Marco Pavone. 2017. Risk-constrained reinforce- +ment learning with percentile risk criteria. The Journal of Machine Learning Research 18(1) 6070–6120. +Clement, Julien, Christian Kroer. 2021a. First-order methods for Wasserstein distributionally robust MDP. +International Conference on Machine Learning. PMLR, 2010–2019. +Clement, Julien Grand, Christian Kroer. 2021b. First-order methods for wasserstein distributionally robust +mdp. International Conference on Machine Learning. PMLR, 2010–2019. +Delage, Erick, Shie Mannor. 2007. +Percentile optimization in uncertain Markov decision processes with +application to efficient exploration. +Proceedings of the 24th International Conference on Machine +Learning. PMLR, 225–232. +Delage, Erick, Shie Mannor. 2010. Percentile optimization for Markov decision processes with parameter +uncertainty. Operations Research 58(1) 203–213. +Delgado, Karina, Leliane De Barros, Daniel Dias, Scott Sanner. 2016. Real-time dynamic programming for +Markov decision processes with imprecise probabilities. Artificial Intelligence 230 192–223. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +20 +Derman, Esther, Shie Mannor. 2020. Distributional robustness and regularization in reinforcement learning. +arXiv preprint arXiv:2003.02894 . +Erdo˘gan, Emre, Garud Iyengar. 2006. Ambiguous chance constrained problems and robust optimization. +Mathematical Programming 107(1) 37–61. +Gao, Rui, Anton Kleywegt. 2016. Distributionally robust stochastic optimization with Wasserstein distance. +arXiv preprint arXiv:1604.02199 . +Gao, Rui, Anton Kleywegt. 2022. Distributionally robust stochastic optimization with Wasserstein distance. +Mathematics of Operations Research . +Gilbert, Hugo, Paul Weng, Yan Xu. 2017. Optimizing quantiles in preference-based Markov decision pro- +cesses. Proceedings of the AAAI Conference on Artificial Intelligence, vol. 31. +Hanasusanto, Grani, Vladimir Roitch, Daniel Kuhn, Wolfram Wiesemann. 2015. A distributionally robust +perspective on uncertainty quantification and chance constrained programming. Mathematical Pro- +gramming 151(1) 35–62. +Hanasusanto, Grani, Vladimir Roitch, Daniel Kuhn, Wolfram Wiesemann. 2017. Ambiguous joint chance +constraints under mean and dispersion information. Operations Research 65(3) 751–767. +Haskell, William, Rahul Jain. 2015. A convex analytic approach to risk-aware Markov decision processes. +SIAM Journal on Control and Optimization 53(3) 1569–1598. +Ho, Jonathan, Stefano Ermon. 2016. Generative adversarial imitation learning. Advances in neural infor- +mation processing systems 29. +Hogg, Robert V, Allen T Craig. 1995. Introduction to mathematical statistics.(5”” edition). Englewood +Hills, New Jersey . +Huang, Wenjie, William Haskell. 2017. Risk-aware q-learning for Markov decision processes. 2017 IEEE +56th Annual Conference on Decision and Control (CDC). IEEE, 4928–4933. +Huang, Yonghui, Xianping Guo. 2016. Minimum average value-at-risk for finite horizon semi-Markov decision +processes in continuous time. SIAM Journal on Optimization 26(1) 1–28. +Iyengar, Garud. 2005. Robust dynamic programming. Mathematics of Operations Research 30(2) 257–280. +Jaimungal, Sebastian, Silvana M Pesenti, Ye Sheng Wang, Hariom Tatsat. 2022. Robust risk-aware rein- +forcement learning. SIAM Journal on Financial Mathematics 13(1) 213–226. +Jiang, Daniel R, Warren B Powell. 2018. Risk-averse approximate dynamic programming with quantile-based +risk measures. Mathematics of Operations Research 43(2) 554–579. +Kaiser, Lukasz, Mohammad Babaeizadeh, Piotr Milos, Blazej Osinski, Roy H Campbell, Konrad Czechowski, +Dumitru Erhan, Chelsea Finn, Piotr Kozakowski, Sergey Levine, et al. 2019. Model-based reinforcement +learning for atari. arXiv preprint arXiv:1903.00374 . + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +21 +Kruschke, John K. 2010. Bayesian data analysis. Wiley Interdisciplinary Reviews: Cognitive Science 1(5) +658–676. +Lagoudakis, Michail G, Ronald Parr. 2003. Least-squares policy iteration. The Journal of Machine Learning +Research 4 1107–1149. +Lobo, Elita A, Mohammad Ghavamzadeh, Marek Petrik. 2020. Soft-robust algorithms for batch reinforce- +ment learning. arXiv preprint arXiv:2011.14495 . +Mannor, Shie, Ofir Mebel, Huan Xu. 2016. Robust MDPs with k-rectangular uncertainty. Mathematics of +Operations Research 41(4) 1484–1509. +Mannor, Shie, Duncan Simester, Peng Sun, John Tsitsiklis. 2007. Bias and variance approximation in value +function estimates. Management Science 53(2) 308–322. +Ng, Andrew Y, Stuart J Russell, et al. 2000. Algorithms for inverse reinforcement learning. Icml, vol. 1. 2. +Nocedal, Jorge, Stephen Wright. 2006. Numerical optimization. Springer Science & Business Media. +Osa, Takayuki, Joni Pajarinen, Gerhard Neumann, J Andrew Bagnell, Pieter Abbeel, Jan Peters. 2018. An +algorithmic perspective on imitation learning. arXiv preprint arXiv:1811.06711 . +Panaganti, Kishan, Dileep Kalathil. 2022. Sample complexity of robust reinforcement learning with a gen- +erative model. International Conference on Artificial Intelligence and Statistics. PMLR, 9582–9602. +Petrik, Marek. 2010. Optimization-based approximate dynamic programming. University of Massachusetts +Amherst. +Petrik, Marek, Ronny Luss. 2016. Interpretable policies for dynamic product recommendations. UAI . +Pflug, Georg, David Wozabal. 2007. Ambiguity in portfolio selection. Quantitative Finance 7(4) 435–442. +Postek, Krzysztof, Aharon Ben-Tal, Dick Den Hertog, Bertrand Melenberg. 2018. Robust optimization with +ambiguous stochastic constraints under mean and dispersion information. Operations Research 66(3) +814–833. +Pr´ekopa, Andr´as. 2013. Stochastic programming, vol. 324. Springer Science & Business Media. +Puterman, Martin. 2014. Markov decision processes: discrete stochastic dynamic programming. John Wiley +& Sons. +Rashidinejad, Paria, Banghua Zhu, Cong Ma, Jiantao Jiao, Stuart Russell. 2021. Bridging offline reinforce- +ment learning and imitation learning: A tale of pessimism. Advances in Neural Information Processing +Systems 34. +Regan, Kevin, Craig Boutilier. 2010. Robust policy computation in reward-uncertain MDPs using nondom- +inated policies. Proceedings of the AAAI Conference on Artificial Intelligence, vol. 24. +Regan, Kevin, Craig Boutilier. 2011a. Eliciting additive reward functions for Markov decision processes. +Twenty-Second International Joint Conference on Artificial Intelligence. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +22 +Regan, Kevin, Craig Boutilier. 2011b. Robust online optimization of reward-uncertain MDPs. Twenty-Second +International Joint Conference on Artificial Intelligence. +Regan, Kevin, Craig Boutilier. 2012. Regret-based reward elicitation for Markov decision processes. arXiv +preprint arXiv:1205.2619 . +Rosen, Laura, Orly Manor, Dan Engelhard, David Zucker. 2006. In defense of the randomized controlled +trial for health promotion research. American journal of public health 96(7) 1181–1186. +Roy, Aurko, Huan Xu, Sebastian Pokutta. 2017. Reinforcement learning under model mismatch. Advances +in neural information processing systems 30. +Sarykalin, Sergey, Gaia Serraino, Stan Uryasev. 2008. +Value-at-risk vs. conditional value-at-risk in risk +management and optimization. State-of-the-art decision-making tools in the information-intensive age. +Informs, 270–294. +Shapiro, Alexander. 2016. Rectangular sets of probability measures. Operations Research 64(2) 528–541. +Shapiro, Alexander. 2021. Distributionally robust optimal control and mdp modeling. Operations Research +Letters 49(5) 809–814. +Shefi, Ron, Marc Teboulle. 2014. +Rate of convergence analysis of decomposition methods based on the +proximal method of multipliers for convex minimization. SIAM Journal on Optimization 24(1) 269– +297. +Smith, James, Robert L Winkler. 2006. +The optimizer’s curse: skepticism and postdecision surprise in +decision analysis. Management Science 52(3) 311–322. +Sutton, Richard S, Andrew G Barto. 2018. Reinforcement learning: An introduction. MIT press. +van de Berg, Damien, Thomas Savage, Panagiotis Petsagkourakis, Dongda Zhang, Nilay Shah, Ehecatl Anto- +nio del Rio-Chanona. 2022. Data-driven optimization for process systems engineering applications. +Chemical Engineering Science 248 117135. +Wang, Yue, Shaofeng Zou. 2021. Online robust reinforcement learning with model uncertainty. Advances in +Neural Information Processing Systems 34 7193–7206. +Wang, Yue, Shaofeng Zou. 2022. Policy gradient method for robust reinforcement learning. arXiv preprint +arXiv:2205.07344 . +Xie, Weijun. 2021. +On distributionally robust chance constrained programs with Wasserstein distance. +Mathematical Programming 186(1) 115–155. +Xu, Huan, Shie Mannor. 2010. +Distributionally robust Markov decision processes. +Advances in Neural +Information Processing Systems 23 2505–2513. +Xu, Huan, Shie Mannor. 2012. Distributionally robust markov decision processes. Mathematics of Operations +Research 37(2). + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +23 +Yamai, Yasuhiro, Toshinao Yoshiba, et al. 2002. Comparative analyses of expected shortfall and value-at- +risk: their estimation error, decomposition, and optimization. Monetary and economic studies 20(1) +87–121. +Yang, Insoon. 2020. Wasserstein distributionally robust stochastic control: A data-driven approach. IEEE +Transactions on Automatic Control 66(8) 3863–3870. +Yu, Pengqian, Huan Xu. 2015. Distributionally robust counterpart in markov decision processes. IEEE +Transactions on Automatic Control 61(9) 2538–2543. +Zheng, Kan, Zhe Yang, Kuan Zhang, Periklis Chatzimisios, Kan Yang, Wei Xiang. 2016. Big data-driven +optimization for mobile networks toward 5g. IEEE network 30(1) 44–51. +Zymler, Steve, Daniel Kuhn, Ber¸c Rustem. 2013. +Distributionally robust joint chance constraints with +second-order moment information. Mathematical Programming 137(1) 167–198. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +24 +A. +Proof of Results +A.1. +Proofs of Results in Section 3 +Proof of Proposition 1. +It is sufficient to rewrite the objective of (4) as follows: +inf +P∈F(θ)EP[˜r⊤x] = − sup +P∈F(θ) +EP[−˜r⊤x] += −min +λ≥0 +� +λθ − +� +RSA inf +ξ∈RSA(λ∥ξ − r∥ + ξ⊤x) dˆPr +� += − min +λ≥∥x∥∗ +� +λθ − +� +RSA r⊤x dˆPr +� += EˆP[˜r⊤x] − θ∥x∥∗, +where the second identity follows from theorem 1 in Gao and Kleywegt (2016) and the third +identity follows from strong conic duality +inf +ξ∈RK(λ∥ξ − r∥ + ξ⊤x) = +� +� +� +r⊤x +λ ≥ ∥x∥∗ +−∞ +λ ∈ [0,∥x∥∗). +Substituting the above reexpression then concludes the proof. +Q.E.D. +A.2. +Proofs of Results in Section 4 +Proof of Lemma 1. +Notice that (6) is equivalent to +sup +P∈F(θ) +P +�˜r⊤x < y +� +≤ ε ⇐⇒ sup +P∈F(θ) +P +�˜r⊤x ≤ y +� +≤ ε, +where it is equivalent if we replace the strict inequality on the left-hand side with a weak one on +the right-hand side; see proposition 3 in Gao and Kleywegt (2016). Exploring the definition of VaR, +we note that +sup +P∈F(θ) +P +�˜r⊤x ≤ y +� +≤ ε ⇐⇒ sup +P∈F(θ) +P-VaR1−ε +� +y − ˜r⊤x +� +≤ 0. +By corollary 4.9 in Chen and Xie (2021) and the assumption of Mahalanobis norm, it holds that +sup +P∈F(θ) +P-VaR1−ε +� +y − ˜r⊤x +� += P(µ,Σ,g)-VaR1−ε +� +y − ˜r⊤x +� +. +In other words, the worst-case VaR around the elliptical distribution P(µ,Σ,g) with the risk threshold +ε is equal to the nominal elliptical VaR with a small risk threshold ε ≤ ε (which, would correspond +to a higher risk level). We thus obtain +sup +P∈F(θ) +P-VaR1−ε +� +y − ˜r⊤x +� +≤ 0 ⇐⇒ P(µ,Σ,g)-VaR1−ε [y − ˜r⊤x] ≤ 0 +⇐⇒ P(µ,Σ,g) +�˜r⊤x ≤ y +� +≤ ε +⇐⇒ P(µ,Σ,g) +�˜r⊤x ≥ y +� +≥ 1 − ε, + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +25 +where the last equivalence follows from P(µ,Σ,g) being a continuous distribution. +Q.E.D. +Proof of Proposition 2. +By Lemma 1, the first constraint in (5) is the same as +P(µ,Σ,g) +�˜r⊤x ≥ y +� +≥ 1 − ε, +where ε = 1 − Φ(¯η⋆) ≤ ε and ¯η⋆ is the smallest η ≥ Φ−1(1 − ε) that satisfies +η(Φ(η) − (1 − ε)) − +� η2/2 +(Φ−1(1−ε)) +2/2 +kg(z)dz ≥ θ. +The constraint can then be further written as +P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ Φ((µ⊤x − y)/ +√ +x⊤Σx) ≥ 1 − ε +⇐⇒ µ⊤x − y ≥ Φ−1(1 − ε) +√ +x⊤Σx +⇐⇒ µ⊤x − y ≥ ∥Φ−1(1 − ε)Σ1/2x∥2, +where the first equivalence holds by the linearity of elliptical distributions, the second one is because +that Φ(·) is non-decreasing, and the last one is due to the fact that 1−ε ≥ 0.5 (which follows from +ε ≤ ε < 0.5). Observe that the optimum is achieved at y⋆ = µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2, plugging +this in the objective of problem (5) then concludes our proof. +Q.E.D. +A.3. +Proofs of Results in Section 5 +Proof of Proposition 3. +By Proposition 1 and Proposition 2, we have +inf +P∈F(θ)EP[˜r⊤x] = −θ∥x∥2 + EˆP[˜r⊤x] +and +inf +P∈F′(θ)P-VaR1−ε[˜r⊤x] = µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2 +with ε as claimed. Substituting the above two equations into (7) and rearranging the terms then +concludes our proof. +Q.E.D. +Proof of Proposition 4. +By the definition of ˆT, problem (9) can be rewritten as: +max +π∈(∆A)S ψ +� +i∈[N] +wi · g(π, ˆP i) + (1 − ψ)max +η∈R +� +� +�η − +1 +1 − ι +� +i∈[N] +wi(η − g(π, ˆP i))+ +� +� +�. +By introducing auxiliary decision variables y ∈ RN, it can be further reformulated as: +max ψ +� +i∈[N] +wi · g(π, ˆP i) + (1 − ψ) +� +�η − +1 +1 − ι +� +i∈[N] +yi +� +� +s.t. yi ≥ wi(η − g(π, ˆP i)) +∀i ∈ [N] +π ∈ (∆A)S,y ∈ RN ++,η ∈ R. +(13) + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +26 +Here we can express +wi · g(π,P ) = max µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2 +s.t. xs,a = πs,a · +� +a′∈A +xs,a′ +∀(s,a) ∈ S × A +(E − γ · ¯P )x = wi · p0 +x ∈ RSA ++ +(14) +as in Lobo et al. (2020). We can then, by combining (13) and (14), reformulate problem (9) as: +max ψ +� +i∈[N] +(µ⊤xi − αθ · ∥xi∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2) + (1 − ψ)(η − +1 +1 − ι +� +i∈[N] +yi) +s.t. yi − wiη ≥ αθ · ∥xi∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2 − µ⊤xi +∀i ∈ [N] +xi +s,a = πs,a · +� +a′∈A +xi +s,a′ +∀i ∈ [N],(s,a) ∈ S × A +(E − γ · ¯P i)xi = wi · p0 +∀i ∈ [N] +π ∈ (∆A)S,η ∈ R,xi ∈ RSA ++ ,y ∈ RN ++ +∀i ∈ [N]. +Now it is sufficient to focus on the second set of constraints +xi +s,a = πs,a · +� +a′∈A +xi +s,a′ ∀i ∈ [N],(s,a) ∈ S × A. +(15) +Since we only consider deterministic policy π ∈ {0,1}SA and � +a∈A xi +s,a ∈ [0,wi/(1 − γ)] (see, e.g., +lemma C.10 in Petrik (2010)), we have the McCormick relaxation (see, e.g., Petrik and Luss (2016)) +of (15) as: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +xi +s,a ≤ +� +a′∈A +xi +s,a′ +xi +s,a ≤ +wi +1 − γ πs,a +xi +s,a ≥ 0 +xi +s,a ≥ +wi +1 − γ (πs,a − 1) + +� +a′∈A +xi +s,a′ +for all i ∈ [N],(s,a) ∈ S × A. Our conclusion then follows from the fact that the McCormick +relaxation is precise when π ∈ {0,1} (i.e., the extreme values of the interval [0,1]). +Q.E.D. +A.4. +Proofs of Results in Section 6 +Proof of Proposition 5. +By (11), it is sufficient to focus on solving ProjBℓΣ(·)(x). By eigenvalue +decomposition, we have Σ = G⊤DG4 with D = diag(d1,··· ,dSA), thus we have: +ProjBℓΣ(·)(x) = arg min 1 +2 · ∥v − x∥2 +2 +s.t. +v⊤G⊤DGv ≤ 1 +v ∈ RSA. +4 The eigenvalue decomposition here is not counted in the time complexity of the bisection method (or the AD-LPMM +algorithm), since this process is carried out for computing Σ1/2 in (8) (before we solve (8)). + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +27 +By change of variable u = Gv and let b = Gx, it is sufficient to focus on the equivalent problem: +arg min 1 +2 · ∥u − b∥2 +2 +s.t. +u⊤Du ≤ 1 +u ∈ RSA, +(16) +where we can retrieve v⋆ = G⊤u⋆. The Lagrangian function of(16) (with the introduced dual +variable ζ ∈ R+) is +L(u;ζ) = 1 +2 · ∥u − b∥2 +2 + ζ(u⊤Du − 1). +Since (16) is a convex optimization problem, the KKT condition is the sufficient condition for the +optimality of the primal and dual solutions: +� +� +� +� +� +� +� +� +� +� +� +� +� +u⊤Du ≤ 1 +ζ ≥ 0 +ζ(u⊤Du − 1) = 0 +∇uL(u;ζ) = u − b + 2ζ · Du = 0, +where for ζ = 0, we have +� +� +� +u⊤Du ≤ 1 +u − b = 0; +while when ζ > 0, we have +� +� +� +u⊤Du = 1 +(I + 2ζ · D)u − b = 0. +Therefore, if b⊤Db ≤ 1, we have u⋆ = b; if b⊤Db > 1, it is sufficient to solve the equation g(ζ) = 1 +where +g(ζ) = +� +i∈[SA] +dib2 +i +(1 + 2ζdi)2 . +The function g is monotonically decreasing function on [0,+∞) and limζ→+∞ g(ζ) = 0, thus we can +apply the bisection method to search on the interval [0, ¯ζ] (where ¯ζ : g(¯ζ) ≤ 1 is the upper bound +for the search which we provide in Lemma 2) to locate ζ⋆ and retrieve u⋆ +i = bi/(1+2ζ⋆di) ∀i ∈ [SA]. +The pseudocode is provided in Algorithm 2. +The time complexity of solving Py(x,ξ;c) is dominated by the bisection method, which has +time complexity O(log(1/δ′)). Our conclusion follows from the fact that the computation in each +iteraion of the bisection takes time O(SA). +Q.E.D. +Lemma 2. The inequality g(ζ) ≤ 1 holds for all ζ ≥ (1/(2di′′))(bi′√SAdi′ − 1), where i′ ∈ +arg maxi∈[SA] dib2 +i and i′′ ∈ arg mini∈[SA] di + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +28 +Algorithm 2: Bisection for Problem (16) +Input: Desired precision δ′, initial lower bound ζ ← 0 and upper bound ζ > 0 +if g(0) ≤ 1 then +u ← b; +end +else +while |ζ − ζ| ≥ δ′ do +ζ ← 0.5(ζ + ζ); +if +g(ζ) >= 1 then +ζ ← ζ; +end +else +ζ ← ζ; +end +end +for i = 1,··· ,SA do +ui = bi/(1 + 2ζdi); +end +end +Output: Solution u +Proof. Observe that, +g(ζ) ≤ +� +i∈[SA] +di′b2 +i′ +(1 + 2ζdi)2 +≤ +SAdi′b2 +i′ +(1+2ζdi′′)2 , +from which we have +SAdi′b2 +i′ +(1 + 2ζdi′′)2 ≤ 1 ⇒ g(ζ) ≤ 1. +Our conclusion thus follows by rearranging the terms of the inequality on the left-hand side. +Q.E.D. +By Lemma 2, one can choose ζ = (1/(2di′′))(bi′√SAdi′ − 1), where i′ ∈ arg maxi∈[SA] dib2 +i and +i′′ ∈ arg mini∈[SA] di for Algorithm 2. +Proof of Proposition 6. +Notice that, it is sufficient to solve the ith subproblem: +arg min +z≥0 +c +2z2 − (cxi + µi + ηi)z = max +� +0, 1 +c(cxi + µi + ηi) +� +for all i ∈ [SA], where our conclusion follows. +Q.E.D. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +29 +Proof of Proposition 7. +By the definition of Q(·,·), we have +Px(y,z,λ,ξ,η;c,ν, ˆx) += arg min +x +αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c +2 · +�������� +(E − γ · ¯P )(x − ˆx) + (E − γ · ¯P )ˆx − p0 +x − ˆx + ˆx − y +x − ˆx + ˆx − z +�������� +2 +2 ++ 1 +2 · ℓ2 +Q(c,ν)(x − ˆx) += arg min +x +αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c +2 · +�������� +(E − γ · ¯P )(x − ˆx) +x − ˆx +x − ˆx +�������� +2 +2 ++c · x⊤ � +(E − γ · ¯P )⊤ � +(E − γ · ¯P )ˆx − p0 +� ++ 2 · ˆx − y − z +� ++ 1 +2 · ℓ2 +Q(c,ν)(x − ˆx) += arg min +x +αθ +cν · ∥x∥2 + x⊤w + 1 +2 · ∥x − ˆx∥2 +2 += arg min +x +αθ +cν · ∥x∥2 + 1 +2 · ∥x − (ˆx − w)∥2 +2 += +� +1 − +αθ +cν +max{∥w∥2, αθ +cν } +� +· (ˆx − w) +where +we +denote +w += +1 +cν +· +�� +E − γ · ¯P +�⊤ λ + ξ + η +� ++ +1 +ν +· +�� +E − γ · ¯P +�⊤ �� +E − γ · ¯P +� ˆx − p0 +� ++ 2 · ˆx − y − z +� +, and the last equality holds by, e.g., exam- +ple 6.1.9 in Beck (2017). +The computation time is dominated by computing ∥w∥2, which is O(SA). +Q.E.D. +B. +Evaluation of VaR and CVaR of Student’s t-Distribution +The VaR of a Student’s t-distribution with threshold ε is in fact the lower-ε percentile of its +probability density function (PDF), which can be looked up in table in, e.g., Hogg and Craig (1995) +(under some common values of ε < 0.5). We provide the calculation of CVaR as follows (with degree +of freedom δ > 1 and v := Pt-dist-VaRε(˜r) assumed known): +Pt-dist-CVaRε(˜r) = 1 +ε · +Γ( δ+1 +2 +) +(πδ) +1 +2 Γ( δ +2 ) +� v +−∞ +r +(1+ r2 +δ ) +δ+1 +2 dr += 1 +ε · +δ +1 +2 ·Γ( δ+1 +2 +) +2π +1 +2 Γ( δ +2 ) +� 1+ v2 +δ +−∞ +u− k+1 +2 du += − +δ +1 +2 ·Γ( δ+1 +2 +) +επ +1 +2 (δ−1)Γ( δ +2 ) · +� +1 + v2 +δ +�− k−1 +2 , +where the first equality follows from the definition of the CVaR and the PDF of the t-distribution +herein, the second equality holds by the technique of integration by substitution. +C. +Preliminaries on Elliptical Distributions +The probability density distribution of an elliptical reference distribution P(µ,Σ,g) is given by +f(r) = k · g +�1 +2(r − µ)⊤Σ−1(r − µ) +� +, + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +30 +where k is a positive normalization scalar, µ is a mean vector, Σ is a positive definite matrix and g +is a generating function. Elliptical distribution is a broad family of distributions that includes for +example, the multivariate normal distribution, multivariate t-distribution and multivariate logistic +distribution, as special cases. One notable property of the elliptical distribution is the linearity: any +linear combination of elliptically distributed random variables still follows an elliptical distribution. +That is, for any random vector ˜r ∼ P(µ,Σ,g), it holds that ˜r⊤x ∼ P(µx,σ2x,g) with µx = µ⊤x and +σx = +√ +x⊤Σx. Indeed, we can express the combination as ˜r⊤x = µx + σx˜z, where ˜z ∼ P(0,1,g) is a +standard elliptically distributed random variable whose probability density function and cumulative +distribution function are φ(z) = k·g (z2/2) and Φ(x) = +� x +−∞ k·g(z2/2)dz, respectively. For a concrete +example we take a closer look at a standard normal distribution, for which the normalization scalar +and generating function are k = 1/ +√ +2π and g(x) = exp(−x), respectively. +D. +Distributionally Optimistic MDPs +In contrast to the robust model, sometimes the decision maker prefers exploration over exploitation +if she would like to learn more information about the MDP. As such, we could instead adopt an +optimistic counterpart where we focus on the best case, motivating the following distributionally +optimistic MDP: +ℓO(θ) = max +x∈X +sup +P∈F(θ) +EP[˜r⊤x]. +(17) +In contrast to the robust case, here our decision depends instead on the best possible (expected) +outcome, which exactly embodies optimism. We summarize the reformulation of (17) as follows. +Proposition 8. The distributionally optimistic MDP (17) is equivalent to an optimization prob- +lem +ℓO(θ) = max +x∈X EˆP[˜r⊤x] + θ∥x∥∗. +Proof. It is sufficient to rewrite the objective of (17) as follows: +sup +P∈F(θ) +EP[˜r⊤x] = − inf +P∈F(θ)EP[−˜r⊤x] = −(EˆP[−˜r⊤x] − θ∥x∥∗) = EˆP[˜r⊤x] + θ∥x∥∗, +where the second identity follows similar lines as in the proof of Proposition 1. +Q.E.D. +The reformulation in Proposition 8 is a reverse conic program that is, in general, non-convex. +However, it can be recast as a mixed-integer linear program, provided that ∥ · ∥∗ is the commonly +used L1-norm or L∞-norm. Such a mixed-integer linear program can be solved by the state-of-the- +art approaches. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +31 +E. +Distributionally Optimistic Chance-Constrained Model +In a distributionally optimistic chance-constrained MDP model, where we focus on the best case +that with high probability, the reward is no smaller than some lower bound that we maximize. +Formally, the distributionally optimistic chance-constrained MDP model is formulated as follows: +ℓDOCC(θ,ε) = +� +� +� +� +� +� +� +� +� +max y +s.t. +sup +P∈F(θ) +P[˜r⊤x ≥ y] ≥ 1 − ε +x ∈ X, y ∈ R. +(18) +The optimistic chance-constrained model (18) is also equivalent to a nominal chance-constrained +model, however, at a less risky level. Before formally establishing this argument, two lemmas are +introduced as follows. +Lemma 3. The worst (largest) probability of the random vector ˜r attaining a value in the set R, +sup +P∈F(θ) +P[˜r ∈ R], +(19) +is equivalent to +min +λ≥0 +� +λθ + +� +r∈RSA(λ · dist(r,R) − 1)−dˆPr +� +. +Here, we use dist(r,R) = inf{∥r − ˆr∥ | ˆr ∈ R} to denote the distance from the vector r ∈ RSA to +the set R ⊆ RSA. +Proof. Using theorem 1 in Gao and Kleywegt (2016) or theorem 1 in Blanchet and Murthy (2019), +the uncertainty quantification problem (19) is equal to +min +λ≥0 +� +λθ − +� +r∈RSA +inf +w∈RSA{λ∥w − r∥ − I[w ∈ R]}dˆPr +� +, +(20) +where I is the 0-1 indicator function. Consider the second term in the objective of the above +minimization problem, we have +inf +w∈RSA{λ∥w − r∥ − I[w ∈ R]} = −(λ · dist(r,R) − 1)−. +(21) +Indeed, if r ∈ R (for which, dist(r,R) = 0), then by choosing w = v, it holds that +inf +w∈RSA{λ∥w − r∥ − I[w ∈ R]} = −1 = −(λ · dist(r,R) − 1); +whereas if r /∈ R, then it holds that +inf +w∈RSA{λ∥w − r∥ − I[w ∈ R]} = min +� +inf +w∈R{λ∥w − r∥ − 1}, inf +w /∈Rλ∥w − r∥ +� += min +� +inf +w∈R{λ∥w − r∥ − 1},0 +� += −(λ · dist(r,R) − 1)−. +Plugging expression (21) into problem (20) gives the desired result, which, by proposition 3 in Gao +and Kleywegt (2016), holds regardless of whether R is open or closed. +Q.E.D. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +32 +Lemma 4. The distributionally optimistic chance constraint +inf +P∈F(θ)P[˜r ∈ R] ≤ ε +(22) +with a risk threshold ε ∈ (0,1) is satisfiable if and only if +P-CVaRε[−dist(˜r, ¯R)] ≥ − +θ +1 − ε, +where ¯R = RSA \ R is the complement of the set of undesired events R. +Proof. We first re-express (22) as +sup +P∈F(θ) +P[˜r ∈ ¯R] ≥ 1 − ε. +Using Lemma 3, the above constraint is equivalent to +min +λ≥0 +� +λθ + +� +r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr +� +≥ 1 − ε. +(23) +The left-hand side problem can be presented by +min +� +min +λ>0 +� +λθ + +� +r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr +� +,1 +� +. +Since 1 ≥ 1 − ε, the above re-expression implies that constraint (23) is equivalent to +min +λ>0 +� +λθ + +� +r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr +� +≥ 1 − ε. +Multiplying both sides by (λ(1 − ε))−1 > 0, we arrive at +min +τ<0 +� +1 +1 − ε +� +r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ +� +≥ − +θ +1 − ε, +which, together with the fact +min +τ≥0 +� +1 +1 − ε +� +r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ +� +≥ 0 ≥ − +θ +1 − ε, +is equivalent to +min +τ∈R +� +1 +1 − ε +� +r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ +� +≥ − +θ +1 − ε, +where the left-hand side is essentially ˆP-CVaRε[−dist(˜r, ¯R)]. +Q.E.D. +Now we are ready to establish the equivalence between the chance-constrained model and its +optimistic counterpart (with an adjusted risk threshold). +Lemma 5. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an ellip- +tical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm +associated with the positive definite matrix Σ. The distributionally optimistic robust chance con- +straint +∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε +is satisfiable if and only if P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε, where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the +smallest η ≤ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) + +� (Φ−1(1−ε)) +2/2 +η2/2 +kg(z)dz ≤ θ. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +33 +Proof. We first look at the individual distributionally optimistic robust chance constraint +∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε +for some generic coefficient vector x ∈ RSA. The above chance constraint is equivalent to +sup +P∈F(θ) +P[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ +sup +P∈F(θ) +P[˜r⊤x > y] ≥ 1 − ε ⇐⇒ +inf +P∈F(θ)P[˜r⊤x ≤ y] ≤ ε, +where for the first equivalence, by using proposition 3 in Gao and Kleywegt (2016) , it is indifferent +to replace the strict inequality with a weak one. Exploring the definition of VaR, we note that +inf +P∈F(θ)P[˜r⊤x ≤ y] ≤ ε ⇐⇒ inf +P∈F(θ)P-VaR1−ε[y − ˜r⊤x] ≤ 0. +Hence, with the translation invariance of VaR, it is sufficient to show that +inf +P∈F(θ)P-VaR1−ε[−˜r⊤x] ≜ inf +v∈R +� +v | +inf +P∈F(θ)P[−˜r⊤x > v] ≤ ε +� +. +(24) +By Lemma 4 and the assumption of Mahalanobis norm, we have +inf +P∈F(θ)P +� +−˜r⊤x > v +� +≤ ε ⇐⇒ P(µ,Σ,g)-CVaRε[−dist(˜r, ¯R)] ≥ − +θ +1 − ε +⇐⇒ −P(µ,Σ,g)-CVaRε[−(−˜r⊤x − v)+] ≤ θ∥x∥Σ−1 +1 − ε +, +where ¯R = +� +r | − r⊤x ≤ v +� +and we leverage the closed form solution +dist(˜r, ¯R) = +� +−˜r⊤x − v +�+ /∥x∥Σ−1; +see, e.g., lemma 2 in Chen et al. (2018). +Let PS = P(µ,Σ,g) for simplicity. By the property of elliptical distribution, for ˜r ∼ PS and any real +vector x, we have −˜r⊤x ∼ P(µS,σ2 +S,g) = P(−µ⊤x,x⊤Σx,g). We denote its probability density function +as +h(z) = k +σS +· g +� +(z − µS) +2 +2σ2 +S +� +. +The left-hand side of the constraint can be further transformed as +−PS-CVaRε[−(−˜r⊤x − v)+] += −EPS[−(−˜r⊤x − v)+ | − (−˜r⊤x − v)+ ≥ PS-VaRε[−(−˜r⊤x − v)+]] += − +1 +1 − ε +� sup{z|−(z−v)+≥PS-VaRε[−(−˜r⊤x−v)+]} +−∞ +−(z − v)+h(z)dz += +1 +1 − ε +� sup{z|−(z−v)+≥PS-VaRε[−(−˜r⊤x−v)+]} +v +(z − v)h(z)dz += +1 +1 − ε +� PS-VaR1−ε[−˜r⊤x] +v +(z − v)h(z)dz, + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +34 +in which the last equality holds from +sup{z | − (z − v)+ ≥ PS-VaRε[−(−˜r⊤x − v)+]} += sup{z | min{v − z,0} ≥ PS-VaRε[min{v + ˜r⊤x,0}]} += sup{z | min{−z,−v} ≥ PS-VaRε[min{˜r⊤x,−v}]} += sup{z | − z ≥ PS-VaRε[min{˜r⊤x,−v}]} += sup{z | z ≤ PS-VaR1−ε[max{−˜r⊤x,v}]} += sup{z | z ≤ PS-VaR1−ε[−˜r⊤x]} += PS-VaR1−ε[−˜r⊤x]. +Here, the second equality is due to the translation invariance of VaR, the third one follows from +−v ≥ PS-VaRε[min{˜r⊤x,−v}], the fifth one is because that for any ε ∈ (0,1), the distributionally +optimistic robust VaR satisfies +v = inf +P∈F(θ)P-VaR1−ε[−˜r⊤x] ≤ PS-VaR1−ε[−˜r⊤x], +(25) +thus the 1 − ε quantiles of −˜r⊤x and max{−˜r⊤x,v} coincide. +Let us denote q1−ε = PS-VaR1−ε[−˜r⊤x], which, by its definition, satisfies +q1−ε − µS +σS += PS-VaR1−ε +�−˜r⊤x − µS +σS +� += P0 +(0,1,g)-VaR1−ε[˜z] = Φ−1(1 − ε), +Here, the first equality holds for the translation invariance and the positive homogeneity of VaR, +while the last one follows from the definition of VaR under the standard elliptical distribution +P(0,1,g). +Following the last reformulation of the constraint, we further have +1 +1 − ε +� q1−ε +v +(z−v)h(z)dz = +1 +1 − ε +� q1−ε +v +z· k +σS +·g +� +(z − µS) +2 +2σ2 +S +� +dz− +v +1 − ε +� q1−ε +v +k +σS +·g +� +(z − µS) +2 +2σ2 +S +� +dz. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +35 +For its first component, we have +1 +1 − ε +� q1−ε +v +z · k +σS +· g +� +(z − µS) +2 +2σ2 +S +� +dz += +1 +1 − ε +� q1−ε +v +z − µS +σS +· k · g +� +(z − µS) +2 +2σ2 +S +� +dz + +1 +1 − ε +� q1−ε +v +µS +σS +· k · g +� +(z − µS) +2 +2σ2 +S +� +dz += +σS +1 − ε +� q1−ε +v +z − µS +σS +· k · g +� +(z − µS) +2 +2σ2 +S +� +d +�z − µS +σS +� ++ +µS +1 − ε +� +Φ +�q1−ε − µS +σS +� +− Φ +�v − µS +σS +�� += +σS +1 − ε +� +q1−ε−µS +σS +v−µS +σS +t · k · g +�t2 +2 +� +d +�z − µS +σS +� ++ µS +1 − ε +� +Φ +�q1−ε − µS +σS +� +− Φ +�v − µS +σS +�� += +σS +1 − ε +� +(q1−ε−µS)2 +2σ2 +S +(v−µS)2 +2σ2 +S +k · g(z)dz + µS +1 − ε +� +Φ +�q1−ε − µS +σS +� +− Φ +�v − µS +σS +�� +, +while for the second component, it holds that +v +1 − ε +� q1−ε +v +k +σS +· g +� +(z − µS) +2 +2σ2 +S +� +dz = +v +1 − ε +� +q1−ε−µS +σS +v−µS +σS +k · g +�z2 +2 +� +dz += +v +1 − ε +� +Φ +�q1−ε − µS +σS +� +− Φ +�v − µS +σS +�� +. +Hence, combine the constraint with (25), we have the following equivalent expression for prob- +lem (24): +inf v +s.t. +� +(q1−ε−µS)2 +2σ2 +S +(v−µS)2 +2σ2 +S +k · g(z)dz + µS − v +σS +� +Φ +�q1−ε − µS +σS +� +− Φ +�v − µS +σS +�� +≤ θ∥x∥Σ−1 +σS += θ +v ≤ PS-VaR1−ε[−˜r⊤x] +v ∈ R, +where the equality follows from the definition of the Mahalanobis norm. Let η = (v − µS)/σS, the +best-case VaR now becomes +inf µS + σSη +s.t. +� (Φ−1(1−ε))2/2 +η2/2 +k · g(z)dz − η · (1 − ε − Φ(η)) ≤ θ +η ≤ Φ−1(1 − ε) +η ∈ R. +(26) +The function +V (η) ≜ +� (Φ−1(1−ε))2/2 +η2/2 +k · g(z)dz − η · (1 − ε − Φ(η)) + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +36 +is monotonically decreasing on (−∞,Φ−1(1 − ε)) since for any η < Φ−1(1 − ε), it holds that +V ′(η) = −η · k · g +�η2 +2 +� +− (1 − ε) + Φ(η) + ηφ(η) = Φ(η) − (1 − ε) < 0. +Thus problem (26) can be efficiently solved be a bisection algorithm and the optimal η⋆ as claimed +can be obtained. Finally the result can be obtained as follows: +∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ −y ≥ σSη⋆ + µS +⇐⇒ −y − µS +σS +≥ η⋆ +⇐⇒ Φ +�−y − µS +σS +� +≥ Φ(η⋆) +⇐⇒ P(µ,Σ,g) +� ˜r⊤x − µS +σS +≥ y − µS +σS +� +≥ 1 − ¯ε +⇐⇒ P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε. +Q.E.D. +With ¯ε in Lemma 5, we are now ready to derive a second-order cone reformulation of the +distributionally optimistic chance-constrained model (18). +Proposition 9. Suppose in the Wasserstein ambiguity set (3), the reference distribution is an +elliptical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis +norm associated with the positive definite matrix Σ. If the risk threshold satisfies ε ≤ ¯ε < 0.5, then +the distributionally optimistic chance-constrained MDP (18) is equivalent to the second-order cone +program +ℓDOCC(θ,ε) = max +x∈X µ⊤x − ∥Φ−1(1 − ¯ε)Σ1/2x∥2, +where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies +η(Φ(η) − (1 − ε)) + +� (Φ−1(1−ε)) +2/2 +η2/2 +kg(z)dz ≤ θ. +Proof. By Lemma 5, the first constraint in (18) is equivalent to +P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε, +where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies +η(Φ(η) − (1 − ε)) + +� (Φ−1(1−ε)) +2/2 +η2/2 +kg(z)dz ≤ θ, +which can be further transformed as follows: +P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε ⇐⇒ Φ((µ⊤x − y)/ +√ +x⊤Σx) ≥ 1 − ¯ε +⇐⇒ µ⊤x − y ≥ Φ−1(1 − ¯ε) +√ +x⊤Σx +⇐⇒ µ⊤x − y ≥ ∥Φ−1(1 − ¯ε)Σ1/2x∥2, + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +37 +where the first equivalence holds by the linearity of elliptical distributions, the second one holds +because of the non-decreasing cumulative distribution function Φ(·), and the third one holds as +¯ε < 0.5. Since the optimal value is achieved with y = µ⊤x − ∥Φ−1(1 − ¯ε)Σ1/2x∥2, plugging this +equation in the objective of (18) then concludes our proof. +Q.E.D. +F. +Additional Details on Robust MDPs +As introduced in Delage and Mannor (2010), robust MDPs maximizes the total expected return +considering the worst-case realization of the uncertain parameter within a predefined ambiguity +set: +max +π∈Π +min +r0∈R,r1∈R,···E +� ∞ +� +t=0 +γtrt(st) | s0 ∝ p0,π +� +, +(27) +where Π is the set of all the stationary randomized policies, rt and st are the reward and state at +time stage t, respectively. As in Delage and Mannor (2010), we set R to be the 99% confidence +ellipsoid of the random reward vector as the uncertainty set. +G. +Additional Details on BROIL +Similar to our return-risk model, BROIL (Brown et al. 2020) also seeks a policy that maximizes +the weighted average of the mean and percentile performances: +max +π∈Π λ · E +� ∞ +� +t=0 +γtrt(st) | s0 ∝ p0,π +� ++ (1 − λ) · CVaRε +� ∞ +� +t=0 +γtrt(st) | s0 ∝ p0.π +� +, +(28) +where λ ∈ [0,1] is the weight. Given R ∈ RSA×n as the matrix of (n) reward samples, BROIL can +be expressed as a linear program as follows: +max +x∈X,y∈Rλ · 1 +ne⊤R⊤x + (1 − λ) · +� +y − 1 +ε · 1 +ne⊤(y · e − R⊤x) +� +. +Observe that, there are two major differences between BROIL and our return-risk model: first, +BROIL use CVaR as its risk measure, while VaR is applied in our return-risk model; second, while +distributionally robustness is considered in (both the mean and VaR of return in) our objective +function, BROIL only computes the nominal mean and CVaR of the return. +H. +Additional Details and Results on the Experiments +H.1. +Additional Details of Parameter Selection +We use cross validation for parameter selection in both the simulation and empirical studies. +For DRMDPs (4), the candidate set for θ is {0,2,··· ,18}; for CC (2), the candidate set for ε +is {iε′/5}i∈[5]; for RR (7), we select θ such that ε varies among {iε′/5}i∈[5], and we select α ∈ +{0,0.25,0.5,0.75,1}; for BROIL (28), we select λ × ε ∈ {0,0.25,0.5,0.75,1} × {0.05,0.1,0.15}; for +RMDPs (27), as in Delage and Mannor (2010), we set R to be the 99% confidence ellipsoid of the +random reward vector as the uncertainty set. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +38 +Figure 6 +A machine replacement problem with fixed Gaussian rewards. +H.2. +Additional Details of the Simulation Study +We consider S = 10 states, A = 10 actions, a uniform initial state distribution, and a discount +factor γ = 0.95. For each state s ∈ [S], the number of reachable next-state is ⌈log S⌉. We sample +the true reward from a multivariate normal distribution N(µ′,Σ′), where for each k ∈ [SA], µ′ +k +is generated as follows: first we sample a number (0 or 1) from a discrete uniform distribution in +{0,1}. If the result is 0, we generate µ′ +k from the normal distribution N(50,100); otherwise we +generate it from N(90,100). Standard deviations of rewards are generated in the same manner +with another two normal distributions N(3,9) and N(18,9). Both standard deviations and means +are trimmed to be non-negative after the above procedure. The correlation matrix of rewards +is generated as follows: we first sample a matrix R ∈ RSA×SA with all its entries independently +sampled in [0.25,1] uniformly, and then obtain our correlation matrix diag(d)V diag(d), where +V = R⊤R and d = {di}i∈[SA] = {1/√Vii}i∈[SA]. +H.3. +Additional Details of the Empirical Study +In this experiment, each machine is subject to the same underlying MDP with a state set S = [S] +with S = 50 and an action set with only two actions: repair the machine or not. The transition is +deterministic and the discount factor is 0.8. The reward depends on both the current state and +action, and all the rewards are independently and normally distributed. Figure 6 illustrates the +true underlying distribution that generates the random rewards. +H.4. +Additional Results of the Simulation Study +H.5. +Additional Results of the Empirical Study + +130,1) +N(-130,1) +-130,1) +N(-130,20) +2 +N(0,10) +N(0,10 +N(0,10-4) +V0.10 +- Repair +N(-100,800) +. Not RepairRuan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +39 +100 +200 +300 +400 +500 +Sample size +1500 +1600 +1700 +VaR ( '=0.05) + +DRMDP +CC +RR +BROIL +RMDP +100 +200 +300 +400 +500 +Sample size +1550 +1600 +1650 +1700 +1750 +VaR ( '=0.1) + +DRMDP +CC +RR +BROIL +RMDP +Figure 7 +Simulation. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk thresh- +old ε′ ∈ {5%,10%}). The upper and lower edges of the shaded areas are respectively the 95% and 5% +percentiles of the 100 performances, while the solid lines are the medians. +100 +200 +300 +400 +500 +Sample size +15.5 +15.0 +14.5 +14.0 +13.5 +VaR ( '=0.05) + +DRMDP +CC +RR +BROIL +RMDP +100 +200 +300 +400 +500 +Sample size +15.5 +15.0 +14.5 +14.0 +13.5 +VaR ( '=0.1) + +DRMDP +CC +RR +BROIL +RMDP +Figure 8 +Empirical. Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold ε′ ∈ +{5%,10%}). The upper and lower edges of the shaded areas are respectively the 95% and 5% percentiles +of the 100 performances, while the solid lines are the medians. +I. +Related Works +Table 2 summarizes literature that is related to our work. We remark that, compared to its related +works in Table 2, our return-risk model is the only one that considers risk ambiguity, and we have +also designed a fast first-order algorithm to obtain its solution, which enhance the practicality of +our model for large-scale problems. + +Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity +40 +Table 2 +Related works. +Paper +Uncertainty +Robustness +Ambiguity set +Risk measure Soft-robustness +Delage and Mannor (2010) +Rewards +and +transition kernel +- +- +VaR +No +Xu and Mannor (2010) +Rewards +and +transition kernel +DRO +Nested +- +No +Yu and Xu (2015) +Rewards +and +transition kernel +DRO +(General) Nested +- +No +Brown et al. (2020) +Rewards +- +- +CVaR +Yes +Gilbert et al. (2017) +Rewards +- +- +VaR +No +Lobo et al. (2020) +Transition kernel +- +- +CVaR +Yes +Yang (2020) +Transition kernel +DRO +Wasserstein +- +No +This paper +Rewards +DRO +Wasserstein +VaR +Yes + diff --git a/GtAzT4oBgHgl3EQfHftK/content/tmp_files/load_file.txt b/GtAzT4oBgHgl3EQfHftK/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..58caa9cf5bdc661e1338504072f25910b7dab8d9 --- /dev/null +++ b/GtAzT4oBgHgl3EQfHftK/content/tmp_files/load_file.txt @@ -0,0 +1,1285 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf,len=1284 +page_content='Risk-Averse MDPs under Reward Ambiguity Haolin Ruan School of Data Science, City University of Hong Kong, Kowloon Tong, Hong Kong haolin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ruan@my.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='hk Zhi Chen Department of Management Sciences, College of Business, City University of Hong Kong, Kowloon Tong, Hong Kong zhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='chen@cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='hk Chin Pang Ho School of Data Science, City University of Hong Kong, Kowloon Tong, Hong Kong clint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ho@cityu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='hk We propose a distributionally robust return-risk model for Markov decision processes (MDPs) under risk and reward ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The proposed model optimizes the weighted average of mean and percentile performances, and it covers the distributionally robust MDPs and the distributionally robust chance-constrained MDPs (both under reward ambiguity) as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By considering that the unknown reward distribution lies in a Wasserstein ambiguity set, we derive the tractable reformulation for our model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In particular, we show that that the return-risk model can also account for risk from uncertain transition kernel when one only seeks deterministic policies, and that a distributionally robust MDP under the percentile criterion can be reformulated as its nominal counterpart at an adjusted risk level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A scalable first-order algorithm is designed to solve large-scale problems, and we demonstrate the advantages of our proposed model and algorithm through numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Introduction Markov decision processes (MDPs) provide a powerful modeling framework for sequential decision- making problems and reinforcement learning in stochastic dynamic environments (Puterman 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Obtaining the model parameters of MDPs that perfectly reflect the environments, however, has always been a challenge in practice, as these parameters are estimated from limited data that are potentially contaminated (Mannor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Moreover, these parameters, such as transition kernel and reward function, are often time-dependent or even uncertain, but they are approximated as fixed values in an overly simplified setting (Mannor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Therefore, the output policies of MDPs are often disappointing in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust MDPs address the aforementioned issues of parameter ambiguity, by allowing the unknown values of transition kernels and reward functions to lie in a given ambiguity set (Behza- dian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021, Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019, Clement and Kroer 2021a, Delgado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Then, robust MDPs seek for policies that maximize the worst-case expected return over all transition kernels 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='01045v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='LG] 3 Jan 2023 Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 2 and reward functions in the ambiguity sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By specifying ambiguity sets that contain the unknown transition kernels with high confidence, the optimal policies of robust MDPs are robust to param- eter ambiguity (Iyengar 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In this paper, we focus on the case where the reward function is ambiguous, which sometimes is referred to as imprecise-reward MDPs (Alizadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2015, Regan and Boutilier 2010, 2011a,b, 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' This particular setting is also closely related to imitation learning, which trains an agent to learn a certain behavior of an expert, while only some demonstrated trajectories of her is available (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020, Ho and Ermon 2016, Osa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018, Rashidinejad et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' When applying inverse reinforcement learning approach to learn the reward function that completely represents the expert’s preference (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020, Choi and Kim 2012, Ng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2000), the yielded policies, which suffer from reward ambiguity, may perform poorly in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To handle reward ambiguity, we utilize techniques from distributionally robust optimization (DRO) (Derman and Mannor 2020) and distributionally robust chance-constrained program (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2007, Postek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018), assuming that the true reward distribution resides in an ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' This approach does not require the reward function to be precisely specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Instead, only the descriptions of common distribution information such as support, moments and shape in the ambiguity set are needed, which are often much easier to be obtained/estimated (Hanasusanto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2015, 2017, Zymler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In this paper, we consider a Wasserstein ambiguity set for our distributionally robust models as in Abdullah et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2019), Calafiore and Ghaoui (2006), Xie (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Unlike phi-divergence ambiguity sets which may contain too extreme member distributions, the closeness between points in the support set is incorporated in Wasserstein sets, thus their member distributions may be more reasonable (Gao and Kleywegt 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' on the other hand, Wasserstein sets are often a better choice than moment-based ambiguity sets when the number of samples is too small to obtain a reliable estimation on moments (Yang 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We choose Wasserstein sets for these reasons, although other types of ambiguity sets such as nested ambiguity sets (Xu and Mannor 2010, 2012) and the ambiguity sets based on Prohorov metric (Erdo˘gan and Iyengar 2006) are also considered in literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For our distributionally robust chance-constrained MDPs, we will furthermore show its equivalence with the nominal counterparts with an adjusted risk level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To the best of our knowledge, this is the first result in MDPs that establishes the mutual transformation between distributional ambiguity and risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our return-risk model (RR) is a risk-averse MDP model that not only takes into account reward ambiguity, but also considers both the average and risk of the return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' MDPs that minimize the risk of the return instead of the expected cost are called risk-aware MDPs (also called risk-sensitive or risk-averse MDPs) (Ahmadi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021, B¨aauerle and Rieder 2017, Carpin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016, Haskell and Jain 2015, Huang and Haskell 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In risk-aware optimization, the objective function is taken as Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 3 a risk measure, such as value-at-risk (VaR) (Delage and Mannor 2007, 2010, Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017), conditional value-at-risk (CVaR) (B¨auerle and Ott 2011, Chow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017, Huang and Guo 2016) and other spectral risk measures (B¨auerle and Glauner 2021), and variants of expected utility (Bernard et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022, Jaimungal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022, Pflug and Wozabal 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Among these risk measures, VaR and CVaR are arguably the most popular ones and have attracted the attention of many researchers (B¨auerle and Ott 2011, Chow et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017, Delage and Mannor 2007, 2010, Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017, Huang and Guo 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By using CVaR, one aims to give a precise depiction of the extreme tail of the distribution (of the uncertain rewards), while VaR does not reflect the extreme scenerios exceeding VaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' It is well-known that CVaR is a coherent risk measure, which can be efficiently optimized by convex optimization tools (Chen and Xie 2021);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' in contrast, VaR is a more challenging risk measure because it is not a coherent one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' One remarkable advantage of VaR is its stability of estimation (especially under fat-tailed reward distribution (Sarykalin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2008)), which is particularly important under data-driven settings where the number of samples are limited and decision makers evaluate models based on their out-of-sample performances (Bertsimas and Thiele 2006, van de Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022, Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To demonstrate, we provide an example where we consider a one-step MDP with only 1 state s and 2 actions a1 and a2 (Sutton and Barto 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In this one-step MDP, the decision maker only makes one decision in each episode, and she aims to maximize her VaR/CVaR of rewards for these episodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We consider uncertain rewards ˜rs,a1 ∼ Pt-dist and ˜rs,a2 = ˜rs,a1 + ρ|s| where Pt-dist is a Student’s t-distribution and we vary its degree of freedom δ ∈ {2,3,4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We set the shift ratios ρ = {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05i}i∈[5], and for testing the estimation accuracy w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' VaR (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', CVaR) (where we choose the risk threshold 10%), we set the shift quantity s as Pt-dist-VaR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1[˜rs,a1] (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', Pt-dist-CVaR0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1[˜rs,a1]), where both risk measures can be efficiently calculated (see Appendix B for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We evaluate the decision maker’s accuracy rate as the proportion of testing samples where she has chosen the action with a higher VaR/CVaR of rewards (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', action a2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' for each pair of accuracy rate and shift ratio, following Yamai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2002), 1000 random reward samples for each state-action pair are available for the decision maker, and we test her accuracy rate based on 10000 testing samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As illustrated in Figure 1, the accuracy rate increases with the shift ratio ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As δ decreases, F becomes more fat-tailed, and the accuracy rate of VaR is remarkably higher than that of CVaR, which indicates that the statistical inference on VaR would be more accurate than on CVaR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Therefore, VaR may be a more preferable choice when only small sample sets are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our return-risk model is motivated by the soft-robust criterion/model, which optimizes a convex combination of the mean and a robust performance in the optimization literature (Ben-Tal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' MDPs with soft-robustness are also popular in recent years, where decision makers aim to Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='25 Shift ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 Accuracy rate VaR CVaR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='25 Shift ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 VaR CVaR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='25 Shift ratio 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 VaR CVaR Figure 1 The accuracy rates of the decision maker choosing the correct action (so that the VaR/CVaR of her rewards is maximized): δ = 4 (left), δ = 3 (middle) and δ = 2 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' maximize a weighted average of the mean and percentile performances (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020, Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Unlike these existing soft-robust MDPs, however, the proposed return-risk model is fundamentally different in two aspects: first, these existing soft-robust models have no consideration for reward ambiguity, while we utilize distributionally robustness to account for reward ambiguity, by which we can hedge against the most adversarial realization of the distribution of rewards (within the ambiguity set), thus our model is more robust to reward uncertainty (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019, Xu and Mannor 2010);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' second, we choose VaR as the risk measure which has a direct interpretation to percentile performances, and, as illustrated above, tends to be more advantageous in data-driven optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our work concentrates on model-based setting, where our proposed models are motivated by the classical (dual formulation of) nominal MDPs (Puterman 2014) and the chance-constrained MDPs (Delage and Mannor 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' It is worth noting that, beyond model-based setting, there are other inspiring and innovative researches on robust reinforcement learning, such as robust TDC algorithms and robust Q-learning (Roy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017, Wang and Zou 2021), robust policy gradient (Wang and Zou 2022), least squares policy iteration (Lagoudakis and Parr 2003) and sample complexity analysis (Panaganti and Kalathil 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Note that, though model-free reinforcement learning can be used to learn satisfactory policies for complex environment, the requirement of large amounts of interaction (with environment) may render the learning process slow (Kaiser et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019), while high sample efficiency is one strong advantage of model-based learning (Sutton and Barto 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We also note that MDPs with transition kernel ambiguity is another active research line where distributionally robustness is widely employed (Clement and Kroer 2021b, Shapiro 2016, 2021, Xu and Mannor 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We may summarize our contributions as follows (and we also compare our contributions to those of related works in Table 2 in Appendix I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 5 (i) We show that the distributionally robust model of optimizing expected rewards can be reformulated as a convex conic program, which is equivalent to the nominal MDP with a convex regularization in the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (ii) For distributionally robust chance-constrained MDPs (DCC), we show that it can be refor- mulated as nominal chance-constrained MDPs at adjusted risk levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' This observation bridges the gap between risk and parameter ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (iii) Combining the proposed models in (i) and (ii), we propose the return-risk MDP that maximizes the weighted average of the expectation and VaR of reward (both under distributionally robustness to reward uncertainty), which is flexible and can perform well under the criteria of mean and percentile returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (iv) When only considering deterministic policies, we show that our return-risk model can also account for risk from uncertain transition kernel, and we derive its equivalent reformulation as a mixed-integer second-order cone program (MISOCP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (v) To solve the proposed return-risk model, we design a first-order method that is more scalable than the MOSEK solver, thus is faster with large-size problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (vi) In the simulation and empirical experiments, we adopt a data-driven setting, where the decision maker aims at maximizing the expectation and VaR of the random reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We compare the performances of distributionally robust MDPs (DRMDPs), DCC, RR, robust MDPs (RMDPs) (Delage and Mannor 2010) and BROIL (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020), and results show that the third one performs the best under both expectation and different VaR’s (with risk thresholds 5%, 10% and 15%), which showcases its advantages and adjustability to the decision makers’ changeable preferences between return and risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The remainder of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We introduce the background in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In Sections 3 and 4, we study DRMDPs as well as the DCC model, respectively, and we derive their tractable reformulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Combining these proposed models, we propose the RR model in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The designed first-order algorithm for the RR model is detailed in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We compare the performances of DRMDP, DCC, RR, RMDP and BROIL, and demonstrate the advantage of our proposed algorithm in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Conclusion is drawn in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Background We consider an infinite-horizon MDP with a finite state space S = {1,··· ,S} and a finite action space A = {1,··· ,A}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Let P ∈ RS×A×S be the transition probability kernel such that ps,a,s′ is denoted to be the transition probability of transiting to state s′ ∈ S when action a ∈ A is chosen in state s ∈ S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' thus, ps,a ∈ ∆S is the transition probability distribution for every (s,a) ∈ S × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Given the state-action pair (s,a), an agent will receive an expected reward rs,a ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To simplify our notation, we denote the reward function as a vector r = {rs,a}(s,a)∈S×A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 6 We seek for the optimal stationary randomized policy π = {πs}s∈S with πs ∈ ∆A for all s ∈ S, where an action a ∈ A will be taken in state s ∈ S with probability πs,a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A nominal MDP that maximizes the expected reward can be formulated (Puterman 2014) as ℓN = max x∈X r⊤x, (1) where the feasible set X is given by X = � x ∈ RSA + �� (E − γ · ¯P )x = p0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Here the coefficient matrices E = diag(e⊤,··· ,e⊤) ∈ RS×SA with S all-ones vectors e ∈ RA and ¯P = (¯p1,··· , ¯pS)⊤ ∈ RS×SA with ¯ps = {ps′,a,s}(s′,a)∈S×A for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For each (s,a) ∈ S × A, we denote the sth sub- vector of x as xs = {xi}i∈{(s−1)A+1,··· ,sA};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' its ath component xs,a can be interpreted as the total discounted probability one occupying state s and choosing action a when applying the policy π⋆ s,a = x⋆ s,a/(� a∈A x⋆ s,a) ∀(s,a) ∈ S × A (Puterman 2014)1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We have a discount factor γ ∈ (0,1) and the initial distribution p0 ∈ RS ++ of the initial states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Problem (1) is a linear program that can be efficiently solved by simplex method and interior-point method (Nocedal and Wright 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' One can also compute the optimal policy efficiently by applying value iteration or policy iteration to solve the associated Bellman equation of this problem (Bertsekas and Tsitsiklis 1995, Puterman 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The nominal MDP (1) does not account for uncertainty in either rewards or transition kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To account for reward uncertainty, Delage and Mannor (2010) assume that the random reward vector ˜r follows a known Gaussian distribution P and propose a chance-constrained MDP model as follows: ℓCC(ε) = � � � � � � � max y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' P[˜r⊤x ≥ y] ≥ 1 − ε x ∈ X, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2) In fact, the above chance-constrained model maximizes the VaR (at the risk level 1 − ε) of the reward with respect to the distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Since P is assumed Gaussian, by theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 in Pr´ekopa (2013), one can reformulate problem (2) as a second-order cone program as follows: ℓCC(ε) = max x∈X EP[˜r⊤x] − ∥F−1(1 − ε)Σ1/2x∥2, where F−1(·) is the inverse of the cumulative density function of the Gaussian distribution P and Σ is the covariance matrix of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Second-order cone programs allow efficient solutions by state-of-the-art commercial solvers such as CPLEX, Gurobi and MOSEK (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', Ben-Tal and Nemirovski (2001)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Despite its tractability, the chance-constrained MDP (2) requires the precise underlying reward distribution as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Moreover, the above reformulation does not hold for generic distribution P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1 By Puterman (2014), any x ∈ X admits such interpretation, thus we can retrieve our policies of all the proposed models in this paper in this way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally Robust MDPs In many real-world situations, the true distribution of the uncertain reward is hard (if not impossi- ble) to obtain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Instead, we may have some firm knowledge, such as moments and shape about it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As one of the most efficacious treatments for such situations, the DRO approach models uncertainty as a random variable governed by an unknown probability distribution residing in an ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Facing distributional ambiguity, a decision maker seeks for solutions that hedge against the most adversarial distribution from within the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To be specific, in our context, we assume that the true distribution of the uncertain reward resides in a Wasserstein ball of radius θ ≥ 0 around some reference distribution ˆP: F(θ) = {P ∈ P(RSA) | dW � P, ˆP � ≤ θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (3) Here P(RSA) is the set of all probability distributions on RSA, and the Wasserstein distance between two distributions P1 and P2, equipped with a general norm ∥ · ∥ in RSA, is given by dW (P1,P2) = infP∈Q(P1,P2) EP[∥˜r1 − ˜r2∥], where Q(P1,P2) is the set of all joint distributions with marginal distributions P1 and P2 that govern ˜r1 and ˜r2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The random parameter in the nominal MDP (1) is the expectation of reward, which in practice, is often estimated by the average of historical samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' However, when the sample size is small, such a sample average is not close to the expectation but rather, is known to be optimistically biased (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', Smith and Winkler (2006)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Hence, the nominal MDP (1) based on samples may yield an unsatisfactory policy that does not perform well out-of-sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For this reason, a possible alternative is to maximize instead the worst-case expected reward as in the following distributionally robust MDP: ℓDRMDP(θ) = max x∈X inf P∈F(θ)EP[˜r⊤x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (4) The following proposition offers an equivalent conic program for (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The distributionally robust MDP (4) can be reformulated a conic program ℓDRMDP(θ) = max x∈X EˆP[˜r⊤x] − θ · ∥x∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' It is not hard to observe that the distributionally robust MDPs can be viewed as a convex reg- ularization of the nominal MDP (4) under the reference distribution ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In particular, the convex regularizing term in the distributionally robust MDP is θ∥x∥∗, which is sized by the Wasserstein radius θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Interestingly, we have also found that an (distributionally) optimistic MDP can be refor- mulated as a reverse conic program with a (concave) regularization term −θ∥x∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We relegate this result to Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 8 Figure 2 Values of ε with respect to different θ’s: ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05 (left), ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 (middle), and ε = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15 (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We remark that, problem (4) is indeed a special case of the robust optimization problem consid- ered in Jaimungal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2022), where we consider the expected utility framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Compared to the policy gradient methods provided in Jaimungal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2022) where convergence is not established, we have derived its equivalent reformulation as a tractable conic program which can be efficiently solved by state-of-the-art commercial solvers such as Gurobi, Mosek and CPLEX, and can also be seamlessly incorporated in the tractable reformulation of our proposed return-risk model in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally Robust Chance-Constrained MDPs In this section, we turn from optimizing the expectation of reward to its tailed performance, by exploring chance-constrained MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In particular, we still consider Wasserstein ambiguity sets (3) to account for distributional ambiguity, meanwhile specifying the reference distribution ˆP and the norm ∥ · ∥ in the definition of the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For the former, we focus on an elliptical reference distribution ˆP = P(µ,Σ,g) 2 throughout this section, whose probability density distribution is given by f(r) = k · g � 1 2(r − µ)⊤Σ−1(r − µ) � , where k is a positive normalization scalar, µ is a mean vector, Σ is a positive definite matrix and g is a generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We emphasize that this assumption on ˆP is mild as this is only the center of the ambiguity set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In particular, our proposed distributionally robust chance-constrained MDPs can account for all types of distributions (as long as they are inside the ambiguity set) and they are not restricted to be all elliptical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As we shall see, such specifications lead to tractable reformulation of our proposed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Preliminaries on elliptical distributions are relegated to Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For the latter, we adopt the Mahalanobis norm associated with the positive definite matrix Σ, captured by ∥x∥Σ = √ x⊤Σ−1x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Note that the dual norm of a Mahalanobis norm ∥ · ∥Σ is another Mahalanobis norm ∥ · ∥Σ−1 that is defined by the inverse matrix Σ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2 Note that results in Section 3 hold for a general reference distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1e-03 8e-04 6e-04 4e-04 2e-04 0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='040 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0501e-03 8e-04 6e-04 4e-04 2e-04 0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='090 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1001e-03 8e-04 6e-04 4e-04 2e-04 0e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='130 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='150Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 9 In a distributionally robust chance-constrained MDP, we hope that even in the worst-case, with a high confidence the reward is no less than a lower bound, and we aim at maximizing such a lower bound by solving ℓDCC(θ,ε) = � � � � � � � � � max y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' inf P∈F(θ)P[˜r⊤x ≥ y] ≥ 1 − ε x ∈ X, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (5) Quite notably, the worst-case chance constraint in the pessimistic chance-constrained MDP (5) is equivalent to a nominal chance constraint in (2) with a higher risky level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Suppose in the Wasserstein ambiguity set (3), the reference distribution is an ellip- tical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm associated with the positive definite matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The distributionally robust chance constraint ∀ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε (6) is satisfiable if and only if P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ε, where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ that can be computed via bisection method which searches for the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) − � η2/2 (Φ−1(1−ε)) 2/2 kg(z)dz ≥ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Equipped with Lemma 1, it then turns out that the distributionally robust chance-constrained MDP (5) is equivalent to a nominal chance-constrained MDP (2) at a higher risky level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Conse- quently, the distributionally robust chance-constrained MDP (5) can be reformulated into a conic program, or more precisely, a second-order cone program owing to our choice of the Mahalanobis norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Suppose in the Wasserstein ambiguity set (3), the reference distribution is an elliptical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm associated with the positive definite matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' If the risk threshold satisfies ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5, then the distributionally robust chance-constrained MDP (5) is equivalent to the second-order cone program ℓDCC(θ,ε) = max x∈X µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2, where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ being the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) − � η2/2 (Φ−1(1−ε)) 2/2 kg(z)dz ≥ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Similar to the distributionally robust MDPs in Section 3, the distributionally robust chance- constrained MDPs also admit an optimistic counterpart, which is equivalent to the nominal chance- constrained MDPs with a larger risk threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We relegate this result to Appendix E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To conclude this section, we present in Figure 2 the relations between ε and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Indeed, for any fixed ε, there is a one-to-one correspondence between the risk threshold ε and the Wasserstein radius θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Following from this fact, for the chance-constrained model in our numerical experiments (Section 7), we only calibrate the risk threshold rather than the Wasserstein radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 10 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Return-Risk MDP For rational decision makers, two types of rewards are their chief concerns: the average and the worst-case rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' However, the risk-averse models often can not achieve decent average return on which the model put no emphasis (Carpin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016, Delage and Mannor 2010, Jiang and Powell 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To take both concerns into considerations, we leverage the established DRMDPs and DCC model in Sections 3 and 4 as ingredients and propose the return-risk MDP that maximizes the weighted average of the worst-case expectation and VaR of reward as follows: ℓRR(α,θ,ε) = max x∈X α inf P∈F(θ)EP[˜r⊤x] + (1 − α) inf P∈F′(θ)P-VaRε[˜r⊤x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (7) Here the Wasserstein ball F(θ) is assumed equipped with a general reference distribution and an L2-norm in the definition of the Wasserstein distance, while an elliptical reference distribution ˆP = P(µ,Σ,g) and a Mahalanobis norm associated with the positive definite matrix Σ are assumed for F ′(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' It is not hard to see that the return-risk MDP (7) takes the distributionally robust MDP (4) and the distributionally robust chance-constrained MDP (5) in as special cases by varying ε, θ and α ∈ {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Furthermore, by choosing a fractional α, the return-risk model enables one to tailor a balance between risk and return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 3 below provides an equivalent second-order cone program for the return-risk MDP (7) under these assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Suppose in (7) the Wasserstein ball F(θ) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', F ′(θ)) is equipped with a general distribution (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', an elliptical reference distribution ˆP = P(µ,Σ,g)) and the norms in the definitions of the Wasserstein distances of F(θ) and F ′(θ) are an L2-norm and the Mahalanobis norm associated with Σ ≻ 0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Assume that the risk threshold satisfies ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5, then the return-risk MDP (7) is equivalent to a second-order cone program ℓRR(α,θ,ε) = max x∈X µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2, (8) where ε = 1 − Φ(¯η⋆) ≤ ε with ¯η⋆ being the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) − � η2/2 (Φ−1(1−ε)) 2/2 kg(z)dz ≥ θ, and it could be computed via bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Risk-Awareness for Uncertain Transition Kernel By adopting the static soft-robust framework in Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2020), one can indeed also account for the uncertainty in transition kernel in our return-risk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As in Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2020), suppose we have finite samples of transition kernel { ˆP i}i∈[N] with weights w ∈ ∆N := {w ∈ RN + | e⊤w = 1} that are generated by MCMC (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', Kruschke (2010)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our proposed model is then as follows: max π∈(∆A)S ψ · EˆP[g(π, ˜P )] + (1 − ψ) · ˆP-CVaRι[g(π, ˜P )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (9) Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 11 max (1 − ψ)(η − 1 1 − ι � i∈[N] yi) + ψ · � i∈[N] (µ⊤xi − αθ · ∥xi∥2 − (1 − α)∥Φ−1(1 − ε)Σ1/2xi∥2) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' yi − wiη ≥ αθ · ∥xi∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2 − µ⊤xi ∀i ∈ [N] (E − γ · ¯P i)xi = wi · p0 ∀i ∈ [N] xi ≤ wi 1−γπ ∀i ∈ [N] xi s,a ≥ wi 1 − γ (πs,a − 1) + � a′∈A xi s,a′ ∀(i,s,a) ∈ N × S × A π ∈ (∆A)S ∩ {0,1}SA,η ∈ R,xi ∈ RSA + ,y ∈ RN + ∀i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Figure 3 Reformulation of (9) as an MISOCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Here the objective function in (9) is again soft-robust against the uncertainty (in transition kernel), with the weight ψ ∈ [0,1] as the controller for the robustness and ι ∈ [0,1] is the risk threshold (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' the uncertain transition kernel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The weighted empirical distribution ˆP[ ˜P = ˆP i] = wi ∀i ∈ [N] and the function g(π,P ) = max µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' xs,a = πs,a · � a′∈A xs,a′ ∀(s,a) ∈ S × A (E − γ · ¯P )x = p0 x ∈ RSA + represents the optimal value of the return-risk model with the additional constraint that the optimal policy should be the input π ∈ (∆A)S and with ¯P as the coefficient matrix corresponding to the input transition kernel P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Quite notably, when focusing on deterministic policies, one can reformulate (9) as an MISOCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' If π is restricted to be a deterministic policy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', π ∈ (∆A)S ∩{0,1}SA), prob- lem (9) has an equivalent MISOCP reformulation as in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We remark that, though deterministic policies seem to be restricted compared to the randomized ones, they actually are more favored under some situations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' for example, they may be a more suitable choice in some medical domains where randomized policies are unworkable for practical and philosophical reasons (Rosen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Also, randomized policies may be difficult to be evaluated after they have been deployed and may have poor reproducibility (Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' First-Order Method In this section, we introduce an efficient first-order algorithm to solve the equivalent formulation (8) of our return-risk model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our algorithm is based on an alternating direction linearized proximal Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 12 method of multipliers (AD-LPMM) algorithm (Beck 2017, Shefi and Teboulle 2014), which is a variant of the alternating direction method of multiplier (ADMM) algorithm and also has a con- vergence rate of O(1/N) (here N is the number of iterations) proved by Beck (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The proposed splitting allows efficient update of variables in AD-LPMM (where the solutions are analytical or can be retrieved by an efficient bisection method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For the primal update of the ADMM algorithm, one needs to solve minimization problems with a quadratic term involved (in its objective function);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' in AD-LPMM, this quadratic term can be linearized by adding a proximity term to the objective function, which could render the primal update much easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' To implement our AD-LPMM algorithm, first we will introduce auxiliary variables and rewrite (8) (as a minimization problem) as follows: min αθ · ∥x∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2y∥2 − µ⊤z s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (E − γ · ¯P )x = p0 x = y x = z x ∈ RSA,y ∈ RSA,z ∈ RSA + , (10) where, in the spirit of AD-LPMM, we can split the decision variables into two groups and update them separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The augmented Lagrangian function of (10) is: L(x,y,z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='λ,ξ,η) = αθ · ∥x∥2 + (1 − α)Φ−1(1 − ε) · ∥Σ1/2y∥2 − µ⊤z + λ⊤((E − γ · ¯P )x − p0) + ξ⊤(x − y) +η⊤(x − z) + c 2 · �������� (E − γ · ¯P )x − p0 x − y x − z �������� 2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Based on our splitting method, we will update the two groups of variables (y,z) and x separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For the update of (y,z), we define two primal update operators Py(x,ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) = arg min y (1 − α)Φ−1(1 − ε) · ∥Σ1/2y∥2 − ξ⊤y + c 2 · ∥x − y∥2 2 and Pz(x,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) = arg min z≥0 −z⊤(µ + η) + c 2 · ∥x − z∥2 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' while for the update of x (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', the second group of variables), we define Px(y,z,λ,ξ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' ˆx) = arg min x αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c 2 · �������� (E − γ · ¯P )x − p0 x − y x − z �������� 2 2 + 1 2 · ℓ2 Q(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ν)(x − ˆx),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ho: Risk-Averse MDPs under Reward Ambiguity 13 Algorithm 1: AD-LPMM for Problem (10) Input: Frobenius norm ν = ∥(E − γ · ¯P )⊤(E − γ · ¯P ) + 2 · I∥F,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' initial stepsize c0 > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' stepsize growth rate β > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' desired precision δ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' y0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' λ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' ξ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' η0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' k ← 0 while �������� (E − γ · ¯P )xk − p0 xk − yk xk − zk �������� ∞ ≥ δ do // Primal update step 1: yk+1 ← Py(xk,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ξk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ck);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' step 2: zk+1 ← Pz(xk,ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ck);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' step 3: xk+1 ← Px(yk+1,zk+1,λk,ξk,ηk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ck,ν,xk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' // Dual update step 4: λk+1 ← λk + ck · ((E − γ · ¯P )xk+1 − p0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' step 5: ξk+1 ← ξk + ck · (xk+1 − yk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' step 6: ηk+1 ← ηk + ck · (xk+1 − zk+1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' // Increase stepsize step 7: ck+1 ← ck + βc0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' step 8: k ← k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' end Output: Solution xk where Q(c,ν) = c · ((ν − 2) · I − (E − γ · ¯P )⊤(E − γ · ¯P )) and ℓQ(·) (equipped with a positive semi-definite matrix Q) is a weighted vector norm such that ℓQ(x) = � x⊤Qx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As we shall see in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='3, the update of x is fast (where an analytical solution is available) with the proximity term (1/2)·ℓ2 Q(c,ν)(x− ˆx) added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Note that when Q(c,ν) ≡ 0, the update in AD-LPMM degenerates to an ADMM’s one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We now introduce our AD-LPMM in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Basically, the most time-consuming computa- tions lie in the primal update phase, where the updates are carried out by solving a minimization problem with other variables fixed at values after their last updates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As shall be detailed soon, owing to our variable splitting method, the primal updates are also quite fast, where analytical solu- tions or solutions obtained by bisection are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Here we choose a stepsize that is increasing in every iteration (with a growth rate β > 0), which in practice accelerates the convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 14 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Subproblem in Step 1: Proximal Mapping and Projection To solve Py(x,ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c), first we would utilize the technique of proximal mapping and establish the following equivalences: Py(x,ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) = Prox (1−α)Φ−1(1−ε) c ∥·∥Σ(x + 1 c · ξ) = x + 1 c · ξ − (1−α)Φ−1(1−ε) c ProjBℓΣ−1 (·) � 1 (1−α)Φ−1(1−ε) · (c · x + ξ) � , (11) where Proxf(·)(x) = arg minv f(v) + 1 2 · ∥v − x∥2 2 is the proximal mapping operator and ProjBℓΣ(·)(x) = arg min v:ℓΣ(v)≤1 1 2 · ∥v − x∥2 2 (12) is the operator of projection on the unit ball BℓΣ(·) = {x ∈ RSA | ℓΣ(x) ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Here, the first equality in (11) holds by the definition of the proximal mapping operator, and the second equality follows from,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='7 in Beck (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Indeed, problem (12) allows an efficient solution obtained by a bisection method to locate its optimal dual solution λ⋆ ≥ 0 (after which the optimal primal solution can be retrieved immediately), where the upper bound of the bisection is provided in Lemma 2 relegated to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The time complexity of the solution process (11), as well as the pseudocode for the bisection method, are provided in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Problem Py(x,ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) can be solved in time O(SAlog(1/δ′)), where δ′ is the desired precision of the bisection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Subproblem is Step 2: Componentwise Update Problem Pz(x,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) can be decomposed into SA single-variable quadratic programming problems, each allowing an analytical solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We summarize the time complexity and details in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Problem Pz(x,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) can be solved in time O(SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Subproblem in Step 3: Linearization and Proximal Mapping Compared to the update in ADMM, in our AD-LPMM, a proximity term (1/2) · ℓ2 Q(c,ν)(x − ˆx) is added to the objective function of the update in step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By choosing Q(·,·) as mentioned in Section 6, we can linearize all the quadratic terms in Px(y,z,λ,ξ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c,ν, ˆx), thus the solution can be obtained analytically by the technique of proximal mapping (meanwhile assuring the positive semi-definiteness of Q(ck,ν) in every iteration of Algorithm 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' This solution process, as well as its time complexity, is provided in the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Problem Px(y,z,λ,ξ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c,ν, ˆx) can be solved in time O(SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=" Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 15 100 200 300 400 500 Sample size 15 14 13 VaR ( '=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15) DRMDP CC RR BROIL RMDP 100 200 300 400 500 Sample size 14 12 10 Mean DRMDP CC RR BROIL RMDP Figure 4 Empirical study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold ε′ = 15%) and mean of reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The upper and lower edges of the shaded areas are respectively the 95% and 5% percentiles of the 100 performances, while the solid lines are the medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Numerical Experiments In this section, we conduct two numerical experiments to compare the performances of DRMDPs (4), CC (2)3, RR (7), RMDPs (Delage and Mannor 2010) and BROIL (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020) (please see Appendices F and G for more details for the last two models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In both experi- ments, we train our reward functions with different sample sizes (100,200,300,400,500).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For each sample size, performance of each model is evaluated for 100 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The performance of each model is evaluated by expectation and VaR with risk thresholds ε′ ∈ {5%,10%,15%}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Cross validations are conducted for parameter selection (please see Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1, we conduct a simulation study where MDPs are generated randomly as in Regan and Boutilier (2012);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2, we study a machine replacement problem introduced in Delage and Mannor (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As implied in our proofs, in this section, the Wasserstein ambiguity set of DRMDPs (4) will be equipped with a general reference distribution and an L2-norm for the Wasserstein distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' as for RR (7), we use a general reference distribution and an L2-norm in the definition of the Wasserstein distance for the Wasserstein ambiguity set F(θ), while for F ′(θ), we use an elliptical reference distribution ˆP = P(µ,Σ,g) and the Mahalanobis norm associated with the positive definite matrix Σ for the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' All optimization problems are solved by MOSEK on a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='3GHz processor with 32GB memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=" Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 16 100 200 300 400 500 Sample size 1600 1650 1700 1750 VaR ( '=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15) DRMDP CC RR BROIL RMDP 100 200 300 400 500 Sample size 1750 1800 1850 Mean DRMDP CC RR BROIL RMDP Figure 5 Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold ε′ = 15%) and mean of reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The upper and lower edges of the shaded areas are respectively the 95% and 5% percentiles of the 100 performances, while the solid lines are the medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Simulation Study In this experiment, we follow the experiment setup in Regan and Boutilier (2012) where the number of reachable next-states and the transition kernel are randomly generated (both of which are known to decision makers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' More details of the experiment setting are relegated to Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As illustrated in Figures 5 and 7 (where the latter for VaR with ε′ ∈ {5%,10%} is relegated to Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4), when the decision maker aims to optimize her tailed performances, CC is a preferable choice compared to DRMDPs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' on the contrary, when pursuing optimizing the average return, DRMDPs perform much better than CC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Observe that the RR model, which includes both DRMDPs and the DCC model as special cases, remains as the best model under all criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In particular, one can observe that, RR achieves higher percentile returns than BROIL (that is a model without robustness), which demonstrates the benefits of distributionally robustness and the advantage of the risk measure VaR for percentile performance optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As expected, RMDPs end up yielding over-conservative policies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' as a result, it performs poorly in most instances under all criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Machine Replacement Problem In this experiment, we follow the experiment setup in Delage and Mannor (2010) and consider the case where a factory holds an extensive amount of machines, each of which is subject to the same underlying MDP (more details of the experiment setting can be found in Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our setting is similar to Delage and Mannor (2010) except for the follows: we use a data-driven setting 3 As we demonstrated in Section 4, a DCC is equivalent to a nominal chance-constrained one with an adjusted risk level, thus here we simply choose the latter as the benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 17 as described above, and we evaluate our (policies of) models by looking at the various performance measures as in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We report the overall performances of the five models in Figures 4 and 8 (where the latter for VaR with ε′ ∈ {5%,10%} is relegated to Appendix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Similar to the previous experiment, RR always performs better than or equal to the best model between CC and DRMDPs, and it provides the best performance under all criteria, which again manifest the merit of taking both the expected and worst-case performances into consideration and distributionally robustness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Computation Times of Different Algorithms Table 1 The average of the runtimes of the MOSEK solver and the AD-LPMM algorithm in seconds and the relative gaps (%) to the optimal values computed by MOSEK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' S=A Runtimes Relative gaps MOSEK AD-LPMM 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='60 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='79 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 % 70 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='81 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 % 100 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='50 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2 % 130 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='54 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='17 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 % 160 444.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='06 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4 % In this section, we compare the computation times of our AD-LPMM algorithm with the state- of-the-art solver MOSEK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Table 1 reports the runtimes of the the AD-LPMM and MOSEK when solving problem (8) at different problem sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Results indicate that, though our AD-LPMM is slower than the MOSEK solver when problem size is small, it showcases its strong scalability and become much faster than MOSEK with large-size problems (while always maintaining high solution quality), where the advantage is more notable when the problem scales up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Conclusion We consider risk-aware MDPs with ambiguous reward functions and propose the return-risk model, which is versatile and can optimize any weighted combination of the average and quantile perfor- mances of a policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' This model generalizes and combines the advantage of distributionally robust MDPs and distributionally robust chance-constrained MDPs, thus is powerful in both average and percentile performances optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In particular, risk from uncertain transition kernel can also be captured by the return-risk model when output policies are deterministic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Tractable refor- mulations are provided for all our proposed models, and we design an AD-LPMM algorithm for Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 18 the return-risk model, which is well scalable and faster than the MOSEK solver with large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Experimental results showcase the versatility of the return-risk model as well as the scalability of the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In the future, we believe that it would be important to explore more efficient methods for obtaining solution of RR, where function approximation and policy gradient (Sutton and Barto 2018) are possible choices to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' References Abdullah, Mohammed Amin, Hang Ren, Haitham Bou Ammar, Vladimir Milenkovic, Rui Luo, Mingtian Zhang, Jun Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Wasserstein robust reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='13196 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ahmadi, Mohamadreza, Ugo Rosolia, Michel Ingham, Richard Murray, Aaron Ames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Constrained risk-averse Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The 35th AAAI Conference on Artificial Intelligence (AAAI-21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Alizadeh, Pegah, Yann Chevaleyre, Jean-Daniel Zucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Approximate regret based elicitation in Markov decision process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The 2015 IEEE RIVF International Conference on Computing & Communication Technologies-Research, Innovation, and Vision for Future (RIVF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE, 47–52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' B¨aauerle, Nicole, Ulrich Rieder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Partially observable risk-sensitive Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathe- matics of Operations Research 42(4) 1180–1196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' B¨auerle, Nicole, Alexander Glauner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Minimizing spectral risk measures applied to Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematical Methods of Operations Research 94(1) 35–69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' B¨auerle, Nicole, Jonathan Ott.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Markov decision processes with average-value-at-risk criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathe- matical Methods of Operations Research 74(3) 361–379.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Beck, Amir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' First-order methods in optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' SIAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Behzadian, Bahram, Reazul Russel, Marek Petrik, Chin Pang Ho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Optimizing percentile criterion using robust MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' International Conference on Artificial Intelligence and Statistics 1009–1017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ben-Tal, Aharon, Dimitris Bertsimas, David B Brown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A soft robust model for optimization under ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations research 58(4-part-2) 1220–1234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ben-Tal, Aharon, Arkadi Nemirovski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lectures on modern convex optimization: analysis, algorithms, and engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' SIAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bernard, Carole, Silvana M Pesenti, Steven Vanduffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust distortion risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='08850 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bertsekas, Dimitri, John Tsitsiklis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Neuro-dynamic programming: an overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proceedings of 1995 34th IEEE Conference on Decision and Control, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE, 560–564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bertsimas, Dimitris, Aur´elie Thiele.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust and data-driven optimization: modern decision making under uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Models, methods, and applications for innovative decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' INFORMS, 95– 122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 19 Blanchet, Jose, Karthyek Murthy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Quantifying distributional model risk via optimal transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Math- ematics of Operations Research 44(2) 565–600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Brown, Daniel, Scott Niekum, Marek Petrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bayesian robust optimization for imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33 2479–2491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Calafiore, Carlo, L El Ghaoui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' On distributionally robust chance-constrained linear programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Journal of Optimization Theory and Applications 130(1) 1–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Carpin, Stefano, Yin-Lam Chow, Marco Pavone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Risk aversion in finite Markov decision processes using total cost criteria and average value at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016 IEEE International Conference on Robotics and Automation (ICRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE, 335–342.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen, Xin, Melvyn Sim, Peng Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A robust optimization perspective on stochastic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research 55(6) 1058–1071.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen, Xinyue, Zijian Zhou, Zheng Wang, Che Wang, Yanqiu Wu, Keith Ross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bail: Best-action imitation learning for batch deep reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33 18353–18363.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen, Zhi, Daniel Kuhn, Wolfram Wiesemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Data-driven chance constrained programs over Wasser- stein balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:1809.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='00210 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen, Zhi, Melvyn Sim, Huan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust optimization with infinitely constrained ambiguity sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research 67(5) 1328–1344.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen, Zhi, Weijun Xie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Sharing the value-at-risk under distributional ambiguity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematical Finance 31(1) 531–559.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Choi, Jaedeug, Kee-Eung Kim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Nonparametric Bayesian inverse reinforcement learning for multiple reward functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chow, Yinlam, Mohammad Ghavamzadeh, Lucas Janson, Marco Pavone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Risk-constrained reinforce- ment learning with percentile risk criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The Journal of Machine Learning Research 18(1) 6070–6120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Clement, Julien, Christian Kroer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' First-order methods for Wasserstein distributionally robust MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' PMLR, 2010–2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Clement, Julien Grand, Christian Kroer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' First-order methods for wasserstein distributionally robust mdp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' PMLR, 2010–2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Delage, Erick, Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Percentile optimization in uncertain Markov decision processes with application to efficient exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proceedings of the 24th International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' PMLR, 225–232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Delage, Erick, Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Percentile optimization for Markov decision processes with parameter uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research 58(1) 203–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Delgado, Karina, Leliane De Barros, Daniel Dias, Scott Sanner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Real-time dynamic programming for Markov decision processes with imprecise probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Artificial Intelligence 230 192–223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 20 Derman, Esther, Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributional robustness and regularization in reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='02894 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Erdo˘gan, Emre, Garud Iyengar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ambiguous chance constrained problems and robust optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematical Programming 107(1) 37–61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Gao, Rui, Anton Kleywegt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust stochastic optimization with Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:1604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='02199 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Gao, Rui, Anton Kleywegt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust stochastic optimization with Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematics of Operations Research .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Gilbert, Hugo, Paul Weng, Yan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Optimizing quantiles in preference-based Markov decision pro- cesses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Hanasusanto, Grani, Vladimir Roitch, Daniel Kuhn, Wolfram Wiesemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A distributionally robust perspective on uncertainty quantification and chance constrained programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematical Pro- gramming 151(1) 35–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Hanasusanto, Grani, Vladimir Roitch, Daniel Kuhn, Wolfram Wiesemann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ambiguous joint chance constraints under mean and dispersion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research 65(3) 751–767.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Haskell, William, Rahul Jain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A convex analytic approach to risk-aware Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' SIAM Journal on Control and Optimization 53(3) 1569–1598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ho, Jonathan, Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Generative adversarial imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in neural infor- mation processing systems 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Hogg, Robert V, Allen T Craig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Introduction to mathematical statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (5”” edition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Englewood Hills, New Jersey .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Huang, Wenjie, William Haskell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Risk-aware q-learning for Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017 IEEE 56th Annual Conference on Decision and Control (CDC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE, 4928–4933.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Huang, Yonghui, Xianping Guo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Minimum average value-at-risk for finite horizon semi-Markov decision processes in continuous time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' SIAM Journal on Optimization 26(1) 1–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Iyengar, Garud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematics of Operations Research 30(2) 257–280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Jaimungal, Sebastian, Silvana M Pesenti, Ye Sheng Wang, Hariom Tatsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust risk-aware rein- forcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' SIAM Journal on Financial Mathematics 13(1) 213–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Jiang, Daniel R, Warren B Powell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Risk-averse approximate dynamic programming with quantile-based risk measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematics of Operations Research 43(2) 554–579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Kaiser, Lukasz, Mohammad Babaeizadeh, Piotr Milos, Blazej Osinski, Roy H Campbell, Konrad Czechowski, Dumitru Erhan, Chelsea Finn, Piotr Kozakowski, Sergey Levine, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Model-based reinforcement learning for atari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='00374 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 21 Kruschke, John K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bayesian data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Wiley Interdisciplinary Reviews: Cognitive Science 1(5) 658–676.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lagoudakis, Michail G, Ronald Parr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Least-squares policy iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The Journal of Machine Learning Research 4 1107–1149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lobo, Elita A, Mohammad Ghavamzadeh, Marek Petrik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Soft-robust algorithms for batch reinforce- ment learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='14495 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mannor, Shie, Ofir Mebel, Huan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust MDPs with k-rectangular uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematics of Operations Research 41(4) 1484–1509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mannor, Shie, Duncan Simester, Peng Sun, John Tsitsiklis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bias and variance approximation in value function estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Management Science 53(2) 308–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ng, Andrew Y, Stuart J Russell, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Algorithms for inverse reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Icml, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Nocedal, Jorge, Stephen Wright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Numerical optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Osa, Takayuki, Joni Pajarinen, Gerhard Neumann, J Andrew Bagnell, Pieter Abbeel, Jan Peters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' An algorithmic perspective on imitation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='06711 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Panaganti, Kishan, Dileep Kalathil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Sample complexity of robust reinforcement learning with a gen- erative model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' International Conference on Artificial Intelligence and Statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' PMLR, 9582–9602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Petrik, Marek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Optimization-based approximate dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' University of Massachusetts Amherst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Petrik, Marek, Ronny Luss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Interpretable policies for dynamic product recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' UAI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Pflug, Georg, David Wozabal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ambiguity in portfolio selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Quantitative Finance 7(4) 435–442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Postek, Krzysztof, Aharon Ben-Tal, Dick Den Hertog, Bertrand Melenberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust optimization with ambiguous stochastic constraints under mean and dispersion information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research 66(3) 814–833.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Pr´ekopa, Andr´as.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Stochastic programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Puterman, Martin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Markov decision processes: discrete stochastic dynamic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' John Wiley & Sons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Rashidinejad, Paria, Banghua Zhu, Cong Ma, Jiantao Jiao, Stuart Russell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Bridging offline reinforce- ment learning and imitation learning: A tale of pessimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Regan, Kevin, Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust policy computation in reward-uncertain MDPs using nondom- inated policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proceedings of the AAAI Conference on Artificial Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Regan, Kevin, Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2011a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Eliciting additive reward functions for Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Twenty-Second International Joint Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 22 Regan, Kevin, Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2011b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Robust online optimization of reward-uncertain MDPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Twenty-Second International Joint Conference on Artificial Intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Regan, Kevin, Craig Boutilier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Regret-based reward elicitation for Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:1205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2619 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Rosen, Laura, Orly Manor, Dan Engelhard, David Zucker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In defense of the randomized controlled trial for health promotion research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' American journal of public health 96(7) 1181–1186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Roy, Aurko, Huan Xu, Sebastian Pokutta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Reinforcement learning under model mismatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in neural information processing systems 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Sarykalin, Sergey, Gaia Serraino, Stan Uryasev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Value-at-risk vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' conditional value-at-risk in risk management and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' State-of-the-art decision-making tools in the information-intensive age.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Informs, 270–294.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Shapiro, Alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Rectangular sets of probability measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research 64(2) 528–541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Shapiro, Alexander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust optimal control and mdp modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Operations Research Letters 49(5) 809–814.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Shefi, Ron, Marc Teboulle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Rate of convergence analysis of decomposition methods based on the proximal method of multipliers for convex minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' SIAM Journal on Optimization 24(1) 269– 297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Smith, James, Robert L Winkler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The optimizer’s curse: skepticism and postdecision surprise in decision analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Management Science 52(3) 311–322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Sutton, Richard S, Andrew G Barto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Reinforcement learning: An introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' MIT press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' van de Berg, Damien, Thomas Savage, Panagiotis Petsagkourakis, Dongda Zhang, Nilay Shah, Ehecatl Anto- nio del Rio-Chanona.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Data-driven optimization for process systems engineering applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chemical Engineering Science 248 117135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Wang, Yue, Shaofeng Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Online robust reinforcement learning with model uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 34 7193–7206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Wang, Yue, Shaofeng Zou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Policy gradient method for robust reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='07344 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Xie, Weijun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' On distributionally robust chance constrained programs with Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematical Programming 186(1) 115–155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Xu, Huan, Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust Markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 23 2505–2513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Xu, Huan, Shie Mannor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematics of Operations Research 37(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 23 Yamai, Yasuhiro, Toshinao Yoshiba, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Comparative analyses of expected shortfall and value-at- risk: their estimation error, decomposition, and optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Monetary and economic studies 20(1) 87–121.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Yang, Insoon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Wasserstein distributionally robust stochastic control: A data-driven approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE Transactions on Automatic Control 66(8) 3863–3870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Yu, Pengqian, Huan Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust counterpart in markov decision processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE Transactions on Automatic Control 61(9) 2538–2543.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Zheng, Kan, Zhe Yang, Kuan Zhang, Periklis Chatzimisios, Kan Yang, Wei Xiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Big data-driven optimization for mobile networks toward 5g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' IEEE network 30(1) 44–51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Zymler, Steve, Daniel Kuhn, Ber¸c Rustem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally robust joint chance constraints with second-order moment information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Mathematical Programming 137(1) 167–198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof of Results A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proofs of Results in Section 3 Proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' It is sufficient to rewrite the objective of (4) as follows: inf P∈F(θ)EP[˜r⊤x] = − sup P∈F(θ) EP[−˜r⊤x] = −min λ≥0 � λθ − � RSA inf ξ∈RSA(λ∥ξ − r∥ + ξ⊤x) dˆPr � = − min λ≥∥x∥∗ � λθ − � RSA r⊤x dˆPr � = EˆP[˜r⊤x] − θ∥x∥∗, where the second identity follows from theorem 1 in Gao and Kleywegt (2016) and the third identity follows from strong conic duality inf ξ∈RK(λ∥ξ − r∥ + ξ⊤x) = � � � r⊤x λ ≥ ∥x∥∗ −∞ λ ∈ [0,∥x∥∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Substituting the above reexpression then concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proofs of Results in Section 4 Proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Notice that (6) is equivalent to sup P∈F(θ) P �˜r⊤x < y � ≤ ε ⇐⇒ sup P∈F(θ) P �˜r⊤x ≤ y � ≤ ε, where it is equivalent if we replace the strict inequality on the left-hand side with a weak one on the right-hand side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' see proposition 3 in Gao and Kleywegt (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Exploring the definition of VaR, we note that sup P∈F(θ) P �˜r⊤x ≤ y � ≤ ε ⇐⇒ sup P∈F(θ) P-VaR1−ε � y − ˜r⊤x � ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='9 in Chen and Xie (2021) and the assumption of Mahalanobis norm, it holds that sup P∈F(θ) P-VaR1−ε � y − ˜r⊤x � = P(µ,Σ,g)-VaR1−ε � y − ˜r⊤x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' In other words, the worst-case VaR around the elliptical distribution P(µ,Σ,g) with the risk threshold ε is equal to the nominal elliptical VaR with a small risk threshold ε ≤ ε (which, would correspond to a higher risk level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We thus obtain sup P∈F(θ) P-VaR1−ε � y − ˜r⊤x � ≤ 0 ⇐⇒ P(µ,Σ,g)-VaR1−ε [y − ˜r⊤x] ≤ 0 ⇐⇒ P(µ,Σ,g) �˜r⊤x ≤ y � ≤ ε ⇐⇒ P(µ,Σ,g) �˜r⊤x ≥ y � ≥ 1 − ε, Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 25 where the last equivalence follows from P(µ,Σ,g) being a continuous distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By Lemma 1, the first constraint in (5) is the same as P(µ,Σ,g) �˜r⊤x ≥ y � ≥ 1 − ε, where ε = 1 − Φ(¯η⋆) ≤ ε and ¯η⋆ is the smallest η ≥ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) − � η2/2 (Φ−1(1−ε)) 2/2 kg(z)dz ≥ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The constraint can then be further written as P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ Φ((µ⊤x − y)/ √ x⊤Σx) ≥ 1 − ε ⇐⇒ µ⊤x − y ≥ Φ−1(1 − ε) √ x⊤Σx ⇐⇒ µ⊤x − y ≥ ∥Φ−1(1 − ε)Σ1/2x∥2, where the first equivalence holds by the linearity of elliptical distributions, the second one is because that Φ(·) is non-decreasing, and the last one is due to the fact that 1−ε ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5 (which follows from ε ≤ ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Observe that the optimum is achieved at y⋆ = µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2, plugging this in the objective of problem (5) then concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proofs of Results in Section 5 Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By Proposition 1 and Proposition 2, we have inf P∈F(θ)EP[˜r⊤x] = −θ∥x∥2 + EˆP[˜r⊤x] and inf P∈F′(θ)P-VaR1−ε[˜r⊤x] = µ⊤x − ∥Φ−1(1 − ε)Σ1/2x∥2 with ε as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Substituting the above two equations into (7) and rearranging the terms then concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By the definition of ˆT, problem (9) can be rewritten as: max π∈(∆A)S ψ � i∈[N] wi · g(π, ˆP i) + (1 − ψ)max η∈R � � �η − 1 1 − ι � i∈[N] wi(η − g(π, ˆP i))+ � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By introducing auxiliary decision variables y ∈ RN, it can be further reformulated as: max ψ � i∈[N] wi · g(π, ˆP i) + (1 − ψ) � �η − 1 1 − ι � i∈[N] yi � � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' yi ≥ wi(η − g(π, ˆP i)) ∀i ∈ [N] π ∈ (∆A)S,y ∈ RN +,η ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (13) Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 26 Here we can express wi · g(π,P ) = max µ⊤x − αθ · ∥x∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2x∥2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' xs,a = πs,a · � a′∈A xs,a′ ∀(s,a) ∈ S × A (E − γ · ¯P )x = wi · p0 x ∈ RSA + (14) as in Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We can then, by combining (13) and (14), reformulate problem (9) as: max ψ � i∈[N] (µ⊤xi − αθ · ∥xi∥2 − (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2) + (1 − ψ)(η − 1 1 − ι � i∈[N] yi) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' yi − wiη ≥ αθ · ∥xi∥2 + (1 − α) · ∥Φ−1(1 − ε)Σ1/2xi∥2 − µ⊤xi ∀i ∈ [N] xi s,a = πs,a · � a′∈A xi s,a′ ∀i ∈ [N],(s,a) ∈ S × A (E − γ · ¯P i)xi = wi · p0 ∀i ∈ [N] π ∈ (∆A)S,η ∈ R,xi ∈ RSA + ,y ∈ RN + ∀i ∈ [N].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Now it is sufficient to focus on the second set of constraints xi s,a = πs,a · � a′∈A xi s,a′ ∀i ∈ [N],(s,a) ∈ S × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (15) Since we only consider deterministic policy π ∈ {0,1}SA and � a∈A xi s,a ∈ [0,wi/(1 − γ)] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='10 in Petrik (2010)), we have the McCormick relaxation (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', Petrik and Luss (2016)) of (15) as: � � � � � � � � � � � � � � � � � � � � � xi s,a ≤ � a′∈A xi s,a′ xi s,a ≤ wi 1 − γ πs,a xi s,a ≥ 0 xi s,a ≥ wi 1 − γ (πs,a − 1) + � a′∈A xi s,a′ for all i ∈ [N],(s,a) ∈ S × A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our conclusion then follows from the fact that the McCormick relaxation is precise when π ∈ {0,1} (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', the extreme values of the interval [0,1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proofs of Results in Section 6 Proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By (11), it is sufficient to focus on solving ProjBℓΣ(·)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By eigenvalue decomposition, we have Σ = G⊤DG4 with D = diag(d1,··· ,dSA), thus we have: ProjBℓΣ(·)(x) = arg min 1 2 · ∥v − x∥2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' v⊤G⊤DGv ≤ 1 v ∈ RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 4 The eigenvalue decomposition here is not counted in the time complexity of the bisection method (or the AD-LPMM algorithm), since this process is carried out for computing Σ1/2 in (8) (before we solve (8)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 27 By change of variable u = Gv and let b = Gx, it is sufficient to focus on the equivalent problem: arg min 1 2 · ∥u − b∥2 2 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' u⊤Du ≤ 1 u ∈ RSA, (16) where we can retrieve v⋆ = G⊤u⋆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The Lagrangian function of(16) (with the introduced dual variable ζ ∈ R+) is L(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ζ) = 1 2 · ∥u − b∥2 2 + ζ(u⊤Du − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Since (16) is a convex optimization problem, the KKT condition is the sufficient condition for the optimality of the primal and dual solutions: � � � � � � � � � � � � � u⊤Du ≤ 1 ζ ≥ 0 ζ(u⊤Du − 1) = 0 ∇uL(u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ζ) = u − b + 2ζ · Du = 0, where for ζ = 0, we have � � � u⊤Du ≤ 1 u − b = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' while when ζ > 0, we have � � � u⊤Du = 1 (I + 2ζ · D)u − b = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Therefore, if b⊤Db ≤ 1, we have u⋆ = b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' if b⊤Db > 1, it is sufficient to solve the equation g(ζ) = 1 where g(ζ) = � i∈[SA] dib2 i (1 + 2ζdi)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The function g is monotonically decreasing function on [0,+∞) and limζ→+∞ g(ζ) = 0, thus we can apply the bisection method to search on the interval [0, ¯ζ] (where ¯ζ : g(¯ζ) ≤ 1 is the upper bound for the search which we provide in Lemma 2) to locate ζ⋆ and retrieve u⋆ i = bi/(1+2ζ⋆di) ∀i ∈ [SA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The pseudocode is provided in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The time complexity of solving Py(x,ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c) is dominated by the bisection method, which has time complexity O(log(1/δ′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our conclusion follows from the fact that the computation in each iteraion of the bisection takes time O(SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The inequality g(ζ) ≤ 1 holds for all ζ ≥ (1/(2di′′))(bi′√SAdi′ − 1), where i′ ∈ arg maxi∈[SA] dib2 i and i′′ ∈ arg mini∈[SA] di Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 28 Algorithm 2: Bisection for Problem (16) Input: Desired precision δ′, initial lower bound ζ ← 0 and upper bound ζ > 0 if g(0) ≤ 1 then u ← b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' end else while |ζ − ζ| ≥ δ′ do ζ ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5(ζ + ζ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' if g(ζ) >= 1 then ζ ← ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' end else ζ ← ζ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' end end for i = 1,··· ,SA do ui = bi/(1 + 2ζdi);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' end end Output: Solution u Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Observe that, g(ζ) ≤ � i∈[SA] di′b2 i′ (1 + 2ζdi)2 ≤ SAdi′b2 i′ (1+2ζdi′′)2 , from which we have SAdi′b2 i′ (1 + 2ζdi′′)2 ≤ 1 ⇒ g(ζ) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Our conclusion thus follows by rearranging the terms of the inequality on the left-hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By Lemma 2, one can choose ζ = (1/(2di′′))(bi′√SAdi′ − 1), where i′ ∈ arg maxi∈[SA] dib2 i and i′′ ∈ arg mini∈[SA] di for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Notice that, it is sufficient to solve the ith subproblem: arg min z≥0 c 2z2 − (cxi + µi + ηi)z = max � 0, 1 c(cxi + µi + ηi) � for all i ∈ [SA], where our conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 29 Proof of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By the definition of Q(·,·), we have Px(y,z,λ,ξ,η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ν,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' ˆx) = arg min x αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c 2 · �������� (E − γ · ¯P )(x − ˆx) + (E − γ · ¯P )ˆx − p0 x − ˆx + ˆx − y x − ˆx + ˆx − z �������� 2 2 + 1 2 · ℓ2 Q(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ν)(x − ˆx) = arg min x αθ · ∥x∥2 + x⊤((E − γ · ¯P )⊤λ + ξ + η) + c 2 · �������� (E − γ · ¯P )(x − ˆx) x − ˆx x − ˆx �������� 2 2 +c · x⊤ � (E − γ · ¯P )⊤ � (E − γ · ¯P )ˆx − p0 � + 2 · ˆx − y − z � + 1 2 · ℓ2 Q(c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='ν)(x − ˆx) = arg min x αθ cν · ∥x∥2 + x⊤w + 1 2 · ∥x − ˆx∥2 2 = arg min x αθ cν · ∥x∥2 + 1 2 · ∥x − (ˆx − w)∥2 2 = � 1 − αθ cν max{∥w∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' αθ cν } � (ˆx − w) where we denote w = 1 cν �� E − γ · ¯P �⊤ λ + ξ + η � + 1 ν �� E − γ · ¯P �⊤ �� E − γ · ¯P � ˆx − p0 � + 2 · ˆx − y − z � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' and the last equality holds by,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', exam- ple 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='9 in Beck (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The computation time is dominated by computing ∥w∥2, which is O(SA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Evaluation of VaR and CVaR of Student’s t-Distribution The VaR of a Student’s t-distribution with threshold ε is in fact the lower-ε percentile of its probability density function (PDF), which can be looked up in table in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', Hogg and Craig (1995) (under some common values of ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We provide the calculation of CVaR as follows (with degree of freedom δ > 1 and v := Pt-dist-VaRε(˜r) assumed known): Pt-dist-CVaRε(˜r) = 1 ε · Γ( δ+1 2 ) (πδ) 1 2 Γ( δ 2 ) � v −∞ r (1+ r2 δ ) δ+1 2 dr = 1 ε · δ 1 2 ·Γ( δ+1 2 ) 2π 1 2 Γ( δ 2 ) � 1+ v2 δ −∞ u− k+1 2 du = − δ 1 2 ·Γ( δ+1 2 ) επ 1 2 (δ−1)Γ( δ 2 ) · � 1 + v2 δ �− k−1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' where the first equality follows from the definition of the CVaR and the PDF of the t-distribution herein,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' the second equality holds by the technique of integration by substitution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Preliminaries on Elliptical Distributions The probability density distribution of an elliptical reference distribution P(µ,Σ,g) is given by f(r) = k · g �1 2(r − µ)⊤Σ−1(r − µ) � , Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 30 where k is a positive normalization scalar, µ is a mean vector, Σ is a positive definite matrix and g is a generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Elliptical distribution is a broad family of distributions that includes for example, the multivariate normal distribution, multivariate t-distribution and multivariate logistic distribution, as special cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' One notable property of the elliptical distribution is the linearity: any linear combination of elliptically distributed random variables still follows an elliptical distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' That is, for any random vector ˜r ∼ P(µ,Σ,g), it holds that ˜r⊤x ∼ P(µx,σ2x,g) with µx = µ⊤x and σx = √ x⊤Σx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Indeed, we can express the combination as ˜r⊤x = µx + σx˜z, where ˜z ∼ P(0,1,g) is a standard elliptically distributed random variable whose probability density function and cumulative distribution function are φ(z) = k·g (z2/2) and Φ(x) = � x −∞ k·g(z2/2)dz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For a concrete example we take a closer look at a standard normal distribution, for which the normalization scalar and generating function are k = 1/ √ 2π and g(x) = exp(−x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally Optimistic MDPs In contrast to the robust model, sometimes the decision maker prefers exploration over exploitation if she would like to learn more information about the MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As such, we could instead adopt an optimistic counterpart where we focus on the best case, motivating the following distributionally optimistic MDP: ℓO(θ) = max x∈X sup P∈F(θ) EP[˜r⊤x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (17) In contrast to the robust case, here our decision depends instead on the best possible (expected) outcome, which exactly embodies optimism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We summarize the reformulation of (17) as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The distributionally optimistic MDP (17) is equivalent to an optimization prob- lem ℓO(θ) = max x∈X EˆP[˜r⊤x] + θ∥x∥∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' It is sufficient to rewrite the objective of (17) as follows: sup P∈F(θ) EP[˜r⊤x] = − inf P∈F(θ)EP[−˜r⊤x] = −(EˆP[−˜r⊤x] − θ∥x∥∗) = EˆP[˜r⊤x] + θ∥x∥∗, where the second identity follows similar lines as in the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The reformulation in Proposition 8 is a reverse conic program that is, in general, non-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' However, it can be recast as a mixed-integer linear program, provided that ∥ · ∥∗ is the commonly used L1-norm or L∞-norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Such a mixed-integer linear program can be solved by the state-of-the- art approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 31 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Distributionally Optimistic Chance-Constrained Model In a distributionally optimistic chance-constrained MDP model, where we focus on the best case that with high probability, the reward is no smaller than some lower bound that we maximize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Formally, the distributionally optimistic chance-constrained MDP model is formulated as follows: ℓDOCC(θ,ε) = � � � � � � � � � max y s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' sup P∈F(θ) P[˜r⊤x ≥ y] ≥ 1 − ε x ∈ X, y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (18) The optimistic chance-constrained model (18) is also equivalent to a nominal chance-constrained model, however, at a less risky level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Before formally establishing this argument, two lemmas are introduced as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The worst (largest) probability of the random vector ˜r attaining a value in the set R, sup P∈F(θ) P[˜r ∈ R], (19) is equivalent to min λ≥0 � λθ + � r∈RSA(λ · dist(r,R) − 1)−dˆPr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Here, we use dist(r,R) = inf{∥r − ˆr∥ | ˆr ∈ R} to denote the distance from the vector r ∈ RSA to the set R ⊆ RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Using theorem 1 in Gao and Kleywegt (2016) or theorem 1 in Blanchet and Murthy (2019), the uncertainty quantification problem (19) is equal to min λ≥0 � λθ − � r∈RSA inf w∈RSA{λ∥w − r∥ − I[w ∈ R]}dˆPr � , (20) where I is the 0-1 indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Consider the second term in the objective of the above minimization problem, we have inf w∈RSA{λ∥w − r∥ − I[w ∈ R]} = −(λ · dist(r,R) − 1)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (21) Indeed, if r ∈ R (for which, dist(r,R) = 0), then by choosing w = v, it holds that inf w∈RSA{λ∥w − r∥ − I[w ∈ R]} = −1 = −(λ · dist(r,R) − 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' whereas if r /∈ R, then it holds that inf w∈RSA{λ∥w − r∥ − I[w ∈ R]} = min � inf w∈R{λ∥w − r∥ − 1}, inf w /∈Rλ∥w − r∥ � = min � inf w∈R{λ∥w − r∥ − 1},0 � = −(λ · dist(r,R) − 1)−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Plugging expression (21) into problem (20) gives the desired result, which, by proposition 3 in Gao and Kleywegt (2016), holds regardless of whether R is open or closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 32 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The distributionally optimistic chance constraint inf P∈F(θ)P[˜r ∈ R] ≤ ε (22) with a risk threshold ε ∈ (0,1) is satisfiable if and only if P-CVaRε[−dist(˜r, ¯R)] ≥ − θ 1 − ε, where ¯R = RSA \\ R is the complement of the set of undesired events R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We first re-express (22) as sup P∈F(θ) P[˜r ∈ ¯R] ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Using Lemma 3, the above constraint is equivalent to min λ≥0 � λθ + � r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr � ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (23) The left-hand side problem can be presented by min � min λ>0 � λθ + � r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr � ,1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Since 1 ≥ 1 − ε, the above re-expression implies that constraint (23) is equivalent to min λ>0 � λθ + � r∈RSA(λ · dist(r, ¯R) − 1)−dˆPr � ≥ 1 − ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Multiplying both sides by (λ(1 − ε))−1 > 0, we arrive at min τ<0 � 1 1 − ε � r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ � ≥ − θ 1 − ε, which, together with the fact min τ≥0 � 1 1 − ε � r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ � ≥ 0 ≥ − θ 1 − ε, is equivalent to min τ∈R � 1 1 − ε � r∈RSA(−dist(r, ¯R) − τ)+dˆPr + τ � ≥ − θ 1 − ε, where the left-hand side is essentially ˆP-CVaRε[−dist(˜r, ¯R)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Now we are ready to establish the equivalence between the chance-constrained model and its optimistic counterpart (with an adjusted risk threshold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Suppose in the Wasserstein ambiguity set (3), the reference distribution is an ellip- tical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm associated with the positive definite matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The distributionally optimistic robust chance con- straint ∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε is satisfiable if and only if P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε, where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) + � (Φ−1(1−ε)) 2/2 η2/2 kg(z)dz ≤ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 33 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We first look at the individual distributionally optimistic robust chance constraint ∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε for some generic coefficient vector x ∈ RSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The above chance constraint is equivalent to sup P∈F(θ) P[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ sup P∈F(θ) P[˜r⊤x > y] ≥ 1 − ε ⇐⇒ inf P∈F(θ)P[˜r⊤x ≤ y] ≤ ε, where for the first equivalence, by using proposition 3 in Gao and Kleywegt (2016) , it is indifferent to replace the strict inequality with a weak one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Exploring the definition of VaR, we note that inf P∈F(θ)P[˜r⊤x ≤ y] ≤ ε ⇐⇒ inf P∈F(θ)P-VaR1−ε[y − ˜r⊤x] ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Hence, with the translation invariance of VaR, it is sufficient to show that inf P∈F(θ)P-VaR1−ε[−˜r⊤x] ≜ inf v∈R � v | inf P∈F(θ)P[−˜r⊤x > v] ≤ ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (24) By Lemma 4 and the assumption of Mahalanobis norm, we have inf P∈F(θ)P � −˜r⊤x > v � ≤ ε ⇐⇒ P(µ,Σ,g)-CVaRε[−dist(˜r, ¯R)] ≥ − θ 1 − ε ⇐⇒ −P(µ,Σ,g)-CVaRε[−(−˜r⊤x − v)+] ≤ θ∥x∥Σ−1 1 − ε , where ¯R = � r | − r⊤x ≤ v � and we leverage the closed form solution dist(˜r, ¯R) = � −˜r⊤x − v �+ /∥x∥Σ−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=', lemma 2 in Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Let PS = P(µ,Σ,g) for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By the property of elliptical distribution, for ˜r ∼ PS and any real vector x, we have −˜r⊤x ∼ P(µS,σ2 S,g) = P(−µ⊤x,x⊤Σx,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We denote its probability density function as h(z) = k σS g � (z − µS) 2 2σ2 S � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The left-hand side of the constraint can be further transformed as −PS-CVaRε[−(−˜r⊤x − v)+] = −EPS[−(−˜r⊤x − v)+ | − (−˜r⊤x − v)+ ≥ PS-VaRε[−(−˜r⊤x − v)+]] = − 1 1 − ε � sup{z|−(z−v)+≥PS-VaRε[−(−˜r⊤x−v)+]} −∞ −(z − v)+h(z)dz = 1 1 − ε � sup{z|−(z−v)+≥PS-VaRε[−(−˜r⊤x−v)+]} v (z − v)h(z)dz = 1 1 − ε � PS-VaR1−ε[−˜r⊤x] v (z − v)h(z)dz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ho: Risk-Averse MDPs under Reward Ambiguity 34 in which the last equality holds from sup{z | − (z − v)+ ≥ PS-VaRε[−(−˜r⊤x − v)+]} = sup{z | min{v − z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0} ≥ PS-VaRε[min{v + ˜r⊤x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0}]} = sup{z | min{−z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='−v} ≥ PS-VaRε[min{˜r⊤x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='−v}]} = sup{z | − z ≥ PS-VaRε[min{˜r⊤x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='−v}]} = sup{z | z ≤ PS-VaR1−ε[max{−˜r⊤x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='v}]} = sup{z | z ≤ PS-VaR1−ε[−˜r⊤x]} = PS-VaR1−ε[−˜r⊤x].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Here, the second equality is due to the translation invariance of VaR, the third one follows from −v ≥ PS-VaRε[min{˜r⊤x,−v}], the fifth one is because that for any ε ∈ (0,1), the distributionally optimistic robust VaR satisfies v = inf P∈F(θ)P-VaR1−ε[−˜r⊤x] ≤ PS-VaR1−ε[−˜r⊤x], (25) thus the 1 − ε quantiles of −˜r⊤x and max{−˜r⊤x,v} coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Let us denote q1−ε = PS-VaR1−ε[−˜r⊤x], which, by its definition, satisfies q1−ε − µS σS = PS-VaR1−ε �−˜r⊤x − µS σS � = P0 (0,1,g)-VaR1−ε[˜z] = Φ−1(1 − ε), Here, the first equality holds for the translation invariance and the positive homogeneity of VaR, while the last one follows from the definition of VaR under the standard elliptical distribution P(0,1,g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Following the last reformulation of the constraint, we further have 1 1 − ε � q1−ε v (z−v)h(z)dz = 1 1 − ε � q1−ε v z· k σS g � (z − µS) 2 2σ2 S � dz− v 1 − ε � q1−ε v k σS g � (z − µS) 2 2σ2 S � dz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ho: Risk-Averse MDPs under Reward Ambiguity 35 For its first component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' we have ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� q1−ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='z · k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='(z − µS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� q1−ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='z − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='k · g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='(z − µS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='dz + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� q1−ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='k · g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='(z − µS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� q1−ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='v ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='z − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='k · g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='(z − µS) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�z − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�q1−ε − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='− Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�v − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='q1−ε−µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='v−µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t · k · g ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�z − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='+ µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�q1−ε − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='− Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�v − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='(q1−ε−µS)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='(v−µS)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='k · g(z)dz + µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1 − ε ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�q1−ε − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='− Φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�v − µS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='σS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' while for the second component,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' it holds that v 1 − ε � q1−ε v k σS g � (z − µS) 2 2σ2 S � dz = v 1 − ε � q1−ε−µS σS v−µS σS k · g �z2 2 � dz = v 1 − ε � Φ �q1−ε − µS σS � − Φ �v − µS σS �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Hence, combine the constraint with (25), we have the following equivalent expression for prob- lem (24): inf v s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' � (q1−ε−µS)2 2σ2 S (v−µS)2 2σ2 S k · g(z)dz + µS − v σS � Φ �q1−ε − µS σS � − Φ �v − µS σS �� ≤ θ∥x∥Σ−1 σS = θ v ≤ PS-VaR1−ε[−˜r⊤x] v ∈ R, where the equality follows from the definition of the Mahalanobis norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Let η = (v − µS)/σS, the best-case VaR now becomes inf µS + σSη s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' � (Φ−1(1−ε))2/2 η2/2 k · g(z)dz − η · (1 − ε − Φ(η)) ≤ θ η ≤ Φ−1(1 − ε) η ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (26) The function V (η) ≜ � (Φ−1(1−ε))2/2 η2/2 k · g(z)dz − η · (1 − ε − Φ(η)) Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 36 is monotonically decreasing on (−∞,Φ−1(1 − ε)) since for any η < Φ−1(1 − ε), it holds that V ′(η) = −η · k · g �η2 2 � − (1 − ε) + Φ(η) + ηφ(η) = Φ(η) − (1 − ε) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Thus problem (26) can be efficiently solved be a bisection algorithm and the optimal η⋆ as claimed can be obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Finally the result can be obtained as follows: ∃ P ∈ F(θ) : P[˜r⊤x ≥ y] ≥ 1 − ε ⇐⇒ −y ≥ σSη⋆ + µS ⇐⇒ −y − µS σS ≥ η⋆ ⇐⇒ Φ �−y − µS σS � ≥ Φ(η⋆) ⇐⇒ P(µ,Σ,g) � ˜r⊤x − µS σS ≥ y − µS σS � ≥ 1 − ¯ε ⇐⇒ P(µ,Σ,g)[˜r⊤x ≥ y] ≥ 1 − ¯ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' With ¯ε in Lemma 5, we are now ready to derive a second-order cone reformulation of the distributionally optimistic chance-constrained model (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Suppose in the Wasserstein ambiguity set (3), the reference distribution is an elliptical distribution ˆP = P(µ,Σ,g) and the Wasserstein distance is equipped with a Mahalanobis norm associated with the positive definite matrix Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' If the risk threshold satisfies ε ≤ ¯ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5, then the distributionally optimistic chance-constrained MDP (18) is equivalent to the second-order cone program ℓDOCC(θ,ε) = max x∈X µ⊤x − ∥Φ−1(1 − ¯ε)Σ1/2x∥2, where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) + � (Φ−1(1−ε)) 2/2 η2/2 kg(z)dz ≤ θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' By Lemma 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' the first constraint in (18) is equivalent to P(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g)[˜r⊤x ≥ y] ≥ 1 − ¯ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' where ¯ε = 1 − Φ(η⋆) ≥ ε with η⋆ being the smallest η ≤ Φ−1(1 − ε) that satisfies η(Φ(η) − (1 − ε)) + � (Φ−1(1−ε)) 2/2 η2/2 kg(z)dz ≤ θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' which can be further transformed as follows: P(µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='g)[˜r⊤x ≥ y] ≥ 1 − ¯ε ⇐⇒ Φ((µ⊤x − y)/ √ x⊤Σx) ≥ 1 − ¯ε ⇐⇒ µ⊤x − y ≥ Φ−1(1 − ¯ε) √ x⊤Σx ⇐⇒ µ⊤x − y ≥ ∥Φ−1(1 − ¯ε)Σ1/2x∥2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Chen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ho: Risk-Averse MDPs under Reward Ambiguity 37 where the first equivalence holds by the linearity of elliptical distributions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' the second one holds because of the non-decreasing cumulative distribution function Φ(·),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' and the third one holds as ¯ε < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Since the optimal value is achieved with y = µ⊤x − ∥Φ−1(1 − ¯ε)Σ1/2x∥2, plugging this equation in the objective of (18) then concludes our proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Details on Robust MDPs As introduced in Delage and Mannor (2010), robust MDPs maximizes the total expected return considering the worst-case realization of the uncertain parameter within a predefined ambiguity set: max π∈Π min r0∈R,r1∈R,···E � ∞ � t=0 γtrt(st) | s0 ∝ p0,π � , (27) where Π is the set of all the stationary randomized policies, rt and st are the reward and state at time stage t, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' As in Delage and Mannor (2010), we set R to be the 99% confidence ellipsoid of the random reward vector as the uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Details on BROIL Similar to our return-risk model, BROIL (Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 2020) also seeks a policy that maximizes the weighted average of the mean and percentile performances: max π∈Π λ · E � ∞ � t=0 γtrt(st) | s0 ∝ p0,π � + (1 − λ) · CVaRε � ∞ � t=0 γtrt(st) | s0 ∝ p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='π � , (28) where λ ∈ [0,1] is the weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Given R ∈ RSA×n as the matrix of (n) reward samples, BROIL can be expressed as a linear program as follows: max x∈X,y∈Rλ · 1 ne⊤R⊤x + (1 − λ) · � y − 1 ε · 1 ne⊤(y · e − R⊤x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Observe that, there are two major differences between BROIL and our return-risk model: first, BROIL use CVaR as its risk measure, while VaR is applied in our return-risk model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' second, while distributionally robustness is considered in (both the mean and VaR of return in) our objective function, BROIL only computes the nominal mean and CVaR of the return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Details and Results on the Experiments H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Details of Parameter Selection We use cross validation for parameter selection in both the simulation and empirical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For DRMDPs (4), the candidate set for θ is {0,2,··· ,18};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' for CC (2), the candidate set for ε is {iε′/5}i∈[5];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' for RR (7), we select θ such that ε varies among {iε′/5}i∈[5], and we select α ∈ {0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='25,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='75,1};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' for BROIL (28), we select λ × ε ∈ {0,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='25,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='75,1} × {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='15};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' for RMDPs (27), as in Delage and Mannor (2010), we set R to be the 99% confidence ellipsoid of the random reward vector as the uncertainty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 38 Figure 6 A machine replacement problem with fixed Gaussian rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Details of the Simulation Study We consider S = 10 states, A = 10 actions, a uniform initial state distribution, and a discount factor γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' For each state s ∈ [S], the number of reachable next-state is ⌈log S⌉.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We sample the true reward from a multivariate normal distribution N(µ′,Σ′), where for each k ∈ [SA], µ′ k is generated as follows: first we sample a number (0 or 1) from a discrete uniform distribution in {0,1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' If the result is 0, we generate µ′ k from the normal distribution N(50,100);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' otherwise we generate it from N(90,100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Standard deviations of rewards are generated in the same manner with another two normal distributions N(3,9) and N(18,9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Both standard deviations and means are trimmed to be non-negative after the above procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The correlation matrix of rewards is generated as follows: we first sample a matrix R ∈ RSA×SA with all its entries independently sampled in [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='25,1] uniformly, and then obtain our correlation matrix diag(d)V diag(d), where V = R⊤R and d = {di}i∈[SA] = {1/√Vii}i∈[SA].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Details of the Empirical Study In this experiment, each machine is subject to the same underlying MDP with a state set S = [S] with S = 50 and an action set with only two actions: repair the machine or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The transition is deterministic and the discount factor is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The reward depends on both the current state and action, and all the rewards are independently and normally distributed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Figure 6 illustrates the true underlying distribution that generates the random rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Results of the Simulation Study H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Additional Results of the Empirical Study 130,1) N(-130,1) 130,1) N(-130,20) 2 N(0,10) N(0,10 N(0,10-4) V0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='10 Repair N(-100,800) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=" Not RepairRuan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 39 100 200 300 400 500 Sample size 1500 1600 1700 VaR ( '=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content="05) DRMDP CC RR BROIL RMDP 100 200 300 400 500 Sample size 1550 1600 1650 1700 1750 VaR ( '=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1) DRMDP CC RR BROIL RMDP Figure 7 Simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk thresh- old ε′ ∈ {5%,10%}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The upper and lower edges of the shaded areas are respectively the 95% and 5% percentiles of the 100 performances, while the solid lines are the medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' 100 200 300 400 500 Sample size 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content="5 VaR ( '=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='05) DRMDP CC RR BROIL RMDP 100 200 300 400 500 Sample size 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content="5 VaR ( '=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content='1) DRMDP CC RR BROIL RMDP Figure 8 Empirical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Models DRMDP (4), CC (2), RR (7), RMDP and BROIL evaluated by VaR (risk threshold ε′ ∈ {5%,10%}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' The upper and lower edges of the shaded areas are respectively the 95% and 5% percentiles of the 100 performances, while the solid lines are the medians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Related Works Table 2 summarizes literature that is related to our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' We remark that, compared to its related works in Table 2, our return-risk model is the only one that considers risk ambiguity, and we have also designed a fast first-order algorithm to obtain its solution, which enhance the practicality of our model for large-scale problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Ruan, Chen, Ho: Risk-Averse MDPs under Reward Ambiguity 40 Table 2 Related works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' Paper Uncertainty Robustness Ambiguity set Risk measure Soft-robustness Delage and Mannor (2010) Rewards and transition kernel VaR No Xu and Mannor (2010) Rewards and transition kernel DRO Nested No Yu and Xu (2015) Rewards and transition kernel DRO (General) Nested No Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2020) Rewards CVaR Yes Gilbert et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2017) Rewards VaR No Lobo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} +page_content=' (2020) Transition kernel CVaR Yes Yang (2020) Transition kernel DRO Wasserstein No This paper Rewards DRO Wasserstein VaR Yes' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/GtAzT4oBgHgl3EQfHftK/content/2301.01045v1.pdf'} diff --git a/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf b/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..057a0b9c46a2730fe66ed081747874de80a1cce2 --- /dev/null +++ b/H9AzT4oBgHgl3EQfU_x9/content/2301.01276v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:2961027b5d983fb8fd9a405c057b38e15e2686f21ebe6902f925f704df755eb7 +size 172712 diff --git a/H9AzT4oBgHgl3EQfU_x9/vector_store/index.faiss b/H9AzT4oBgHgl3EQfU_x9/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..cd4c1094585ed701619e2c025917c921f11ae4c4 --- /dev/null +++ b/H9AzT4oBgHgl3EQfU_x9/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:74a3d7dd9fc2249f1c93f7bb9d187e713d21bbfe772d9a858b0378e6ce02611f +size 2424877 diff --git a/H9AzT4oBgHgl3EQfU_x9/vector_store/index.pkl b/H9AzT4oBgHgl3EQfU_x9/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..3089000c783f988b2f0bea305da9f3c9e22c2be5 --- /dev/null +++ b/H9AzT4oBgHgl3EQfU_x9/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7e44dd90602580b1dcee43bfe3ec1a31616a5f7ceed3ff919604903c028faa04 +size 92004 diff --git a/HNFAT4oBgHgl3EQfth7Z/content/tmp_files/2301.08665v1.pdf.txt b/HNFAT4oBgHgl3EQfth7Z/content/tmp_files/2301.08665v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3ee231032689b97d638e49304b52a9b8c2767450 --- /dev/null +++ b/HNFAT4oBgHgl3EQfth7Z/content/tmp_files/2301.08665v1.pdf.txt @@ -0,0 +1,730 @@ +MTGP: Combining Metamorphic Testing and +Genetic Programming +Dominik Sobania, Martin Briesch, Philipp Röchner, and Franz Rothlauf +Johannes Gutenberg University, Mainz, Germany +{dsobania,briesch,proechne,rothlauf}@uni-mainz.de +Abstract. Genetic programming is an evolutionary approach known +for its performance in program synthesis. However, it is not yet ma- +ture enough for a practical use in real-world software development, since +usually many training cases are required to generate programs that gen- +eralize to unseen test cases. As in practice, the training cases have to be +expensively hand-labeled by the user, we need an approach to check the +program behavior with a lower number of training cases. Metamorphic +testing needs no labeled input/output examples. Instead, the program +is executed multiple times, first on a given (randomly generated) in- +put, followed by related inputs to check whether certain user-defined +relations between the observed outputs hold. In this work, we suggest +MTGP, which combines metamorphic testing and genetic programming +and study its performance and the generalizability of the generated pro- +grams. Further, we analyze how the generalizability depends on the num- +ber of given labeled training cases. We find that using metamorphic test- +ing combined with labeled training cases leads to a higher generalization +rate than the use of labeled training cases alone in almost all studied +configurations. Consequently, we recommend researchers to use meta- +morphic testing in their systems if the labeling of the training data is +expensive. +Keywords: Program Synthesis · Metamorphic Testing · Genetic Pro- +gramming. +1 +Introduction +Genetic programming (GP) [7,21] is an evolutionary algorithm-based approach +to automatically generate programs in a given programming language that meet +user-defined requirements. In GP-based program synthesis, these specifications +are usually given as input/output examples, which define the expected output +from a generated program for a given input, and are used as training data during +the evolutionary search. +With the introduction of new selection methods [17,31], variation opera- +tors [16] and grammar design techniques [10], GP-based program synthesis has +made great progress in the last years. Recently, it has been shown that GP- +based approaches for program synthesis are even competitive in performance +with state-of-the-art neural network-based approaches [26]. +arXiv:2301.08665v1 [cs.SE] 20 Jan 2023 + +2 +Sobania et al. +Unfortunately, GP-based program synthesis is not yet mature enough for +a practical use in real-world software development, since many input/output +examples are usually required (up to 200 are regularly used in the literature [13]) +to generate programs that not only work on the training cases, but also generalize +to previously unseen test cases. In practice, however, the training cases have to be +labeled manually by the user which is very expensive and time consuming. So it +is necessary to reduce the number of required training cases in order to minimize +the user’s manual effort. However, simply reducing the number of training cases +is not sufficient, since a small training set can easily be overfitted which will on +average lead to a poor generalization of the generated programs. Consequently, +we need a supplementary approach to specify and check the desired program +behavior without adding more manually defined training cases. +With metamorphic testing [6], we do not need additional hand-labeled train- +ing cases as we execute a program multiple times (starting with a random input, +followed by related inputs) and check whether the relations between the observed +outputs logically fit the user’s domain knowledge. E.g., a function that assigns a +grade based on the score achieved in an exam could be executed with a random +score. If we then increase this score, the function must return an equal or better +grade, otherwise the metamorphic relation is violated and the function must be +incorrect. Such metamorphic relations, where labeling is not necessary, could be +used together with a classical (but smaller) hand-labeled training set. We ex- +pect that these additional relations help to improve the generalization ability of +GP-generated programs. +Therefore, in this work, we suggest an approach that combines metamorphic +testing and genetic programming (MTGP) and study its performance and the +generalization ability of the generated programs on a set of common program +synthesis benchmark problems. To analyze how GP’s generalization ability de- +pends on the number of given training cases, we perform experiments for different +(labeled) training set sizes. +MTGP is based on a grammar-guided GP approach which uses lexicase [31] +for the selection of individuals during evolution. Since lexicase selection is not +based on an aggregated fitness value, but considers the performance on individ- +ual cases, it is well suited to take hand-labeled training cases in combination +with metamorphic relations into account during selection. To study this combi- +nation, we analyze different sizes of the hand-labeled training set and add further +tests based on the metamorphic relations defined for the considered benchmark +problem. More specific, the tests based on the metamorphic relations can be +constructed automatically based on random inputs. A candidate program is ex- +ecuted first on the random input and then on a follow-up input (based on the +random input). After that, the outputs of the candidate program on the random +input and the follow-up input are compared regarding to a pre-defined meta- +morphic relation. If the relation holds, the test is passed, otherwise it is failed. +So the outcome of a metamorphic test can therefore be treated in the same way +as that of a test based on labeled input/output examples, with the difference +that no expensive manual labeling is necessary for an automatically generated + +MTGP: Combining Metamorphic Testing and Genetic Programming +3 +metamorphic test case. In our experiments, we find that incorporating metamor- +phic testing in combination with hand-labeled training cases leads to a higher +generalization rate than the use of hand-labeled training cases alone (as usual +in GP-based program synthesis) in almost all studied configurations. +Following this introduction, we present in Sect. 2 recent work related to GP- +based program synthesis and work on metamorphic testing. In Sect. 3 we describe +metamorphic testing as well as its integration into GP in detail. Furthermore, +we present the used program synthesis benchmark problems together with their +associated metamorphic relations. In Sect. 4 we describe our experimental setup +and discuss the results before concluding the paper in Sect. 5. +2 +Related Work +The main approaches in GP-based program synthesis are stack-based GP and +grammar-guided GP [30]. These approaches differ primarily in their program +representation and their techniques used to support different data types. +Stack-based GP approaches use different stacks for the separation of different +data types [33]. During program execution, data is taken as input from the +appropriate stacks and the results are pushed back to the associated stack. In +current systems, the individual program instructions are also on their own stack, +which allows changes to the program flow at runtime [32]. +Grammar-guided GP approaches use a context-free grammar to represent +the supported control structures and statements in their relationship to each +other and to distinguish different data types [11,34]. In principle, this technique +can be used to create programs in any programming language. In recent years, +however, using grammar-guided GP approaches, mainly Python programs have +been evolved [10,24,27]. +Regardless of the GP approach used, the program synthesis results have +been significantly improved in recent years, primarily through the use of lexicase +selection and its variants [10,12,15,17,19]. In contrast to selection methods such +as tournament selection, lexicase selection is not based on an aggregated fitness +value (see evaluation bottleneck [22]), but selects on the basis of the results on +individual training cases [31] which allows to include also the structure of the +given training data. +Also independent of the used approach, mainly input/output examples are +used for training in GP-based program synthesis. However, there exists also work +which uses additional information, such as the textual description of the problem +[20] or formal constraints [3] to improve the program synthesis performance. +Since the input/output examples given as training data are only an incom- +plete problem definition, it is important that the generated programs not only +work on the training data, but also produce correct results on previously unseen +inputs. In order to improve the generalization ability of the programs generated +by GP, the literature knows approaches that generate smaller programs, either +by post-simplification [14] or by a selection at the end of a run [25]. Another +option is to use batch lexicase selection [1] to improve generalizability [29]. In + +4 +Sobania et al. +addition to that, recently a method has been presented that can be used to pre- +dict whether the programs generated by GP will generalize to unseen data or +not [28]. +Metamorphic testing introduced by Chen et al. [6] is a method from soft- +ware development that allows to check certain properties in the program under +test without the need of explicitly specifying the expected output of a test (see +Sect. 3.1 for a detailed description). In the field of evolutionary computation, +metamorphic testing was used, e.g., for the genetic improvement of existing +software [23]. +However, to the best of our knowledge, no work so far studied the impact of +combining metamorphic testing and GP on the program synthesis performance +and the generalizability of the generated programs. +3 +Methodology +In this section, we describe the basics of metamorphic testing and show how it +works with some illustrative examples. Furthermore, we present the selected pro- +gram synthesis benchmark problems together with their metamorphic relations. +Lastly, we describe in detail how metamorphic testing and GP-based program +synthesis can be combined. +3.1 +Metamorphic Testing +Metamorphic testing [6] is a method from software development to check if cer- +tain logic properties hold in a given function f. These properties are defined by +so-called metamorphic relations which describe the logic connection between a +given (random) base input I with its observed output f(I) and a further follow- +up input I′ with its corresponding output f(I′).1 The key advantage of meta- +morphic testing compared to classical test methods is that we need no expensive +labeled input/output examples as we are just interested if the metamorphic re- +lation between the observed outputs f(I) and f(I′) holds. +As a first intuitive example (based on the example given in [2]), we can think +of a simple web search. E.g., as base input I we search for the exact term "Genetic +Programming" without filters in a scientific search engine and find f(I) results. +As follow-up input I′ we search for the same term "Genetic Programming" but +limit the results to publications since 2022 and get f(I′) results. As metamorphic +relation, we define that f(I) ≥ f(I′), as additional filters should lead to an equal +or lower number of results. Figure 1, shows screenshots from Google Scholar +illustrating this example. We see that f(I) = 246, 000 and f(I′) = 6, 650, so the +metamorphic relation holds as f(I) ≥ f(I′). +As a second, more technical, example, we choose the sine function, where we +expect that it is 2π periodic. Consequently, for the inputs I and I′ = I + 2π, a +1 More than one follow-up test is also possible, but in this work we focus on exactly +one follow-up test. + +MTGP: Combining Metamorphic Testing and Genetic Programming +5 +(a) Search result for base input I (no filter). +(b) Search result for follow-up input I′ (active filter). +Fig. 1: Screenshots from Google Scholar illustrating a simple example of meta- +morphic testing. For the base input I (a), we see the corresponding output +f(I) = 246, 000 and for the follow-up input I′ (b) the output is f(I′) = 6, 650, +so the defined relation f(I) ≥ f(I′) holds. +metamorphic relation could be defined as f(I) = f(I′) [4,5]. Figure 2 illustrates +this example for I = −3.7. We see that constructed follow-up input I′ = −3.7 + +2π leads to the same result, so the relation f(I) = f(I′) holds. +3.2 +Benchmark Problems and Metamorphic Relations +For the evaluation, we selected three problems from the program synthesis bench- +mark suite by Helmuth and Spector [18] which differ both in difficulty, according +to a recent meta study [30], as well as in the data types used for input and output. +Furthermore, we define two metamorphic relations for each of the benchmark +problems for better comparison. However, more metamorphic relations are con- +ceivable. We focused on relatively simple metamorphic relations which can be +formulated by a user even with basic domain knowledge about the problem. The +benchmark problems and metamorphic relations are defined as follows: + +GoogleScholar +"Genetic Programming" +Articles +Abot 246.000 results 0,04 sec) +Any time +Since 2022 +Since 2021 +Since 2018 +Custom range..Google Scholar +"Genetic Programming" +Articles +Abo6.650 results0,05 sec) +Any time +Since 2022 +Since 2021 +Since 2018 +Custom range..6 +Sobania et al. +-2π +-π +0 +π +2π +−1 +0 +1 +I = -3.7 +I' = -3.7 + 2π +Fig. 2: An example of the sine function. We see that the result of the base input +I = −3.7 and the follow-up input I′ = −3.7 + 2π is identical, so the defined +metamorphic relation f(I) = f(I′) is satisfied. +Count Odds Problem: +Definition: A program should be generated that returns the number of odd +values in a given list of integers. More formal, we search for a function f that +maps the given vector of integers I = (x1, . . . , xn) ∈ Zn to the number of odds +f(I) = c ∈ N0 as return value. +Metamorphic Relations: 1) As first metamorphic relation, we require that the +output of the program under test does not change when the input list is extended +by an arbitrary number of even integers. For a given input I = (x1, . . . , xn) and +the program f we calculate f(I) = cI. As follow-up test, we create an extended +input I′ = (x1, . . . , xn, 2, 4, 6, . . . , 2, 4, 6) by extending the input I with a random +repetition of the vector (2, 4, 6). The corresponding output of the manipulated +input is f(I′) = cI′. This relation holds if cI = cI′. We chose 2, 4, and 6 +as we expect that these values are known to be even numbers for users with +basic domain knowledge about the problem (no knowledge about the modulo +function needed). 2) Analogously, we require for a second metamorphic rela- +tion that the output value increases if we extend the input by an arbitrary +number of odd integers. Contrarily to the first relation, we extend the input +I′ = (x1, . . . , xn, 1, 3, 5, . . . , 1, 3, 5) by a random number of repetitions of the +vector (1, 3, 5). Consequently, the corresponding output is f(I′) = cI′. This re- +lation holds if cI < cI′. This time, we chose 1, 3, and 5 as trivial odd numbers. + +MTGP: Combining Metamorphic Testing and Genetic Programming +7 +Grade Problem: +Definition: We search a program that maps the numeric score achieved by a +student to a discrete grade based on given thresholds. Specifically, we search for a +function f that maps the input I to its grade f(I) = g, where I = (t1, t2, t3, t4, s) +with ti, s ∈ N0, ti, s ≤ 100 for i ∈ {1, 2, 3, 4} and ti < tj for i > j and +i, j ∈ {1, 2, 3, 4} where ti are the thresholds and s is the score achieved by +a student and g = {A, B, C, D, F} is the corresponding grade with the order +A ≻ B ≻ C ≻ D ≻ F. +Metamorphic Relations: 1) As first relation, we require that a better nu- +meric score leads to an equal or better discrete grade. For a given valid random +input I = (t1, t2, t3, t4, s) and the program f we calculate f(I) = gI. After +that, we create a manipulated input I′ = (t1, t2, t3, t4, s + k) based on I with +k ∈ {0, . . . , 100−s} and grade f(I′) = gI′ as follow-up test. The metamorphic re- +lation holds if gI′ ⪰ gI. 2) Second, we define the opposite relation which requires +that a lower numeric score leads to an equal or worse discrete grade. We create +a manipulated input I′ = (t1, t2, t3, t4, s − k) based on I with k ∈ {0, . . . , s} and +grade f(I′) = gI′ as follow-up test. This second metamorphic relation holds if +gI′ ⪯ gI. +Small or Large Problem: +Definition: The generated program should classify a given integer either as +small, large, or in between. More precise, we search for a function f that maps a +given integer I = n ∈ Z to its label f(I) = l, where l = {"small", "", "large"} +with the order "small" ≺ "" ≺ "large". The function f should return "small" +if n < 1, 000, "large" if n ≥ 2, 000, and an empty string ("") otherwise. +Metamorphic Relations: 1) First, we require that for an increased input the +label also has to stay equal or increase according to the defined label ordering. +For a given integer I = n we compute the resulting label f(I) = lI. Following +this, we manipulate the input I′ = n + k with a random k ∈ N with the cor- +responding label f(I′) = lI′. The relation holds if lI′ ⪰ lI. 2) Consequently, as +second relation, we define that for an decreased input the resulting label has to +stay equal or decrease according to the label ordering. For the given input I = n +we calculate the label f(I) = lI. Then, we manipulate the input I′ = n − k with +k ∈ N and f(I′) = lI′. If lI′ ⪯ lI, the relation is satisfied. +3.3 +Incorporating Metamorphic Testing in GP +MTGP benefits from the use of lexicase selection, since lexicase considers the +individual cases instead of an aggregated fitness value [31]. Consequently, dif- +ferent types of (training) cases can be used simultaneously for the selection. In +MTGP, these are hand-labeled training cases for which we know the input and +the corresponding output, as well as cases based on metamorphic relations. + +8 +Sobania et al. +For the hand-labeled training cases, both the inputs and the outputs are +known. In order to check whether a candidate program solves a training case or +not, we run the program with the given input I and check whether the generated +output f(I) matches the expected (in a real-world scenario hand-labeled) output +O. If f(I) = O, then the test is passed, otherwise not. +To test the metamorphic relations defined for a benchmark problem, we pro- +vide random inputs as base inputs for which the outputs do not need to be +known. Thus, any number of random entries can be generated at low cost. In +our experiments, we have ensured that these random inputs do not already ex- +ist in the training or test set. In practice, they could be chosen freely. To check +whether a candidate program satisfies a metamorphic relation or not, we execute +it with a given base input I (one of the randomly generated inputs) and save the +output f(I). We then change the input randomly according to the manipulation +rule of the metamorphic relation to create the follow-up input I′ and run the +program again. The test is passed if the observed outputs f(I) and f(I′) satisfy +the metamorphic relation, otherwise it is failed. Since we have defined two meta- +morphic relations for all benchmark problems considered in the experiments, we +determine at random, with probability p = 0.5, which of the two relations is +used each time a base input is selected. +One of the key advantages of using metamorphic testing this way is that +during a GP run always many different metamorphic tests are executed, since +the manipulation of the given basic input happens randomly. Our hope is, that +this further supports the generalizability of the generated programs. +4 +Experiments and Discussion +In this section, we study the performance of MTGP and analyze how well the +generated programs generalize to previously unseen tests cases. To analyze how +the generalization ability depends on the number of given training cases, we +perform experiments for different labeled training set sizes. Below, we describe +the experimental setup and discuss our results. +4.1 +Experimental Setup +For the implementation of MTGP, we use the PonyGE2 framework [9]. Our +used grammars are based on the program synthesis grammars provided by the +PonyGE2 framework for the problems from the benchmark suite [18] and fol- +low the principle suggested by Forstenlechner et al. [10] according to which the +grammar of a problem is restricted to the data types (and dependent functions) +that are defined for the input and output of the considered problem, in addition +to required base data types (e.g., like integer and Boolean). This allows to +keep the used grammars small and effective but still expressive. +We initialize a run with position independent grow [8] and use a population +size of 1, 000. The maximum initial tree depth is set to 10 and the maximum over- +all tree depth is limited to 17. As variation operators, we use sub-tree crossover + +MTGP: Combining Metamorphic Testing and Genetic Programming +9 +and sub-tree mutation. For the sub-tree crossover we set the probability to 0.9 +and for the sub-tree mutation we set the number of mutation steps to 1. As +mentioned above, we use lexicase [31] as selection method. A GP run is stopped +either after a program is found that solves all labeled training cases and all +defined metamorphic tests or after 300 generations. +For every considered benchmark problem, we have 200 labeled training cases +that we can use in the experiments. The test set consisting of 1, 000 labeled cases +is used to check if a candidate program also generalizes to unseen cases. Further +we provide a large set of randomly generated inputs (800 for each benchmark +problem) which we use as base inputs for checking if the defined metamorphic +relations hold for a candidate program or not. In the experiments, we choose +from these available training cases and randomly generated inputs depending on +the considered configuration as specified below. +4.2 +Results and Discussion +As we investigate the impact of using metamorphic testing in GP, we compare a +standard GP approach, which only uses the training data, and the novel MTGP +approach, which also includes metamorphic tests. In addition, we examine how +standard GP and MTGP perform on different training set sizes |Ttraining|. There- +Table 1: Success rates on the training set straining and the test set stest achieved +by standard GP and MTGP for all studied labeled training set sizes |Ttraining| +and benchmark problems. For MTGP, we use in addition 200 − |Ttraining| meta- +morphic tests. Best success rates achieved on the test set stest are printed in +bold font. +Standard GP +MTGP +Problem +|Ttraining| +straining +stest +straining +stest +Count Odds +25 +43 +28 +32 +20 +50 +59 +46 +43 +35 +100 +70 +64 +61 +59 +200 +83 +81 +- +- +Grade +25 +83 +4 +74 +14 +50 +86 +12 +76 +15 +100 +81 +23 +84 +33 +200 +93 +45 +- +- +Small or Large +25 +94 +2 +58 +7 +50 +79 +11 +77 +25 +100 +91 +35 +86 +29 +200 +89 +43 +- +- + +10 +Sobania et al. +fore, we analyze for both approaches the labeled training set sizes 25, 50, 100, +and 200. MTGP also uses 200 − |Ttraining| metamorphic tests so that MTGP +considers exactly 200 tests during the training phase (e.g., for |Ttraining| = 25, +MTGP uses 175 metamorphic tests). +As a first step, we study the achieved success rates on the training set straining +and the test set stest as a performance indicator. The success rate on the training +set straining measures the percentage of runs in which a program is found that is +successful on all given training cases (including metamorphic tests for MTGP). +To determine the success rate on the test set stest, we take from each run the +candidate program that performs best on the training data and measure the +percentage of candidate programs that solve all previously unseen test cases. +For every studied configuration, we performed 100 runs. +Table 1 shows the success rates on the training set straining and the test set +stest achieved by standard GP and MTGP for all studied labeled training set +sizes |Ttraining| (and corresponding metamorphic tests) and benchmark problems. +Best success rates achieved on the test set stest are printed in bold font. +For most configurations, we see that standard GP achieves a higher success +rate on the training data straining compared to MTGP. Our assumption is that +this is because the additional metamorphic tests prevent an overfitting to the +training data. On the test data, MTGP performs best for the Grade problem +and for most of the configurations of the Small or Large problem (compared to +the corresponding standard GP runs). For the Count Odds problem best results +for stest are achieved with standard GP. +More important than the pure success rates, however, is how well the pro- +grams found on the training data generalize to unseen test cases. In practice, +a program synthesis approach which is known for its high generalization can +simply be executed again if no solution was found in the first run. If the gen- +eralization is expected to be poor, it is unclear whether solutions found on the +training data also work on previously unseen test cases, regardless of the suc- +cess rate achieved on the training data. Large sets of additional test cases (to +check for generalizability) are not available in a real-world scenario as manually +labeling additional input/output examples is far too expensive. Therefore, in the +second step, we analyze the generalization rate +G = +stest +straining +· 100. +Table 2 shows the generalization rate G achieved by standard GP and MTGP +for all studied labeled training set sizes |Ttraining| (and corresponding metamor- +phic tests) and benchmark problems. Again, best generalization rates G are +printed in bold font. +We see that on average, best generalization rates G are achieved with MTGP, +as MTGP performed best in 7 out of 9 configurations where we have results for +standard GP as well as for MTGP. The differences are particularly obvious for +the Grade and the Small or Large problem when only a small labeled training set +(|Ttraining| = 25) is used. MTGP achieves for the Grade problem a generalization + +MTGP: Combining Metamorphic Testing and Genetic Programming +11 +Table 2: Generalization rate G achieved by standard GP and MTGP for all +studied labeled training set sizes |Ttraining| and benchmark problems. For MTGP, +we use 200 − |Ttraining| metamorphic tests in addition to the considered labeled +training cases. Best results are printed in bold font. +Generalization rate G +Problem +|Ttraining| +Standard GP +MTGP +Count Odds +25 +65.116 +62.5 +50 +77.966 +81.395 +100 +91.429 +96.721 +200 +97.59 +- +Grade +25 +4.819 +18.919 +50 +13.953 +19.737 +100 +28.395 +39.286 +200 +48.387 +- +Small or Large +25 +2.128 +12.069 +50 +13.924 +32.468 +100 +38.462 +33.721 +200 +48.315 +- +rate G of 18.919 and for the Small or Large problem of 12.069 while standard +GP achieves only 4.819 and 2.128, respectively. +So far we have only studied MTGP with a reduced labeled training set. But +can the generalization rate even be increased compared to standard GP even if +the complete labeled training set (|Ttraining| = 200) is used during the run? To +answer this question, we run MTGP this time with 800 metamorphic tests. +Table 3: Generalization rate G achieved by standard GP and MTGP for all +considered benchmark problems. All configurations use as labeled training set +size |Ttraining| = 200. For MTGP, we use 800 metamorphic tests in addition to +the considered labeled training cases. Best results are printed in bold font. +Generalization rate G +Problem +|Ttraining| +Standard GP +MTGP +Count Odds +200 +97.59 +100.0 +Grade +200 +48.387 +61.538 +Small or Large +200 +48.315 +62.069 + +12 +Sobania et al. +Table 3 shows the achieved generalization rates G for standard GP and +MTGP for all considered benchmark problems. As before, best generalization +rates G are printed in bold font. +Wee see that even when the complete labeled training set Ttraining is used +during a run, using metamorphic testing can improve the generalization rate G. +MTGP performs best on all studied benchmark problems. For the Count Odds +problem, we even achieve a perfect generalization rate G of 100. +In summary, the generalization ability of the generated programs can be +increased by using metamorphic testing. On average, best generalization rates +are achieved with MTGP. The major advantage of metamorphic tests is that they +do not require the expensive manual calculation of the expected outputs, since +they work exclusively with random inputs which can be generated automatically. +5 +Conclusion +GP [7,21] is an evolutionary approach that is well known for its performance in +automatic program synthesis. Even if GP is competitive in performance to other +state-of-the-art program synthesis approaches [26], it is not yet mature enough +for a practical use in real-world software development, as many input/output +examples are usually required during the training process to generate programs +that also generalize to unseen test cases. As in practice, the training cases have to +be labeled manually by the user which is very expensive, we need a supplemen- +tary approach to check the program behavior with a lower number of manually +defined training cases. +With metamorphic testing [6], we do not need labeled input/output exam- +ples. The program is executed multiple times, first on a given input (which can +be generated randomly) and followed by related inputs to check whether certain +user-defined metamorphic relations hold between the observed outputs. +Therefore, in this work we suggested MTGP, an approach that combines +metamorphic testing and GP and studied its performance and the generalization +ability of the generated programs on common program synthesis benchmark +problems. Further, we analyzed how the generalization ability depends on the +number of given training cases and performed experiments for different labeled +training set sizes. +We found that incorporating metamorphic testing in combination with hand- +labeled training cases leads to a higher generalization rate than the exclusive use +of hand-labeled training cases in almost all configurations studied in our exper- +iments, including those using smaller labeled training sets as usual in GP-based +program synthesis. Even with the largest considered labeled training set, the gen- +eralization rate could be increased by a large margin on all studied benchmark +problems with the use of metamorphic testing. Consequently, we recommend +researchers to use metamorphic testing in their GP approaches if the labeling of +the training data is an expensive process in the considered application domain. +In future work, we will study MTGP on additional program synthesis bench- +mark problems and further analyze the usage of the metamorphic tests as well + +MTGP: Combining Metamorphic Testing and Genetic Programming +13 +as the given labeled training cases during a run to gain a deeper understanding +of the implications of incorporating metamorphic testing in GP. +References +1. Aenugu, S., Spector, L.: Lexicase selection in learning classifier systems. In: Pro- +ceedings of the Genetic and Evolutionary Computation Conference. pp. 356–364 +(2019) +2. Arrieta, A.: Multi-objective metamorphic follow-up test case selection for deep +learning systems. In: Proceedings of the Genetic and Evolutionary Computation +Conference. pp. 1327–1335 (2022) +3. Błądek, I., Krawiec, K., Swan, J.: Counterexample-driven genetic programming: +heuristic program synthesis from formal specifications. Evolutionary computation +26(3), 441–469 (2018) +4. Chen, T.Y., Kuo, F.C., Liu, H., Poon, P.L., Towey, D., Tse, T., Zhou, Z.Q.: Meta- +morphic testing: A review of challenges and opportunities. ACM Computing Sur- +veys (CSUR) 51(1), 1–27 (2018) +5. Chen, T.Y., Kuo, F.C., Liu, Y., Tang, A.: Metamorphic testing and testing with +special values. In: SNPD. pp. 128–134 (2004) +6. Chen, T., Cheung, S., Yiu, S.: Metamorphic testing: A new approach for generating +next test cases. Department of Computer Science, The Hong Kong University of +Science and Technology, Tech. Rep (1998) +7. Cramer, N.L.: A representation for the adaptive generation of simple sequential +programs. In: proceedings of an International Conference on Genetic Algorithms +and the Applications. pp. 183–187 (1985) +8. Fagan, D., Fenton, M., O’Neill, M.: Exploring position independent initialisation +in grammatical evolution. In: 2016 IEEE Congress on Evolutionary Computation +(CEC). pp. 5060–5067. IEEE (2016) +9. Fenton, M., McDermott, J., Fagan, D., Forstenlechner, S., Hemberg, E., O’Neill, +M.: PonyGE2: Grammatical evolution in Python. In: Proceedings of the Genetic +and Evolutionary Computation Conference Companion. pp. 1194–1201 (2017) +10. Forstenlechner, S., Fagan, D., Nicolau, M., O’Neill, M.: A grammar design pattern +for arbitrary program synthesis problems in genetic programming. In: European +Conference on Genetic Programming. pp. 262–277. Springer (2017) +11. Forstenlechner, S., Nicolau, M., Fagan, D., O’Neill, M.: Grammar design for deriva- +tion tree based genetic programming systems. In: European Conference on Genetic +Programming. pp. 199–214. Springer (2016) +12. Helmuth, T., Abdelhady, A.: Benchmarking parent selection for program synthesis +by genetic programming. In: Proceedings of the 2020 Genetic and Evolutionary +Computation Conference Companion. pp. 237–238 (2020) +13. Helmuth, T., Kelly, P.: PSB2: the second program synthesis benchmark suite. In: +Proceedings of the Genetic and Evolutionary Computation Conference. pp. 785– +794 (2021) +14. Helmuth, T., McPhee, N.F., Pantridge, E., Spector, L.: Improving generalization of +evolved programs through automatic simplification. In: Proceedings of the Genetic +and Evolutionary Computation Conference. pp. 937–944 (2017) +15. Helmuth, T., McPhee, N.F., Spector, L.: Lexicase selection for program synthesis: +a diversity analysis. In: Genetic Programming Theory and Practice XIII, pp. 151– +167. Springer (2016) + +14 +Sobania et al. +16. Helmuth, T., McPhee, N.F., Spector, L.: Program synthesis using uniform muta- +tion by addition and deletion. In: Proceedings of the Genetic and Evolutionary +Computation Conference. pp. 1127–1134 (2018) +17. Helmuth, T., Pantridge, E., Spector, L.: On the importance of specialists for +lexicase selection. Genetic Programming and Evolvable Machines 21(3), 349–373 +(2020) +18. Helmuth, T., Spector, L.: General program synthesis benchmark suite. In: Proceed- +ings of the 2015 Annual Conference on Genetic and Evolutionary Computation. +pp. 1039–1046 (2015) +19. Helmuth, T., Spector, L.: Explaining and exploiting the advantages of down- +sampled lexicase selection. In: ALIFE 2020: The 2020 Conference on Artificial +Life. pp. 341–349. MIT Press (2020) +20. Hemberg, E., Kelly, J., O’Reilly, U.M.: On domain knowledge and novelty to im- +prove program synthesis performance with grammatical evolution. In: Proceedings +of the Genetic and Evolutionary Computation Conference. pp. 1039–1046 (2019) +21. Koza, J.R.: Genetic programming: on the programming of computers by means of +natural selection, vol. 1. MIT press (1992) +22. Krawiec, K.: Behavioral program synthesis with genetic programming, vol. 618. +Springer (2016) +23. Langdon, W.B., Krauss, O.: Evolving sqrt into 1/x via software data maintenance. +In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference +Companion. pp. 1928–1936 (2020) +24. Schweim, D., Sobania, D., Rothlauf, F.: Effects of the training set size: A com- +parison of standard and down-sampled lexicase selection in program synthesis. In: +2022 IEEE Congress on Evolutionary Computation (CEC). pp. 1–8. IEEE (2022) +25. Sobania, D.: On the generalizability of programs synthesized by grammar-guided +genetic programming. In: European Conference on Genetic Programming (Part of +EvoStar). pp. 130–145. Springer (2021) +26. Sobania, D., Briesch, M., Rothlauf, F.: Choose your programming copilot: a com- +parison of the program synthesis performance of Github Copilot and genetic pro- +gramming. In: Proceedings of the Genetic and Evolutionary Computation Confer- +ence. pp. 1019–1027 (2022) +27. Sobania, D., Rothlauf, F.: Challenges of program synthesis with grammatical evo- +lution. In: European Conference on Genetic Programming (Part of EvoStar). pp. +211–227. Springer (2020) +28. Sobania, D., Rothlauf, F.: A generalizability measure for program synthesis with +genetic programming. In: Proceedings of the Genetic and Evolutionary Computa- +tion Conference. pp. 822–829 (2021) +29. Sobania, D., Rothlauf, F.: Program synthesis with genetic programming: The in- +fluence of batch sizes. In: European Conference on Genetic Programming (Part of +EvoStar). pp. 118–129. Springer (2022) +30. Sobania, D., Schweim, D., Rothlauf, F.: A comprehensive survey on program syn- +thesis with evolutionary algorithms. IEEE Transactions on Evolutionary Compu- +tation (2022) +31. Spector, L.: Assessment of problem modality by differential performance of lexicase +selection in genetic programming: a preliminary report. In: Proceedings of the 14th +annual conference companion on Genetic and evolutionary computation. pp. 401– +408 (2012) +32. Spector, L., Klein, J., Keijzer, M.: The Push3 execution stack and the evolution of +control. In: Proceedings of the 7th annual conference on Genetic and evolutionary +computation. pp. 1689–1696 (2005) + +MTGP: Combining Metamorphic Testing and Genetic Programming +15 +33. Spector, L., Robinson, A.: Genetic programming and autoconstructive evolution +with the Push programming language. Genetic Programming and Evolvable Ma- +chines 3(1), 7–40 (2002) +34. Whigham, P.A., et al.: Grammatically-based genetic programming. In: Proceedings +of the workshop on genetic programming: from theory to real-world applications. +vol. 16, pp. 33–41. Citeseer (1995) + diff --git a/HNFAT4oBgHgl3EQfth7Z/content/tmp_files/load_file.txt b/HNFAT4oBgHgl3EQfth7Z/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b7c7fa21282d48d320360df162b993f95bfe942 --- /dev/null +++ b/HNFAT4oBgHgl3EQfth7Z/content/tmp_files/load_file.txt @@ -0,0 +1,548 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf,len=547 +page_content='MTGP: Combining Metamorphic Testing and Genetic Programming Dominik Sobania, Martin Briesch, Philipp Röchner, and Franz Rothlauf Johannes Gutenberg University, Mainz, Germany {dsobania,briesch,proechne,rothlauf}@uni-mainz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='de Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Genetic programming is an evolutionary approach known for its performance in program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' However, it is not yet ma- ture enough for a practical use in real-world software development, since usually many training cases are required to generate programs that gen- eralize to unseen test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As in practice, the training cases have to be expensively hand-labeled by the user, we need an approach to check the program behavior with a lower number of training cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Metamorphic testing needs no labeled input/output examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Instead, the program is executed multiple times, first on a given (randomly generated) in- put, followed by related inputs to check whether certain user-defined relations between the observed outputs hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In this work, we suggest MTGP, which combines metamorphic testing and genetic programming and study its performance and the generalizability of the generated pro- grams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Further, we analyze how the generalizability depends on the num- ber of given labeled training cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We find that using metamorphic test- ing combined with labeled training cases leads to a higher generalization rate than the use of labeled training cases alone in almost all studied configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Consequently, we recommend researchers to use meta- morphic testing in their systems if the labeling of the training data is expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Keywords: Program Synthesis · Metamorphic Testing · Genetic Pro- gramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1 Introduction Genetic programming (GP) [7,21] is an evolutionary algorithm-based approach to automatically generate programs in a given programming language that meet user-defined requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In GP-based program synthesis, these specifications are usually given as input/output examples, which define the expected output from a generated program for a given input, and are used as training data during the evolutionary search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' With the introduction of new selection methods [17,31], variation opera- tors [16] and grammar design techniques [10], GP-based program synthesis has made great progress in the last years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Recently, it has been shown that GP- based approaches for program synthesis are even competitive in performance with state-of-the-art neural network-based approaches [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='08665v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='SE] 20 Jan 2023 2 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Unfortunately, GP-based program synthesis is not yet mature enough for a practical use in real-world software development, since many input/output examples are usually required (up to 200 are regularly used in the literature [13]) to generate programs that not only work on the training cases, but also generalize to previously unseen test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In practice, however, the training cases have to be labeled manually by the user which is very expensive and time consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' So it is necessary to reduce the number of required training cases in order to minimize the user’s manual effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' However, simply reducing the number of training cases is not sufficient, since a small training set can easily be overfitted which will on average lead to a poor generalization of the generated programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Consequently, we need a supplementary approach to specify and check the desired program behavior without adding more manually defined training cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' With metamorphic testing [6], we do not need additional hand-labeled train- ing cases as we execute a program multiple times (starting with a random input, followed by related inputs) and check whether the relations between the observed outputs logically fit the user’s domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', a function that assigns a grade based on the score achieved in an exam could be executed with a random score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' If we then increase this score, the function must return an equal or better grade, otherwise the metamorphic relation is violated and the function must be incorrect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Such metamorphic relations, where labeling is not necessary, could be used together with a classical (but smaller) hand-labeled training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We ex- pect that these additional relations help to improve the generalization ability of GP-generated programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Therefore, in this work, we suggest an approach that combines metamorphic testing and genetic programming (MTGP) and study its performance and the generalization ability of the generated programs on a set of common program synthesis benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To analyze how GP’s generalization ability de- pends on the number of given training cases, we perform experiments for different (labeled) training set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MTGP is based on a grammar-guided GP approach which uses lexicase [31] for the selection of individuals during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Since lexicase selection is not based on an aggregated fitness value, but considers the performance on individ- ual cases, it is well suited to take hand-labeled training cases in combination with metamorphic relations into account during selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To study this combi- nation, we analyze different sizes of the hand-labeled training set and add further tests based on the metamorphic relations defined for the considered benchmark problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' More specific, the tests based on the metamorphic relations can be constructed automatically based on random inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' A candidate program is ex- ecuted first on the random input and then on a follow-up input (based on the random input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' After that, the outputs of the candidate program on the random input and the follow-up input are compared regarding to a pre-defined meta- morphic relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' If the relation holds, the test is passed, otherwise it is failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' So the outcome of a metamorphic test can therefore be treated in the same way as that of a test based on labeled input/output examples, with the difference that no expensive manual labeling is necessary for an automatically generated MTGP: Combining Metamorphic Testing and Genetic Programming 3 metamorphic test case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In our experiments, we find that incorporating metamor- phic testing in combination with hand-labeled training cases leads to a higher generalization rate than the use of hand-labeled training cases alone (as usual in GP-based program synthesis) in almost all studied configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Following this introduction, we present in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2 recent work related to GP- based program synthesis and work on metamorphic testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 3 we describe metamorphic testing as well as its integration into GP in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Furthermore, we present the used program synthesis benchmark problems together with their associated metamorphic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 4 we describe our experimental setup and discuss the results before concluding the paper in Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2 Related Work The main approaches in GP-based program synthesis are stack-based GP and grammar-guided GP [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' These approaches differ primarily in their program representation and their techniques used to support different data types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Stack-based GP approaches use different stacks for the separation of different data types [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' During program execution, data is taken as input from the appropriate stacks and the results are pushed back to the associated stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In current systems, the individual program instructions are also on their own stack, which allows changes to the program flow at runtime [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Grammar-guided GP approaches use a context-free grammar to represent the supported control structures and statements in their relationship to each other and to distinguish different data types [11,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In principle, this technique can be used to create programs in any programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In recent years, however, using grammar-guided GP approaches, mainly Python programs have been evolved [10,24,27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Regardless of the GP approach used, the program synthesis results have been significantly improved in recent years, primarily through the use of lexicase selection and its variants [10,12,15,17,19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In contrast to selection methods such as tournament selection, lexicase selection is not based on an aggregated fitness value (see evaluation bottleneck [22]), but selects on the basis of the results on individual training cases [31] which allows to include also the structure of the given training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Also independent of the used approach, mainly input/output examples are used for training in GP-based program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' However, there exists also work which uses additional information, such as the textual description of the problem [20] or formal constraints [3] to improve the program synthesis performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Since the input/output examples given as training data are only an incom- plete problem definition, it is important that the generated programs not only work on the training data, but also produce correct results on previously unseen inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In order to improve the generalization ability of the programs generated by GP, the literature knows approaches that generate smaller programs, either by post-simplification [14] or by a selection at the end of a run [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Another option is to use batch lexicase selection [1] to improve generalizability [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In 4 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' addition to that, recently a method has been presented that can be used to pre- dict whether the programs generated by GP will generalize to unseen data or not [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Metamorphic testing introduced by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' [6] is a method from soft- ware development that allows to check certain properties in the program under test without the need of explicitly specifying the expected output of a test (see Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='1 for a detailed description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In the field of evolutionary computation, metamorphic testing was used, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', for the genetic improvement of existing software [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' However, to the best of our knowledge, no work so far studied the impact of combining metamorphic testing and GP on the program synthesis performance and the generalizability of the generated programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 3 Methodology In this section, we describe the basics of metamorphic testing and show how it works with some illustrative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Furthermore, we present the selected pro- gram synthesis benchmark problems together with their metamorphic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Lastly, we describe in detail how metamorphic testing and GP-based program synthesis can be combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='1 Metamorphic Testing Metamorphic testing [6] is a method from software development to check if cer- tain logic properties hold in a given function f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' These properties are defined by so-called metamorphic relations which describe the logic connection between a given (random) base input I with its observed output f(I) and a further follow- up input I′ with its corresponding output f(I′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='1 The key advantage of meta- morphic testing compared to classical test methods is that we need no expensive labeled input/output examples as we are just interested if the metamorphic re- lation between the observed outputs f(I) and f(I′) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As a first intuitive example (based on the example given in [2]), we can think of a simple web search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', as base input I we search for the exact term "Genetic Programming" without filters in a scientific search engine and find f(I) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As follow-up input I′ we search for the same term "Genetic Programming" but limit the results to publications since 2022 and get f(I′) results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As metamorphic relation, we define that f(I) ≥ f(I′), as additional filters should lead to an equal or lower number of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Figure 1, shows screenshots from Google Scholar illustrating this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We see that f(I) = 246, 000 and f(I′) = 6, 650, so the metamorphic relation holds as f(I) ≥ f(I′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As a second, more technical, example, we choose the sine function, where we expect that it is 2π periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Consequently, for the inputs I and I′ = I + 2π, a 1 More than one follow-up test is also possible, but in this work we focus on exactly one follow-up test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MTGP: Combining Metamorphic Testing and Genetic Programming 5 (a) Search result for base input I (no filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' (b) Search result for follow-up input I′ (active filter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1: Screenshots from Google Scholar illustrating a simple example of meta- morphic testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For the base input I (a), we see the corresponding output f(I) = 246, 000 and for the follow-up input I′ (b) the output is f(I′) = 6, 650, so the defined relation f(I) ≥ f(I′) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' metamorphic relation could be defined as f(I) = f(I′) [4,5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Figure 2 illustrates this example for I = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We see that constructed follow-up input I′ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='7 + 2π leads to the same result, so the relation f(I) = f(I′) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='2 Benchmark Problems and Metamorphic Relations For the evaluation, we selected three problems from the program synthesis bench- mark suite by Helmuth and Spector [18] which differ both in difficulty, according to a recent meta study [30], as well as in the data types used for input and output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Furthermore, we define two metamorphic relations for each of the benchmark problems for better comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' However, more metamorphic relations are con- ceivable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We focused on relatively simple metamorphic relations which can be formulated by a user even with basic domain knowledge about the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The benchmark problems and metamorphic relations are defined as follows: GoogleScholar "Genetic Programming" Articles Abot 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='000 results 0,04 sec) Any time Since 2022 Since 2021 Since 2018 Custom range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='.Google Scholar "Genetic Programming" Articles Abo6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='650 results0,05 sec) Any time Since 2022 Since 2021 Since 2018 Custom range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='.6 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2π π 0 π 2π −1 0 1 I = -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content="7 I' = -3." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='7 + 2π Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2: An example of the sine function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We see that the result of the base input I = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='7 and the follow-up input I′ = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='7 + 2π is identical, so the defined metamorphic relation f(I) = f(I′) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Count Odds Problem: Definition: A program should be generated that returns the number of odd values in a given list of integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' More formal, we search for a function f that maps the given vector of integers I = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , xn) ∈ Zn to the number of odds f(I) = c ∈ N0 as return value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Metamorphic Relations: 1) As first metamorphic relation, we require that the output of the program under test does not change when the input list is extended by an arbitrary number of even integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For a given input I = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , xn) and the program f we calculate f(I) = cI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As follow-up test, we create an extended input I′ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , xn, 2, 4, 6, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , 2, 4, 6) by extending the input I with a random repetition of the vector (2, 4, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The corresponding output of the manipulated input is f(I′) = cI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' This relation holds if cI = cI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We chose 2, 4, and 6 as we expect that these values are known to be even numbers for users with basic domain knowledge about the problem (no knowledge about the modulo function needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2) Analogously, we require for a second metamorphic rela- tion that the output value increases if we extend the input by an arbitrary number of odd integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Contrarily to the first relation, we extend the input I′ = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , xn, 1, 3, 5, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , 1, 3, 5) by a random number of repetitions of the vector (1, 3, 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Consequently, the corresponding output is f(I′) = cI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' This re- lation holds if cI < cI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' This time, we chose 1, 3, and 5 as trivial odd numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MTGP: Combining Metamorphic Testing and Genetic Programming 7 Grade Problem: Definition: We search a program that maps the numeric score achieved by a student to a discrete grade based on given thresholds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Specifically, we search for a function f that maps the input I to its grade f(I) = g, where I = (t1, t2, t3, t4, s) with ti, s ∈ N0, ti, s ≤ 100 for i ∈ {1, 2, 3, 4} and ti < tj for i > j and i, j ∈ {1, 2, 3, 4} where ti are the thresholds and s is the score achieved by a student and g = {A, B, C, D, F} is the corresponding grade with the order A ≻ B ≻ C ≻ D ≻ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Metamorphic Relations: 1) As first relation, we require that a better nu- meric score leads to an equal or better discrete grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For a given valid random input I = (t1, t2, t3, t4, s) and the program f we calculate f(I) = gI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' After that, we create a manipulated input I′ = (t1, t2, t3, t4, s + k) based on I with k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , 100−s} and grade f(I′) = gI′ as follow-up test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The metamorphic re- lation holds if gI′ ⪰ gI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2) Second, we define the opposite relation which requires that a lower numeric score leads to an equal or worse discrete grade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We create a manipulated input I′ = (t1, t2, t3, t4, s − k) based on I with k ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' , s} and grade f(I′) = gI′ as follow-up test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' This second metamorphic relation holds if gI′ ⪯ gI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Small or Large Problem: Definition: The generated program should classify a given integer either as small, large, or in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' More precise, we search for a function f that maps a given integer I = n ∈ Z to its label f(I) = l, where l = {"small", "", "large"} with the order "small" ≺ "" ≺ "large".' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The function f should return "small" if n < 1, 000, "large" if n ≥ 2, 000, and an empty string ("") otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Metamorphic Relations: 1) First, we require that for an increased input the label also has to stay equal or increase according to the defined label ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For a given integer I = n we compute the resulting label f(I) = lI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Following this, we manipulate the input I′ = n + k with a random k ∈ N with the cor- responding label f(I′) = lI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The relation holds if lI′ ⪰ lI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 2) Consequently, as second relation, we define that for an decreased input the resulting label has to stay equal or decrease according to the label ordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For the given input I = n we calculate the label f(I) = lI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Then, we manipulate the input I′ = n − k with k ∈ N and f(I′) = lI′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' If lI′ ⪯ lI, the relation is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='3 Incorporating Metamorphic Testing in GP MTGP benefits from the use of lexicase selection, since lexicase considers the individual cases instead of an aggregated fitness value [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Consequently, dif- ferent types of (training) cases can be used simultaneously for the selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In MTGP, these are hand-labeled training cases for which we know the input and the corresponding output, as well as cases based on metamorphic relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 8 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For the hand-labeled training cases, both the inputs and the outputs are known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In order to check whether a candidate program solves a training case or not, we run the program with the given input I and check whether the generated output f(I) matches the expected (in a real-world scenario hand-labeled) output O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' If f(I) = O, then the test is passed, otherwise not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To test the metamorphic relations defined for a benchmark problem, we pro- vide random inputs as base inputs for which the outputs do not need to be known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Thus, any number of random entries can be generated at low cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In our experiments, we have ensured that these random inputs do not already ex- ist in the training or test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In practice, they could be chosen freely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To check whether a candidate program satisfies a metamorphic relation or not, we execute it with a given base input I (one of the randomly generated inputs) and save the output f(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We then change the input randomly according to the manipulation rule of the metamorphic relation to create the follow-up input I′ and run the program again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The test is passed if the observed outputs f(I) and f(I′) satisfy the metamorphic relation, otherwise it is failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Since we have defined two meta- morphic relations for all benchmark problems considered in the experiments, we determine at random, with probability p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='5, which of the two relations is used each time a base input is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' One of the key advantages of using metamorphic testing this way is that during a GP run always many different metamorphic tests are executed, since the manipulation of the given basic input happens randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Our hope is, that this further supports the generalizability of the generated programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 4 Experiments and Discussion In this section, we study the performance of MTGP and analyze how well the generated programs generalize to previously unseen tests cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To analyze how the generalization ability depends on the number of given training cases, we perform experiments for different labeled training set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Below, we describe the experimental setup and discuss our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='1 Experimental Setup For the implementation of MTGP, we use the PonyGE2 framework [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Our used grammars are based on the program synthesis grammars provided by the PonyGE2 framework for the problems from the benchmark suite [18] and fol- low the principle suggested by Forstenlechner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' [10] according to which the grammar of a problem is restricted to the data types (and dependent functions) that are defined for the input and output of the considered problem, in addition to required base data types (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', like integer and Boolean).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' This allows to keep the used grammars small and effective but still expressive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We initialize a run with position independent grow [8] and use a population size of 1, 000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The maximum initial tree depth is set to 10 and the maximum over- all tree depth is limited to 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As variation operators, we use sub-tree crossover MTGP: Combining Metamorphic Testing and Genetic Programming 9 and sub-tree mutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For the sub-tree crossover we set the probability to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='9 and for the sub-tree mutation we set the number of mutation steps to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As mentioned above, we use lexicase [31] as selection method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' A GP run is stopped either after a program is found that solves all labeled training cases and all defined metamorphic tests or after 300 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For every considered benchmark problem, we have 200 labeled training cases that we can use in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The test set consisting of 1, 000 labeled cases is used to check if a candidate program also generalizes to unseen cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Further we provide a large set of randomly generated inputs (800 for each benchmark problem) which we use as base inputs for checking if the defined metamorphic relations hold for a candidate program or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In the experiments, we choose from these available training cases and randomly generated inputs depending on the considered configuration as specified below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='2 Results and Discussion As we investigate the impact of using metamorphic testing in GP, we compare a standard GP approach, which only uses the training data, and the novel MTGP approach, which also includes metamorphic tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In addition, we examine how standard GP and MTGP perform on different training set sizes |Ttraining|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' There- Table 1: Success rates on the training set straining and the test set stest achieved by standard GP and MTGP for all studied labeled training set sizes |Ttraining| and benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For MTGP, we use in addition 200 − |Ttraining| meta- morphic tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Best success rates achieved on the test set stest are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Standard GP MTGP Problem |Ttraining| straining stest straining stest Count Odds 25 43 28 32 20 50 59 46 43 35 100 70 64 61 59 200 83 81 Grade 25 83 4 74 14 50 86 12 76 15 100 81 23 84 33 200 93 45 Small or Large 25 94 2 58 7 50 79 11 77 25 100 91 35 86 29 200 89 43 10 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' fore, we analyze for both approaches the labeled training set sizes 25, 50, 100, and 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MTGP also uses 200 − |Ttraining| metamorphic tests so that MTGP considers exactly 200 tests during the training phase (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', for |Ttraining| = 25, MTGP uses 175 metamorphic tests).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As a first step, we study the achieved success rates on the training set straining and the test set stest as a performance indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The success rate on the training set straining measures the percentage of runs in which a program is found that is successful on all given training cases (including metamorphic tests for MTGP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To determine the success rate on the test set stest, we take from each run the candidate program that performs best on the training data and measure the percentage of candidate programs that solve all previously unseen test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For every studied configuration, we performed 100 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Table 1 shows the success rates on the training set straining and the test set stest achieved by standard GP and MTGP for all studied labeled training set sizes |Ttraining| (and corresponding metamorphic tests) and benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Best success rates achieved on the test set stest are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For most configurations, we see that standard GP achieves a higher success rate on the training data straining compared to MTGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Our assumption is that this is because the additional metamorphic tests prevent an overfitting to the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' On the test data, MTGP performs best for the Grade problem and for most of the configurations of the Small or Large problem (compared to the corresponding standard GP runs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For the Count Odds problem best results for stest are achieved with standard GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' More important than the pure success rates, however, is how well the pro- grams found on the training data generalize to unseen test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In practice, a program synthesis approach which is known for its high generalization can simply be executed again if no solution was found in the first run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' If the gen- eralization is expected to be poor, it is unclear whether solutions found on the training data also work on previously unseen test cases, regardless of the suc- cess rate achieved on the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Large sets of additional test cases (to check for generalizability) are not available in a real-world scenario as manually labeling additional input/output examples is far too expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Therefore, in the second step, we analyze the generalization rate G = stest straining 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Table 2 shows the generalization rate G achieved by standard GP and MTGP for all studied labeled training set sizes |Ttraining| (and corresponding metamor- phic tests) and benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Again, best generalization rates G are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We see that on average, best generalization rates G are achieved with MTGP, as MTGP performed best in 7 out of 9 configurations where we have results for standard GP as well as for MTGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The differences are particularly obvious for the Grade and the Small or Large problem when only a small labeled training set (|Ttraining| = 25) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MTGP achieves for the Grade problem a generalization MTGP: Combining Metamorphic Testing and Genetic Programming 11 Table 2: Generalization rate G achieved by standard GP and MTGP for all studied labeled training set sizes |Ttraining| and benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For MTGP, we use 200 − |Ttraining| metamorphic tests in addition to the considered labeled training cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Best results are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Generalization rate G Problem |Ttraining| Standard GP MTGP Count Odds 25 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='116 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='5 50 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='966 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='395 100 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='429 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='721 200 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='59 Grade 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='819 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='919 50 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='953 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='737 100 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='395 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='286 200 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='387 Small or Large 25 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='128 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='069 50 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='924 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='468 100 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='462 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='721 200 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='315 rate G of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='919 and for the Small or Large problem of 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='069 while standard GP achieves only 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='819 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='128, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' So far we have only studied MTGP with a reduced labeled training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' But can the generalization rate even be increased compared to standard GP even if the complete labeled training set (|Ttraining| = 200) is used during the run?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' To answer this question, we run MTGP this time with 800 metamorphic tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Table 3: Generalization rate G achieved by standard GP and MTGP for all considered benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' All configurations use as labeled training set size |Ttraining| = 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For MTGP, we use 800 metamorphic tests in addition to the considered labeled training cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Best results are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Generalization rate G Problem |Ttraining| Standard GP MTGP Count Odds 200 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='59 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='0 Grade 200 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='387 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='538 Small or Large 200 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='315 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='069 12 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Table 3 shows the achieved generalization rates G for standard GP and MTGP for all considered benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As before, best generalization rates G are printed in bold font.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Wee see that even when the complete labeled training set Ttraining is used during a run, using metamorphic testing can improve the generalization rate G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MTGP performs best on all studied benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' For the Count Odds problem, we even achieve a perfect generalization rate G of 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In summary, the generalization ability of the generated programs can be increased by using metamorphic testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' On average, best generalization rates are achieved with MTGP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The major advantage of metamorphic tests is that they do not require the expensive manual calculation of the expected outputs, since they work exclusively with random inputs which can be generated automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 5 Conclusion GP [7,21] is an evolutionary approach that is well known for its performance in automatic program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Even if GP is competitive in performance to other state-of-the-art program synthesis approaches [26], it is not yet mature enough for a practical use in real-world software development, as many input/output examples are usually required during the training process to generate programs that also generalize to unseen test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' As in practice, the training cases have to be labeled manually by the user which is very expensive, we need a supplemen- tary approach to check the program behavior with a lower number of manually defined training cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' With metamorphic testing [6], we do not need labeled input/output exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' The program is executed multiple times, first on a given input (which can be generated randomly) and followed by related inputs to check whether certain user-defined metamorphic relations hold between the observed outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Therefore, in this work we suggested MTGP, an approach that combines metamorphic testing and GP and studied its performance and the generalization ability of the generated programs on common program synthesis benchmark problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Further, we analyzed how the generalization ability depends on the number of given training cases and performed experiments for different labeled training set sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' We found that incorporating metamorphic testing in combination with hand- labeled training cases leads to a higher generalization rate than the exclusive use of hand-labeled training cases in almost all configurations studied in our exper- iments, including those using smaller labeled training sets as usual in GP-based program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Even with the largest considered labeled training set, the gen- eralization rate could be increased by a large margin on all studied benchmark problems with the use of metamorphic testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Consequently, we recommend researchers to use metamorphic testing in their GP approaches if the labeling of the training data is an expensive process in the considered application domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In future work, we will study MTGP on additional program synthesis bench- mark problems and further analyze the usage of the metamorphic tests as well MTGP: Combining Metamorphic Testing and Genetic Programming 13 as the given labeled training cases during a run to gain a deeper understanding of the implications of incorporating metamorphic testing in GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Aenugu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Lexicase selection in learning classifier systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Pro- ceedings of the Genetic and Evolutionary Computation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 356–364 (2019) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Arrieta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Multi-objective metamorphic follow-up test case selection for deep learning systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1327–1335 (2022) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Błądek, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Krawiec, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Swan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Counterexample-driven genetic programming: heuristic program synthesis from formal specifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Evolutionary computation 26(3), 441–469 (2018) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Kuo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Poon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Towey, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Tse, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' : Meta- morphic testing: A review of challenges and opportunities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' ACM Computing Sur- veys (CSUR) 51(1), 1–27 (2018) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Kuo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Tang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Metamorphic testing and testing with special values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: SNPD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 128–134 (2004) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Chen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Cheung, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Yiu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Metamorphic testing: A new approach for generating next test cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Department of Computer Science, The Hong Kong University of Science and Technology, Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Rep (1998) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Cramer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' : A representation for the adaptive generation of simple sequential programs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: proceedings of an International Conference on Genetic Algorithms and the Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 183–187 (1985) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Fagan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Fenton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', O’Neill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Exploring position independent initialisation in grammatical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: 2016 IEEE Congress on Evolutionary Computation (CEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 5060–5067.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' IEEE (2016) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Fenton, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', McDermott, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Fagan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Forstenlechner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Hemberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', O’Neill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': PonyGE2: Grammatical evolution in Python.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Conference Companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1194–1201 (2017) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Forstenlechner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Fagan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Nicolau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', O’Neill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': A grammar design pattern for arbitrary program synthesis problems in genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: European Conference on Genetic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 262–277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2017) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Forstenlechner, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Nicolau, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Fagan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', O’Neill, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Grammar design for deriva- tion tree based genetic programming systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: European Conference on Genetic Programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 199–214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2016) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Abdelhady, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Benchmarking parent selection for program synthesis by genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 237–238 (2020) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Kelly, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': PSB2: the second program synthesis benchmark suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 785– 794 (2021) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', McPhee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Pantridge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Improving generalization of evolved programs through automatic simplification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 937–944 (2017) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', McPhee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Lexicase selection for program synthesis: a diversity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Genetic Programming Theory and Practice XIII, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 151– 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2016) 14 Sobania et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', McPhee, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Program synthesis using uniform muta- tion by addition and deletion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1127–1134 (2018) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Pantridge, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': On the importance of specialists for lexicase selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Genetic Programming and Evolvable Machines 21(3), 349–373 (2020) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': General program synthesis benchmark suite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceed- ings of the 2015 Annual Conference on Genetic and Evolutionary Computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1039–1046 (2015) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Helmuth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Explaining and exploiting the advantages of down- sampled lexicase selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: ALIFE 2020: The 2020 Conference on Artificial Life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 341–349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MIT Press (2020) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Hemberg, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Kelly, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', O’Reilly, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' : On domain knowledge and novelty to im- prove program synthesis performance with grammatical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1039–1046 (2019) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Koza, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Genetic programming: on the programming of computers by means of natural selection, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' MIT press (1992) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Krawiec, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Behavioral program synthesis with genetic programming, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 618.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2016) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Langdon, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Krauss, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Evolving sqrt into 1/x via software data maintenance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1928–1936 (2020) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Schweim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Effects of the training set size: A com- parison of standard and down-sampled lexicase selection in program synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: 2022 IEEE Congress on Evolutionary Computation (CEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' IEEE (2022) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': On the generalizability of programs synthesized by grammar-guided genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: European Conference on Genetic Programming (Part of EvoStar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 130–145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2021) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Briesch, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Choose your programming copilot: a com- parison of the program synthesis performance of Github Copilot and genetic pro- gramming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computation Confer- ence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1019–1027 (2022) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Challenges of program synthesis with grammatical evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: European Conference on Genetic Programming (Part of EvoStar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 211–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2020) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': A generalizability measure for program synthesis with genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the Genetic and Evolutionary Computa- tion Conference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 822–829 (2021) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Program synthesis with genetic programming: The in- fluence of batch sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: European Conference on Genetic Programming (Part of EvoStar).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 118–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Springer (2022) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Sobania, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Schweim, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Rothlauf, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': A comprehensive survey on program syn- thesis with evolutionary algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' IEEE Transactions on Evolutionary Compu- tation (2022) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Assessment of problem modality by differential performance of lexicase selection in genetic programming: a preliminary report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the 14th annual conference companion on Genetic and evolutionary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 401– 408 (2012) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Klein, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Keijzer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': The Push3 execution stack and the evolution of control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the 7th annual conference on Genetic and evolutionary computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 1689–1696 (2005) MTGP: Combining Metamorphic Testing and Genetic Programming 15 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Spector, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', Robinson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=': Genetic programming and autoconstructive evolution with the Push programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Genetic Programming and Evolvable Ma- chines 3(1), 7–40 (2002) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Whigham, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' : Grammatically-based genetic programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' In: Proceedings of the workshop on genetic programming: from theory to real-world applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 16, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' 33–41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} +page_content=' Citeseer (1995)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HNFAT4oBgHgl3EQfth7Z/content/2301.08665v1.pdf'} diff --git a/JdA0T4oBgHgl3EQfCf9p/content/tmp_files/2301.01990v1.pdf.txt b/JdA0T4oBgHgl3EQfCf9p/content/tmp_files/2301.01990v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..9a4740fbf9315e6eb31cead0b1d4366ef7d5bea4 --- /dev/null +++ b/JdA0T4oBgHgl3EQfCf9p/content/tmp_files/2301.01990v1.pdf.txt @@ -0,0 +1,3255 @@ +arXiv:2301.01990v1 [math.DG] 5 Jan 2023 +Witten deformation for non-Morse functions and +gluing formula for analytic torsions +Junrong Yan ∗ +January 6, 2023 +Abstract +In this paper, we provide a novel analytic proof of the gluing formula for the +analytic torsion of flat vector bundles in complete generality. It’s quite interesting +that in this paper, the gluing formula could be interpreted as the Bismut-Zhang +theorem [1] for some non-Morse functions. +Contents +1 +Introduction +2 +1.1 +Overview +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +1.2 +Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +1.3 +Main ideas +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +1.4 +Organization +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +5 +2 +Preliminary +6 +2.1 +A brief review on Hodge theory . . . . . . . . . . . . . . . . . . . . . +6 +2.2 +Analytic torsion for flat vector bundles . . . . . . . . . . . . . . . . . +7 +2.2.1 +On closed manifolds . . . . . . . . . . . . . . . . . . . . . . . +7 +2.2.2 +On manifolds with boundary +. . . . . . . . . . . . . . . . . . +8 +2.3 +Analytic torsion for Witten Laplacian and Weighted Laplacian +. . . +9 +2.3.1 +Witten Laplacian v.s. Weighted Laplacian . . . . . . . . . . . +10 +2.3.2 +Absolute/Relative boundary conditions for weighted Laplacian 10 +3 +Intermidiate Results +11 +4 +Convergence of Eigenvalues +13 +∗Beijing International Center for Mathematical Research, Peking University, Beijing, China 100871, +j yan@bicmr.pku.edu.cn. Supported by Boya Postdoctoral Fellowship at Peking University. +1 + +5 +Large Time Contributions +19 +6 +A Gluing Formula for Heat Trace in Small Time +20 +6.1 +Several heat kernels and Laplacians . . . . . . . . . . . . . . . . . . . +20 +6.2 +Gluing heat kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +6.3 +Heat trace expansion for e−t(∆1⊕∆2) +. . . . . . . . . . . . . . . . . . +24 +6.4 +Heat trace expansion for e−t∆T +. . . . . . . . . . . . . . . . . . . . . +24 +6.5 +Proof of Theorem 3.3 when M is even dimensional . . . . . . . . . . +25 +7 +One-dimensional Model and Coupling Techniques +25 +7.1 +Heat trace estimate when t and T are coupled . . . . . . . . . . . . . +26 +7.2 +Partial proof of Theorem 3.3 when M is odd dimensional +. . . . . . +30 +8 +Ray-Singer metrics on S1 and [−2, 2] +31 +8.1 +Ray-Singer metrics on S1 +. . . . . . . . . . . . . . . . . . . . . . . . +31 +8.2 +Ray-Singer metric on [−2, 2] . . . . . . . . . . . . . . . . . . . . . . . +32 +9 +Proof of Theorem 3.4 +34 +1 +Introduction +1.1 +Overview +Let (M, g) be a closed Riemannian manifold associated with a flat complex vector +bundle F → M, and suppose that F has a hermitian metric hF . The correspond- +ing Ray-Singer analytic torsion [14] is the determinant of the Hodge-Laplacian on +F-valued differential forms. The Ray-Singer metric [1] on det H∗(M; F) is the prod- +uct of the Ray-Singer analytic torsion and the Hodge metric induced by F-valued +harmonic forms. The Ray-Singer metric has a well-known topological counterpart, +the Reidemeister metric [15]. The famous Ray-Singer conjecture [14] states that the +two metrics for a unitary flat vector bundle coincide, which was proved by Cheeger +[4] and M¨uller [10] independently. M¨uller [11] then extended the theorem to uni- +modular flat bundles. Bismut and Zhang [1] extended the theorem to general flat +vector bundles. +Now suppose there is a hypersurface Y ⊂ M that divides M into two parts, M1 +and M2. As a preliminary step toward the Ray-Singer conjecture, Ray and Singer +[14] proposed that there should be a gluing formula relating the analytic torsion of +M and the analytic torsions of M1 and M2. However, the conjecture was proved by +other methods, and the gluing formula follows from the conjecture [8, 3]. There are +also purely analytic proofs, i.e., without using Cheeger-M¨uller theorem [16, 13, 7]. +The proof of gluing formula in this paper is based on the Witten deformation for +non-Morse functions and some coupling techniques (see the last paragraph in §1.3 +for a description of coupling techniques). Roughly, we choose a family of smooth +functions fT such that the limit limT→∞ fT has crucial loci M1 and M2 with “Morse +2 + +indices” of 0 and 1, respectively. Then, based on the philosophy of Witten defor- +mation, letting T range from 0 to ∞, the relationship between analytic torsion on +M and analytic torsion on the two pieces above can be understood. This method +could be applied to analytic torsion forms [17]. +See [12] for another proof. It’s +also possible that this method could be applied to the proof of gluing formulas for +other global invariants (such as eta-invariant, partition functions from QFT, e.t.c.). +Finally, this paper’s exploration of Witten deformation for non-Morse functions is +intended to provide clues toward proving the family version of Cheeger-M¨uller the- +orem for general flat bundles. +Acknowledgment: The author is appreciative of Professor Xianzhe Dai’s consis- +tently stimulating conversation and encouragement. The author also appreciates the +insightful discussion with Martin Puchol and Yeping Zhang. +1.2 +Main Results +Let (M, gTM) be a closed Riemannian manifold and Y ⊂ M be a hypersurface that +divides M into two pieces M1 and M2. +Let F → M be a flat complex vector bundle, and ∇F be a flat connection on +F. Let dF be the covariant differential on F-valued differentials Ω(M; F), which is +induced by ∇F (c.f. [18]). +Let hF be a Hermitian metric on F. Let dF,∗ be the (formal) adjoint operator +of dF associated with gTM and hF . Then D := dF + dF,∗ is a first-order self-adjoint +elliptic operator acting on Ω(M; F). +Let N be the number operator on Ω (M; F), i.e., Nω = pω for ω ∈ Ωp (M; F). +The zeta function ζ for ∆ := D2 is defined as follows, for z ∈ {C : Re(z) > 1 +2 dim M +� +ζ(z) := +1 +Γ(z) +� ∞ +0 +tz−1 Trs +� +Ne−t∆′� +dt. +See §2.2.1 for more details. The function ζ(s) admits a meromorphic continuation +to the whole complex plane, which is holomorphic at 0 ∈ C. +Then the Ray-Singer analytic torsion T (gTM, hF , ∇F ) := e +1 +2ζ′(0). +Let Fi be the restriction of F on Mi, dF +i be the restriction of dF on Mi, dF,∗ +i +be +the formal adjoint of dF +i (i=1,2). Let ∆1 := (dF +1 + dF,∗ +1 )2 and ∆2 = (dF +2 + dF,∗ +2 )2 +act on Ωrel (M1; F1) and Ωabs (M2; F2) respectively (see §2.2.2 for the definition of +Ωabs (M1; F1) and Ωrel (M2; F2)). Let ζi be the zeta functions for ∆i. Then similarly, +one can define Ray-Singer analytic torsion Ti(gTMi, hFi, ∇Fi) = e +1 +2 ζ′ +i(0), where gTMi, +Fi, hFi and ∇Fi are the restriction of gTM, F, hF and ∇F on Mi respectively(i = 1, 2). +Lastly, we have the following Mayer-Vietoris exact sequence +· · · → Hp +rel (M2; F2) → Hp (M; F) → Hp +abs (M1; F1) → · · · . +We denote by T the analytic torsion for the exact sequence above equipped with +L2-metrics (induced by Hodge theory). +3 + +Theorem 1.1. +log T (gTM, hF , ∇F ) − log T1(gTM1, hF1, ∇F1) − log T2(gTM2, hF2, ∇F2) += log T + 1 +2χ(Y )rank(F) log 2 + (−1)dim(M)rank(F) +� +Y +B +� +gTM� +, +where B +� +gTM� +is the secondary characteristic form introduced in [2], which is zero +if Y is totally geodesic in +� +M, gTM� +. +Remark 1.1. Based on [1, Theorem 4.7] and [3, Theorem 3.4], we only need to +consider the situation in which gTM and hF are product-type near Y . In this case, +� +Y B +� +gTM� += 0. +1.3 +Main ideas +Let Y ⊂ M be a hypersurface cutting M into two pieces M1 and M2. Let U be +a neighbor of Y , such that U ∩ M1 is diffeomorphic to (−2, −1] × Y and identify +∂M1 with {−1} × Y , U ∩ M2 is diffeomorphic to [1, 2) × Y and identify ∂M2 with +{1} × Y . Moreover, assume that on U, gTM and hF are product-type. +We then glue M1, M2 and [−1, 1] × Y naturally, we get a manifold ¯ +M, which is +diffeomorphic to the original manifold M. Let fT be a smooth function on ¯ +M, such +that +1. fT |M1 ≡ −T/2, +2. fT |M2 ≡ T/2, +3. fT |[−1,0]×Y (s, y) ≈ T(s + 1)2/2 − T/2, +4. fT |[−1,0]×Y (s, y) ≈ −T(s − 1)2/2 + T/2. +Let dF +T := dF + dfT∧, dF,∗ +T +be the formal adjoint of dF +T . Set DT := dF +T + dF,∗ +T , +∆T := D2 +T . +Let λk be the k-th eigenvalue (counted with multiplicities) of ∆1 ⊕ ∆2 (acting +on Ωrel(M1; F1)⊕Ωabs(M2; F2)) . Let {λk(T)} be the k-th eigenvalue (counted with +multiplicities) of ∆T (acting on Ω( ¯ +M; ¯F)). +One has a nice observation that +lim +T→∞ λk(T) = λk. +(1) +We temporarily assume that dim Hk(M) = dim Hk(M1) + dim Hk(M2, ∂M2). +Let T (gT ¯ +M, hF , ∇F )(T) be the analytic torsion with respect to ∆F +T . Then based on +(1), naively, one should expect that +lim +T→∞ log T (gT ¯ +M, hF , ∇F )(T) = log T1(gTM1, hF1, ∇F1) + log T2(gTM2, hF2, ∇F2), +(2) +and +lim +T→0 log T (gT ¯ +M, hF , ∇F )(T) = log T (gT ¯ +M, hF , ∇F ). +4 + +As a result, the relationship between analytic torsion on M and analytic torsion on +the two pieces above could be seen, proving Theorem 1.1. +Although the idea is quite simple, the devil is in the details. Let ζT , ζ1 and ζ2 +be the zeta functions for ∆T , ∆1 and ∆2 respectively. Let +ζL +T (z) := +1 +Γ(z) +� ∞ +1 +tz−1 Trs +� +Ne−t∆′ +T +� +dt, +and +ζS +T (z) := +1 +Γ(z) +� 1 +0 +tz−1 Trs +� +Ne−t∆′ +T +� +dt. +Similarly, ζL +i +and ζS +i (i = 1, 2) can be defined. +By (1), it should be clear that +(ζL +T )′(0) → (ζL +1 )′(0) + (ζL +1 )′(0) as T → ∞ (Theorem 3.2). To get (2), the next step +is to show the convergence of (ζS +T )′(0). The main difficulty for this method is that +when t ∈ (0, 1), the convergence of Trs(Ne−t∆′ +T ) as T → ∞ is unclear. To resolve +a similar issue, [13] considers the adiabatic limit of analytic torsion. While in this +paper, we observe that when t and T are coupled, Trs(Ne−t∆′ +t−5T ) behaves nicely +as T → ∞. That is, (see §7.1), +Trs(Ne−t∆′ +t−5T ) → Trs(Ne−t∆′ +1) + Trs(Ne−t∆′ +2). +Similar ideas appear in the author’s previous joint work with Xianzhe Dai [6]. +Note that the space generated by the eigenforms of ∆T for small eigenvalues, +denoted by Ωsm(M, F)(T), is finite-dimensional for large values of T. +The sec- +ond crucial piece in our proof is the relationship between the analytic torsion for +Ωsm(M, F)(T) and the analytic torsion for the exact sequence (7) as T → ∞. See +§9 for further details. +1.4 +Organization +In §2, we will give a brief review of analytic torsion and establish the basic settings. +In §3, we state and prove several intermediate results and show Theorem 1.1. While +Theorem 3.1, 3.2, 3.3 and 3.4 will be proved in subsequent sections. +In §4, we +investigate the behavior of eigenvalues as T → ∞ and prove Theorem 3.1. In §5, +we analyze the long-time behavior of the zeta function and prove Theorem 3.2. In +§6, the gluing formula of the heat trace for small time t is discussed. In §7, we +investigate the small-time behavior of the zeta function, discovering that if time t +and the deformation parameter T are well-coupled, we understand the behavior of +the heat kernel on the tube well. Then we partially establish Theorem 3.3, leaving +some mysterious terms unknown. Nonetheless, it is worth noting that these terms +are independent of M, Y and F. To calculate these terms, the Ray-Singer metric on +S1 and [−2, 2] is explored in §8. Therefore, thoroughly prove Theorem 3.3. Theorem +3.4 will be proved in the last section. +5 + +2 +Preliminary +From now on, we assume that gTM and hF are product-type near Y . +That is, +there exists a neighborhood U of Y , such that U ∼= (−1, 1) × Y , and let (s, y) +be its coordinate. Then gTM|U = ds ⊗ ds + gTY for some metric gTY on Y. Let +hF +Y := hF |{0}×Y . +For any v ∈ F(s,y), let Pγ ∈ End(F(s,y), F(0,y)) be the parallel +transport associated with ∇F w.r.t. +the path γ(t) = (st, y), t ∈ [0, 1], then we +require that hF (v, v) = hF +Y (Pγv, Pγv). Hence on U, ∇F +∂ +∂s hF = 0. +If T is sufficiently large, all constants appearing in this paper are at least inde- +pendent of T. The notations C and c, et cetera, denote constants that may vary +based on context. +2.1 +A brief review on Hodge theory +Let (X, gTX) be a compact manifold with boundaries Y = ∂X, where Y could be +an empty set. Let F → X be a flat vector bundle with flat connection ∇F, and +hF be a Hermitian metric on F. We identify a neighborhood of ∂X to (−1, 0] × Y , +and let (s, y) be its coordinates. Let Ω(X; F) denote the space of smooth F-valued +differential forms. +Set +Aabs(X; F) := +� +ω ∈ Ω(X; F) : i ∂ +∂s ω = 0 on Y +� +, +Arel(X; F) := {ω ∈ Ω(X; F) : ds ∧ ω = 0 on Y } ; +Ωabs(X; F) := +� +ω ∈ Ω(X; F) : i ∂ +∂s ω = 0, i ∂ +∂s dF ω = 0 on Y +� +, +Ωrel(X; F) := +� +ω ∈ Ω(X; F) : ds ∧ ω = 0, ds ∧ dF,∗ω = 0 on Y +� +. +For the sake of convenience, “bd” will be adopted to represent “abs” or “rel”, when +it is not necessary to distinguish the boundary conditions. +Let dF : Ω∗(X; F) → Ω∗+1(X; F) denote the covariant derivative with respect +to ∇F. Assuming that {ek} is a local orthonormal frame of TX and {ek} is its +dual frame, then dF = � +k ek ∧ ∇F +ei locally. Let ∇F,∗ be the dual connection of +∇F, i.e., for any s1, s2 ∈ Γ(F), dh(s1, s2) = h(∇F s1, s2) + h(s1, ∇F,∗s2). Then set +dF,∗ := − � +k iek∇F,∗ +ek . +Let dF +bd, dF,∗ +bd and ∆bd be the restrictions of dF , dF,∗ and ∆ to Abd(X; F) re- +spectively. Let (·, ·)L2(X) be the L2-inner product induced by hF and gTX, then +integration by parts implies that +(dF +bdα, β)L2 = (α, dF,∗ +bd β)L2, ∀α, β ∈ Abd(X; F). +(3) +For a closable operator S : H → H on a Hilbert space (H, (·, ·)H) with a dense +domain Dom(S), define an inner product (·, ·)S on Dom(S) by +(α, β)S := (α, β)H + (S α, S β)H, ∀α, β ∈ Dom(S). +6 + +Let Wmin be the completion of Dom(S) with respect to the norm ∥· ∥S, then one +can extend S to Smin naturally with Dom(Smin) = Wmin. +Assume that there is another closable operator ˜S : H → H, such that Dom(˜S) = +Dom(S) and +(S α, β)H = (α, ˜Sβ)H, ∀α, β ∈ Dom(S). +Let +Wmax := {α ∈ H : |(α, ˜Sβ)H| ≤ Mα∥β∥H for some constant Mα > 0, ∀β ∈ Dom(˜S)}. +Because Dom(S) is dense, the Riesz representation theorem states that γ ∈ H exists +such that (γ, β)H = (α, ˜Sβ)H. Then we define Smax α = γ. One can see easily that +Smax is nothing but the adjoint of ˜S. +By (3), [5, Proposition A.3] and the discussion above, one has +Im(dF +bd,min) ⊕ Im(dF,∗ +bd,min) = Im(dF +bd,max) ⊕ Im(dF,∗ +bd,max). +Together with [3, Theorem 1.1], one has +Theorem 2.1 (Hodge decomposition). +(1) We have +ker ∆bd = ker +� +dF � +∩ ker +� +dF,∗� +∩ Ωbd(X; F). +(2) The vector space ker ∆bd is finite-dimensional. +(3) We have the following orthogonal decomposition +Abd(X; F) = ker(∆bd) ⊕ ImdF +bd ⊕ ImdF,∗ +bd . +L2Abd(X; F) = ker(∆bd) ⊕ ImdF +bd,min ⊕ ImdF,∗ +bd,min += ker(∆bd) ⊕ ImdF +bd,max ⊕ ImdF,∗ +bd,max. +Here L2Abd(X; F) is the completion of Abd(X, F) with respect to (·, ·)L2. +2.2 +Analytic torsion for flat vector bundles +2.2.1 +On closed manifolds +Let (M, gTM) be a closed Riemannian manifold, F → M be a flat vector bundle +with a Hermitian metric hF , and ∇F be a flat connection on F. Let dF be the +covariant differential on Ω(M; F) induced by ∇F, then we have a complex +0 → Ω0(M; F) dF +→ Ω1(M; F) dF +→ · · · dF +→ Ωdim(M)(M; F) → 0. +(4) +Denote H(M; F) to be the cohomology of this complex. One can see that gTM and +hF induce an L2-inner product (·, ·)L2(M) on Ω∗(M; F). +7 + +Let dF,∗ be the formal adjoint of dF with respect to metric gTM and hF . Then +the Hodge Laplacian ∆ : Ω(M; F) → Ω(M; F) is defined as +∆ := (dF + dF,∗)2. +Ω(M; F) has a natural Z2 grading: +Ω+ := ⊕k is even Ωk(M; F), Ω− := ⊕k is oddΩk(M; F). +For a trace class operator A : Ω(M; F) → Ω(M; F), Trs(A) denotes its supertrace. +If A has an integral kernel a, the pointwise supertrace of a is denoted by trs(a). +Let N : Ω(M; F) → Ω(M; F) be a linear operator, such that for α ∈ Ωk(M; F), +Nα = kα. N is called the number operator. Let P : Ω(M; F) → Ω(M; F) be the +orthogonal projection to ker(∆), ∆′ := ∆(1 − P). +Definition 2.1. The zeta function ζ for ∆ is defined as +ζ(z) := +1 +Γ(z) +� ∞ +0 +tz−1 Trs +� +Ne−t∆′� +dt. +where Γ is the Gamma function. The zeta function is well defined whenever Re(z) is +large enough. And it could be extended to a meromorphic function on C. Moreover, +ζ is holomorphic at 0. +The Ray-Singer analytic torsion T (gTM, hF , ∇F ) for the complex (4) is defined +as +T (gTM, hF , ∇F ) := e +1 +2 ζ′(0). +2.2.2 +On manifolds with boundary +Let (X, g) be a compact manifold with boundaries Y = ∂X, gTX be a Riemannian +metric on X. Let F → X be a flat vector bundle with flat connection ∇F, and hF +be a Hermitian metric on F. We identify a neighborhood of ∂X to (−1, 0] × Y . Let +(s, y) be its coordinates. +Let dF,∗ be the formal adjoint of the de Rham operator dF with respect to the +L2 metric (·, ·)L2(X) induced from hF and gTX. +Set +Ωabs(X; F) := +� +ω ∈ Ω(X; F) : i ∂ +∂u ω = 0, i ∂ +∂u dF ω = 0 on Y +� +, +Ωrel(X; F) := +� +ω ∈ Ω(X; F) : du ∧ ω = 0, du ∧ dF,∗ω = 0 on Y +� +. +We write Ωbd(X; F) for short if the choice of abs/rel is clear. +Let ∆bd := (dF + dF,∗)2 act on Ωbd(X; F). +According to the Hodge theory, +ker(∆bd) ∼= Hbd(X; F). +Here Hrel(X; F) := H(X, ∂X; F), and Habs(X; F) := +H(X; F). +Let ζbd be the zeta functions for ∆bd. +The analytic torsion Tbd(gTX, hF , ∇F) is defined by e +1 +2 ζ′ +bd(0). +8 + +In particular, let Y ⊂ M be a hypersurface cutting M into two pieces M1 and +M2. Denote gTMi, Fi, hFi and ∇Fi to be the restriction of gTM, F, hF and ∇F to Mi +respectively(i = 1, 2). One can see that gTMi and hFi induce an L2-inner product +(·, ·)L2(Mi) on Ω∗(Mi; Fi). +Let ∆1 := (dF1 + dF1,∗)2 act on Ωabs(M1; F1), and ∆2 := (dF2 + dF2,∗)2 act on +Ωrel(M2; F2). +Definition 2.2. We set ζi to be the zeta function for ∆i. And let +T1(gM1, hF1, ∇F1) := Tabs(gM1, hF1, ∇F1) = e +1 +2ζ′ +1(0), +and +T2(gM2, hF2, ∇F2) := Trel(gM2, hF2, ∇F2) = e +1 +2ζ′ +2(0). +2.3 +Analytic torsion for Witten Laplacian and Weighted +Laplacian +Let Y ⊂ M be a hypersurface cutting M into two pieces M1 and M2. Denote gTMi, +Fi, hFi and ∇Fi to be the restriction of gTM, F, hF and ∇F to Mi respectively(i = +1, 2). We identify a neighborhood of Y = ∂M1 in M1 to (−2, −1] × Y , and ∂M1 is +identified with {−1} × Y . Similarly, we identify a neighborhood of Y = ∂M2 in M2 +to [1, 2) × Y , and ∂M2 is identified with {1} × Y . Let ¯ +M = M1 ∪ [−1, 1] × Y ∪ M2, +and ¯F, h ¯F , gT ¯ +M and ∇ ¯F be the natural extensions of F, hF , gTM and ∇F to ¯ +M. One +can see that gT ¯ +M and h ¯F induce an L2-inner product (·, ·)L2( ¯ +M) on Ω∗( ¯ +M; ¯F). +Let fT be a family of odd smooth functions on [−2, 2], such that +(a) fT |[1,2] ≡ T/2, +(b) fT |[1/2,1](s) = −Tρ +� +eT 2(1 − s) +� +(s − 1)2/2+ T/2 , where ρ ∈ C∞ +c ([0, ∞)), such +that 0 ≤ ρ ≤ 1, ρ[0,1/2] ≡ 0, ρ[3/4,∞] ≡ 1, |ρ′| ≤ δ1 and |ρ′′| ≤ δ2 for some +universal constant δ1 and δ2. +(c) C1T ≤ |f ′ +T|(s) ≤ 2C1T, |f ′′ +T | ≤ C2T for some universal constants C1 and C2 +whenever s ∈ [0, 1/2]. +Then one can see that C3T ||s| − 1| ≤ |f ′ +T |(s) ≤ 2C3T ||s| − 1| and |f ′′ +T |(s) ≤ C4T +whenever ||s| − 1| ≤ e−T 2 for some universal constant C3 and C4. +We could think fT as a function on ¯ +M. Let dT := d ¯F + dfT ∧ . Then the Witten +Laplacian ∆T is the Hodge Laplacian with respect to dT . +Definition 2.3. We have a complex +0 → Ω0( ¯ +M, ¯F) +dT +→ Ω1( ¯ +M, ¯F) +dT +→ · · · +dT +→ Ωdim(M)( ¯ +M, ¯F) → 0. +(5) +Denote H( ¯ +M, ¯F)(T) to be the cohomology of this complex, ∆T to be the Hodge +Laplacian for dT , and |·|gT ¯ +M ,h ¯ +F ,∇ ¯ +F (T) to be the Hodge metric on det +� +H( ¯ +M, ¯F)(T) +� +. +9 + +Let ζT be the zeta function for ∆T . Similarly, one could define Ray-Singer analytic +torsion T (gT ¯ +M, h ¯F , ∇ ¯F )(T) := e +1 +2 ζ′ +T (0) for the complex (5). +Lastly, for the sake of convenience, (·, ·)L2 (resp. ∥ · ∥L2 := +� +(·, ·)L2) will be +adopted to represent (·, ·)L2(M) (resp. ∥ · ∥L2(M) := +� +(·, ·)L2(M)) , (·, ·)L2( ¯ +M) (resp. +∥ · ∥L2( ¯ +M) := +� +(·, ·)L2( ¯ +M)) or (·, ·)L2(Mi)(resp. ∥ · ∥L2(Mi) := +� +(·, ·)L2(Mi)) (i = 1, 2), +when the context is clear. +2.3.1 +Witten Laplacian v.s. Weighted Laplacian +Instead of deforming the de Rham differential d ¯F , we could also deform the metric +h ¯F : let h ¯F +T := e−2fT h ¯F . Similarly, g ¯ +M and h ¯F +T induce an L2-norm (·, ·)L2( ¯ +M),T on +Ω( ¯ +M; ¯F). Let L2Ω( ¯ +M; ¯F)(T) be the completion of Ω( ¯ +M; ¯F) w.r.t (·, ·)L2( ¯ +M),T . +Then the formal adjoint d +¯F,∗ +T +of d ¯F w.r.t. +the (·, ·)L2( ¯ +M),T is then given by +efT d∗ +T e−fT . +The Weighted Laplacian ˜∆T := d ¯F d +¯F ,∗ +T ++ d +¯F,∗ +T d ¯F , one can see that +˜∆T = efT ∆T e−fT . Let lk(T) be the k-th eigenvalue of ˜∆T, then lk(T) = λk(T). +Moreover, if u is an eigenform of ∆T w.r.t. eigenvalue λ, then efT u is an eigenform +of ˜∆T w.r.t. eigenvalue λ. +As a result, Trs(Ne−t ˜∆T ) = Trs(Ne−t∆T ). +2.3.2 +Absolute/Relative boundary conditions for weighted Lapla- +cian +Let ¯ +M1 := M1 ∪ [−1, 0] × Y , ¯ +M2 := M2 ∪ [0, 1] × Y , and ¯Fi be the restriction of ¯F +on ¯ +Mi (i = 1, 2). Set +Aabs( ¯ +M1; ¯F1) := +� +ω ∈ Ω( ¯ +M1; ¯F1) : i ∂ +∂s ω = 0 on {0} × Y +� +, +Arel( ¯ +M2; ¯F2) := +� +ω ∈ Ω( ¯ +M2; ¯F2) : ds ∧ ω = 0 on {0} × Y +� +; +Ωabs( ¯ +M1; ¯F1) := +� +ω ∈ Ω( ¯ +M1; ¯F1) : i ∂ +∂s ω = 0, i ∂ +∂s d +¯F1ω = 0 on {0} × Y +� +, +Ωrel( ¯ +M2; ¯F2)T := +� +ω ∈ Ω( ¯ +M2; ¯F2) : ds ∧ ω = 0, ds ∧ d +¯Fi,∗ +T +ω = 0 on {0} × Y +� +. +One can see that gT ¯ +Mi and h +¯Fi +T (The resriection of gT ¯ +M and h ¯Fi on ¯ +Mi) induce +an L2-inner product (·, ·)L2( ¯ +Mi),T on Ω∗( ¯ +Mi; ¯Fi). +Let ˜∆T,i be the restriction of ˜∆T acting on Ωbd( ¯ +Mi; ¯Fi). Then by Hodge theory, +ker( ˜∆T,i) ∼= Hbd( ¯ +Mi; ¯Fi). +Lastly, for the sake of convenience, (·, ·)L2,T (resp. +∥ · ∥L2,T := +� +(·, ·)L2,T) +will be adopted to represent (·, ·)L2( ¯ +M),T (resp. ∥ · ∥L2( ¯ +M),T := +� +(·, ·)L2( ¯ +M),T ) , or +(·, ·)L2( ¯ +Mi),T (resp. ∥ · ∥L2( ¯ +Mi),T := +� +(·, ·)L2( ¯ +Mi),T ) (i = 1, 2), when the context is +clear. +10 + +3 +Intermidiate Results +In this section, we will state and prove some intermediate results to prove Theorem +1.1. Let λk(T) be the k-th eigenvalue for ∆T, λk be the k-th eigenvalue of ∆1 ⊕ ∆2 +acting on Ωabs(M1; F1) ⊕ Ωrel(M2; F2). and ˜λk(T) be the k-th eigenvalue of ˜∆T,1 ⊕ +˜∆T,2 acting on Ωabs( ¯ +M1; ¯F1) ⊕ Ωrel( ¯ +M2; ¯F2)T . +Theorem 3.1. Then limT→∞ λk(T) = limT→∞ ˜λk(T) = λk. +Let δ > 0 denote half of the first nonzero eigenvalue of ∆1 ⊕ ∆2. Then by The- +orem 3.1, all eigenvalues of ∆T inside [0, δ] converge to 0 as T → ∞, and all eigen- +values of ˜∆T,1 ⊕ ˜∆T,2 inside [0, δ] are 0 when T is large enough. Let Ωsm( ¯ +M, ¯F)(T) +be the space generated by eigenforms for eigenvalues of ∆T inside [0, δ], and Pδ(T) +be the orthogonal projection from L2Ω( ¯ +M, ¯F) to Ωsm( ¯ +M, ¯F)(T). +Let +ζT,la := +1 +Γ(z) +� ∞ +0 +tz−1 Trs +� +Ne−t∆′ +T +� +1 − Pδ(T) +�� +dt, +ζT,sm := +1 +Γ(z) +� ∞ +0 +tz−1 Trs +� +Ne−t∆′ +T Pδ(T) +� +dt. +For i = 1, 2, let +ζL +i (z) := +1 +Γ(z) +� ∞ +1 +tz−1 Trs(Ne−t∆′)dt, +ζS +i (z) := +1 +Γ(z) +� 1 +0 +tz−1 Trs(Ne−t∆′)dt. +Then it is clear that ζi = ζL +i + ζS +i . +Similarly, one can define ζL +T,i(z), ζS +T,i(z), ζL +T,la(z), and ζS +T,la e.t.c. +Theorem 3.2. +lim +T→∞(ζL +T,la)′(0) = lim +T→∞ +2 +� +i=1 +(ζL +T,i)′(0) = +2 +� +i=1 +(ζL +i )′(0). +That is, +lim +T→∞ +� ∞ +1 +t−1 Trs +� +Ne−t∆′ +T (1 − Pδ) +� +dt += lim +T→∞ +2 +� +i=1 +� ∞ +1 +t−1 Trs(Ne−t∆′ +T,i)dt = +2 +� +i=1 +� ∞ +1 +t−1 Trs(Ne−t∆′ +i)dt. +Theorem 3.3. As T → ∞, +(ζS +T,la)′(0) = +2 +� +i=1 +(ζS +i )′(0) − (T − log(2)) χ(Y )rank(F) + o(1), +11 + +(ζS +i,T )′(0) = (ζS +i )′(0) − Tχ(Y )rank(F)/2 + o(1). +Thus, as T → ∞, +(ζS +T,la)′(0) − +2 +� +i=1 +(ζS +i,T )′(0) = log(2)χ(Y )rank(F) + o(1). +Next, we have the following Mayer-Vietoris exact sequence (c.f. [3, (0.16)]) +MV : · · · +∂k−1 +→ Hk +rel +� ¯ +M2; ¯F2 +� ek +→ Hk � ¯ +M; ¯F +� rk +→ Hk +rel +� ¯ +M1; ¯F1 +� ∂k +→ · · · . +(6) +Let H( ¯ +M; ¯F)(T) := ker( ˜∆T ), and H( ¯ +Mi; ¯Fi)(T) := ker( ˜∆T,i). We also have the +following Mayer-Vietoris exact sequence induced by Hodge theory and (6) +MV(T) : · · · +∂k−1,T +→ +Hk � ¯ +M2; ¯F2 +� +(T) +ek,T +→ Hk � ¯ +M; ¯F +� +(T) +rk,T +→ Hk � ¯ +M1; ¯F1 +� +(T) +∂k,T +→ · · · +(7) +with metric induced by gT ¯ +M and h ¯F +T . Let T (T) be the analytic torsion for this +complex. +Recall that +ζT,sm := +1 +Γ(z) +� ∞ +0 +tz−1 Trs(Ne−t∆′ +T Pδ(T))dt. +And set Tsm(gT ¯ +M, h ¯F , ∇ ¯F )(T) := e +1 +2 ζ′ +T,sm(0). +Moreover, the following proposition will be proved in §9, +Theorem 3.4. limT→∞ log Tsm(gT ¯ +M, h ¯F , ∇ ¯F)(T) − log T (T) = 0. +Let Ti(gT ¯ +Mi, h ¯Fi, ∇ ¯Fi)(T) be the analytic torsion w.r.t. ˜∆T,i, then it follows from +anomaly formula [1, Theorem 0.1] and [2, Theorem 0.1] that +Theorem 3.5. +log T (gT ¯ +M, h +¯F , ∇ +¯F )(T) − +2 +� +i=1 +log Ti(gT ¯ +Mi, h +¯Fi, ∇ +¯Fi)(T) − log T (T) += log T (gTM, hF , ∇F ) − +2 +� +i=1 +log Ti(gTMi, hFi, ∇Fi) − log T . +Proof of Theorem 1.1. It follows from Theorem 3.2, Theorem 3.3 and Proposition +3.4 that +log T (gT ¯ +M, h +¯F , ∇ +¯F )(T) − +2 +� +i=1 +log Ti(gT ¯ +Mi, h +¯Fi, ∇ +¯Fi)(T) − log T (T) += log(2)χ(Y )rank(F)/2 + o(1). +Hence, by Theorem 3.5, Theorem 1.1 follows. +12 + +4 +Convergence of Eigenvalues +For simplicity, in this section, let d := d ¯F and d∗ := d ¯F ,∗. Let {ei} be a local +frame of TM, and{ei} its dual frame. Set LfT := Hess(ei, ej)fT c(ei)ˆc(ej), where +c(ei) := ei ∧ −iei, ˆc(ei) := ei ∧ +iei. Recall that λk(T) is the k-th eigenvalue of +∆T , λk is the k-th eigenvalue of ∆1 ⊕ ∆2. Let ∇ be the connection on Ω( ¯ +M, ¯F) +induced by ∇ ¯F and gT ¯ +M, and ∇Y = ∇|Y . In this section, we are going to show that +limT→∞ λk(T) = limT→∞ ˜λk(T) = λk, i.e. prove Theorem 3.1. +First, one observes that λk(T) has uniform upper bounds: +Lemma 4.1. Fix k ∈ Z+. There exists an increasing sequence {Λk}∞ +k=1, such that +λk(T) ≤ Λk. +Proof. Choose k disjoint balls Bk in M◦ +1 := M1 − {−1} × Y, k nonzero smooth +functions ηk with support supp(ηk) ⊂⊂ Bk. Let Vk be the linear space generated +by {ηk}. Then by the min-max principle, it’s easy to check that +λk(T) ≤ Λk := sup +ψ∈Vk +� +¯ +M |dψ|2 + |d∗ψ|2dvol ¯ +M +� +¯ +M |ψ|2dvol ¯ +M += sup +ψ∈Vk +� +¯ +M |dT ψ|2 + |d∗ +T ψ|2dvol ¯ +M +� +¯ +M |ψ|2dvol ¯ +M +. +Next, by the trace theorem: +Lemma 4.2. Let u ∈ Ω(M; F), such that +� +¯ +M |u|2dvol ¯ +M = 1 and +� +¯ +M |dT u|2 + +|d∗ +T u|dvol ¯ +M ≤ λ. Then for s ∈ +� +−2, −1 + +� +2 +T +� +∪ +� +1 − +� +2 +T , 2 +� +� +Y +|u|2(s, y)dvolY ≤ C(λ + 1) +if T is large enough. +Proof. First, by trace formula, +� +Y +|u|2(−1, y)dvolY ≤ C +� +M1 +|u|2 + |∇u|2dvolM1 ≤ C +� +M1 +|u|2 + |(d + d∗)u|2dvolM1 += C +� +M1 +|u|2 + |(dT + d∗ +T )u|2dvolM1 +≤ C +� +¯ +M +|u|2 + |(dT + d∗ +T )u|2dvol ¯ +M ≤ C(λ + 1). +Similarly, one still have for s ∈ [−2, −1] ∪ [1, 2], +� +Y +|u|2(y, s)dvolY ≤ C(λ + 1). +13 + +Next for γ ∈ (0, 1) to be determined, suppose s0 ∈ +� +−1, −1 + +� +γ +T +� +achieves the +supreme of +AT := +sup +s∈[−1,−1+√ γ +T ]∪[1−√ γ +T ,1] +� +Y +|u|2(y)dvolY , +then +� +Y +|u(s0, y) − u(−1, y)|2dvolY ≤ +� +Y +����� +� −1+√ γ +T +−1 +| ∂ +∂s′ u(y, s′)|ds′ +����� +2 +dvolY +≤ +� γ +T +� +Y +� −1+√ γ +T +−1 +|du(y, s′)|2 + |d∗u(y, s′)|2ds′dvolY . +(8) +Integration by parts, +λ ≥ +� +¯ +M +|dT u|2 + |d∗ +T u|2dvol ¯ +M +≥ +� −1+√ γ +T +−1 +� +Y +|dT u|2 + |d∗ +T u|2dvolY ds +≥ +� −1+√ γ +T +−1 +� +Y +|du|2 + |d∗u|2 + (LfT u, u) + |∇fT|2|u|2dvolY ds +− +� +Y +|dfT ||u|2 +� +−1 + +� γ +T , y +� +dvolY +≥ +� −1+√ γ +T +−1 +� +Y +|du|2 + |d∗u|2 − C4T|u|2dvolY ds − +√ +TAT +≥ +� −1+√ γ +T +−1 +� +Y +|du|2 + |d∗u|2dvolY ds − (1 + C4 +√γ) +√ +TAT . +(9) +See §2.3 for the definition of C4. By (8) and (9), one can see that +AT ≤ (λ + 1) +�� γ +T + C +� ++ √γ(1 + C4 +√γ)AT . +Fix γ ∈ (0, 1), such that √γ(1+C4√γ) ≤ 1/2. Thus, whenever s ∈ +� +−1, −1 + +� +γ +T +� +, +� +Y +|u(s, y)|2dvolY ≤ 3C(λ + 1). +Similarly, one can show that for s ∈ +� +−1 + +� +γ +T , −1 + 2 +� +γ +T +� +� +Y +|u(s, y)|2dvolY ≤ 32C(λ + 1). +14 + +Let m = [ 2 +γ ] + 1, then repeating the arguments above for m times, one can see that +� +Y +|u(s, y)|2dvolY ≤ 3mC(λ + 1) +whenever s ∈ +� +−1, −1 + +� +2 +T +� +. +Lemma 4.3. Assume u meets the same conditions as in Lemma 4.2. Then +� 1/2 +−1/2 +� +Y +|u(s, y)|2dvolY ds ≤ C(λ + 1) +T 3/2 +if T is large enough. +Proof. Just notice that |f ′ +T |2(s)−f ′′ +T(s) ≥ 0 when s /∈ +� +−1, −1 + +� +2 +T +� +∪ +� +1 − +� +2 +T , 1 +� +, +� 1/2 +−1/2 +� +Y +T 2|u(s, y)|2dvolY ds +≤ C +� 1/2 +−1/2 +� +Y +|du|2 + |d∗u|2 + (LfT u, u) + |∇fT|2|u2|dvolY ds +≤ C +� +¯ +M +|du|2 + |d∗u|2 + (LfT u, u) + |∇fT |2|u2|dvol ¯ +M ++ C′T +� −1+ +� +2 +T +−1 +� +Y +|u|2dvolY ds + C′T +� 1 +1− +� +2 +T +� +Y +|u|2dvolY ds +≤ Cλ + C′√ +T. +Lemma 4.4. Assume u meets the same conditions as in Lemma 4.2. Moreover, if +u|[−2,2] = v1(y, s) + v2(y, s)ds, define Pa(u)(y) = v2(y, −1), Pb(u)(y) = v1(y, 1). +� +Y +|Pa(u)|2dvolY ≤ C(λ + 1) +√ +T +, +(10) +and +� +Y +|Pb(u)|2dvolY ≤ C(λ + 1) +√ +T +. +(11) +Moreover, if u is an eigenform with respect to an eigenvalue µ ≤ λ, then we also +have +� +Y +|Pa(du)|2dvolY ≤ C(µ + 1)2 +√ +T +≤ C(λ + 1)2 +√ +T +, +(12) +and +� +Y +|Pb(d∗u)|2dvolY ≤ C(µ + 1)2 +√ +T +≤ C(λ + 1)2 +√ +T +. +(13) +15 + +Proof. Let E(T) := +� +Y |v2(−1, y)|2dvolY . Notice that on Ω(Y ; F|Y )du, ✷T = − ∂2 +∂s2 + +∆Y + | ∂ +∂sfT |2 + ∂2 +∂s2fT . By repeating previous steps, +inf +s′∈[0, +� +2 +T ] +� +Y +|v2(y, s′)|2dvolY ≥ cE(T) − Cλ +√ +T +. +(14) +Let η ∈ C∞ +c [−2, 2) be a bump function, such that η|[−2,−1/2] ≡ 1, η[−3/4,2) ≡ 0. +Since f ′′ +T = T ≥ 0 in [−1 + e−T 2, −3/4], |f ′ +T (−1 + e−T 2)| ≤ 1, by (14) and +integration by parts, when T is big enough, +c +√ +TE(T) − Cλ ≤ +� −1+ +� +2 +T +−1+e−T 2 +� +Y +T|v2(y, s)|2dvolY ds +≤ +� −1+ +� +2 +T +−1+e−T 2 +� +Y +|dv2(y, s)|2 + |d∗v2(y, s)|2 + f ′′ +T |v2|2 + |∇fT |2u2dvolY ds +≤ +� 2 +−1+e−T 2 +� +Y +|dηv2(y, s)|2 + |d∗ηv2(y, s)|2 + f ′′ +T|ηv2|2 + |∇fT |2η2u2dvolY ds +≤ +� 2 +−1+e−T 2 +� +Y +|dT ηv2(y, s)|2 + |d∗ +T ηv2(y, s)|2dvolY ds ++ +� +Y +|dfT||v2|2(s, y)dvolY |s=−1+e−T 2 +≤ +� +¯ +M +|d∗ +T u|2 + |dT u|2 + |η′||∂sv2||v2|dvol ¯ +M + C(λ + 1). +(15) +Let DT = dT + d∗ +T , +� +¯ +M +|η′||∂sv2||v2|dvol ¯ +M ≤ +� +¯ +M +|η′||DT u||u| + |η′||dfT u||u|dvol ¯ +M +≤ +� +¯ +M +|DT u|2 + |u|2 + CT|η′||u|2dvol ¯ +M ≤ C(1 + λ) + C +� +¯ +M +T|η′||u|2 +≤ C(λ + 1) (By Lemma 4.3). +(16) +According to (15) and (16), E(T) ≤ C(λ+1) +√ +T +. Similarly, one has (11). +Replace u with dT u and notice that dT u = du on {−1} × Y , (12) follows. +Similarly, one has (13). +Lemma 4.5. Suppose u meets the same condition as Lemma 4.2. Then +� 1 +−1 +� +Y +|u|2(y, s)dvolY ds ≤ C(λ + 1) +√ +T +. +16 + +Proof. By Lemma 4.2, one can show that +� +||s|−1|≤ +� +2 +T +� +Y +|u|2(y, s)dvolY ds ≤ C(λ + 1) +√ +T +. +Notice that on +� +−1 + +� +2 +T , 1 − +� +2 +T +� +, f ′′ +T + |f ′ +T|2 ≥ T, |f ′ +T | +� +± +� +1 − +� +2 +T +�� += +√ +2 +√ +T, +hence, integration by parts, +� 1− +� +2 +T +−1+ +� +2 +T +� +Y +Tu2(y, s)dvolY ds +≤ +� 1− +� +2 +T +−1+ +� +2 +T +� +Y +|∇u|2 + |∇fTl|2u2+ < LfT u, u > dvolY ds +≤ +� 1− +� +2 +T +−1+ +� +2 +T +� +Y +|(dT + d∗ +T )u|2dvolY ds + C +� +Y +|(dfT + df ∗ +T )u||u|dvolY |s=±(−1+ +� +2 +T ) +≤ +� +M +|(dT + d∗ +T )u|2dvolM + C +√ +T +� +Y +|u|2dvolY |s=±(−1+ +� +2 +T ) +≤ C +√ +T(λ + 1) (By Lemma 4.2). +Thus, +� 1− +� +2 +T +−1+ +� +2 +T +� +Y +|u(s, y)|2dvolY du ≤ C(λ + 1) +√ +T +. +Proof of Theorem 3.1. +• lim supT→∞ λk(T) ≤ λk. +Let ui = (ui,1, ui,2) be the i-th eigenvalue of ∆1 ⊕∆2 on Ωrel(M1; F1)⊕Ωabs(M2; F2) +(1 ≤ i ≤ k). +Let η ∈ C∞ +c ([0, 1]), such that η[0,1/4] ≡ 0, η|[1/2,1] ≡ 1. +For any u = (u1, u2) ∈ Ωrel(M1; F1) ⊕ Ωabs(M2; F2) , let QT : Ωabs(M1; F1) ⊕ +Ωrel(M2, F2) → Ω( ¯ +M; ¯F), s.t., +QT (u)(x) = + + + + + +ui(x), if x ∈ Mi; +η(−s)u(−1, y)e−fT (s)−T , if x = (s, y) ∈ [−1, 0] × Y ; +η(s)u(1, y)efT (s)−T , if x = (s, y) ∈ [0, 1] × Y . +Let ¯u = QT (u), then one can see that dim span{¯ui}k +i=1 = k. Moreover, by trace +formula, one can show that for any u ∈ span{ui}, there exists C1 > 0 +� +Y +|u(0, y)|2 + |∇Y u(0, y)|2dvolY ≤ C1(1 + λ2 +k) +� +M1∪M2 +|u|2dvol ¯ +M. +(17) +17 + +One computes +� +¯ +M +|¯u|2dvol ¯ +M ≥ +� +M1∪M2 +|u|2dvol ¯ +M. +Then by (17) and the construction of ¯u, one has +� +¯ +M +|∇¯u|2 + |∇fT|2|¯u|2 + (LfT ¯u, ¯u)dvolM += +� +M1∪M2 +|∇u|2dvol + +� 0 +−1 +� +Y +|∇¯u|2 + |∇fT|2|¯u|2 + (LfT ¯u, ¯u)dvolY ds ++ +� 1 +0 +� +Y +|∇¯u|2 + |∇fT |2|¯u|2 + (LfT ¯u, ¯u)dvolY ds += I + II + III. +First, notice that I ≤ λk +� +M1∪M2 |u|2dvol. +By a straightforward computation and Lemma 4.4, +II ≤ +� 0 +−1 +� +Y +|∇Y u1|2e−2fT (s)−T dvolY ds + C +� −1/2 +−1/4 +� +Y +T 2|e−T/8u1|2dvolY ds +≤ C2(1 + λ2 +k) +√ +T +� +M2∪M2 +|¯u|2dvol ¯ +M +≤ C2(1 + λ2 +k) +√ +T +� +¯ +M +|¯u|2dvol ¯ +M. +Similarly, III ≤ C2(1+λ2 +k) +√ +T +� +¯ +M |¯u|2dvol ¯ +M. Hence, λk(T) ≤ λk + C2(1+λ2 +k) +√ +T +. +Consequently, as T → ∞, lim supT→∞ λk(T) ≤ λk. +• lim infT→∞ λk(T) ≥ λk. +Let {Tl} be a sequence, such that limk→∞ λk(Tl) = lim infT→∞ λk,T. Let ui,l be +an eigenform of ∆Tl with norm 1, w. r. t. λi(Tl)(1 ≤ i ≤ k). By Lemma 4.1, +λi(Tl) ≤ Λi for some Λi > 0. Since +∥ui,l |M1∪M2 ∥W N,2(M1)⊕W N,2(M2) ≤ C3(1 + Λk)N∥ui,l∥L2( ¯ +M) = C3(1 + Λk)N +for any N > 0, by Sobolev’s embedding theorem, we may as well assume that +{ui,l|M1∪M2} converges in W 2,2(M1) ⊕ W 2,2(M2)-topology, and assume ui,∞ := +liml→∞ ui,l|Mj(j = 1, 2). +Moreover, one can also have dim span{ui,∞}k +i=1 = k. Hence, for any u∞ ∈ +span{ui,∞}k +i=1 with +� +M1∪M2 |u∞|2 = 1, one can find ul ∈ span{ui,l}k +i=1, such that +ul → u∞ in W 2,2(M1) ⊕ W 2,2(M2) topology. +By Lemma 4.4, we can see that u∞ ∈ Ωrel(M1; F1) ⊕ Ω∗ +abs(M2; F2). +By Lemma 4.5, one has +lim +l→∞ +� +¯ +M +|ul|2dvol ¯ +M = lim +l→∞ +� +M1∪M2 +|ul|2dvol ¯ +M = +� +M1∪M2 +|u∞|2dvol ¯ +M = 1. +(18) +18 + +As a result, we +lim +l→∞ λk(Tl) ≥ lim +l→∞ +� +¯ +M +(dT ul, dT ul) + (d∗ +T ul, d∗ +T ul)dvol +≥ lim +l→∞ +� +M1∪M2 +(dul, dul) + (d∗ul, d∗ul)dvol ¯ +M += +� +M1∪M2 +(du∞, du∞) + (d∗u∞, d∗u∞)dvol ¯ +M. +Hence lim infT→∞ λk,T ≥ λk. +• Similarly, one can show that limT→∞ ˜λk,T = λk. +5 +Large Time Contributions +We will prove Theorem 3.2 in this section. +Let M3 = [−1, 0]×Y , M4 = [0, 1]×Y , then ¯ +M = M1∪M2∪M3∪M4. Let ∆i,T,bd +be the restriction of ∆T on Mi with some boundary conditions(i=1,2,3,4). For bd = +rel, abs, D and N, we mean relative, absolute, Dirichlet, and Neumann boundary +conditions. Let λk,bd(T) be the k-the eigenvalues of ∆1,T,bd ⊕ ∆2,T,bd ⊕ ∆3,T,bd ⊕ +∆4,T,bd (acting on Ω∗ +bd(M1; F1)⊕Ω∗ +bd(M2; F2)⊕Ω∗ +bd(M3; ¯F|M3)⊕Ω∗ +bd(M4; ¯F|M4)⊕), +it follows from domain monotonicity of eigenvalues that +λk,D(T) ≥ λk(T) ≥ λk,N(T). +(19) +Before moving on, let’s study the following one-dimensional model problem first. +Let ∆R +T,N,± := −∂2 +s + |f ′ +T |2 ± f ′′ +T on [−1, 0] with Neumann boundary conditions, and +λR +k,T,N,± be the k-th eigenvalues of ∆R +T,N,±. +Lemma 5.1. For k ≥ 2, one has λR +k,T,N,± ≥ vk. Here {vk(T)}∞ +k=1 is the collection +of {T max{c1l−c2, 0}}∞ +l=1 ∪{c3l2}∞ +l=1, listed in the increasing order and counted with +multiplicity, and constants c1, c2 and c3 are independent of T. +Proof. Let I1 := [−1, −1+ +1 +√ +T ], I2 := [−1+ +1 +√ +T , 0]. It follows from the constructions +of fT that when T is large enough ∆R +T,N,± ≥ −∂2 +s on I2. That is, for all φ ∈ C∞(I2) +with φ′(−1 + +1 +√ +T ) = φ′(0) = 0 +� +I2 +∆R +T,N,±φ · φds ≥ +� +I2 +−∂2 +sφ · φds. +(20) +For I1, changing the variable +√ +T(s + 1) → ˜s, and suppose ∆R +T,N,± → ˜∆R +T,N,±. +Then direct computations yields, on [0, 1], +19 + +˜∆R +T,N,± ≥ T(−∂2 +˜s + ˜s2 − C). +(21) +The lemma then follows from (20), (21) and domain monotonicity of eigenvalues. +It follows from (19) and Weyl’s law that +Lemma 5.2. λk(T) ≥ uk(T). +Here {uk(T)}∞ +k=1 is the collection of 4 copies of +{vl(T) + c4m2/(dim M−1)}∞ +l=1,m=1 and 2 copy of {c5l2/ dim M}, listed in the increasing +order and counted with multiplicities. Moreover, constants c4 and c5 are independent +of T. +Proof of Theorem 3.2. By Lemma 5.2, λk(T) ≥ uk(T) ≥ uk(1). Let +F(t) := dim(M) +∞ +� +k=1 +e−t max{uk(1),δ}. +Then F(t)/t ∈ L1((1, ∞)). Moreover, +t−1| Trs +� +N exp +� +−t∆′ +T +� +(1 − Pδ) +� +| ≤ t−1F(t). +(22) +Hence by dominated convergence theorem, +lim +T→∞ +� ∞ +1 +t−1 Trs +� +Ne−t∆′ +T (1 − Pδ) +� +dt = +2 +� +i=1 +� ∞ +1 +t−1 Trs(Ne−t∆′ +i)dt. +Similarly, one can show that +2 +� +i=1 +lim +T→∞ +� ∞ +1 +t−1 Trs(Ne−t∆′ +i,T )dt = +2 +� +i=1 +� ∞ +1 +t−1 Trs(Ne−t∆′ +i)dt. +6 +A Gluing Formula for Heat Trace in Small +Time +6.1 +Several heat kernels and Laplacians +To show the gluing formula for heat trace, we introduce several heat kernels and +Laplacians. +Let KT be the heat kernel for ∆T. Let K1⊕2 be the heat kernel for ∆1 ⊕ ∆2, Ki +be the heat kernel for ∆i(i = 1, 2). +Let ∆B,1 be the Hodge Laplacian on [−2, −1] with absolute boundary condi- +tions, and kB,1 be the heat kernel for ∆B,1. +It’s easy to see that ker(∆B,1) is +20 + +one-dimensional and generated by constant functions. Since nonzero eigenvalues of +∆B come in pairs, +Trs(e−t∆B,1) ≡ 1. +(23) +Let ∆B,2 be the usual Hodge Laplacian on [1, 2] with relative boundary condi- +tions, and kB,2 be the heat kernel of ∆B,2. It can be checked that ker(∆B,2) is one +dimensional and generated by the constant one forms. Since nonzero eigenvalues of +∆B,2 come in pairs, +Trs(e−t∆B,2) ≡ −1. +(24) +Let ¯∆B be the usual Hodge Laplacian on [−2, 2] with absolute boundary condi- +tion on −2, relative boundary condition on 2, and ¯kB be its heat kernel. +We can also regards fT as a smooth function in (−2, 2), and let ∆R +T be the +Witten Laplacian on (−2, 2) with respect to fT , with absolute boundary condition +on −2, and relative boundary condition on 2, and kT be the heat kernel for ∆R +T . +On the restriction of F|Y → Y , let ∆Y be the induced Hodge Laplacian on +Ω(Y ; F|Y ). Let KY be the heat kernel of ∆Y . +6.2 +Gluing heat kernels +Then the following lemmas will be needed in the proof of gluing formula for heat +kernels. +It follows from a standard argument that +Lemma 6.1 (Finite Propagation Speed). Let s0 be a smooth section of Ω(M; F) +with compact support C, st := exp +�√−1tDT +� +s0, where DT = dT + (dT )∗. +Let +Ct := {x′ ∈ M : d(x, x′) ≤ 2t}, then the support of st is inside Ct. +For a, b ∈ (−2, 2)(a < b), let Ma,b = [a, b] × Y. +Lemma 6.2. Suppose x /∈ M−9/8,−7/8 ∪ M7/8,9/8 and d(x, x′) ≥ 4δ for some δ > +1/16. Then there exists ck(X), Ck(X) > 0, such that +|∇kKT (t, x, x′)| ≤ Cke−ck/t +if T ≥ 16. +Proof. Choose a bump function φ on R such that +φ(λ) = 1, |λ| ⩽ δ; +φ(λ) = 0, |λ| ⩾ 2δ. +Let f1, f2 ∈ S(R) such that +ˆf1(λ) = (4πt)− 1 +2 exp +� +−λ2/4t +� +φ(λ), +ˆf2(λ) = (4πt)− 1 +2 exp +� +−λ2/4t +� +(1 − φ(λ)), +where S(R) is the Schwartz space on R. +Since D2 +T = ∆T , one can see that +f1(DT ) + f2(DT ) = exp (−t∆T) . +21 + +Denote K1 and K2 to be the integral kernel of f1(DT ) and f2(DT ) respectively, then +KT (t, x, x′) = K1(t, x, x′) + K2(t, x, x′). +By Lemma 6.1, a standard argument shows that K1(t, x, x′) = 0 if d(x, x′) ≥ 4δ. +Next, let +˜st(x) := +� +¯ +M +K2(t, x, x′)s(x′)dx′ = +1 +√ +4πt +� +|λ|⩾δ +exp +� +−λ2/4t +� +[1 − φ(λ)] exp(iλDT )s(x)dλ. +Next, we may assume that ∆Ts = µs for some µ ∈ R then +|2k∆k +T ˜st(x)| = 2k +���� +� +¯ +M +∆k +T,xK2(t, x, x′)s(x′)dx′ +���� += +����� +1 +√ +4πt +� +|λ|⩾δ +exp +� +−λ2/4t +� +[1 − φ(λ)] exp(iλDT )D2k +T s(x)dλ +����� |s(x)| +≤ C exp +� +−δ2/8t − 2tµ2� +µ2k|s(x)| +≤ Ck exp +� +−ckδ2/t +� +|s(x)|. +(25) +Here ∆T,x means that the derivative is taken with respect to x, C, Ck and ck are +independent of µ and T. +As a consequence, +� +¯ +M +|∆k +T,xK(t, x, x′)|2dx ≤ Ck exp +� +−ckδ2/t +� +. +(26) +Let ρ ∈ C∞ +c (M) be a bump function, such that in +¯ +M − M− 33 +32 ,− 31 +32 ∪ M 33 +32 , 31 +32 , +ρ ≡ 1; in M− 17 +16 ,− 15 +16 ∪ M 15 +16 , 17 +16 , ρ ≡ 0. Then one still have +� +¯ +M +|∆k +T,xρ(x)K(t, x, x′)|2dx ≤ Ck exp +� +−ckδ2/t +� +(27) +for some other constant Ck and ck. +Notice that when restricted in ¯ +M −M− 9 +8 ,− 7 +8 −M 7 +8 , 9 +8 , ∆T ≥ ∆ if T is big enough. +Here ∆T ≥ ∆ on some open subset U ⊂ ¯ +M means that for all φ ∈ Ω(U; F) with +compact support inside U, +⟨∆T φ, φ⟩L2 ≥ ⟨∆φ, φ⟩L2. +It follows from G˚arding’s inequality and Sobolev’s embedding theorem that +|∇kρ(x)KT (t, x, x′)| ≤ Cke−ckδ2/t, +where Ck, ck are independent of δ. +22 + +Let ηi(i = 1, 2) be a smooth function on (−∞, ∞) satisfying +1. 0 ≤ ηi ≤ 1. +2. η1 ≡ 1 in (−∞, −3/2),η1 ≡ 0 in (−5/4, ∞); +3. η2 ≡ 1 in (3/2, ∞),η2 ≡ 0 in (−∞, 5/4); +Let η be an even function, such that η|[0,∞) ≡ η2. We can think ηi as a functions on +Mi(i = 1, 2) and η as a function on ¯ +M. +Lemma 6.3. There exists T-independent C, c > 0, such that for t ∈ (0, 1), +| Trs(N(1 − η)e−t∆T ) − Trs(N(1 − η)e−t∆R +T ⊗ e−t∆Y )| ≤ Ce−c/t, +| Trs(Nηie−t∆R +T ⊗ e−t∆Y ) − Trs(Nηie−t ¯∆B ⊗ e−t∆Y )| ≤ Ce−c/t. +Proof. Let f1 and f2 be functions constructed in Lemma 6.2 for some T-independent +and small δ > 0 . +Let uk be a normal eigenform of λk(T) w.r.t. ∆T, then for any j ∈ Z+, |f2(s)| ≤ +Cje−c/t +(|s|+1)j , hence +|(f2(∆T )uk, uk)L2| ≤ +Cje−c/t +(|λk(T)| + 1)j . +(28) +By Lemma 5.2 and (28), +| Trs(N(1 − η)e−t∆T )| ≤ Ce−c/t. +(29) +Similarly, +| Trs(N(1 − η)e−t∆R +T ⊗ e−t∆Y )| ≤ Ce−c/t. +(30) +Let M′ +1 = M1 − (−2, −1] × Y and M′ +2 = M2 − [1, 2) × Y . Since L2( ¯ +M) = L2(M′ +1) ⊕ +L2([−2, 2] × Y ) ⊕ L2(M′ +2), let {uk}, {wk} and {vk} be an orthonormal basis of +L2(M′ +1), L2([−2, 2] × Y ) and L2(M′ +2) respectively. Then by Lemma 6.1 and the +maximal principle, if δ is small enough, +(f1(∆T )(1 − η)uk, uk) = (f1(∆T )(1 − η)vk, vk) = 0, +(31) +and +(f1(∆T )(1 − η)wk, wk) = (f1(∆R +T ⊕ ∆Y )(1 − η)wk, wk). +(32) +Since e−ts2 = f1(s) + f2(s), by (29), (30), (31) and (32) +| Trs(N(1 − η)e−t∆T ) − Trs(N(1 − η)e−t∆R +T ⊗ e−t∆Y )| ≤ Ce−c/t. +Similarly, +| Trs(Nηie−t∆R +T ⊗ e−t∆Y ) − Trs(Nηie−t ¯∆B ⊗ e−t∆Y )| ≤ Ce−c/t. +23 + +Similarly, +Lemma 6.4. There exists T-independent C, c > 0, such that for t ∈ (0, 1), +| Trs(N(1 − ηi)e−t∆i) − Trs(N(1 − ηi)e−t∆B,i ⊗ e−t∆Y )| ≤ Ce−c/t, +| Trs(Nηie−t∆B,i ⊗ e−t∆Y ) − Trs(Nηie−t ¯∆B ⊗ e−t∆Y )| ≤ Ce−c/t. +6.3 +Heat trace expansion for e−t(∆1⊕∆2) +It follows from Lemma 6.4 that for some C, c > 0, t ∈ (0, 1], +�����Trs(Ne−t(∆1⊕∆2)) − +2 +� +i=1 +Trs(Nηie−t(∆i)) ++ +2 +� +i=1 +� +Trs(Nηi(x)e−t ¯∆B⊗e−t∆Y ) − Trs(Ne−t∆B,i ⊗ e−t∆Y ) +������ +≤ C exp(−c/t). +(33) +Next, notice that on M−2,−1, the number operator can be decomposed as N = +N Y + N R canonically (Here N Y and N R are the number operator on Y and R +components respectively), +Trs(Ne−t∆B,i ⊗ e−t∆Y ) += Trs(N Re−t∆B,i ⊗ e−t∆Y ) + Trs(e−t∆B,i ⊗ N Y e−t∆Y ) += Trs(N Y e−t∆Y ) Trs(e−t∆B,i) + Trs(N Re−t∆R +B,1)χ(Y )rank(F). +(34) +As a result, by (23) and (24) +2 +� +i=1 +Trs(Ne−t∆B,i ⊗ e−t∆Y ) += Trs(N Re−t∆R +B,1)χ(Y )rank(F) + Trs(N Re−t∆R +B,2)χ(Y )rank(F) += Trs(N Re−t∆R +B,1)χ(Y )rank(F) + Trs +� +(N R − 1)e−t∆R +B,2 +� +χ(Y )rank(F) − χ(Y )rank(F). +(35) +6.4 +Heat trace expansion for e−t∆T +It follows from Lemma 6.3 that +�����Trs(Ne−t(∆T )) − +2 +� +i=1 +Trs(Nηie−t(∆i)) ++ +2 +� +i=1 +� +Trs(Nηi(x)e−t ¯∆B⊗e−t∆Y ) − Trs(Ne−t∆R +T ⊗ e−t∆Y ) +������ +≤ C exp(−c/t). +(36) +24 + +Hence +Trs(Ne−t∆R +T ⊗ e−t∆Y ) += Trs(N Y e−t∆Y ) Trs(e−t∆R +T ) + Trs(N Re−t∆R +T ) Trs(e−t∆Y ) += Trs(N Y e−t∆Y ) Trs(e−t∆R +T ) + Trs(N Re−t∆R +T )χ(Y )rank(F). +(37) +Since nonzero eigenforms comes in pairs, Trs(e−t∆R +T ) = dim(ker(∆R +T )0)−dim(ker(∆R +T )1). +Here ker(∆R +T )i denotes the space of harmonic i-forms(i = 0, 1). Since fT is odd, one +can see easily that if u(s) ∈ ker(∆R +T )0, then u(−s)ds ∈ ker(∆R +T )1. +As a result, Trs(e−t∆R +T ) = 0, which implies that +Trs(Ne−t∆R +T ⊗ e−t∆Y ) = Trs(N Re−t∆R +T )χ(Y )rank(F). +(38) +6.5 +Proof of Theorem 3.3 when M is even dimensional +First, by Theorem 3.1 +Trs(Ne−t∆T ) − Trs(Ne−t(∆1⊕∆2)) += Trs(Ne−t∆′ +T (1 − P δ)) − Trs(Ne−t(∆1⊕∆2)′) + o(t). +(39) +When M is even dimensional, χ(Y ) = 0. It follows from (33), (35), (36), (38), +(39) and dominated convergence theorem that +lim +T→∞ +� 1 +0 +t−1| Trs(Ne−t∆T ) − Trs(Ne−t(∆1⊕∆2))|dt = 0. +Hence, +(ζS +T,la)′(0) = +2 +� +i=1 +(ζS +i )′(0) + o(1). +Similarly, +(ζS +i,T)′(0) = (ζS +i )′(0) + o(1). +7 +One-dimensional Model and Coupling Tech- +niques +To show Theorem 3.3 when M is odd dimensional, it suffices to compare Trs(N Re−t∆R +T ) +and Trs(N Re−t(∆R +B,1⊕∆R +B,2)).To this end, we explore the one-dimensional model first. +Let VT,± = |f ′ +T |2 ± f ′′ +T . +Recall that kT (t, s, s′) is the heat kernel for Witten +Laplacian ∆R +T on (−2, 2) with the absolute boundary condition on −2 and the +relative boundary condition on 2. Then restricted on functions, ∆R +T = ∆R +T,− := +−∂2 +s+VT,− with the Neumann boundary condition on −2 and the Dirichlet boundary +condition on 2. Restricted on 1-forms ∆R +T = ∆R +T,+ := −∂2 +s + VT,+ with Dirichlet +boundary condition on −2 and Neumann boundary condition on 2. +25 + +7.1 +Heat trace estimate when t and T are coupled +Let λk,T,± be the k-th eigenvalue with respect to ∆R +T,±. Assume φk,T,± is a k-th +eigenfunction, such that ∥φk,T,±∥L2 = 1 and {φk,T,±} forms a complete orthonormal +basis of L2([−2, 2]). +Let ˜T = t−5T, kT,± be the heat kernel for ∆R +T,±. In this section, we assume that +t ∈ (0, 1]. +Proposition 7.1. If t ∈ (0, 1], +� 1 +−1 +|k ˜T,±(t, s, s)|ds ≤ Ct +√ +T +for some constant C that doesn’t depends on T. +Proof. By Lemma 4.5, +� 1 +−1 +|φk, ˜T,±|2(s)ds ≤ +C(1 + λk, ˜T,±) +� +˜T +. +(40) +It follows from (40), Lemma 5.2 and the domain monotonicity of eigenvalues that +for fixed a ∈ (0, 1) +� 1 +−1 +k ˜T,±(t, s, s)ds = +� 1 +−1 +� +k +e−tλk, ˜ +T ,+|φk, ˜T,+|2ds +≤ C +� +k≥1 +e−tλk, ˜ +T ,+ 1 + λk, ˜T,+ +� +˜T +≤ C +� +k≥1 +e−atλk, ˜ +T ,+ +1 +t +� +˜T +≤ Ct +√ +T +, +where C is independent of T. +Lemma 7.2. For s ∈ (−2, −1), t ∈ (0, 1), and T is large, +|kT,+|(t, s, −1) ≤ min{ +C +tT 1/4 + +C +t3/2T 5/4 , C +√ +t}; +for s ∈ (1, 2), +|kT,−|(t, s, 1) ≤ min{ +C +tT 1/4 + +C +t3/2T 5/4 , C +√ +t}. +Here the constant C doesn’t depend on T. +Proof. +• |kT,+|(t, s, −1) ≤ C +√ +t : +It follows from the construction of fT that there exists C > 0, s.t. +−∂2 +s − CT ≤ ∆R +T ≤ −∂2 +s + CT 2. +26 + +Let ¯kB,1 be the restriction of ¯kB on 1-foms, then it follows from the maximal principle +that +0 ≤ e−CT 2t¯kB,1 ≤ kT,+ ≤ eCtT ¯kB,1. +(41) +Let ρ ∈ C∞ +c [−2, ∞) be a nonnegative function, s.t. ρ ≡ 1 on (−∞, −1/8), ρ ≡ 0 +on (−1/16, ∞). Since +(∂t + ∆R +T,s′)ρ(s′)¯kB,1(t, s, s′) += −2ρ′(s′)∂s′¯kB,1(t, s, s′) − ρ′′(s′)¯kB,1(t, s, s′) + ρ(s′)VT,+(s′)¯kB,1(T, s, s′). +Here ∆R +T,s′ means that the derivative is taken w.r.t. s′. +Set h(t, s, s′) = −2ρ′(s′)∂s′¯kB,1(t, s, s′)−ρ′′(s′)¯kB,1(t, s, s′)+ρ(s′)VT,+(s′)¯kB,1(T, s, s′). +Since VT,+ ≥ 0 on [−1 + eT 2, −1/16], it follows from Duhamel’s principle, Lemma +6.2 and (41) that for s ∈ (−2, −1), +0 ≤ kT,+(t, s, −1) = ¯kB,1(t, s, −1) − +� t +0 +� 2 +−2 +kT,+(t − t′, s, s′)h(t′, s′, −1)ds′dt′ +≤ ¯kB,1(t, s, −1) + +� t +0 +� −1+e−T 2 +−1 +eCtT ¯kB,1(t − t′, s, s′)¯kB,1(t′, s′, −1)ds′dt′ ++ +� t +0 +� −1/16 +−1/8 +Ce−c/(t−t′)(¯kB,1(t′, s′, −1) + ∂s′¯kB,1(t′, s′, −1))ds′dt′ +≤ ¯kB,1(t, s, −1) + C +� t +0 +e−T 2eCtT +1 +√ +t − t′√ +t′ ds′dt′ ++ C +� t +0 +e−c/(t−t′)e−c/t′dt′ ≤ C′ +√ +t. +• |kT,+|(t, s, −1) ≤ +C +tT 1/4 + +C +t3/2T 5/4 : +Let φk,T,+ be a united eigenfunction for λk,T,+, then φk,T,+ satisfies +� +−φ′′ +k,T,+ = λk,T,+, in (−2, −1) +φk,T,+(−1) = 0. +Hence φk,T,+(s) = ck sin(λk,T,+(s + 2)) in (−2, −1) for some ck. +2c2 +kλk,T,+ + sin(2λk,T,+) +4λk,T,+ += +� −1 +−2 +|φk,T,+|2 ≤ +� −2 +−2 +|φk,T,+|2 = 1 +implies that ck ≤ C for some constant C that doesn’t depend on T. As a result, +|φk,T,+|(s) ≤ C if s ∈ (−2, −1). +Moreover, Lemma 4.4 implies that |φk,T,+|2(−1) ≤ +C(λk,T,++1) +√ +T +. +Hence, by +Lemma 5.2, for a fixed a ∈ (0, 1), +27 + +kT,+(t, s, −1) = +� +k +e−tλk,T,+φk,T,+(s)φk,T,+(−1) +≤ C +� +k +e−tλk,T,+ +� +λk,T,+ + 1 +T 1/4 +≤ C +� +k +e−atλk,T,+ +1 +√ +tT 1/4 +≤ +C +tT 1/4 + +C +t3/2T 5/4 . +The second inequality could be proved similarly. +Recall that ∆R +B,1 is the usual Hodge Laplacian on [−2, −1] with absolute bound- +ary conditions, and kB,1 is the heat kernel of ∆R +B,1. Let kB,1,+ be the restriction of +kB,1 on 1-forms. Then kB,1,+(t, s, −1) = 0. While ∆R +B,2 is the usual Hodge Lapla- +cian on [1, 2] with relative boundary conditions, kB,2 be the heat kernel of ∆R +B,2. +Let kB,2,− be the restriction of kB,2 on functions. Then kB,2,−(t, s, 1) = 0. +Proposition 7.3. For t ∈ (0, 1], T is large enough, +� −1 +−2 +|k ˜T ,+ − kB,1,+|(t, s, s)ds ≤ Ct0.1 +T 0.05 + Ce−c/t; +� 2 +1 +|k ˜T,− − kB,2,−|(t, s, s)ds ≤ Ct0.1 +T 0.05 + Ce−c/t. +Here constants C and c are independent of T. +Proof. By Lemma 6.3 and Lemma 6.4, it suffices to estimate +� −1 +−9/8 |k ˜T,+−kB,+|(t, s, s)ds. +First, one can compute directly that |∂s′kB,1,+(t, s, s′)|s′=−1| ≤ C(s+1)e−(s+1)2/t +t3/2 +. +It follows from Duhamel principle and Lemma 6.2 that for s ∈ (−9/8, −1) +|k ˜T ,+ − kB,1,+|(t, s, s) ≤ +� t +0 +���∂s′kB,1,+(t − t′, s, s′)|s′=−1kT,+(t′, s, −1) +���dt′ +≤ C′ +� t +0 +(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +��kT,+(t′, s, −1) +�� dt′ = J1(t, s). +Here constants C and C′ are independent of T. +28 + +Hence, +� −1 +−9/8 +J1(t, s)ds ≤ C +� −1 +−9/8 +� t +0 +(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +|k ˜T ,+(t′, s, −1)|dt′ds +≤ C +� t +0 +� −1 +−9/8 +(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +|k ˜T ,+(t′, s, −1)|dsdt′ +≤ C +� t1.2/T 0.2 +0 +� −1 +−9/8 +(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +|k ˜T ,+(t′, s, −1)|dsdt′ ++ C +� t +t1.2/T 0.2 +� −1 +−9/8 +(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +|k ˜T,+(t′, s, −1)|dsdt′ += I1 + I2. +While by Lemma 7.2, +I1 ≤ C +� t1.2/T 0.2 +0 +� −1 +−9/8 +(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +1 +√ +t′ dsdt′ +≤ C +� t1.2/T 0.2 +0 +1 +√ +t − t′√ +t′ dt′ +≤ C +√ +t +� t1.2/T 0.2 +0 +1 +√ +t′ dt′ = Ct0.1 +T 0.1 . +By Lemma 7.2 again, +I2 ≤ C +� t +t1.2/T 0.2 +� −1 +−9/8 +s(s + 1)e−(s+1)2/t′ +(t − t′)3/2 +� +1 +t′ ˜T 1/4 + +1 +t′3/2 ˜T 5/4 +� +dsdt′ +≤ C +� t +t1.2/T 0.2 +1 +√ +t − t′ +� +1 +t′ ˜T 1/4 + +1 +t′3/2 ˜T 5/4 +� +dt′ +≤ C( t0.05 +T 0.05 + t4.45 +T 0.95 ) +� t +t1.2/T 0.2 +1 +√ +t − t′ dt ≤ Ct0.55 +T 0.05 . +Similarly, +� 2 +1 +|k ˜T,− − kB,1,−|(t, s, s)ds ≤ Ct0.55 +T 0.05 . +Lemma 7.4. +���� +� 2 +1 +trs(k ˜T (t, s, s))ds + 1 +���� ≤ Ce−ct + Ct +√ +T +for some constants C and c that don’t depend on T. +Proof. Let pT be a family of smooth functions on (−∞, 2], such that +(a) pT |[1,2] ≡ T/2, +29 + +(b) pT |(−∞,1](s) = −Tρ(eT 2(1 − s))(s − 1)2/2 + T/2 , where ρ ∈ C∞ +c ([0, ∞)), such +that 0 ≤ ρ ≤ 1, ρ[0,1/2] ≡ 0, ρ[3/4,∞] ≡ 1, |ρ′| ≤ δ1 and |ρ′′| ≤ δ2 for some +universal constant δ1 and δ2. +Let ¯∆T be the Witten Laplacian w.r.t. pT , ¯kT be its heat kernel. Let ¯λk(T) be the +k-th eigenvalue of ¯∆T , and λB,2,k be the k-th eigenvalue of ∆B,2.Then repeating +what we did before, we still have ¯λk(T) → λB,2,k. Thus when T is big enough, +ker( ¯∆T ) is one dimensional and generated by a 1-form. Since nonzero eigenvalues +of ¯∆T come in pairs, +Trs(e−t ¯∆T ) = +� 2 +−∞ +trs(¯kT (t, s, s))ds = −1 +(42) +if T is large. +Moreover, one still have Proposition 7.1 for ¯∆T , i.e., +� 1 +−∞ ¯k ˜T (t, s, s)ds ≤ Ct +√ +T for +some constant C that doesn’t depend on T. As a result, by (42) +� 2 +1 +| trs(¯k ˜T )(t, s, s) + 1|ds ≤ Ct +√ +T +. +Moreover, since pT = fT on [1/2, 2], it follows from Lemma 6.2 and Duhamel’s +principle that for s ∈ [1/2, 2], |¯kT (t, s, s) − kT (t, s, s)| ≤ Ce−c/t for some constants +c and C that don’t depend on T. Thus, the lemma follows. +7.2 +Partial proof of Theorem 3.3 when M is odd dimen- +sional +By Proposition 7.1, Proposition 7.3, Lemma 7.4 and dominated convergence theo- +rem, +lim +T→∞ +� 1 +0 +t−1| Trs(N Re−t∆R +t−5T ) − Trs(N Re−t(∆R +B,1⊕∆R +B,2))|dt = 0. +(43) +Let +˜ζT (z) := +1 +Γ(z) +� 1 +0 +tz−1(Trs(N Re−t∆R +T ) − Trs(N Re−t∆R +t−5T ))dt. +By (33), (35), (36), (38), (39), and (43), one can see that +Theorem 7.1. As T → ∞, +(ζS +T,la)′(0) = +2 +� +i=1 +(ζS +i )′(0) + ˜ζ′ +T(0)χ(Y )rank(F) + o(1). +In particular, if dim(Hk(M; F)) = dim(Hk +abs(M1; F1)) + dim(Hk +rel(M2; F2)) for all +k, then ζT,la = ζT whenever T is large enough. As a result, by Theorem 3.2 and the +equality above, +log T (gT ¯ +M, h +¯F , ∇ +¯F)(T) += log Tabs(gTM1, hF1, ∇F1) + log Trel(gTM2, hF2, ∇F2) + χ(Y )rank(F)˜ζ′ +T (0)/2 + o(1). +30 + +Similarly, let +˜ζT,i(z) := +1 +Γ(z) +� 1 +0 +tz−1(Trs(N Re−t∆R +T,i) − Trs(N Re +−t∆R +t−5T,i))dt. +Theorem 7.2. As T → ∞, +log Ti(gT ¯ +Mi, h +¯Fi, ∇ +¯Fi)(T) = log Ti(gTMi, hFi, ∇Fi) + χ(Y )rank(F)˜ζ′ +T,i(0)/2 + o(1). +To prove Theorem 3.3, it is necessary to show that ˜ζ′ +T (0) = log(2) − T + o(1) +and ˜ζ′ +T,i(0) = −T/2 + o(1), which will be accomplished in the next section. More +precisely, see Proposition 8.2 and Proposition 8.5. +8 +Ray-Singer metrics on S1 and [−2, 2] +8.1 +Ray-Singer metrics on S1 +First, let pT ∈ C∞(R) be a smooth function with period 8 as follows +1. pT (s) = fT(s), ∀s ∈ [−2, 2]; +2. pT (s) = fT(4 − s), ∀s ∈ [2, 6]. +Let S1(8) denote the circle with length 8. Then we regard S1(8) as the interval +[−2, 6] with −2 and 6 identified, so we could think pT as a smooth function on S1(8). +Let dT = d + dpT ∧ be the Witten deformation of de Rham differentials on +S1(8), and ∆S1 +T := dT d∗ +T + d∗ +T dT be its Witten Laplacian. Let H(S1(8), dT ) be the +cohomology for dT , and | · |RS,T be the L2-metric on det +� +H∗(S1(8), dT ) +� +induced by +∆S1 +T -harmonic forms. +Notice that τ : H∗(S1(8), d) → H∗(S1(8), dT ), [w] �→ [e−pT w] is an isomorphism. +Let T (S1)(T) be the analytic torsion for ∆S1 +T , ∥ · ∥RS,T = | · |RS,TT (S1)(T) be +the associated Ray-Singer metric on det +� +H∗(S1(8), dT ) +� +. +Lemma 8.1. log T (S1)(0) = log T (S1)(T) − 2 log(2) + T + o(1) as T → ∞. +Proof. It follows from [1, Theorem 4.7] that ∥ · ∥RS,0 = τ ∗∥ · ∥RS,T . +To show +log T (S1)(0) = log T (S1)(T) − 2 log(2) + T + o(1), it suffices to show log | · |RS,0 = +log τ ∗| · |RS,T + 2 log(2) − T − o(1). +Let +α(T) := +� +S1(8) +e2pT ds, +then [e2pT ds] = [α(T)ds/8] in H∗(S1(8), d). Moreover, α(T)ds is ∆S1 +0 -harmonic. +Let ρT := [1] ⊗ [e2pT ds]−1 ∈ det +� +H∗(S1, d) +� +, then one computes +|ρT |RS,0 = +8 +α(T). +31 + +On the other hand, τ[1] = [e−pT ], τ[e2pT ds] = [epT ds]. Since e−pT and epT ds are +both ∆S1 +T -harmonic, +τ ∗|ρT |RS,T = |τ(ρT )|RS,T = 1. +Here the last inequality follows from the fact that fT is an odd function on [−2, 2], +thus +� +S1(8) e−2pT ds = +� +S1(8) e2pT ds. +Hence, +log | · |RS,0 = log τ ∗| · |RS,T − log(α(T)) + 3 log 2. +A simple calculation yields α(T) = 2eT (1 + o(1)). The lemma follows. +Let ∆[0,2] +1 +be the usual Hodge Laplacian on Ω([0, 2]) with absolute boundary con- +ditions, ∆[0,2] +2 +be the usual Hodge Laplacian on Ω([0, 2]) with relative boundary +conditions. Let T ([0, 2], i) be the analytic torsion with respect to ∆[0,2] +i +(i = 1, 2). +By a straightforward computation (recall that S1has length 8), +log T (S1)(0) = −3 log(2), +log T ([0, 2], i) = − log(2). +(44) +Hence, by Lemma 8.1 and (44), log T (S1)(T) = − log(2) − T + o(1). +While by Theorem 7.1 and (44) +log T (S1)(T) = −2 log(2) + ˜ζ′ +T(0). +Hence, +Proposition 8.2. ˜ζ′ +T (0) = log(2) − T + o(1). +8.2 +Ray-Singer metric on [−2, 2] +Let qT be a smooth even function on [−2, 2], such that for s ∈ [−2, 0], pT (s) = +fT(s − 2). Let dT,2 = d + dqT ∧ be the Witten deformation of de Rham differentials +on [−2, 2], and ∆[−2,2] +T,2 +be its Witten Laplacian with relative boundary conditions. +Let Hrel([−2, 2], dT,2) be the cohomology w.r.t. dT,2, and | · |RS,T,2 be the L2-metric +on det(H∗ +rel([−2, 2], dT,2)) induced by ∆R +T,2-harmonic forms. +Notice that τ2 : H∗ +rel([−2, 2], d) → H∗ +rel([−2, 2], dT ), [w] �→ [e−qT w] is an isomor- +phism. +Let T2(T) be the analytic torsion for ∆R +T,2, ∥ · ∥RS,T,2 = | · |RS,T,2T2(T) be the +associated Ray-Singer metric on det(H∗ +rel([0, 2], dT )). +Similarly, for dT,1 := d − dqT , let ∆R +T,2 be its Witten Laplacian with absolute +boundary conditions. +Let T1(T) be the analytic torsion for ∆R +T,1, ∥ · ∥RS,T,1 = +| · |RS,T,1T1(T) be the associated Ray-Singer metric on det(H∗ +abs([−2, 0], dT )). +Notice that τ1 : H∗ +abs([−2, 2], d) → H∗ +abs([−2, 2], dT ), [w] �→ [eqT w] is an isomor- +phism. +We have the following lemma, the proof is an easy exercise for an expert (Just +notice that fT(2) = fT (−2) = 0), +32 + +Lemma 8.3. +∂T log ∥ · ∥RS,T,i = 0, i = 1, 2. +Lemma 8.4. log Ti(0) = log Ti(T) + T/2 − log(2)/2 + o(1), i = 1, 2 as T → ∞. +Proof. By Lemma 8.3, it suffices to show �2 +i=1 log | · |RS,0,i = �2 +i=1 log τ ∗| · |RS,T,i − +2T + o(1). +• When i = 2: +Let +α(T) := +� 2 +−2 +e2qT ds, +then [e2qT ds] = [α(T)ds/4] in H∗ +rel([−2, 2], d). Moreover, α(T)ds is ∆[−2,2] +2 +-harmonic. +Here ∆[−2,2] +2 +is the restriction of −∂2 +s on [−2, 2] with relative boundary conditions. +Let ρT,2 := [e2qT ds]−1 ∈ det(H∗ +rel([−2, 2], d)), then one computes +|ρT,2|RS,0,2 = +2 +α(T). +On the other hand, τ2[e2qT ds] = [eqT ds]. Since eqT ds is ∆[−2,2] +T,2 +-harmonic, +τ ∗ +2 |ρT,2|RS,T,2 = |τ2(ρT,2)|RS,T,2 = +1 +�� 2 +−2(eqT )2ds += +1 +� +α(T) +. +Hence, +log | · |RS,0,2 = log τ ∗ +2 | · |RS,T,2 − log(α(T))/2 + log(2). +A simple calculation yields α(T) = 2eT (1 + o(1)). Hence, +log | · |RS,0,2 = log τ ∗ +2 | · |RS,T,2 − T/2 + log(2)/2 + o(1). +• When i = 1: +Notice that the constant function 1 is ∆[−2,2] +1 +-harmonic. +Here ∆[−2,2] +1 +is the +restriction of −∂2 +s on [−2, 2] with relative boundary conditions. +Let ρT,1 := [1] ∈ det(H∗ +abs([−2, 2], d)), then one computes +|ρT,1|RS,0,1 = 2. +On the other hand, τ1[1] = [epT ]. Since epT is ∆[−2,2] +T,1 +-harmonic, +τ ∗ +1 |ρT,1|RS,T,1 = |τ(ρT,1)|RS,T,1 = +�� 2 +−2 +(eqT )2ds = +� +α(T). +Hence, +log | · |RS,0,1 = log τ ∗| · |RS,T,1 − log(α(T))/2 + log(2). +33 + +A simple calculation yields α(T) = 2eT (1 + o(1)). Hence, +log | · |RS,0,1 = log τ ∗| · |RS,T,1 − T/2 + log(2)/2 + o(1). +Let ∆[−2,2] +1 +be the usual Hodge Laplacian on Ω([−2, 2]) with absolute boundary con- +ditions, ∆[−2,2] +2 +be the usual Hodge Laplacian on Ω([−2, 2]) with relative boundary +conditions. Let T ([−2, 2], i) be the analytic torsion with respect to ∆[−2,2] +i +(i = 1, 2). +Similarly, let ∆[−1,1] +1 +be the usual Hodge Laplacian on Ω([−1, 1]) with absolute +boundary conditions, ∆[−1,1] +2 +be the usual Hodge Laplacian on Ω([−1, 1]) with rel- +ative boundary conditions. Let T ([−1, 1], i) be the analytic torsion with respect to +∆[−1,1]i(i = 1, 2). +By a straightforward computation +log T ([−2, 2], i) = −3 log(2)/2, +log T ([−1, 1], i) = − log(2). +(45) +Hence, by Lemma 8.4 and (45), log Ti(T) = −3 log(2)/2 − T/2 + log(2)/2 + o(1). +While by Theorem 7.1 and (45) +log Ti(T) = − log(2) + ˜ζ′ +i,T (0) + o(1). +Hence, +Proposition 8.5. ˜ζ′ +T,i(0) = −T/2 + o(1). +9 +Proof of Theorem 3.4 +Let ˜Ωsm( ¯ +M, ¯F)(T) be the space generated by eigenforms of ˜∆T for eigenvalues inside +[0, δ], then by our discussion above in §2.3.1, +˜Ωsm( ¯ +M, ¯F)(T) = efT Ωsm( ¯ +M, ¯F)(T). +And ˜Pδ(T) := efT Pδ(T)e−fT is the orthogonal projection w.r.t. ˜Ωsm( ¯ +M, ¯F)(T). +Let η ∈ C∞ +c ([0, 1]), such that η|[0,1/4] ≡ 0, η|[1/2,1] ≡ 1. +For u = (u1, u2) ∈ +ker (∆1) ⊕ ker (∆2), recall that QT : Ωabs (M1; F1) ⊕ Ωrel (M2; F2) → Ω( ¯ +M, ¯F) is +QT (u)(x) := + + + +ui(x), if x ∈ Mi, +η(−s)u(−1, y)e−fT (s)−T/2, if x = (s, y) ∈ [−1, 0] × Y, +η(s)u(1, y)efT (s)−T/2, if x = (s, y) ∈ [0, 1] × Y. +And let ˜QT = efT QT . +For any L2-form w support on Mi (or ¯ +Mi), let E(w) be an extension of w, such +that E(w) is an L2-form on ¯ +M and outside Mi (or ¯ +Mi), E(w) = 0. One can see that +34 + +Proposition 9.1. For u ∈ ker (∆1) ⊕ ker (∆2), +∥QT u − E(u)∥2 +L2 ≤ +C +√ +T ∥u∥2 +L2, +��Pδ(T)QT (u) − E(u) +��2 +L2 ≤ +C∥u∥2 +L2 +√ +T +. +for some constant C that is independent of T. +As a result, +��� ˜QT u − E(efT u) +��� +2 +L2,T ≤ +C +√ +T ∥efT u∥2 +L2,T, +��� ˜Pδ(T) ˜QT (u) − E(efT u) +��� +2 +L2,T ≤ +C∥efT u∥2 +L2,T +√ +T +. +Recall that ∥ · ∥L2,T is the norm induced by gT ¯ +M and h ¯F +T := e−2fT h ¯F . +As a result, when T is large enough, ˜Pδ(T) ˜QT (u) spans ˜Ωsm( ¯ +M, ¯F)(T) for u ∈ +ker (∆1) ⊕ ker (∆2). +Proof. For u ∈ ker (∆1) ⊕ ker (∆2), set uT = Pδ(T)QT (u), vT = QT (u) − uT . First, +by trace theorem and G˚arding’s inequality, +� +Y +��ui +� +(−1)i, y +���2 dvolY ≤ C +� +Mi +|ui|2 + |∇ui|2 dvolMi ≤ C′ +� +Mi +|ui|2 dvolMi +(46) +for some constants C, C′ that doesn’t depends on T. By (46) and a straightforward +computation, one can see that +∥QT u − E(u)∥2 +L2 ≤ C +√ +T +∥u∥2 +L2 +and +∥DT QT u∥2 +L2 ≤ C +√ +T +∥u∥2 +L2. +(47) +Here DT := dT + d∗ +T . Moreover, +δ ∥vT ∥2 +L2 ≤ ∥DT vT ∥2 +L2 ≤ ∥DT QT u∥2 +L2 . +(48) +(47) and (48) then imply that +∥vT ∥2 +L2 ≤ +C +δ +√ +T +∥u∥2 +L2 +i.e., +���Pδ(T)QT (u) − QT (u) +��� +2 +L2 ≤ C∥u∥2 +L2 +δ +√ +T +. +Notice that if u ∈ Ωbd(Mi, Fi), then QTu ∈ Ωbd( ¯ +Mi, ¯Fi). By Hodge theory and +Theorem 3.1, when T is big enough, all eigenvalues of ˜∆T,i inside [0, δ] must be 0. +Let ˜PT,i be the orthogonal projection w.r.t. ker( ˜∆T,i). Similarly, one has +35 + +Proposition 9.2. For u ∈ ker (∆i), +��� ˜QT u − E(efT u) +��� +2 +L2,T ≤ +C +√ +T ∥efT u∥2 +L2,T, +��� ˜PT,i ˜QT (u) − E(efT u) +��� +2 +L2,T ≤ +C∥efT u∥2 +L2,T +√ +T +. +As a result, when T is large enough, ˜Pδ(T) ˜QT (u) spans H( ¯ +Mi, ¯Fi)(T) for u ∈ +ker (∆i). +First, notice that when T is large enough, we have a sequence of maps +0 → Hk( ¯ +M2, ¯F2)(T) +˜ek,T +→ ˜Ωk +sm( ¯ +M, ¯F)(T) +˜rk,T +→ Hk( ¯ +M1, ¯F1)(T) → 0. +(49) +Here ˜ek,T is given by u �→ ˜Pδ(T)E(u) for all u ∈ ker( ˜∆T,1). And ˜rk,T is given by +u �→ ˜PT,1(u| ¯ +M1) for all u ∈ ˜Ωk +sm( ¯ +M, ¯F)(T), where ˜PT,i : L2Ω( ¯ +Mi, ¯Fi) → ker(∆T,i) is +the orthogonal projection (i = 1, 2). +Proposition 9.3. ˜ek,T and ˜r∗ +k,T are almost isometric embeddings as T → ∞. That +is, for example, for any u ∈ Hk( ¯ +M2; ¯F2)(T), limT→∞ +∥˜ek,T u∥L2,T +∥uT ∥L2,T += 1. Here ˜r∗ +k,T is +the adjoint of ˜rk,T. +Proof. +•˜ek,T is almost isometric. +For any u ∈ Hk( ¯ +M1, ¯F1)(T), there exists uT ∈ ker(∆1) ∩ Ωk +rel(M1, F1) such that +u = ˜PT,i ˜QT (uT ), then by Proposition 9.2, +∥u∥2 +L2,T ≥ (1 − C +√ +T +)∥efT uT ∥2 +L2,T. +(50) +While +∥˜ek,T u∥2 +L2,T = ∥ ˜Pδ(T)u∥2 +L2,T +≤ ∥ ˜Pδ(T)QT uT ∥2 +L2,T + C∥efT uT ∥2 +L2 +√ +T +(By Proposition 9.2 and the fact that ∥ ˜Pδ(T)∥ = 1) +≤ ∥efT uT ∥2 +L2,T(1 + C′ +√ +T +) (By Proposition 9.1). +(51) +It follows from (50) and (51) that +lim sup +T→∞ +∥˜ek,T u∥2 +∥u∥L2,T += 1. +Similarly, +lim inf +T→∞ +∥˜ek,T u∥2 +∥u∥L2,T += 1. +36 + +•˜r∗ +k,T is almost isometric. +For u ∈ Hk( ¯ +M2, ¯F2)(T), we first show that ˜r∗ +k,Tu = ˜Pδ(T)E(u) : Notice that for any +v ∈ ˜Ωsm( ¯ +M, ¯F)(T), +(˜rk,T v, u)L2( ¯ +M2),T = (v, E(u))L2( ¯ +M),T = (v, ˜Pδ(T)E(u))L2( ¯ +M),T . +Following the same steps as above, one derives that ˜r∗ +k,T is almost isometric. +Theorem 9.1. With maps ˜ek,T and ˜rk,T given above, the sequence (49) is exact. +Proof. Let ⟨·, ·⟩T be the pointwise inner product induced by gT ¯ +M and h ¯F +T . +• In follows from Proposition 9.3 that ˜ek,T and ˜r∗ +k,T are injective when T is large. +• E(ker( ˜∆T,1)) ⊂ ker(d +¯F ,∗ +T +) and E(ker( ˜∆T,2)) ⊂ ker(d ¯F ): +Let u ∈ ker( ˜∆T,2). First, since u satisfies relative boundary conditions, integra- +tion by parts shows that for any β ∈ Ω( ¯ +M, ¯F), +� +¯ +M⟨E(u), d +¯F ,∗ +T +β⟩T dvol = 0. Thus, +E(u) ∈ ker(d ¯F ). Similarly, E(ker( ˜∆T,1)) ⊂ ker(d +¯F ,∗ +T +). +• im ˜ek,T = ker ˜rk,T : +For the dimension reason, it suffices to show that im˜ek,T ⊂ ker ˜rk,T. That is, it +suffices to show im˜ek,T ⊥ im˜r∗ +k,T . First, for any ui ∈ ker( ˜∆T,i), i = 1, 2, it’s clear +that +(E(u1), E(u2))L2,T = 0. +(52) +Since E(u1) ∈ ker(d +¯F ,∗ +T +), E(u2) ∈ ker(d ¯F ), one can see that (1 − ˜Pδ(T))u1 ∈ +im d +¯F ,∗ +T +, (1 − ˜Pδ(T))u1 ∈ im d ¯F , which means +((1 − ˜Pδ(T))E(u1), (1 − ˜Pδ(T))E(u2))L2,T = 0. +(53) +By (52) and (53), +(˜ek,T u2, ˜r∗ +k,Tu1)L2,T = 0. +Moreover, we have the following complexes of finite dimensional vector spaces +0 → H0( ¯ +Mi, ¯Fi) +0→ H1( ¯ +Mi, ¯Fi) 0→ · · · +0→ Hdim M( ¯ +Mi, ¯Fi) → 0 +(54) +0 → ˜Ω0 +sm( ¯ +M, ¯F) d ¯ +F +→ ˜Ω1 +sm( ¯ +M, ¯F) d ¯ +F +→ · · · d ¯ +F +→ ˜Ωdim M +sm +( ¯ +M, ¯F) → 0. +(55) +Integration by parts as in the proof of Theorem 9.1, one can show easily that +Proposition 9.4. d ¯F ◦ ˜ek,T = 0, ˜rk,T ◦ d ¯F = 0. +37 + +Hence, by Theorem 9.1 and Proposition 9.4, we get the following long exact +sequence again +MV(T) : · · · +∂k−1,T +→ +Hk � ¯ +M2; ¯F2 +� +(T) +ek,T +→ Hk � ¯ +M; ¯F +� +(T) +rk,T +→ Hk � ¯ +M1; ¯F1 +� +(T) +∂k,T +→ · · · +(56) +with metric induced by gT ¯ +M and h ¯F +T . +Proof of Theorem 3.4. Since (54) and (55) are complexes of finite dimensional vec- +tor spaces, it follows from Proposition 9.3 and [7, Proposition 2.6] (or [9, Theorem +3.1 and Theorem 3.2]) that +lim +T→∞ log Tsm(gT ¯ +M, h +¯F , ∇ +¯F )(T) − log T (T) = 0. +References +[1] J.-M. Bismut and W. Zhang. An extension of a theorem by Cheeger and M¨uller. +Ast´erisque, 205, 1992. +[2] J. Br¨uning and X. Ma. An anomaly formula for Ray–Singer metrics on mani- +folds with boundary. Geometric & Functional Analysis, 16(4):767–837, 2006. +[3] J. Br¨uning and X. Ma. On the gluing formula for the analytic torsion. Mathe- +matische Zeitschrift, 273(3):1085–1117, 2013. +[4] J. Cheeger. Analytic torsion and the heat equation. Annals of Mathematics, +109(2):259–321, 1979. +[5] X. Dai and J. Yan. +Witten deformation for noncompact manifolds with +bounded geometry. Journal of the Institute of Mathematics of Jussieu, pages +1–38, 2021. +[6] X. Dai and J. Yan. Witten deformation on non-compact manifold: Heat kernel +expansion and local index theorem. Mathematische Zeitschrift, 303(1), 2022. +[7] M. Lesch. A gluing formula for the analytic torsion on singular spaces. Analysis +& PDE, 6(1):221–256, 2013. +[8] W. L¨uck. Analytic and topological torsion for manifolds with boundary and +symmetry. Journal of Differential Geometry, 37(2):263–322, 1993. +[9] J. Milnor. Whitehead torsion. Bulletin of the American Mathematical Society, +72(3):358–426, 1966. +[10] W. M¨uller. Analytic torsion and R-torsion of Riemannian manifolds. Advances +in Mathematics, 28(3):233–305, 1978. +[11] W. M¨uller. +Analytic torsion and R-torsion for unimodular representations. +Journal of the American Mathematical Society, 6(3):721–753, 1993. +38 + +[12] M. Puchol, Y. Zhang, and J. Zhu. Adiabatic limit, Witten deformation and +analytic torsion forms. arXiv:2009.13925. +[13] M. Puchol, Y. Zhang, and J. Zhu. Scattering matrices and analytic torsions. +Analysis & PDE, 14(1):77 – 134, 2021. +[14] D. B. Ray and I. M. Singer. +R-torsion and the Laplacian on Riemannian +manifolds. Advances in Mathematics, 7(2):145–210, 1971. +[15] K. Reidemeister. Homotopieringe und linsenr¨aume. In Abhandlungen aus dem +Mathematischen Seminar der Universit¨at Hamburg, volume 11, pages 102–109. +Springer, 1935. +[16] S. M. Vishik. Generalized Ray-Singer conjecture. i. A manifold with a smooth +boundary. Communications in Mathematical Physics, 167(1):1–102, 1995. +[17] J. Yan. A new proof of gluing formula for analytic torsion forms. In Preparation. +[18] W. Zhang. Lectures on Chern-Weil theory and Witten deformations, volume 4. +World Scientific, 2001. +39 + diff --git a/JdA0T4oBgHgl3EQfCf9p/content/tmp_files/load_file.txt b/JdA0T4oBgHgl3EQfCf9p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a008113b72bfd7de60165a6612e9c6fb27e11ff1 --- /dev/null +++ b/JdA0T4oBgHgl3EQfCf9p/content/tmp_files/load_file.txt @@ -0,0 +1,1533 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf,len=1532 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='01990v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='DG] 5 Jan 2023 Witten deformation for non-Morse functions and gluing formula for analytic torsions Junrong Yan ∗ January 6, 2023 Abstract In this paper, we provide a novel analytic proof of the gluing formula for the analytic torsion of flat vector bundles in complete generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It’s quite interesting that in this paper, the gluing formula could be interpreted as the Bismut-Zhang theorem [1] for some non-Morse functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Contents 1 Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Overview .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Main Results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 Main ideas .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 Organization .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 5 2 Preliminary 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 A brief review on Hodge theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Analytic torsion for flat vector bundles .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 On closed manifolds .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 On manifolds with boundary .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 Analytic torsion for Witten Laplacian and Weighted Laplacian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Witten Laplacian v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Weighted Laplacian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Absolute/Relative boundary conditions for weighted Laplacian 10 3 Intermidiate Results 11 4 Convergence of Eigenvalues 13 ∗Beijing International Center for Mathematical Research, Peking University, Beijing, China 100871, j yan@bicmr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Supported by Boya Postdoctoral Fellowship at Peking University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 1 5 Large Time Contributions 19 6 A Gluing Formula for Heat Trace in Small Time 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Several heat kernels and Laplacians .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Gluing heat kernels .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 21 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 Heat trace expansion for e−t(∆1⊕∆2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 Heat trace expansion for e−t∆T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 24 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 when M is even dimensional .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 25 7 One-dimensional Model and Coupling Techniques 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Heat trace estimate when t and T are coupled .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 26 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Partial proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 when M is odd dimensional .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 30 8 Ray-Singer metrics on S1 and [−2, 2] 31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Ray-Singer metrics on S1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 31 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Ray-Singer metric on [−2, 2] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 32 9 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 34 1 Introduction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Overview Let (M, g) be a closed Riemannian manifold associated with a flat complex vector bundle F → M, and suppose that F has a hermitian metric hF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The correspond- ing Ray-Singer analytic torsion [14] is the determinant of the Hodge-Laplacian on F-valued differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The Ray-Singer metric [1] on det H∗(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) is the prod- uct of the Ray-Singer analytic torsion and the Hodge metric induced by F-valued harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The Ray-Singer metric has a well-known topological counterpart, the Reidemeister metric [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The famous Ray-Singer conjecture [14] states that the two metrics for a unitary flat vector bundle coincide, which was proved by Cheeger [4] and M¨uller [10] independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' M¨uller [11] then extended the theorem to uni- modular flat bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Bismut and Zhang [1] extended the theorem to general flat vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Now suppose there is a hypersurface Y ⊂ M that divides M into two parts, M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a preliminary step toward the Ray-Singer conjecture, Ray and Singer [14] proposed that there should be a gluing formula relating the analytic torsion of M and the analytic torsions of M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' However, the conjecture was proved by other methods, and the gluing formula follows from the conjecture [8, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' There are also purely analytic proofs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=', without using Cheeger-M¨uller theorem [16, 13, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The proof of gluing formula in this paper is based on the Witten deformation for non-Morse functions and some coupling techniques (see the last paragraph in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 for a description of coupling techniques).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Roughly, we choose a family of smooth functions fT such that the limit limT→∞ fT has crucial loci M1 and M2 with “Morse 2 indices” of 0 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then, based on the philosophy of Witten defor- mation, letting T range from 0 to ∞, the relationship between analytic torsion on M and analytic torsion on the two pieces above can be understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' This method could be applied to analytic torsion forms [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' See [12] for another proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It’s also possible that this method could be applied to the proof of gluing formulas for other global invariants (such as eta-invariant, partition functions from QFT, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Finally, this paper’s exploration of Witten deformation for non-Morse functions is intended to provide clues toward proving the family version of Cheeger-M¨uller the- orem for general flat bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Acknowledgment: The author is appreciative of Professor Xianzhe Dai’s consis- tently stimulating conversation and encouragement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The author also appreciates the insightful discussion with Martin Puchol and Yeping Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Main Results Let (M, gTM) be a closed Riemannian manifold and Y ⊂ M be a hypersurface that divides M into two pieces M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let F → M be a flat complex vector bundle, and ∇F be a flat connection on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF be the covariant differential on F-valued differentials Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), which is induced by ∇F (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let hF be a Hermitian metric on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF,∗ be the (formal) adjoint operator of dF associated with gTM and hF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then D := dF + dF,∗ is a first-order self-adjoint elliptic operator acting on Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let N be the number operator on Ω (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=', Nω = pω for ω ∈ Ωp (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The zeta function ζ for ∆ := D2 is defined as follows, for z ∈ {C : Re(z) > 1 2 dim M � ζ(z) := 1 Γ(z) � ∞ 0 tz−1 Trs � Ne−t∆′� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' See §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The function ζ(s) admits a meromorphic continuation to the whole complex plane, which is holomorphic at 0 ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then the Ray-Singer analytic torsion T (gTM, hF , ∇F ) := e 1 2ζ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Fi be the restriction of F on Mi, dF i be the restriction of dF on Mi, dF,∗ i be the formal adjoint of dF i (i=1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆1 := (dF 1 + dF,∗ 1 )2 and ∆2 = (dF 2 + dF,∗ 2 )2 act on Ωrel (M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) and Ωabs (M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2) respectively (see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 for the definition of Ωabs (M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) and Ωrel (M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ζi be the zeta functions for ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then similarly, one can define Ray-Singer analytic torsion Ti(gTMi, hFi, ∇Fi) = e 1 2 ζ′ i(0), where gTMi, Fi, hFi and ∇Fi are the restriction of gTM, F, hF and ∇F on Mi respectively(i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lastly, we have the following Mayer-Vietoris exact sequence · · → Hp rel (M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2) → Hp (M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → Hp abs (M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We denote by T the analytic torsion for the exact sequence above equipped with L2-metrics (induced by Hodge theory).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 3 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' log T (gTM, hF , ∇F ) − log T1(gTM1, hF1, ∇F1) − log T2(gTM2, hF2, ∇F2) = log T + 1 2χ(Y )rank(F) log 2 + (−1)dim(M)rank(F) � Y B � gTM� , where B � gTM� is the secondary characteristic form introduced in [2], which is zero if Y is totally geodesic in � M, gTM� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Based on [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='7] and [3, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4], we only need to consider the situation in which gTM and hF are product-type near Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In this case, � Y B � gTM� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 Main ideas Let Y ⊂ M be a hypersurface cutting M into two pieces M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let U be a neighbor of Y , such that U ∩ M1 is diffeomorphic to (−2, −1] × Y and identify ∂M1 with {−1} × Y , U ∩ M2 is diffeomorphic to [1, 2) × Y and identify ∂M2 with {1} × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, assume that on U, gTM and hF are product-type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We then glue M1, M2 and [−1, 1] × Y naturally, we get a manifold ¯ M, which is diffeomorphic to the original manifold M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let fT be a smooth function on ¯ M, such that 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' fT |M1 ≡ −T/2, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' fT |M2 ≡ T/2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' fT |[−1,0]×Y (s, y) ≈ T(s + 1)2/2 − T/2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' fT |[−1,0]×Y (s, y) ≈ −T(s − 1)2/2 + T/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF T := dF + dfT∧, dF,∗ T be the formal adjoint of dF T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Set DT := dF T + dF,∗ T , ∆T := D2 T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let λk be the k-th eigenvalue (counted with multiplicities) of ∆1 ⊕ ∆2 (acting on Ωrel(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1)⊕Ωabs(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let {λk(T)} be the k-th eigenvalue (counted with multiplicities) of ∆T (acting on Ω( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One has a nice observation that lim T→∞ λk(T) = λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (1) We temporarily assume that dim Hk(M) = dim Hk(M1) + dim Hk(M2, ∂M2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T (gT ¯ M, hF , ∇F )(T) be the analytic torsion with respect to ∆F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then based on (1), naively, one should expect that lim T→∞ log T (gT ¯ M, hF , ∇F )(T) = log T1(gTM1, hF1, ∇F1) + log T2(gTM2, hF2, ∇F2), (2) and lim T→0 log T (gT ¯ M, hF , ∇F )(T) = log T (gT ¯ M, hF , ∇F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 4 As a result, the relationship between analytic torsion on M and analytic torsion on the two pieces above could be seen, proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Although the idea is quite simple, the devil is in the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ζT , ζ1 and ζ2 be the zeta functions for ∆T , ∆1 and ∆2 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ζL T (z) := 1 Γ(z) � ∞ 1 tz−1 Trs � Ne−t∆′ T � dt, and ζS T (z) := 1 Γ(z) � 1 0 tz−1 Trs � Ne−t∆′ T � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, ζL i and ζS i (i = 1, 2) can be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By (1), it should be clear that (ζL T )′(0) → (ζL 1 )′(0) + (ζL 1 )′(0) as T → ∞ (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' To get (2), the next step is to show the convergence of (ζS T )′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The main difficulty for this method is that when t ∈ (0, 1), the convergence of Trs(Ne−t∆′ T ) as T → ∞ is unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' To resolve a similar issue, [13] considers the adiabatic limit of analytic torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' While in this paper, we observe that when t and T are coupled, Trs(Ne−t∆′ t−5T ) behaves nicely as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' That is, (see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1), Trs(Ne−t∆′ t−5T ) → Trs(Ne−t∆′ 1) + Trs(Ne−t∆′ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similar ideas appear in the author’s previous joint work with Xianzhe Dai [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Note that the space generated by the eigenforms of ∆T for small eigenvalues, denoted by Ωsm(M, F)(T), is finite-dimensional for large values of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The sec- ond crucial piece in our proof is the relationship between the analytic torsion for Ωsm(M, F)(T) and the analytic torsion for the exact sequence (7) as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' See §9 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 Organization In §2, we will give a brief review of analytic torsion and establish the basic settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In §3, we state and prove several intermediate results and show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' While Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 will be proved in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In §4, we investigate the behavior of eigenvalues as T → ∞ and prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In §5, we analyze the long-time behavior of the zeta function and prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In §6, the gluing formula of the heat trace for small time t is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In §7, we investigate the small-time behavior of the zeta function, discovering that if time t and the deformation parameter T are well-coupled, we understand the behavior of the heat kernel on the tube well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then we partially establish Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3, leaving some mysterious terms unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Nonetheless, it is worth noting that these terms are independent of M, Y and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' To calculate these terms, the Ray-Singer metric on S1 and [−2, 2] is explored in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Therefore, thoroughly prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 will be proved in the last section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 5 2 Preliminary From now on, we assume that gTM and hF are product-type near Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' That is, there exists a neighborhood U of Y , such that U ∼= (−1, 1) × Y , and let (s, y) be its coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then gTM|U = ds ⊗ ds + gTY for some metric gTY on Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let hF Y := hF |{0}×Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For any v ∈ F(s,y), let Pγ ∈ End(F(s,y), F(0,y)) be the parallel transport associated with ∇F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' the path γ(t) = (st, y), t ∈ [0, 1], then we require that hF (v, v) = hF Y (Pγv, Pγv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence on U, ∇F ∂ ∂s hF = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' If T is sufficiently large, all constants appearing in this paper are at least inde- pendent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The notations C and c, et cetera, denote constants that may vary based on context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 A brief review on Hodge theory Let (X, gTX) be a compact manifold with boundaries Y = ∂X, where Y could be an empty set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let F → X be a flat vector bundle with flat connection ∇F, and hF be a Hermitian metric on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We identify a neighborhood of ∂X to (−1, 0] × Y , and let (s, y) be its coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) denote the space of smooth F-valued differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Set Aabs(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := � ω ∈ Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) : i ∂ ∂s ω = 0 on Y � , Arel(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := {ω ∈ Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) : ds ∧ ω = 0 on Y } ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ωabs(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := � ω ∈ Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) : i ∂ ∂s ω = 0, i ∂ ∂s dF ω = 0 on Y � , Ωrel(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := � ω ∈ Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) : ds ∧ ω = 0, ds ∧ dF,∗ω = 0 on Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For the sake of convenience, “bd” will be adopted to represent “abs” or “rel”, when it is not necessary to distinguish the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF : Ω∗(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → Ω∗+1(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) denote the covariant derivative with respect to ∇F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Assuming that {ek} is a local orthonormal frame of TX and {ek} is its dual frame, then dF = � k ek ∧ ∇F ei locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∇F,∗ be the dual connection of ∇F, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=', for any s1, s2 ∈ Γ(F), dh(s1, s2) = h(∇F s1, s2) + h(s1, ∇F,∗s2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then set dF,∗ := − � k iek∇F,∗ ek .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF bd, dF,∗ bd and ∆bd be the restrictions of dF , dF,∗ and ∆ to Abd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let (·, ·)L2(X) be the L2-inner product induced by hF and gTX, then integration by parts implies that (dF bdα, β)L2 = (α, dF,∗ bd β)L2, ∀α, β ∈ Abd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (3) For a closable operator S : H → H on a Hilbert space (H, (·, ·)H) with a dense domain Dom(S), define an inner product (·, ·)S on Dom(S) by (α, β)S := (α, β)H + (S α, S β)H, ∀α, β ∈ Dom(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 6 Let Wmin be the completion of Dom(S) with respect to the norm ∥· ∥S, then one can extend S to Smin naturally with Dom(Smin) = Wmin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Assume that there is another closable operator ˜S : H → H, such that Dom(˜S) = Dom(S) and (S α, β)H = (α, ˜Sβ)H, ∀α, β ∈ Dom(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Wmax := {α ∈ H : |(α, ˜Sβ)H| ≤ Mα∥β∥H for some constant Mα > 0, ∀β ∈ Dom(˜S)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Because Dom(S) is dense, the Riesz representation theorem states that γ ∈ H exists such that (γ, β)H = (α, ˜Sβ)H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then we define Smax α = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One can see easily that Smax is nothing but the adjoint of ˜S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By (3), [5, Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3] and the discussion above, one has Im(dF bd,min) ⊕ Im(dF,∗ bd,min) = Im(dF bd,max) ⊕ Im(dF,∗ bd,max).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Together with [3, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1], one has Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 (Hodge decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (1) We have ker ∆bd = ker � dF � ∩ ker � dF,∗� ∩ Ωbd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (2) The vector space ker ∆bd is finite-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (3) We have the following orthogonal decomposition Abd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) = ker(∆bd) ⊕ ImdF bd ⊕ ImdF,∗ bd .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' L2Abd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) = ker(∆bd) ⊕ ImdF bd,min ⊕ ImdF,∗ bd,min = ker(∆bd) ⊕ ImdF bd,max ⊕ ImdF,∗ bd,max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here L2Abd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) is the completion of Abd(X, F) with respect to (·, ·)L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Analytic torsion for flat vector bundles 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 On closed manifolds Let (M, gTM) be a closed Riemannian manifold, F → M be a flat vector bundle with a Hermitian metric hF , and ∇F be a flat connection on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF be the covariant differential on Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) induced by ∇F, then we have a complex 0 → Ω0(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) dF → Ω1(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) dF → · · · dF → Ωdim(M)(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (4) Denote H(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) to be the cohomology of this complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One can see that gTM and hF induce an L2-inner product (·, ·)L2(M) on Ω∗(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 7 Let dF,∗ be the formal adjoint of dF with respect to metric gTM and hF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then the Hodge Laplacian ∆ : Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) is defined as ∆ := (dF + dF,∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) has a natural Z2 grading: Ω+ := ⊕k is even Ωk(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), Ω− := ⊕k is oddΩk(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For a trace class operator A : Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), Trs(A) denotes its supertrace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' If A has an integral kernel a, the pointwise supertrace of a is denoted by trs(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let N : Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) be a linear operator, such that for α ∈ Ωk(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), Nα = kα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' N is called the number operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let P : Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) → Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) be the orthogonal projection to ker(∆), ∆′ := ∆(1 − P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The zeta function ζ for ∆ is defined as ζ(z) := 1 Γ(z) � ∞ 0 tz−1 Trs � Ne−t∆′� dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' where Γ is the Gamma function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The zeta function is well defined whenever Re(z) is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' And it could be extended to a meromorphic function on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, ζ is holomorphic at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The Ray-Singer analytic torsion T (gTM, hF , ∇F ) for the complex (4) is defined as T (gTM, hF , ∇F ) := e 1 2 ζ′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 On manifolds with boundary Let (X, g) be a compact manifold with boundaries Y = ∂X, gTX be a Riemannian metric on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let F → X be a flat vector bundle with flat connection ∇F, and hF be a Hermitian metric on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We identify a neighborhood of ∂X to (−1, 0] × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let (s, y) be its coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dF,∗ be the formal adjoint of the de Rham operator dF with respect to the L2 metric (·, ·)L2(X) induced from hF and gTX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Set Ωabs(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := � ω ∈ Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) : i ∂ ∂u ω = 0, i ∂ ∂u dF ω = 0 on Y � , Ωrel(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := � ω ∈ Ω(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) : du ∧ ω = 0, du ∧ dF,∗ω = 0 on Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We write Ωbd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) for short if the choice of abs/rel is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆bd := (dF + dF,∗)2 act on Ωbd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' According to the Hodge theory, ker(∆bd) ∼= Hbd(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here Hrel(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := H(X, ∂X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), and Habs(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) := H(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ζbd be the zeta functions for ∆bd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The analytic torsion Tbd(gTX, hF , ∇F) is defined by e 1 2 ζ′ bd(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 8 In particular, let Y ⊂ M be a hypersurface cutting M into two pieces M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Denote gTMi, Fi, hFi and ∇Fi to be the restriction of gTM, F, hF and ∇F to Mi respectively(i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One can see that gTMi and hFi induce an L2-inner product (·, ·)L2(Mi) on Ω∗(Mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆1 := (dF1 + dF1,∗)2 act on Ωabs(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1), and ∆2 := (dF2 + dF2,∗)2 act on Ωrel(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We set ζi to be the zeta function for ∆i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' And let T1(gM1, hF1, ∇F1) := Tabs(gM1, hF1, ∇F1) = e 1 2ζ′ 1(0), and T2(gM2, hF2, ∇F2) := Trel(gM2, hF2, ∇F2) = e 1 2ζ′ 2(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 Analytic torsion for Witten Laplacian and Weighted Laplacian Let Y ⊂ M be a hypersurface cutting M into two pieces M1 and M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Denote gTMi, Fi, hFi and ∇Fi to be the restriction of gTM, F, hF and ∇F to Mi respectively(i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We identify a neighborhood of Y = ∂M1 in M1 to (−2, −1] × Y , and ∂M1 is identified with {−1} × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, we identify a neighborhood of Y = ∂M2 in M2 to [1, 2) × Y , and ∂M2 is identified with {1} × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ¯ M = M1 ∪ [−1, 1] × Y ∪ M2, and ¯F, h ¯F , gT ¯ M and ∇ ¯F be the natural extensions of F, hF , gTM and ∇F to ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One can see that gT ¯ M and h ¯F induce an L2-inner product (·, ·)L2( ¯ M) on Ω∗( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let fT be a family of odd smooth functions on [−2, 2], such that (a) fT |[1,2] ≡ T/2, (b) fT |[1/2,1](s) = −Tρ � eT 2(1 − s) � (s − 1)2/2+ T/2 , where ρ ∈ C∞ c ([0, ∞)), such that 0 ≤ ρ ≤ 1, ρ[0,1/2] ≡ 0, ρ[3/4,∞] ≡ 1, |ρ′| ≤ δ1 and |ρ′′| ≤ δ2 for some universal constant δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (c) C1T ≤ |f ′ T|(s) ≤ 2C1T, |f ′′ T | ≤ C2T for some universal constants C1 and C2 whenever s ∈ [0, 1/2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then one can see that C3T ||s| − 1| ≤ |f ′ T |(s) ≤ 2C3T ||s| − 1| and |f ′′ T |(s) ≤ C4T whenever ||s| − 1| ≤ e−T 2 for some universal constant C3 and C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We could think fT as a function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dT := d ¯F + dfT ∧ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then the Witten Laplacian ∆T is the Hodge Laplacian with respect to dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We have a complex 0 → Ω0( ¯ M, ¯F) dT → Ω1( ¯ M, ¯F) dT → · · · dT → Ωdim(M)( ¯ M, ¯F) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (5) Denote H( ¯ M, ¯F)(T) to be the cohomology of this complex, ∆T to be the Hodge Laplacian for dT , and |·|gT ¯ M ,h ¯ F ,∇ ¯ F (T) to be the Hodge metric on det � H( ¯ M, ¯F)(T) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 9 Let ζT be the zeta function for ∆T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one could define Ray-Singer analytic torsion T (gT ¯ M, h ¯F , ∇ ¯F )(T) := e 1 2 ζ′ T (0) for the complex (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lastly, for the sake of convenience, (·, ·)L2 (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2 := � (·, ·)L2) will be adopted to represent (·, ·)L2(M) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2(M) := � (·, ·)L2(M)) , (·, ·)L2( ¯ M) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2( ¯ M) := � (·, ·)L2( ¯ M)) or (·, ·)L2(Mi)(resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2(Mi) := � (·, ·)L2(Mi)) (i = 1, 2), when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Witten Laplacian v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Weighted Laplacian Instead of deforming the de Rham differential d ¯F , we could also deform the metric h ¯F : let h ¯F T := e−2fT h ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, g ¯ M and h ¯F T induce an L2-norm (·, ·)L2( ¯ M),T on Ω( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let L2Ω( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F)(T) be the completion of Ω( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t (·, ·)L2( ¯ M),T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then the formal adjoint d ¯F,∗ T of d ¯F w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' the (·, ·)L2( ¯ M),T is then given by efT d∗ T e−fT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The Weighted Laplacian ˜∆T := d ¯F d ¯F ,∗ T + d ¯F,∗ T d ¯F , one can see that ˜∆T = efT ∆T e−fT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let lk(T) be the k-th eigenvalue of ˜∆T, then lk(T) = λk(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, if u is an eigenform of ∆T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' eigenvalue λ, then efT u is an eigenform of ˜∆T w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' eigenvalue λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, Trs(Ne−t ˜∆T ) = Trs(Ne−t∆T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Absolute/Relative boundary conditions for weighted Lapla- cian Let ¯ M1 := M1 ∪ [−1, 0] × Y , ¯ M2 := M2 ∪ [0, 1] × Y , and ¯Fi be the restriction of ¯F on ¯ Mi (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Set Aabs( ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1) := � ω ∈ Ω( ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1) : i ∂ ∂s ω = 0 on {0} × Y � , Arel( ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2) := � ω ∈ Ω( ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2) : ds ∧ ω = 0 on {0} × Y � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ωabs( ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1) := � ω ∈ Ω( ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1) : i ∂ ∂s ω = 0, i ∂ ∂s d ¯F1ω = 0 on {0} × Y � , Ωrel( ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2)T := � ω ∈ Ω( ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2) : ds ∧ ω = 0, ds ∧ d ¯Fi,∗ T ω = 0 on {0} × Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One can see that gT ¯ Mi and h ¯Fi T (The resriection of gT ¯ M and h ¯Fi on ¯ Mi) induce an L2-inner product (·, ·)L2( ¯ Mi),T on Ω∗( ¯ Mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ˜∆T,i be the restriction of ˜∆T acting on Ωbd( ¯ Mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then by Hodge theory, ker( ˜∆T,i) ∼= Hbd( ¯ Mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lastly, for the sake of convenience, (·, ·)L2,T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2,T := � (·, ·)L2,T) will be adopted to represent (·, ·)L2( ¯ M),T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2( ¯ M),T := � (·, ·)L2( ¯ M),T ) , or (·, ·)L2( ¯ Mi),T (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∥ · ∥L2( ¯ Mi),T := � (·, ·)L2( ¯ Mi),T ) (i = 1, 2), when the context is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 10 3 Intermidiate Results In this section, we will state and prove some intermediate results to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let λk(T) be the k-th eigenvalue for ∆T, λk be the k-th eigenvalue of ∆1 ⊕ ∆2 acting on Ωabs(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) ⊕ Ωrel(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' and ˜λk(T) be the k-th eigenvalue of ˜∆T,1 ⊕ ˜∆T,2 acting on Ωabs( ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1) ⊕ Ωrel( ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2)T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then limT→∞ λk(T) = limT→∞ ˜λk(T) = λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let δ > 0 denote half of the first nonzero eigenvalue of ∆1 ⊕ ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then by The- orem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, all eigenvalues of ∆T inside [0, δ] converge to 0 as T → ∞, and all eigen- values of ˜∆T,1 ⊕ ˜∆T,2 inside [0, δ] are 0 when T is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Ωsm( ¯ M, ¯F)(T) be the space generated by eigenforms for eigenvalues of ∆T inside [0, δ], and Pδ(T) be the orthogonal projection from L2Ω( ¯ M, ¯F) to Ωsm( ¯ M, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ζT,la := 1 Γ(z) � ∞ 0 tz−1 Trs � Ne−t∆′ T � 1 − Pδ(T) �� dt, ζT,sm := 1 Γ(z) � ∞ 0 tz−1 Trs � Ne−t∆′ T Pδ(T) � dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For i = 1, 2, let ζL i (z) := 1 Γ(z) � ∞ 1 tz−1 Trs(Ne−t∆′)dt, ζS i (z) := 1 Γ(z) � 1 0 tz−1 Trs(Ne−t∆′)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then it is clear that ζi = ζL i + ζS i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one can define ζL T,i(z), ζS T,i(z), ζL T,la(z), and ζS T,la e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' lim T→∞(ζL T,la)′(0) = lim T→∞ 2 � i=1 (ζL T,i)′(0) = 2 � i=1 (ζL i )′(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' That is, lim T→∞ � ∞ 1 t−1 Trs � Ne−t∆′ T (1 − Pδ) � dt = lim T→∞ 2 � i=1 � ∞ 1 t−1 Trs(Ne−t∆′ T,i)dt = 2 � i=1 � ∞ 1 t−1 Trs(Ne−t∆′ i)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As T → ∞, (ζS T,la)′(0) = 2 � i=1 (ζS i )′(0) − (T − log(2)) χ(Y )rank(F) + o(1), 11 (ζS i,T )′(0) = (ζS i )′(0) − Tχ(Y )rank(F)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Thus, as T → ∞, (ζS T,la)′(0) − 2 � i=1 (ζS i,T )′(0) = log(2)χ(Y )rank(F) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Next, we have the following Mayer-Vietoris exact sequence (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [3, (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='16)]) MV : · · · ∂k−1 → Hk rel � ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2 � ek → Hk � ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F � rk → Hk rel � ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1 � ∂k → · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (6) Let H( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F)(T) := ker( ˜∆T ), and H( ¯ Mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯Fi)(T) := ker( ˜∆T,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We also have the following Mayer-Vietoris exact sequence induced by Hodge theory and (6) MV(T) : · · · ∂k−1,T → Hk � ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2 � (T) ek,T → Hk � ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F � (T) rk,T → Hk � ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1 � (T) ∂k,T → · · · (7) with metric induced by gT ¯ M and h ¯F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T (T) be the analytic torsion for this complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Recall that ζT,sm := 1 Γ(z) � ∞ 0 tz−1 Trs(Ne−t∆′ T Pδ(T))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' And set Tsm(gT ¯ M, h ¯F , ∇ ¯F )(T) := e 1 2 ζ′ T,sm(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, the following proposition will be proved in §9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' limT→∞ log Tsm(gT ¯ M, h ¯F , ∇ ¯F)(T) − log T (T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Ti(gT ¯ Mi, h ¯Fi, ∇ ¯Fi)(T) be the analytic torsion w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ˜∆T,i, then it follows from anomaly formula [1, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1] and [2, Theorem 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1] that Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' log T (gT ¯ M, h ¯F , ∇ ¯F )(T) − 2 � i=1 log Ti(gT ¯ Mi, h ¯Fi, ∇ ¯Fi)(T) − log T (T) = log T (gTM, hF , ∇F ) − 2 � i=1 log Ti(gTMi, hFi, ∇Fi) − log T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 that log T (gT ¯ M, h ¯F , ∇ ¯F )(T) − 2 � i=1 log Ti(gT ¯ Mi, h ¯Fi, ∇ ¯Fi)(T) − log T (T) = log(2)χ(Y )rank(F)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 12 4 Convergence of Eigenvalues For simplicity, in this section, let d := d ¯F and d∗ := d ¯F ,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let {ei} be a local frame of TM, and{ei} its dual frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Set LfT := Hess(ei, ej)fT c(ei)ˆc(ej), where c(ei) := ei ∧ −iei, ˆc(ei) := ei ∧ +iei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Recall that λk(T) is the k-th eigenvalue of ∆T , λk is the k-th eigenvalue of ∆1 ⊕ ∆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∇ be the connection on Ω( ¯ M, ¯F) induced by ∇ ¯F and gT ¯ M, and ∇Y = ∇|Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In this section, we are going to show that limT→∞ λk(T) = limT→∞ ˜λk(T) = λk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, one observes that λk(T) has uniform upper bounds: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Fix k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' There exists an increasing sequence {Λk}∞ k=1, such that λk(T) ≤ Λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Choose k disjoint balls Bk in M◦ 1 := M1 − {−1} × Y, k nonzero smooth functions ηk with support supp(ηk) ⊂⊂ Bk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Vk be the linear space generated by {ηk}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then by the min-max principle, it’s easy to check that λk(T) ≤ Λk := sup ψ∈Vk � ¯ M |dψ|2 + |d∗ψ|2dvol ¯ M � ¯ M |ψ|2dvol ¯ M = sup ψ∈Vk � ¯ M |dT ψ|2 + |d∗ T ψ|2dvol ¯ M � ¯ M |ψ|2dvol ¯ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Next, by the trace theorem: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let u ∈ Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F), such that � ¯ M |u|2dvol ¯ M = 1 and � ¯ M |dT u|2 + |d∗ T u|dvol ¯ M ≤ λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then for s ∈ � −2, −1 + � 2 T � ∪ � 1 − � 2 T , 2 � � Y |u|2(s, y)dvolY ≤ C(λ + 1) if T is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, by trace formula, � Y |u|2(−1, y)dvolY ≤ C � M1 |u|2 + |∇u|2dvolM1 ≤ C � M1 |u|2 + |(d + d∗)u|2dvolM1 = C � M1 |u|2 + |(dT + d∗ T )u|2dvolM1 ≤ C � ¯ M |u|2 + |(dT + d∗ T )u|2dvol ¯ M ≤ C(λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one still have for s ∈ [−2, −1] ∪ [1, 2], � Y |u|2(y, s)dvolY ≤ C(λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 13 Next for γ ∈ (0, 1) to be determined, suppose s0 ∈ � −1, −1 + � γ T � achieves the supreme of AT := sup s∈[−1,−1+√ γ T ]∪[1−√ γ T ,1] � Y |u|2(y)dvolY , then � Y |u(s0, y) − u(−1, y)|2dvolY ≤ � Y ����� � −1+√ γ T −1 | ∂ ∂s′ u(y, s′)|ds′ ����� 2 dvolY ≤ � γ T � Y � −1+√ γ T −1 |du(y, s′)|2 + |d∗u(y, s′)|2ds′dvolY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (8) Integration by parts, λ ≥ � ¯ M |dT u|2 + |d∗ T u|2dvol ¯ M ≥ � −1+√ γ T −1 � Y |dT u|2 + |d∗ T u|2dvolY ds ≥ � −1+√ γ T −1 � Y |du|2 + |d∗u|2 + (LfT u, u) + |∇fT|2|u|2dvolY ds − � Y |dfT ||u|2 � −1 + � γ T , y � dvolY ≥ � −1+√ γ T −1 � Y |du|2 + |d∗u|2 − C4T|u|2dvolY ds − √ TAT ≥ � −1+√ γ T −1 � Y |du|2 + |d∗u|2dvolY ds − (1 + C4 √γ) √ TAT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (9) See §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 for the definition of C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By (8) and (9), one can see that AT ≤ (λ + 1) �� γ T + C � + √γ(1 + C4 √γ)AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Fix γ ∈ (0, 1), such that √γ(1+C4√γ) ≤ 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Thus, whenever s ∈ � −1, −1 + � γ T � , � Y |u(s, y)|2dvolY ≤ 3C(λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one can show that for s ∈ � −1 + � γ T , −1 + 2 � γ T � � Y |u(s, y)|2dvolY ≤ 32C(λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 14 Let m = [ 2 γ ] + 1, then repeating the arguments above for m times, one can see that � Y |u(s, y)|2dvolY ≤ 3mC(λ + 1) whenever s ∈ � −1, −1 + � 2 T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Assume u meets the same conditions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then � 1/2 −1/2 � Y |u(s, y)|2dvolY ds ≤ C(λ + 1) T 3/2 if T is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Just notice that |f ′ T |2(s)−f ′′ T(s) ≥ 0 when s /∈ � −1, −1 + � 2 T � ∪ � 1 − � 2 T , 1 � , � 1/2 −1/2 � Y T 2|u(s, y)|2dvolY ds ≤ C � 1/2 −1/2 � Y |du|2 + |d∗u|2 + (LfT u, u) + |∇fT|2|u2|dvolY ds ≤ C � ¯ M |du|2 + |d∗u|2 + (LfT u, u) + |∇fT |2|u2|dvol ¯ M + C′T � −1+ � 2 T −1 � Y |u|2dvolY ds + C′T � 1 1− � 2 T � Y |u|2dvolY ds ≤ Cλ + C′√ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Assume u meets the same conditions as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, if u|[−2,2] = v1(y, s) + v2(y, s)ds, define Pa(u)(y) = v2(y, −1), Pb(u)(y) = v1(y, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' � Y |Pa(u)|2dvolY ≤ C(λ + 1) √ T , (10) and � Y |Pb(u)|2dvolY ≤ C(λ + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (11) Moreover, if u is an eigenform with respect to an eigenvalue µ ≤ λ, then we also have � Y |Pa(du)|2dvolY ≤ C(µ + 1)2 √ T ≤ C(λ + 1)2 √ T , (12) and � Y |Pb(d∗u)|2dvolY ≤ C(µ + 1)2 √ T ≤ C(λ + 1)2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (13) 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let E(T) := � Y |v2(−1, y)|2dvolY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that on Ω(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F|Y )du, ✷T = − ∂2 ∂s2 + ∆Y + | ∂ ∂sfT |2 + ∂2 ∂s2fT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By repeating previous steps, inf s′∈[0, � 2 T ] � Y |v2(y, s′)|2dvolY ≥ cE(T) − Cλ √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (14) Let η ∈ C∞ c [−2, 2) be a bump function, such that η|[−2,−1/2] ≡ 1, η[−3/4,2) ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since f ′′ T = T ≥ 0 in [−1 + e−T 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' −3/4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' |f ′ T (−1 + e−T 2)| ≤ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' by (14) and integration by parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' when T is big enough,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' c √ TE(T) − Cλ ≤ � −1+ � 2 T −1+e−T 2 � Y T|v2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2dvolY ds ≤ � −1+ � 2 T −1+e−T 2 � Y |dv2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2 + |d∗v2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2 + f ′′ T |v2|2 + |∇fT |2u2dvolY ds ≤ � 2 −1+e−T 2 � Y |dηv2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2 + |d∗ηv2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2 + f ′′ T|ηv2|2 + |∇fT |2η2u2dvolY ds ≤ � 2 −1+e−T 2 � Y |dT ηv2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2 + |d∗ T ηv2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)|2dvolY ds + � Y |dfT||v2|2(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' y)dvolY |s=−1+e−T 2 ≤ � ¯ M |d∗ T u|2 + |dT u|2 + |η′||∂sv2||v2|dvol ¯ M + C(λ + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (15) Let DT = dT + d∗ T , � ¯ M |η′||∂sv2||v2|dvol ¯ M ≤ � ¯ M |η′||DT u||u| + |η′||dfT u||u|dvol ¯ M ≤ � ¯ M |DT u|2 + |u|2 + CT|η′||u|2dvol ¯ M ≤ C(1 + λ) + C � ¯ M T|η′||u|2 ≤ C(λ + 1) (By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (16) According to (15) and (16), E(T) ≤ C(λ+1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one has (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Replace u with dT u and notice that dT u = du on {−1} × Y , (12) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one has (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Suppose u meets the same condition as Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then � 1 −1 � Y |u|2(y, s)dvolY ds ≤ C(λ + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, one can show that � ||s|−1|≤ � 2 T � Y |u|2(y, s)dvolY ds ≤ C(λ + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that on � −1 + � 2 T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 1 − � 2 T � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' f ′′ T + |f ′ T|2 ≥ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' |f ′ T | � ± � 1 − � 2 T �� = √ 2 √ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' integration by parts,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' � 1− � 2 T −1+ � 2 T � Y Tu2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s)dvolY ds ≤ � 1− � 2 T −1+ � 2 T � Y |∇u|2 + |∇fTl|2u2+ < LfT u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' u > dvolY ds ≤ � 1− � 2 T −1+ � 2 T � Y |(dT + d∗ T )u|2dvolY ds + C � Y |(dfT + df ∗ T )u||u|dvolY |s=±(−1+ � 2 T ) ≤ � M |(dT + d∗ T )u|2dvolM + C √ T � Y |u|2dvolY |s=±(−1+ � 2 T ) ≤ C √ T(λ + 1) (By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Thus, � 1− � 2 T −1+ � 2 T � Y |u(s, y)|2dvolY du ≤ C(λ + 1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' lim supT→∞ λk(T) ≤ λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ui = (ui,1, ui,2) be the i-th eigenvalue of ∆1 ⊕∆2 on Ωrel(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1)⊕Ωabs(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2) (1 ≤ i ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let η ∈ C∞ c ([0, 1]), such that η[0,1/4] ≡ 0, η|[1/2,1] ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For any u = (u1, u2) ∈ Ωrel(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) ⊕ Ωabs(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2) , let QT : Ωabs(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) ⊕ Ωrel(M2, F2) → Ω( ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=', QT (u)(x) = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 ui(x), if x ∈ Mi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' η(−s)u(−1, y)e−fT (s)−T , if x = (s, y) ∈ [−1, 0] × Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' η(s)u(1, y)efT (s)−T , if x = (s, y) ∈ [0, 1] × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ¯u = QT (u), then one can see that dim span{¯ui}k i=1 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, by trace formula, one can show that for any u ∈ span{ui}, there exists C1 > 0 � Y |u(0, y)|2 + |∇Y u(0, y)|2dvolY ≤ C1(1 + λ2 k) � M1∪M2 |u|2dvol ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (17) 17 One computes � ¯ M |¯u|2dvol ¯ M ≥ � M1∪M2 |u|2dvol ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then by (17) and the construction of ¯u, one has � ¯ M |∇¯u|2 + |∇fT|2|¯u|2 + (LfT ¯u, ¯u)dvolM = � M1∪M2 |∇u|2dvol + � 0 −1 � Y |∇¯u|2 + |∇fT|2|¯u|2 + (LfT ¯u, ¯u)dvolY ds + � 1 0 � Y |∇¯u|2 + |∇fT |2|¯u|2 + (LfT ¯u, ¯u)dvolY ds = I + II + III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, notice that I ≤ λk � M1∪M2 |u|2dvol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By a straightforward computation and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4, II ≤ � 0 −1 � Y |∇Y u1|2e−2fT (s)−T dvolY ds + C � −1/2 −1/4 � Y T 2|e−T/8u1|2dvolY ds ≤ C2(1 + λ2 k) √ T � M2∪M2 |¯u|2dvol ¯ M ≤ C2(1 + λ2 k) √ T � ¯ M |¯u|2dvol ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, III ≤ C2(1+λ2 k) √ T � ¯ M |¯u|2dvol ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, λk(T) ≤ λk + C2(1+λ2 k) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Consequently, as T → ∞, lim supT→∞ λk(T) ≤ λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' lim infT→∞ λk(T) ≥ λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let {Tl} be a sequence, such that limk→∞ λk(Tl) = lim infT→∞ λk,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ui,l be an eigenform of ∆Tl with norm 1, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' λi(Tl)(1 ≤ i ≤ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, λi(Tl) ≤ Λi for some Λi > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since ∥ui,l |M1∪M2 ∥W N,2(M1)⊕W N,2(M2) ≤ C3(1 + Λk)N∥ui,l∥L2( ¯ M) = C3(1 + Λk)N for any N > 0, by Sobolev’s embedding theorem, we may as well assume that {ui,l|M1∪M2} converges in W 2,2(M1) ⊕ W 2,2(M2)-topology, and assume ui,∞ := liml→∞ ui,l|Mj(j = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, one can also have dim span{ui,∞}k i=1 = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, for any u∞ ∈ span{ui,∞}k i=1 with � M1∪M2 |u∞|2 = 1, one can find ul ∈ span{ui,l}k i=1, such that ul → u∞ in W 2,2(M1) ⊕ W 2,2(M2) topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4, we can see that u∞ ∈ Ωrel(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) ⊕ Ω∗ abs(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5, one has lim l→∞ � ¯ M |ul|2dvol ¯ M = lim l→∞ � M1∪M2 |ul|2dvol ¯ M = � M1∪M2 |u∞|2dvol ¯ M = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (18) 18 As a result, we lim l→∞ λk(Tl) ≥ lim l→∞ � ¯ M (dT ul, dT ul) + (d∗ T ul, d∗ T ul)dvol ≥ lim l→∞ � M1∪M2 (dul, dul) + (d∗ul, d∗ul)dvol ¯ M = � M1∪M2 (du∞, du∞) + (d∗u∞, d∗u∞)dvol ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence lim infT→∞ λk,T ≥ λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one can show that limT→∞ ˜λk,T = λk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 5 Large Time Contributions We will prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let M3 = [−1, 0]×Y , M4 = [0, 1]×Y , then ¯ M = M1∪M2∪M3∪M4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆i,T,bd be the restriction of ∆T on Mi with some boundary conditions(i=1,2,3,4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For bd = rel, abs, D and N, we mean relative, absolute, Dirichlet, and Neumann boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let λk,bd(T) be the k-the eigenvalues of ∆1,T,bd ⊕ ∆2,T,bd ⊕ ∆3,T,bd ⊕ ∆4,T,bd (acting on Ω∗ bd(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1)⊕Ω∗ bd(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2)⊕Ω∗ bd(M3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F|M3)⊕Ω∗ bd(M4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F|M4)⊕), it follows from domain monotonicity of eigenvalues that λk,D(T) ≥ λk(T) ≥ λk,N(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (19) Before moving on, let’s study the following one-dimensional model problem first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆R T,N,± := −∂2 s + |f ′ T |2 ± f ′′ T on [−1, 0] with Neumann boundary conditions, and λR k,T,N,± be the k-th eigenvalues of ∆R T,N,±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For k ≥ 2, one has λR k,T,N,± ≥ vk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here {vk(T)}∞ k=1 is the collection of {T max{c1l−c2, 0}}∞ l=1 ∪{c3l2}∞ l=1, listed in the increasing order and counted with multiplicity, and constants c1, c2 and c3 are independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let I1 := [−1, −1+ 1 √ T ], I2 := [−1+ 1 √ T , 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from the constructions of fT that when T is large enough ∆R T,N,± ≥ −∂2 s on I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' That is, for all φ ∈ C∞(I2) with φ′(−1 + 1 √ T ) = φ′(0) = 0 � I2 ∆R T,N,±φ · φds ≥ � I2 −∂2 sφ · φds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (20) For I1, changing the variable √ T(s + 1) → ˜s, and suppose ∆R T,N,± → ˜∆R T,N,±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then direct computations yields, on [0, 1], 19 ˜∆R T,N,± ≥ T(−∂2 ˜s + ˜s2 − C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (21) The lemma then follows from (20), (21) and domain monotonicity of eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from (19) and Weyl’s law that Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' λk(T) ≥ uk(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here {uk(T)}∞ k=1 is the collection of 4 copies of {vl(T) + c4m2/(dim M−1)}∞ l=1,m=1 and 2 copy of {c5l2/ dim M}, listed in the increasing order and counted with multiplicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, constants c4 and c5 are independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, λk(T) ≥ uk(T) ≥ uk(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let F(t) := dim(M) ∞ � k=1 e−t max{uk(1),δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then F(t)/t ∈ L1((1, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, t−1| Trs � N exp � −t∆′ T � (1 − Pδ) � | ≤ t−1F(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (22) Hence by dominated convergence theorem, lim T→∞ � ∞ 1 t−1 Trs � Ne−t∆′ T (1 − Pδ) � dt = 2 � i=1 � ∞ 1 t−1 Trs(Ne−t∆′ i)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one can show that 2 � i=1 lim T→∞ � ∞ 1 t−1 Trs(Ne−t∆′ i,T )dt = 2 � i=1 � ∞ 1 t−1 Trs(Ne−t∆′ i)dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 6 A Gluing Formula for Heat Trace in Small Time 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Several heat kernels and Laplacians To show the gluing formula for heat trace, we introduce several heat kernels and Laplacians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let KT be the heat kernel for ∆T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let K1⊕2 be the heat kernel for ∆1 ⊕ ∆2, Ki be the heat kernel for ∆i(i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆B,1 be the Hodge Laplacian on [−2, −1] with absolute boundary condi- tions, and kB,1 be the heat kernel for ∆B,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It’s easy to see that ker(∆B,1) is 20 one-dimensional and generated by constant functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since nonzero eigenvalues of ∆B come in pairs, Trs(e−t∆B,1) ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (23) Let ∆B,2 be the usual Hodge Laplacian on [1, 2] with relative boundary condi- tions, and kB,2 be the heat kernel of ∆B,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It can be checked that ker(∆B,2) is one dimensional and generated by the constant one forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since nonzero eigenvalues of ∆B,2 come in pairs, Trs(e−t∆B,2) ≡ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (24) Let ¯∆B be the usual Hodge Laplacian on [−2, 2] with absolute boundary condi- tion on −2, relative boundary condition on 2, and ¯kB be its heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We can also regards fT as a smooth function in (−2, 2), and let ∆R T be the Witten Laplacian on (−2, 2) with respect to fT , with absolute boundary condition on −2, and relative boundary condition on 2, and kT be the heat kernel for ∆R T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' On the restriction of F|Y → Y , let ∆Y be the induced Hodge Laplacian on Ω(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let KY be the heat kernel of ∆Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Gluing heat kernels Then the following lemmas will be needed in the proof of gluing formula for heat kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from a standard argument that Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 (Finite Propagation Speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let s0 be a smooth section of Ω(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) with compact support C, st := exp �√−1tDT � s0, where DT = dT + (dT )∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Ct := {x′ ∈ M : d(x, x′) ≤ 2t}, then the support of st is inside Ct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For a, b ∈ (−2, 2)(a < b), let Ma,b = [a, b] × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Suppose x /∈ M−9/8,−7/8 ∪ M7/8,9/8 and d(x, x′) ≥ 4δ for some δ > 1/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then there exists ck(X), Ck(X) > 0, such that |∇kKT (t, x, x′)| ≤ Cke−ck/t if T ≥ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Choose a bump function φ on R such that φ(λ) = 1, |λ| ⩽ δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' φ(λ) = 0, |λ| ⩾ 2δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let f1, f2 ∈ S(R) such that ˆf1(λ) = (4πt)− 1 2 exp � −λ2/4t � φ(λ), ˆf2(λ) = (4πt)− 1 2 exp � −λ2/4t � (1 − φ(λ)), where S(R) is the Schwartz space on R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since D2 T = ∆T , one can see that f1(DT ) + f2(DT ) = exp (−t∆T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 21 Denote K1 and K2 to be the integral kernel of f1(DT ) and f2(DT ) respectively, then KT (t, x, x′) = K1(t, x, x′) + K2(t, x, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, a standard argument shows that K1(t, x, x′) = 0 if d(x, x′) ≥ 4δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Next, let ˜st(x) := � ¯ M K2(t, x, x′)s(x′)dx′ = 1 √ 4πt � |λ|⩾δ exp � −λ2/4t � [1 − φ(λ)] exp(iλDT )s(x)dλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Next, we may assume that ∆Ts = µs for some µ ∈ R then |2k∆k T ˜st(x)| = 2k ���� � ¯ M ∆k T,xK2(t, x, x′)s(x′)dx′ ���� = ����� 1 √ 4πt � |λ|⩾δ exp � −λ2/4t � [1 − φ(λ)] exp(iλDT )D2k T s(x)dλ ����� |s(x)| ≤ C exp � −δ2/8t − 2tµ2� µ2k|s(x)| ≤ Ck exp � −ckδ2/t � |s(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (25) Here ∆T,x means that the derivative is taken with respect to x, C, Ck and ck are independent of µ and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a consequence, � ¯ M |∆k T,xK(t, x, x′)|2dx ≤ Ck exp � −ckδ2/t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (26) Let ρ ∈ C∞ c (M) be a bump function, such that in ¯ M − M− 33 32 ,− 31 32 ∪ M 33 32 , 31 32 , ρ ≡ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' in M− 17 16 ,− 15 16 ∪ M 15 16 , 17 16 , ρ ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then one still have � ¯ M |∆k T,xρ(x)K(t, x, x′)|2dx ≤ Ck exp � −ckδ2/t � (27) for some other constant Ck and ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that when restricted in ¯ M −M− 9 8 ,− 7 8 −M 7 8 , 9 8 , ∆T ≥ ∆ if T is big enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here ∆T ≥ ∆ on some open subset U ⊂ ¯ M means that for all φ ∈ Ω(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F) with compact support inside U, ⟨∆T φ, φ⟩L2 ≥ ⟨∆φ, φ⟩L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from G˚arding’s inequality and Sobolev’s embedding theorem that |∇kρ(x)KT (t, x, x′)| ≤ Cke−ckδ2/t, where Ck, ck are independent of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 22 Let ηi(i = 1, 2) be a smooth function on (−∞, ∞) satisfying 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 0 ≤ ηi ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' η1 ≡ 1 in (−∞, −3/2),η1 ≡ 0 in (−5/4, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' η2 ≡ 1 in (3/2, ∞),η2 ≡ 0 in (−∞, 5/4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let η be an even function, such that η|[0,∞) ≡ η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We can think ηi as a functions on Mi(i = 1, 2) and η as a function on ¯ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' There exists T-independent C, c > 0, such that for t ∈ (0, 1), | Trs(N(1 − η)e−t∆T ) − Trs(N(1 − η)e−t∆R T ⊗ e−t∆Y )| ≤ Ce−c/t, | Trs(Nηie−t∆R T ⊗ e−t∆Y ) − Trs(Nηie−t ¯∆B ⊗ e−t∆Y )| ≤ Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let f1 and f2 be functions constructed in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 for some T-independent and small δ > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let uk be a normal eigenform of λk(T) w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∆T, then for any j ∈ Z+, |f2(s)| ≤ Cje−c/t (|s|+1)j , hence |(f2(∆T )uk, uk)L2| ≤ Cje−c/t (|λk(T)| + 1)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (28) By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and (28), | Trs(N(1 − η)e−t∆T )| ≤ Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (29) Similarly, | Trs(N(1 − η)e−t∆R T ⊗ e−t∆Y )| ≤ Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (30) Let M′ 1 = M1 − (−2, −1] × Y and M′ 2 = M2 − [1, 2) × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since L2( ¯ M) = L2(M′ 1) ⊕ L2([−2, 2] × Y ) ⊕ L2(M′ 2), let {uk}, {wk} and {vk} be an orthonormal basis of L2(M′ 1), L2([−2, 2] × Y ) and L2(M′ 2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 and the maximal principle, if δ is small enough, (f1(∆T )(1 − η)uk, uk) = (f1(∆T )(1 − η)vk, vk) = 0, (31) and (f1(∆T )(1 − η)wk, wk) = (f1(∆R T ⊕ ∆Y )(1 − η)wk, wk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (32) Since e−ts2 = f1(s) + f2(s), by (29), (30), (31) and (32) | Trs(N(1 − η)e−t∆T ) − Trs(N(1 − η)e−t∆R T ⊗ e−t∆Y )| ≤ Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, | Trs(Nηie−t∆R T ⊗ e−t∆Y ) − Trs(Nηie−t ¯∆B ⊗ e−t∆Y )| ≤ Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 23 Similarly, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' There exists T-independent C, c > 0, such that for t ∈ (0, 1), | Trs(N(1 − ηi)e−t∆i) − Trs(N(1 − ηi)e−t∆B,i ⊗ e−t∆Y )| ≤ Ce−c/t, | Trs(Nηie−t∆B,i ⊗ e−t∆Y ) − Trs(Nηie−t ¯∆B ⊗ e−t∆Y )| ≤ Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 Heat trace expansion for e−t(∆1⊕∆2) It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 that for some C, c > 0, t ∈ (0, 1], �����Trs(Ne−t(∆1⊕∆2)) − 2 � i=1 Trs(Nηie−t(∆i)) + 2 � i=1 � Trs(Nηi(x)e−t ¯∆B⊗e−t∆Y ) − Trs(Ne−t∆B,i ⊗ e−t∆Y ) ������ ≤ C exp(−c/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (33) Next, notice that on M−2,−1, the number operator can be decomposed as N = N Y + N R canonically (Here N Y and N R are the number operator on Y and R components respectively), Trs(Ne−t∆B,i ⊗ e−t∆Y ) = Trs(N Re−t∆B,i ⊗ e−t∆Y ) + Trs(e−t∆B,i ⊗ N Y e−t∆Y ) = Trs(N Y e−t∆Y ) Trs(e−t∆B,i) + Trs(N Re−t∆R B,1)χ(Y )rank(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (34) As a result, by (23) and (24) 2 � i=1 Trs(Ne−t∆B,i ⊗ e−t∆Y ) = Trs(N Re−t∆R B,1)χ(Y )rank(F) + Trs(N Re−t∆R B,2)χ(Y )rank(F) = Trs(N Re−t∆R B,1)χ(Y )rank(F) + Trs � (N R − 1)e−t∆R B,2 � χ(Y )rank(F) − χ(Y )rank(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (35) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 Heat trace expansion for e−t∆T It follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 that �����Trs(Ne−t(∆T )) − 2 � i=1 Trs(Nηie−t(∆i)) + 2 � i=1 � Trs(Nηi(x)e−t ¯∆B⊗e−t∆Y ) − Trs(Ne−t∆R T ⊗ e−t∆Y ) ������ ≤ C exp(−c/t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (36) 24 Hence Trs(Ne−t∆R T ⊗ e−t∆Y ) = Trs(N Y e−t∆Y ) Trs(e−t∆R T ) + Trs(N Re−t∆R T ) Trs(e−t∆Y ) = Trs(N Y e−t∆Y ) Trs(e−t∆R T ) + Trs(N Re−t∆R T )χ(Y )rank(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (37) Since nonzero eigenforms comes in pairs, Trs(e−t∆R T ) = dim(ker(∆R T )0)−dim(ker(∆R T )1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here ker(∆R T )i denotes the space of harmonic i-forms(i = 0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since fT is odd, one can see easily that if u(s) ∈ ker(∆R T )0, then u(−s)ds ∈ ker(∆R T )1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, Trs(e−t∆R T ) = 0, which implies that Trs(Ne−t∆R T ⊗ e−t∆Y ) = Trs(N Re−t∆R T )χ(Y )rank(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (38) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 when M is even dimensional First, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Trs(Ne−t∆T ) − Trs(Ne−t(∆1⊕∆2)) = Trs(Ne−t∆′ T (1 − P δ)) − Trs(Ne−t(∆1⊕∆2)′) + o(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (39) When M is even dimensional, χ(Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from (33), (35), (36), (38), (39) and dominated convergence theorem that lim T→∞ � 1 0 t−1| Trs(Ne−t∆T ) − Trs(Ne−t(∆1⊕∆2))|dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, (ζS T,la)′(0) = 2 � i=1 (ζS i )′(0) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, (ζS i,T)′(0) = (ζS i )′(0) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 7 One-dimensional Model and Coupling Tech- niques To show Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 when M is odd dimensional, it suffices to compare Trs(N Re−t∆R T ) and Trs(N Re−t(∆R B,1⊕∆R B,2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='To this end, we explore the one-dimensional model first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let VT,± = |f ′ T |2 ± f ′′ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Recall that kT (t, s, s′) is the heat kernel for Witten Laplacian ∆R T on (−2, 2) with the absolute boundary condition on −2 and the relative boundary condition on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then restricted on functions, ∆R T = ∆R T,− := −∂2 s+VT,− with the Neumann boundary condition on −2 and the Dirichlet boundary condition on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Restricted on 1-forms ∆R T = ∆R T,+ := −∂2 s + VT,+ with Dirichlet boundary condition on −2 and Neumann boundary condition on 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Heat trace estimate when t and T are coupled Let λk,T,± be the k-th eigenvalue with respect to ∆R T,±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Assume φk,T,± is a k-th eigenfunction, such that ∥φk,T,±∥L2 = 1 and {φk,T,±} forms a complete orthonormal basis of L2([−2, 2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ˜T = t−5T, kT,± be the heat kernel for ∆R T,±.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In this section, we assume that t ∈ (0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' If t ∈ (0, 1], � 1 −1 |k ˜T,±(t, s, s)|ds ≤ Ct √ T for some constant C that doesn’t depends on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5, � 1 −1 |φk, ˜T,±|2(s)ds ≤ C(1 + λk, ˜T,±) � ˜T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (40) It follows from (40), Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and the domain monotonicity of eigenvalues that for fixed a ∈ (0, 1) � 1 −1 k ˜T,±(t, s, s)ds = � 1 −1 � k e−tλk, ˜ T ,+|φk, ˜T,+|2ds ≤ C � k≥1 e−tλk, ˜ T ,+ 1 + λk, ˜T,+ � ˜T ≤ C � k≥1 e−atλk, ˜ T ,+ 1 t � ˜T ≤ Ct √ T , where C is independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For s ∈ (−2, −1), t ∈ (0, 1), and T is large, |kT,+|(t, s, −1) ≤ min{ C tT 1/4 + C t3/2T 5/4 , C √ t};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' for s ∈ (1, 2), |kT,−|(t, s, 1) ≤ min{ C tT 1/4 + C t3/2T 5/4 , C √ t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here the constant C doesn’t depend on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' |kT,+|(t, s, −1) ≤ C √ t : It follows from the construction of fT that there exists C > 0, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' −∂2 s − CT ≤ ∆R T ≤ −∂2 s + CT 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 26 Let ¯kB,1 be the restriction of ¯kB on 1-foms, then it follows from the maximal principle that 0 ≤ e−CT 2t¯kB,1 ≤ kT,+ ≤ eCtT ¯kB,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (41) Let ρ ∈ C∞ c [−2, ∞) be a nonnegative function, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ρ ≡ 1 on (−∞, −1/8), ρ ≡ 0 on (−1/16, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since (∂t + ∆R T,s′)ρ(s′)¯kB,1(t, s, s′) = −2ρ′(s′)∂s′¯kB,1(t, s, s′) − ρ′′(s′)¯kB,1(t, s, s′) + ρ(s′)VT,+(s′)¯kB,1(T, s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here ∆R T,s′ means that the derivative is taken w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Set h(t, s, s′) = −2ρ′(s′)∂s′¯kB,1(t, s, s′)−ρ′′(s′)¯kB,1(t, s, s′)+ρ(s′)VT,+(s′)¯kB,1(T, s, s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since VT,+ ≥ 0 on [−1 + eT 2, −1/16], it follows from Duhamel’s principle, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and (41) that for s ∈ (−2, −1), 0 ≤ kT,+(t, s, −1) = ¯kB,1(t, s, −1) − � t 0 � 2 −2 kT,+(t − t′, s, s′)h(t′, s′, −1)ds′dt′ ≤ ¯kB,1(t, s, −1) + � t 0 � −1+e−T 2 −1 eCtT ¯kB,1(t − t′, s, s′)¯kB,1(t′, s′, −1)ds′dt′ + � t 0 � −1/16 −1/8 Ce−c/(t−t′)(¯kB,1(t′, s′, −1) + ∂s′¯kB,1(t′, s′, −1))ds′dt′ ≤ ¯kB,1(t, s, −1) + C � t 0 e−T 2eCtT 1 √ t − t′√ t′ ds′dt′ + C � t 0 e−c/(t−t′)e−c/t′dt′ ≤ C′ √ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' |kT,+|(t, s, −1) ≤ C tT 1/4 + C t3/2T 5/4 : Let φk,T,+ be a united eigenfunction for λk,T,+, then φk,T,+ satisfies � −φ′′ k,T,+ = λk,T,+, in (−2, −1) φk,T,+(−1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence φk,T,+(s) = ck sin(λk,T,+(s + 2)) in (−2, −1) for some ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2c2 kλk,T,+ + sin(2λk,T,+) 4λk,T,+ = � −1 −2 |φk,T,+|2 ≤ � −2 −2 |φk,T,+|2 = 1 implies that ck ≤ C for some constant C that doesn’t depend on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, |φk,T,+|(s) ≤ C if s ∈ (−2, −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 implies that |φk,T,+|2(−1) ≤ C(λk,T,++1) √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, for a fixed a ∈ (0, 1), 27 kT,+(t, s, −1) = � k e−tλk,T,+φk,T,+(s)φk,T,+(−1) ≤ C � k e−tλk,T,+ � λk,T,+ + 1 T 1/4 ≤ C � k e−atλk,T,+ 1 √ tT 1/4 ≤ C tT 1/4 + C t3/2T 5/4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The second inequality could be proved similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Recall that ∆R B,1 is the usual Hodge Laplacian on [−2, −1] with absolute bound- ary conditions, and kB,1 is the heat kernel of ∆R B,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let kB,1,+ be the restriction of kB,1 on 1-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then kB,1,+(t, s, −1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' While ∆R B,2 is the usual Hodge Lapla- cian on [1, 2] with relative boundary conditions, kB,2 be the heat kernel of ∆R B,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let kB,2,− be the restriction of kB,2 on functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then kB,2,−(t, s, 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For t ∈ (0, 1], T is large enough, � −1 −2 |k ˜T ,+ − kB,1,+|(t, s, s)ds ≤ Ct0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='05 + Ce−c/t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' � 2 1 |k ˜T,− − kB,2,−|(t, s, s)ds ≤ Ct0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='05 + Ce−c/t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here constants C and c are independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4, it suffices to estimate � −1 −9/8 |k ˜T,+−kB,+|(t, s, s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, one can compute directly that |∂s′kB,1,+(t, s, s′)|s′=−1| ≤ C(s+1)e−(s+1)2/t t3/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from Duhamel principle and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 that for s ∈ (−9/8, −1) |k ˜T ,+ − kB,1,+|(t, s, s) ≤ � t 0 ���∂s′kB,1,+(t − t′, s, s′)|s′=−1kT,+(t′, s, −1) ���dt′ ≤ C′ � t 0 (s + 1)e−(s+1)2/t′ (t − t′)3/2 ��kT,+(t′, s, −1) �� dt′ = J1(t, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here constants C and C′ are independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 28 Hence, � −1 −9/8 J1(t, s)ds ≤ C � −1 −9/8 � t 0 (s + 1)e−(s+1)2/t′ (t − t′)3/2 |k ˜T ,+(t′, s, −1)|dt′ds ≤ C � t 0 � −1 −9/8 (s + 1)e−(s+1)2/t′ (t − t′)3/2 |k ˜T ,+(t′, s, −1)|dsdt′ ≤ C � t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 0 � −1 −9/8 (s + 1)e−(s+1)2/t′ (t − t′)3/2 |k ˜T ,+(t′, s, −1)|dsdt′ + C � t t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 � −1 −9/8 (s + 1)e−(s+1)2/t′ (t − t′)3/2 |k ˜T,+(t′, s, −1)|dsdt′ = I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' While by Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, I1 ≤ C � t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 0 � −1 −9/8 (s + 1)e−(s+1)2/t′ (t − t′)3/2 1 √ t′ dsdt′ ≤ C � t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 0 1 √ t − t′√ t′ dt′ ≤ C √ t � t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 0 1 √ t′ dt′ = Ct0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 again, I2 ≤ C � t t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 � −1 −9/8 s(s + 1)e−(s+1)2/t′ (t − t′)3/2 � 1 t′ ˜T 1/4 + 1 t′3/2 ˜T 5/4 � dsdt′ ≤ C � t t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 1 √ t − t′ � 1 t′ ˜T 1/4 + 1 t′3/2 ˜T 5/4 � dt′ ≤ C( t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='05 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='05 + t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='45 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='95 ) � t t1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2/T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 1 √ t − t′ dt ≤ Ct0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='55 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, � 2 1 |k ˜T,− − kB,1,−|(t, s, s)ds ≤ Ct0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='55 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='05 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ���� � 2 1 trs(k ˜T (t, s, s))ds + 1 ���� ≤ Ce−ct + Ct √ T for some constants C and c that don’t depend on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let pT be a family of smooth functions on (−∞, 2], such that (a) pT |[1,2] ≡ T/2, 29 (b) pT |(−∞,1](s) = −Tρ(eT 2(1 − s))(s − 1)2/2 + T/2 , where ρ ∈ C∞ c ([0, ∞)), such that 0 ≤ ρ ≤ 1, ρ[0,1/2] ≡ 0, ρ[3/4,∞] ≡ 1, |ρ′| ≤ δ1 and |ρ′′| ≤ δ2 for some universal constant δ1 and δ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ¯∆T be the Witten Laplacian w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' pT , ¯kT be its heat kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ¯λk(T) be the k-th eigenvalue of ¯∆T , and λB,2,k be the k-th eigenvalue of ∆B,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='Then repeating what we did before, we still have ¯λk(T) → λB,2,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Thus when T is big enough, ker( ¯∆T ) is one dimensional and generated by a 1-form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since nonzero eigenvalues of ¯∆T come in pairs, Trs(e−t ¯∆T ) = � 2 −∞ trs(¯kT (t, s, s))ds = −1 (42) if T is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, one still have Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 for ¯∆T , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=', � 1 −∞ ¯k ˜T (t, s, s)ds ≤ Ct √ T for some constant C that doesn’t depend on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, by (42) � 2 1 | trs(¯k ˜T )(t, s, s) + 1|ds ≤ Ct √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, since pT = fT on [1/2, 2], it follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and Duhamel’s principle that for s ∈ [1/2, 2], |¯kT (t, s, s) − kT (t, s, s)| ≤ Ce−c/t for some constants c and C that don’t depend on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Thus, the lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Partial proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 when M is odd dimen- sional By Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3, Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 and dominated convergence theo- rem, lim T→∞ � 1 0 t−1| Trs(N Re−t∆R t−5T ) − Trs(N Re−t(∆R B,1⊕∆R B,2))|dt = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (43) Let ˜ζT (z) := 1 Γ(z) � 1 0 tz−1(Trs(N Re−t∆R T ) − Trs(N Re−t∆R t−5T ))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By (33), (35), (36), (38), (39), and (43), one can see that Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As T → ∞, (ζS T,la)′(0) = 2 � i=1 (ζS i )′(0) + ˜ζ′ T(0)χ(Y )rank(F) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In particular, if dim(Hk(M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F)) = dim(Hk abs(M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1)) + dim(Hk rel(M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2)) for all k, then ζT,la = ζT whenever T is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and the equality above, log T (gT ¯ M, h ¯F , ∇ ¯F)(T) = log Tabs(gTM1, hF1, ∇F1) + log Trel(gTM2, hF2, ∇F2) + χ(Y )rank(F)˜ζ′ T (0)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 30 Similarly, let ˜ζT,i(z) := 1 Γ(z) � 1 0 tz−1(Trs(N Re−t∆R T,i) − Trs(N Re −t∆R t−5T,i))dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As T → ∞, log Ti(gT ¯ Mi, h ¯Fi, ∇ ¯Fi)(T) = log Ti(gTMi, hFi, ∇Fi) + χ(Y )rank(F)˜ζ′ T,i(0)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' To prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3, it is necessary to show that ˜ζ′ T (0) = log(2) − T + o(1) and ˜ζ′ T,i(0) = −T/2 + o(1), which will be accomplished in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' More precisely, see Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 8 Ray-Singer metrics on S1 and [−2, 2] 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 Ray-Singer metrics on S1 First, let pT ∈ C∞(R) be a smooth function with period 8 as follows 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' pT (s) = fT(s), ∀s ∈ [−2, 2];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' pT (s) = fT(4 − s), ∀s ∈ [2, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let S1(8) denote the circle with length 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Then we regard S1(8) as the interval [−2, 6] with −2 and 6 identified, so we could think pT as a smooth function on S1(8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dT = d + dpT ∧ be the Witten deformation of de Rham differentials on S1(8), and ∆S1 T := dT d∗ T + d∗ T dT be its Witten Laplacian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let H(S1(8), dT ) be the cohomology for dT , and | · |RS,T be the L2-metric on det � H∗(S1(8), dT ) � induced by ∆S1 T -harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that τ : H∗(S1(8), d) → H∗(S1(8), dT ), [w] �→ [e−pT w] is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T (S1)(T) be the analytic torsion for ∆S1 T , ∥ · ∥RS,T = | · |RS,TT (S1)(T) be the associated Ray-Singer metric on det � H∗(S1(8), dT ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' log T (S1)(0) = log T (S1)(T) − 2 log(2) + T + o(1) as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' It follows from [1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='7] that ∥ · ∥RS,0 = τ ∗∥ · ∥RS,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' To show log T (S1)(0) = log T (S1)(T) − 2 log(2) + T + o(1), it suffices to show log | · |RS,0 = log τ ∗| · |RS,T + 2 log(2) − T − o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let α(T) := � S1(8) e2pT ds, then [e2pT ds] = [α(T)ds/8] in H∗(S1(8), d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, α(T)ds is ∆S1 0 -harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ρT := [1] ⊗ [e2pT ds]−1 ∈ det � H∗(S1, d) � , then one computes |ρT |RS,0 = 8 α(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 31 On the other hand, τ[1] = [e−pT ], τ[e2pT ds] = [epT ds].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since e−pT and epT ds are both ∆S1 T -harmonic, τ ∗|ρT |RS,T = |τ(ρT )|RS,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here the last inequality follows from the fact that fT is an odd function on [−2, 2], thus � S1(8) e−2pT ds = � S1(8) e2pT ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, log | · |RS,0 = log τ ∗| · |RS,T − log(α(T)) + 3 log 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' A simple calculation yields α(T) = 2eT (1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' The lemma follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆[0,2] 1 be the usual Hodge Laplacian on Ω([0, 2]) with absolute boundary con- ditions, ∆[0,2] 2 be the usual Hodge Laplacian on Ω([0, 2]) with relative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T ([0, 2], i) be the analytic torsion with respect to ∆[0,2] i (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By a straightforward computation (recall that S1has length 8), log T (S1)(0) = −3 log(2), log T ([0, 2], i) = − log(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (44) Hence, by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 and (44), log T (S1)(T) = − log(2) − T + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' While by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 and (44) log T (S1)(T) = −2 log(2) + ˜ζ′ T(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ˜ζ′ T (0) = log(2) − T + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 Ray-Singer metric on [−2, 2] Let qT be a smooth even function on [−2, 2], such that for s ∈ [−2, 0], pT (s) = fT(s − 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let dT,2 = d + dqT ∧ be the Witten deformation of de Rham differentials on [−2, 2], and ∆[−2,2] T,2 be its Witten Laplacian with relative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let Hrel([−2, 2], dT,2) be the cohomology w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' dT,2, and | · |RS,T,2 be the L2-metric on det(H∗ rel([−2, 2], dT,2)) induced by ∆R T,2-harmonic forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that τ2 : H∗ rel([−2, 2], d) → H∗ rel([−2, 2], dT ), [w] �→ [e−qT w] is an isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T2(T) be the analytic torsion for ∆R T,2, ∥ · ∥RS,T,2 = | · |RS,T,2T2(T) be the associated Ray-Singer metric on det(H∗ rel([0, 2], dT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, for dT,1 := d − dqT , let ∆R T,2 be its Witten Laplacian with absolute boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T1(T) be the analytic torsion for ∆R T,1, ∥ · ∥RS,T,1 = | · |RS,T,1T1(T) be the associated Ray-Singer metric on det(H∗ abs([−2, 0], dT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that τ1 : H∗ abs([−2, 2], d) → H∗ abs([−2, 2], dT ), [w] �→ [eqT w] is an isomor- phism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' We have the following lemma, the proof is an easy exercise for an expert (Just notice that fT(2) = fT (−2) = 0), 32 Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ∂T log ∥ · ∥RS,T,i = 0, i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' log Ti(0) = log Ti(T) + T/2 − log(2)/2 + o(1), i = 1, 2 as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3, it suffices to show �2 i=1 log | · |RS,0,i = �2 i=1 log τ ∗| · |RS,T,i − 2T + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' When i = 2: Let α(T) := � 2 −2 e2qT ds, then [e2qT ds] = [α(T)ds/4] in H∗ rel([−2, 2], d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, α(T)ds is ∆[−2,2] 2 harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here ∆[−2,2] 2 is the restriction of −∂2 s on [−2, 2] with relative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ρT,2 := [e2qT ds]−1 ∈ det(H∗ rel([−2, 2], d)), then one computes |ρT,2|RS,0,2 = 2 α(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' On the other hand, τ2[e2qT ds] = [eqT ds].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since eqT ds is ∆[−2,2] T,2 harmonic, τ ∗ 2 |ρT,2|RS,T,2 = |τ2(ρT,2)|RS,T,2 = 1 �� 2 −2(eqT )2ds = 1 � α(T) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, log | · |RS,0,2 = log τ ∗ 2 | · |RS,T,2 − log(α(T))/2 + log(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' A simple calculation yields α(T) = 2eT (1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, log | · |RS,0,2 = log τ ∗ 2 | · |RS,T,2 − T/2 + log(2)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' When i = 1: Notice that the constant function 1 is ∆[−2,2] 1 harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here ∆[−2,2] 1 is the restriction of −∂2 s on [−2, 2] with relative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ρT,1 := [1] ∈ det(H∗ abs([−2, 2], d)), then one computes |ρT,1|RS,0,1 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' On the other hand, τ1[1] = [epT ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since epT is ∆[−2,2] T,1 harmonic, τ ∗ 1 |ρT,1|RS,T,1 = |τ(ρT,1)|RS,T,1 = �� 2 −2 (eqT )2ds = � α(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, log | · |RS,0,1 = log τ ∗| · |RS,T,1 − log(α(T))/2 + log(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 33 A simple calculation yields α(T) = 2eT (1 + o(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, log | · |RS,0,1 = log τ ∗| · |RS,T,1 − T/2 + log(2)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ∆[−2,2] 1 be the usual Hodge Laplacian on Ω([−2, 2]) with absolute boundary con- ditions, ∆[−2,2] 2 be the usual Hodge Laplacian on Ω([−2, 2]) with relative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T ([−2, 2], i) be the analytic torsion with respect to ∆[−2,2] i (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, let ∆[−1,1] 1 be the usual Hodge Laplacian on Ω([−1, 1]) with absolute boundary conditions, ∆[−1,1] 2 be the usual Hodge Laplacian on Ω([−1, 1]) with rel- ative boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let T ([−1, 1], i) be the analytic torsion with respect to ∆[−1,1]i(i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By a straightforward computation log T ([−2, 2], i) = −3 log(2)/2, log T ([−1, 1], i) = − log(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (45) Hence, by Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 and (45), log Ti(T) = −3 log(2)/2 − T/2 + log(2)/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' While by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 and (45) log Ti(T) = − log(2) + ˜ζ′ i,T (0) + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Hence, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ˜ζ′ T,i(0) = −T/2 + o(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 9 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4 Let ˜Ωsm( ¯ M, ¯F)(T) be the space generated by eigenforms of ˜∆T for eigenvalues inside [0, δ], then by our discussion above in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, ˜Ωsm( ¯ M, ¯F)(T) = efT Ωsm( ¯ M, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' And ˜Pδ(T) := efT Pδ(T)e−fT is the orthogonal projection w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ˜Ωsm( ¯ M, ¯F)(T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let η ∈ C∞ c ([0, 1]), such that η|[0,1/4] ≡ 0, η|[1/2,1] ≡ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For u = (u1, u2) ∈ ker (∆1) ⊕ ker (∆2), recall that QT : Ωabs (M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F1) ⊕ Ωrel (M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' F2) → Ω( ¯ M, ¯F) is QT (u)(x) := \uf8f1 \uf8f2 \uf8f3 ui(x), if x ∈ Mi, η(−s)u(−1, y)e−fT (s)−T/2, if x = (s, y) ∈ [−1, 0] × Y, η(s)u(1, y)efT (s)−T/2, if x = (s, y) ∈ [0, 1] × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' And let ˜QT = efT QT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For any L2-form w support on Mi (or ¯ Mi), let E(w) be an extension of w, such that E(w) is an L2-form on ¯ M and outside Mi (or ¯ Mi), E(w) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' One can see that 34 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For u ∈ ker (∆1) ⊕ ker (∆2), ∥QT u − E(u)∥2 L2 ≤ C √ T ∥u∥2 L2, ��Pδ(T)QT (u) − E(u) ��2 L2 ≤ C∥u∥2 L2 √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' for some constant C that is independent of T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, ��� ˜QT u − E(efT u) ��� 2 L2,T ≤ C √ T ∥efT u∥2 L2,T, ��� ˜Pδ(T) ˜QT (u) − E(efT u) ��� 2 L2,T ≤ C∥efT u∥2 L2,T √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Recall that ∥ · ∥L2,T is the norm induced by gT ¯ M and h ¯F T := e−2fT h ¯F .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, when T is large enough, ˜Pδ(T) ˜QT (u) spans ˜Ωsm( ¯ M, ¯F)(T) for u ∈ ker (∆1) ⊕ ker (∆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For u ∈ ker (∆1) ⊕ ker (∆2), set uT = Pδ(T)QT (u), vT = QT (u) − uT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, by trace theorem and G˚arding’s inequality, � Y ��ui � (−1)i, y ���2 dvolY ≤ C � Mi |ui|2 + |∇ui|2 dvolMi ≤ C′ � Mi |ui|2 dvolMi (46) for some constants C, C′ that doesn’t depends on T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By (46) and a straightforward computation, one can see that ∥QT u − E(u)∥2 L2 ≤ C √ T ∥u∥2 L2 and ∥DT QT u∥2 L2 ≤ C √ T ∥u∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (47) Here DT := dT + d∗ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, δ ∥vT ∥2 L2 ≤ ∥DT vT ∥2 L2 ≤ ∥DT QT u∥2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (48) (47) and (48) then imply that ∥vT ∥2 L2 ≤ C δ √ T ∥u∥2 L2 i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=', ���Pδ(T)QT (u) − QT (u) ��� 2 L2 ≤ C∥u∥2 L2 δ √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Notice that if u ∈ Ωbd(Mi, Fi), then QTu ∈ Ωbd( ¯ Mi, ¯Fi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' By Hodge theory and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, when T is big enough, all eigenvalues of ˜∆T,i inside [0, δ] must be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ˜PT,i be the orthogonal projection w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ker( ˜∆T,i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, one has 35 Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For u ∈ ker (∆i), ��� ˜QT u − E(efT u) ��� 2 L2,T ≤ C √ T ∥efT u∥2 L2,T, ��� ˜PT,i ˜QT (u) − E(efT u) ��� 2 L2,T ≤ C∥efT u∥2 L2,T √ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' As a result, when T is large enough, ˜Pδ(T) ˜QT (u) spans H( ¯ Mi, ¯Fi)(T) for u ∈ ker (∆i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, notice that when T is large enough, we have a sequence of maps 0 → Hk( ¯ M2, ¯F2)(T) ˜ek,T → ˜Ωk sm( ¯ M, ¯F)(T) ˜rk,T → Hk( ¯ M1, ¯F1)(T) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (49) Here ˜ek,T is given by u �→ ˜Pδ(T)E(u) for all u ∈ ker( ˜∆T,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' And ˜rk,T is given by u �→ ˜PT,1(u| ¯ M1) for all u ∈ ˜Ωk sm( ¯ M, ¯F)(T), where ˜PT,i : L2Ω( ¯ Mi, ¯Fi) → ker(∆T,i) is the orthogonal projection (i = 1, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ˜ek,T and ˜r∗ k,T are almost isometric embeddings as T → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' That is, for example, for any u ∈ Hk( ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2)(T), limT→∞ ∥˜ek,T u∥L2,T ∥uT ∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Here ˜r∗ k,T is the adjoint of ˜rk,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ˜ek,T is almost isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For any u ∈ Hk( ¯ M1, ¯F1)(T), there exists uT ∈ ker(∆1) ∩ Ωk rel(M1, F1) such that u = ˜PT,i ˜QT (uT ), then by Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2, ∥u∥2 L2,T ≥ (1 − C √ T )∥efT uT ∥2 L2,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (50) While ∥˜ek,T u∥2 L2,T = ∥ ˜Pδ(T)u∥2 L2,T ≤ ∥ ˜Pδ(T)QT uT ∥2 L2,T + C∥efT uT ∥2 L2 √ T (By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2 and the fact that ∥ ˜Pδ(T)∥ = 1) ≤ ∥efT uT ∥2 L2,T(1 + C′ √ T ) (By Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (51) It follows from (50) and (51) that lim sup T→∞ ∥˜ek,T u∥2 ∥u∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, lim inf T→∞ ∥˜ek,T u∥2 ∥u∥L2,T = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 36 ˜r∗ k,T is almost isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' For u ∈ Hk( ¯ M2, ¯F2)(T), we first show that ˜r∗ k,Tu = ˜Pδ(T)E(u) : Notice that for any v ∈ ˜Ωsm( ¯ M, ¯F)(T), (˜rk,T v, u)L2( ¯ M2),T = (v, E(u))L2( ¯ M),T = (v, ˜Pδ(T)E(u))L2( ¯ M),T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Following the same steps as above, one derives that ˜r∗ k,T is almost isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' With maps ˜ek,T and ˜rk,T given above, the sequence (49) is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Let ⟨·, ·⟩T be the pointwise inner product induced by gT ¯ M and h ¯F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In follows from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 that ˜ek,T and ˜r∗ k,T are injective when T is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' E(ker( ˜∆T,1)) ⊂ ker(d ¯F ,∗ T ) and E(ker( ˜∆T,2)) ⊂ ker(d ¯F ): Let u ∈ ker( ˜∆T,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, since u satisfies relative boundary conditions, integra- tion by parts shows that for any β ∈ Ω( ¯ M, ¯F), � ¯ M⟨E(u), d ¯F ,∗ T β⟩T dvol = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Thus, E(u) ∈ ker(d ¯F ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Similarly, E(ker( ˜∆T,1)) ⊂ ker(d ¯F ,∗ T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' im ˜ek,T = ker ˜rk,T : For the dimension reason, it suffices to show that im˜ek,T ⊂ ker ˜rk,T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' That is, it suffices to show im˜ek,T ⊥ im˜r∗ k,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' First, for any ui ∈ ker( ˜∆T,i), i = 1, 2, it’s clear that (E(u1), E(u2))L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (52) Since E(u1) ∈ ker(d ¯F ,∗ T ), E(u2) ∈ ker(d ¯F ), one can see that (1 − ˜Pδ(T))u1 ∈ im d ¯F ,∗ T , (1 − ˜Pδ(T))u1 ∈ im d ¯F , which means ((1 − ˜Pδ(T))E(u1), (1 − ˜Pδ(T))E(u2))L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (53) By (52) and (53), (˜ek,T u2, ˜r∗ k,Tu1)L2,T = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Moreover, we have the following complexes of finite dimensional vector spaces 0 → H0( ¯ Mi, ¯Fi) 0→ H1( ¯ Mi, ¯Fi) 0→ · · · 0→ Hdim M( ¯ Mi, ¯Fi) → 0 (54) 0 → ˜Ω0 sm( ¯ M, ¯F) d ¯ F → ˜Ω1 sm( ¯ M, ¯F) d ¯ F → · · · d ¯ F → ˜Ωdim M sm ( ¯ M, ¯F) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' (55) Integration by parts as in the proof of Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1, one can show easily that Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' d ¯F ◦ ˜ek,T = 0, ˜rk,T ◦ d ¯F = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 37 Hence, by Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 and Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4, we get the following long exact sequence again MV(T) : · · · ∂k−1,T → Hk � ¯ M2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F2 � (T) ek,T → Hk � ¯ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F � (T) rk,T → Hk � ¯ M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' ¯F1 � (T) ∂k,T → · · · (56) with metric induced by gT ¯ M and h ¯F T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Since (54) and (55) are complexes of finite dimensional vec- tor spaces, it follows from Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='3 and [7, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='6] (or [9, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='1 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='2]) that lim T→∞ log Tsm(gT ¯ M, h ¯F , ∇ ¯F )(T) − log T (T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Bismut and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' An extension of a theorem by Cheeger and M¨uller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ast´erisque, 205, 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Br¨uning and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' An anomaly formula for Ray–Singer metrics on mani- folds with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Geometric & Functional Analysis, 16(4):767–837, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [3] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Br¨uning and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' On the gluing formula for the analytic torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Mathe- matische Zeitschrift, 273(3):1085–1117, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [4] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Cheeger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Analytic torsion and the heat equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Annals of Mathematics, 109(2):259–321, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [5] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Dai and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Witten deformation for noncompact manifolds with bounded geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Journal of the Institute of Mathematics of Jussieu, pages 1–38, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [6] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Dai and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Witten deformation on non-compact manifold: Heat kernel expansion and local index theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Mathematische Zeitschrift, 303(1), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lesch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' A gluing formula for the analytic torsion on singular spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Analysis & PDE, 6(1):221–256, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [8] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' L¨uck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Analytic and topological torsion for manifolds with boundary and symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Journal of Differential Geometry, 37(2):263–322, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [9] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Milnor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Whitehead torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Bulletin of the American Mathematical Society, 72(3):358–426, 1966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [10] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' M¨uller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Analytic torsion and R-torsion of Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Advances in Mathematics, 28(3):233–305, 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [11] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' M¨uller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Analytic torsion and R-torsion for unimodular representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Journal of the American Mathematical Society, 6(3):721–753, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 38 [12] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Puchol, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Adiabatic limit, Witten deformation and analytic torsion forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' arXiv:2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content='13925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [13] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Puchol, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Zhang, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Scattering matrices and analytic torsions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Analysis & PDE, 14(1):77 – 134, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [14] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Ray and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' R-torsion and the Laplacian on Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Advances in Mathematics, 7(2):145–210, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Reidemeister.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Homotopieringe und linsenr¨aume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In Abhandlungen aus dem Mathematischen Seminar der Universit¨at Hamburg, volume 11, pages 102–109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Springer, 1935.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Vishik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Generalized Ray-Singer conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' A manifold with a smooth boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Communications in Mathematical Physics, 167(1):1–102, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' A new proof of gluing formula for analytic torsion forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' In Preparation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' [18] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' Lectures on Chern-Weil theory and Witten deformations, volume 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' World Scientific, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} +page_content=' 39' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdA0T4oBgHgl3EQfCf9p/content/2301.01990v1.pdf'} diff --git a/JdE4T4oBgHgl3EQfhg2P/content/tmp_files/2301.05126v1.pdf.txt b/JdE4T4oBgHgl3EQfhg2P/content/tmp_files/2301.05126v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b950abcab8c2d10326c1ca1d98c523d63b85eab7 --- /dev/null +++ b/JdE4T4oBgHgl3EQfhg2P/content/tmp_files/2301.05126v1.pdf.txt @@ -0,0 +1,1065 @@ +This paper has been accepted for presentation in the 5th Workshop on Accelerated Machine Learning (AccML), co-located with HiPEAC’23. +HEP-BNN: A Framework for Finding Low-Latency +Execution Configurations of BNNs on +Heterogeneous Multiprocessor Platforms +Leonard David Bereholschi, Ching-Chi Lin, Mikail Yayla, Jian-Jia Chen +Technical University of Dortmund, Germany +{leonard.bereholschi, chingchi.lin, mikail.yayla, jian-jia.chen}@tu-dortmund.de +Abstract—Binarized Neural Networks (BNNs) significantly +reduce the computation and memory demands with binarized +weights and activations compared to full-precision NNs. Execut- +ing a layer in a BNN on different devices of a heterogeneous +multiprocessor platform consisting of CPU and GPU can affect +the inference performance, i.e., accuracy and latency. Usually, a +heterogeneous HW platform consisting of a CPU and a GPU +is available to execute the BNN workloads. However, to use +the heterogeneous HW effectively, it is necessary to find an +efficient strategy for BNN workload mapping. In this work, +we propose a framework that generates efficient BNN layer- +to-device mappings (i.e. suitable parallel configuration for each +layer of the model) for execution platforms comprised of CPU +and CUDA-capable GPU. We evaluate our proposed framework +with two BNN architectures using two well-known datasets, +Fashion-MNIST and CIFAR-10, on three hardware platforms +with different characteristics. The results show that compared to +running a fully-parallelized GPU implementation, our framework +generates an efficient configuration up to 2×, 2.6× and 11.8× +faster on our tested hardware respectively. +Index Terms—Binarized Neural Network, inference, GPU, +CUDA +I. INTRODUCTION +Neural Networks (NN) have been applied to various prac- +tical domains in the last decade, e.g., image recognition in +computer vision, prediction of chemical patterns in chemistry, +and cancer detection in medical science. [1] Given a well- +trained NN model, inference is the process of using the model +to make predictions against previously unseen data. Depending +on the structure of the NN model, the inference can be time- +and resource-consuming. +Binarized Neural Network (BNN) [2] is a resource-efficient +variant of NNs. In BNNs, the weights and the activations are +binarized into 1-bit representation. Multiplications and accu- +mulations in a BNN can be computed using the xnor operand +and popcount(), respectively. Therefore, BNNs are signifi- +cantly more resource-efficient compared to full-precision NNs, +which makes BNNs excellent candidates for running AI ap- +plication on resource-constrained edge devices. +For executing BNN workloads, customized hardware accel- +erators (i.e. on FPGAs or ASICs) are still in the early stages +of development [3], while CPUs and GPUs are mature and +readily available. Several recent studies have evaluated the +use of GPUs for BNNs. Hubara et al. [2] evaluate BNNs +using XNOR kernels for GPUs. Xu et al. [4] implement a +computation kernel for BNNs as well. Li et al. [5] run BNNs +on Turing GPUs, focusing on the bit-level parallelism and +strides in memory. Chen et al. [6] develop a BNN acceleration +engine for mobile phones. +Although GPU provides high computing capability, exe- +cuting every layer in a BNN on GPU does not guarantee +good performance. We reveal in this work that executing all +the BNN layers exclusively on the GPU leads to significant +increase in latency compared to running the model on the +CPU sequentially (e.g. model is too small and CPU-overhead +too significant, see Fig. 1). Therefore, a proper layer-to-device +mapping is imperative for achieving efficient BNN inference +on heterogeneous multiprocessor platforms consisting of CPU +and GPU. +Contributions: In this paper, we propose a framework, +HEP-BNN, which automatically generates an efficient layer- +to-device mapping (i.e. the suitable parallel configuration for +each layer) for a given BNN model running on a heterogeneous +multiprocessor platform consisting of CPU and GPU. Given a +trained BNN model, HEP-BNN systematically evaluates the +execution time of the model on CPU and on GPU under +different parallel configurations. The configuration with the +least execution time is highlighted as the efficient CPU/GPU +configuration for the BNN on target platform. Such a frame- +work would lead to a more efficient use of available hard- +ware resources. Furthermore, automatically generating directly +applicable code containing the optimized mapping, would +enable highly efficient BNN inference. It would allow both +researchers and practitioners to fully exploit the capabilities +of their available hardware platforms in applying BNNs effi- +ciently on resource-limited edge devices. +Our contributions are summarized as follows: +• We present our HEP-BNN framework that automatically +generates the efficient layer-to-device mapping for given +BNN models and heterogeneous execution platforms con- +sisting of a CPU and a CUDA-capable GPU. The gener- +ated code, containing the efficient configuration for each +layer of the model, can then be used for applications using +BNN inference in practice. The proposed framework is +published on Github1. +1https://github.com/LeonardDavid/hep-bnn +1 +arXiv:2301.05126v1 [cs.DC] 12 Jan 2023 + +0 +25 +50 +75 +100 +125 +150 +Latency (s) +GPU-only (parallel) +CPU-only (sequential) +Fig. 1. +Fashion-MNIST on Jetson TX2: example from out results for the +difference of total latency between the sequential CPU model and the parallel +GPU model (with a higher CPU-overhead). +• To demonstrate the capabilities of our framework, we +apply our framework on two BNN models with two +commonly used datasets, Fashion-MNIST and CIFAR10, +on three hardware platforms with different characteristics: +Server, Laptop, and Jetson TX2. Across the different +BNN models, our results show that by applying properly +chosen parallel configurations for layers running on GPU, +inference times can be reduced by up to 2×, 2.6× and +11.8× for the three execution platforms, respectively. +II. SYSTEM MODEL +In Section II-A, the basics of BNNs the layers used in our +models are presented. An introduction into GPU computing +is given in Section II-B, where the CUDA framework is +also described, including its features, limitations, and some +implementation details. The different parallel configurations +implemented and used in our framework, as well as their nota- +tions used throughout this paper, are presented in Section II-C. +Finally, the problem definition is elaborated in Section II-D. +A. Basics of Binarized Neural Networks +Binarized Neural Networks (BNNs) [2] are a resource- +efficient variant of NNs. In a BNN model, the weights and +the activations are binarized into 1-bit representation. Unlike +the full-precision NNs, where one matrix multiplication must +be performed for computing the output of each neuron, we +can simply apply xnor operands for computing the outputs of +neurons in BNNs. Specifically, the output of a layer can be +computed with +2 ∗ popcount(xnor(W l +i , Il−1)) − #bits > T, +where W l +i are the weights of layer l, Il−1 are the inputs to +layer l, popcount() is a function which counts the number of +1s in the results of the xnor operand, #bits is the number of +bits in the xnor operand, and T is a learnable threshold pa- +rameter which can be computed with the batch normalization +parameters. The binary outputs depend on the truth value of the +statement, which represents a shifted binarization function [7]. +In this work, we consider convolutional BNNs. There are +four basic types of layers in a convolutional BNN, i.e., convo- +lutional layer, maxpool layer, step layer, and fully-connected +layer, as shown in Figure 2. We also employ the flattening +layer in the BNNs in this work. +The convolutional layer computes a 2D convolution of the +input with filters. We use “Cχ” to denote a convolutional layer, +where χ is a number indicating the amount of neurons in the +layer. For example, “C64” is a convolutional layer with 64 +neurons. In this work, the filter size is fixed at 3 × 3 for all +the convolutional layers. +The maxpool layer downsamples the input by selecting the +maximum value of the input in a given window size, which is +set to 2×2 in our models. We use “MPχ” to denote a maxpool +layer, where χ indicates it’s output size, e.g., “MP16” for a +output of size “16 × 16”. +A step layer performs batch normalization followed by a +binary activation function. Batch normalization [7] is used in +NN models for faster and more stable training, which helps +increase the accuracy. Note that the threshold values in the +batch normalization are still signed integers, even in BNNs. In +our models, we apply Hard-Tanh as our binary activation func- +tion. For inference in a BNN, batch normalization followed by +activation can be computed with binary thresholding [7]. We +denote the step layer as “S” in the rest of this paper. +A fully-connected layer connects every neuron in the current +layer with every neuron in the next layer. We denote a fully- +connected layer as “FCχ”, where χ is the amount of neurons +in the layer. Note that fully-connected layers also have binary +weights as learnable parameters. +We use “FLAT” to denote a flattening layer. The layer +rearranges a high dimension matrix into a lower dimension +matrix, e.g., from a 3-dimension matrix into a 1-dimension +array in our models. The rearrangement can be done with a +simple one-line operation on CPU in C++ code. +B. GPU +Due to their architecture, GPUs are suitable for highly +parallel use cases, such as repetitive matrix multiplication +for graphics-intensive tasks. Most state-of-the-art GPUs have +numerous computation cores, operating in an efficient manner +with large and fast memories. On GPUs, the computations +are performed by threads in parallel. To program the thread +behaviours, specialized GPU code, e.g., frameworks such as +CUDA or OpenCL, needs to be employed. In our work, we use +the CUDA programming language to deploy the computation +workload of a BNN model on Nvidia graphics cards. +In a CUDA program, threads can be arranged into thread +blocks, in which up to 1024 threads are executed in parallel. +Thread blocks can be further organized into a 3D grid struc- +ture, allowing for a greater degree of parallelization on the +GPU. The size of the grid is limited on the different dimension +axes as follows: {x, y, z} → {231−1, 65536, 65536}. Further- +more, the usage of internal CUDA variables, such as threadIDx +and blockIDx, allow each thread to address individual values +from arrays related to its specific computation. +Functions in which computation tasks are executed on +GPU are called kernels in the CUDA programming language. +Before launching a CUDA kernel, memory is allocated on +the GPU memory, after which all the required data for the +computations is transferred from the host (CPU) to the device +(GPU). The sizes of the thread blocks and the grid are also +specified before the kernel launch, while respecting previously +2 + +Fig. 2. Structure of a Convolutional BNN model demonstrating the three major layer types: Convolution, Maxpool, and Step layer. +Fig. 3. Concepts of the parallelism aspects: a) Data-based, b) Window-based, and c) Neuron-based +mentioned limitations. After the GPU finishes executing every +task in the kernel, the results are copied from the device back +to the host, and the previously allocated memory on the GPU +is freed. +Although GPUs provide massive computational power com- +pared to CPU, and are often used as accelerators in many use +cases, running an application on the GPU does not always lead +to performance improvement. In some cases, running parallel +code on GPU can take longer than the sequential CPU code +because of, for example, time overheads in communication. +Therefore, an analysis on the characteristics of an application, +e.g., degree of parallelism, can determine if it is beneficial +to run the application on GPU. For applications that can be +accelerated by GPU, how to organize the workload to achieve +the optimal performance is a crucial issue that needs to be +solved. +C. Data Parallelism Aspects +The workload of a BNN model consists of multiple data +images which are used as input. We define batch size as +the amount of data images in a batch, that are processed +concurrently. To process the workload of BNN inference on a +data set in parallel, we organize the workloads based on three +aspects of data parallelism: +1) Data-based: every data image in a batch is inferred +concurrently. +2) Window-based: a data image is divided into convolution +windows of consecutive pixels, with the windows being +processed concurrently. +3) Neuron-based: the outputs of the neurons in the same +layer in a NN model are calculated concurrently. +In the Data (X) configuration, multiple data images in a +batch are inferred on the GPU concurrently, as shown in +Figure 3 (a). Each data image in a batch is assigned to one +3 + +Convoluted +Convolution Layer +Maxpool Layer +Step Layer +Input Image +Image +neuroni +neuron1 +MAC +POPCOUNT +MAX +HTanH +KNOR +w +w +neuron2 +neuron2 +neuron2 +MAC +BN +POPCOUNT +MAX +HTanH +XNOR +h +h +neuronn +neuronn +neuronn +MAC +BN +POPCOUNT +MAX +HTanH +XNOR +n3 +(images) +BATCH_SIZE +1 +neurons) +a) +image +w +P1.1 +p +>blocks +pixels< +> threads +2 +P1. +h +Y (windows) +b) +c)GPU thread block. If a thread block is assigned with multiple +data images, these images are processed one after another. +Each pixel (and its subsequent operations) in a data image is +processed by one thread in the thread block. +Figure 3 (b) demonstrates the idea for the Window (Y) +configuration, in which a data image is divided into windows +of consecutive pixels in a row-wise manner, with each window +being assigned to one GPU thread block. The workloads re- +lated to the pixels (threads) in the same window are processed +on the GPU concurrently. +For the Neuron (Z) configuration, the outputs of neurons +from the same layer are calculated concurrently, as shown in +Figure 3 (c). The output of a neuron is the weighted sum from +its predecessors after going through an activation function. +Each neuron in a layer is assigned to a GPU thread block, +with threads in a thread block taking the corresponding outputs +from the previous layer as input, and calculating the output for +the neuron. +Using these aspects, we consider the following seven par- +allel configurations and their notations, which will be used +throughout this paper: 1) Data (X), 2) Window (Y), 3) Neuron +(Z), 4) Data + Window (XY), 5) Data + Neuron (XZ), 6) +Window + Neuron (YZ), and 7) Data + Window + Neuron +(XYZ). The configurations composed of multiple aspects are +implemented according to all of the implementations of the +individual aspects at the same time. Note that for all the +parallel configurations, the threads in thread blocks perform +the same operation depending on the layer, e.g., convolution of +pixels in a convolution layer. The blockIDx and threadIDx vari- +ables determine which pixel(s) and/or neuron(s) each thread +is responsible for. +D. Problem Definition +Given a well-trained BNN model, we aim to reduce the in- +ference time of a BNN model with the help of GPU. However, +there are multiple aspects for parallelizing the computation +workloads on GPU. Each layer in the BNN model can have +different suitable parallel configurations. Nevertheless, there +might also be layers with workloads that are not beneficial if +running on GPU due to overheads, e.g., which are caused by +data migration between host and the GPU device. Therefore, +our objective is to generate an efficient layer-to-device map- +ping for a given BNN model, so that the inference time is +minimized. For layers that are mapped to GPU for execution, +we also determine their suitable parallel configurations. +III. FRAMEWORK PRESENTATION +We introduce the proposed framework in details in this +section. The operational steps of our HEP-BNN framework +are outlined in Section III-A. The mapping algorithm is de- +scribed in Section III-B. Information about the folder structure, +important script files, and generated files of the framework, are +detailed in Sections III-C, III-D and III-E, respectively. +A. High-level Overview of Our HEP-BNN Framework +The operational steps performed by our framework are +represented in Figure 4. First, the program receives a BNN +Input: BNN model +L1 +L2 +L3 +Generate and Infer Configurations +t11 +t1m +tn1 +tnm +L1 +Ln +batch size1 +Optimal(1) +... +... +... +... +... +t11 +t1m +tn1 +tnm +L1 +Ln +batch sizeb +Optimal(b) +... +... +... +... +Greedy Mapping Algorithm +Output: Optimal Configurations +Fig. 4. Operational steps our HEP-BNN framework. +model in ONNX format as input, previously trained on a +specific dataset (e.g. Fashion-MNIST, CIFAR10). Then, for +every batch size in a defined range, the appropriate C++ and +CUDA code for the CPU and GPU is generated. After every +layer (L1, . . . , Ln) is implemented using different configura- +tions, the model is inferred for every configuration applied, +which results in different runtimes (with t11, . . . , t1m +∈ +L1 and tn1, . . . , tnm ∈ Ln). These timing information are used +for choosing an efficient configuration in a greedy manner, +according to Alg. 1 described in Section III-B. +B. Mapping Algorithm +In order to determine the suitable parallel configura- +tion which achieves the lowest inference time, the fol- +lowing layer-to-device mapping algorithm is applied (see +Alg. 1). The algorithm profiles each layer of the BNN +model using different batch sizes, both on the CPU and +the GPU. On the GPU, every parallel configuration from +Section II-C are implemented. In total, each layer is pro- +filed on 8 different implementations: 1) CPU, 2) Data (X), +3) Window (Y), 4) Neuron (Z), 5) Data + Window (XY), +6) Data + Neuron (XZ), 7) Window + Neuron (YZ), and 8) +Data + Window + Neuron (XYZ). +The implementation that achieves the lowest inference time +for the profiled layer is mapped to the specific batch size. +Summing up the lowest inference times for every layer, results +in the total runtime for executing the BNN model, using the +efficient implementations for the specific batch size. +After profiling every layer, the minimal total runtime, as +well as the proper batch size for which it is achieved, is +searched. Finally, the proper batch size is used to get the +mapped implementation of each layer of the BNN model. This +creates the efficient parallel configuration, which achieves the +lowest expected inference time. Algorithm 1 represents the +pseudocode of the described mapping algorithm. +4 + +Data: BNN model +Result: Efficient Configuration +1 properbatch size ← 0 +2 result time ← MAX_INT +3 foreach batch size do +4 +summin time ← 0 +5 +foreach layer do +6 +min time ← MAX_INT +7 +foreach implem do +8 +implement layer using implem +9 +(CPUtime, GPUtime) ← +profile(implementedlayer(batch size)) +10 +inference time ← CPUtime + GPUtime +11 +if inference time < min time then +12 +min time ← inference time +13 +MAP implem(layer) to batch size +14 +end +15 +end +16 +summin time ← summin time + min time +17 +end +18 +if summin time < result time then +19 +result time ← summin time +20 +properbatch size ← batch size +21 +end +22 end +23 foreach layer do +24 +get implem(layer) from MAP[properbatch size] +25 +add layerimplem to Efficient Configuration +26 end +27 return Efficient Configuration +Algorithm 1: Mapping algorithm that determines the +efficient configuration of a BNN model that achieves the +lowest inference time (Note: implem is short for implemen- +tation) +C. Folder Structure +Our HEP-BNN framework uses Python to run the imple- +mentations and optimizations of the input model. To exemplify +our implementation, we use the open source machine learning +compiler and code generator Fastinference +[8]. Specifically, +for the generation of the C++ and CUDA code for the CPU and +GPU respectively, the templating language Jinja2 is used. An +overview of the most important part of the folder-tree structure +(’fastinference/’) will be outlined in this section. +Each model, optimizer and implementation is defined in +a separate folder in ’fastinference/’. These are further sepa- +rated into the supported algorithm types such as ’ensemble/’, +’tree/’, ’neuralnet/’. Specifically, in ’fastinference/implemen- +tations/neuralnet’ there are folders for the different target +hardware: ’cpp/’, ’fpga/’, ’iree/’. +This is where the main part of our work is located, namely in +the ’cuda/’ folder, which contains a separate directory for each +parallel configuration. The folder names follow the notations +introduced in Section II-C. +In every folder there are Jinja2 files containing mainly +CUDA code templates for every layer, for parallel execution +of the model on the GPU. There is also a ’cpu/’ folder, that +contains C++ templates for the sequential operation of the +model on the CPU, used for the sequential implementations. +Each ’implement.py’ file found in every folder, contains the +function ’to implementation()’ and is responsible for selecting +the appropriate template files for each layer of the model, +and to generate the necessary C++ and CUDA files for the +implementation. +Additionally, the ’automatic/’ folder contains the code +which runs our mapping algorithm, that automatically reads +the model and data from the appropriate path, generates and +infers all of the selected configurations, and maps the suitable +configuration in a greedy manner. In the following section, we +give a more detailed description of the usage of certain files, +by using the CIFAR10 dataset as an example. +D. Important script files +In test cuda.py, the implementations of the parallel con- +figurations which can be used, are specified. Their notations +are consistent with the ones described in Section II-C. After- +wards, the HEP-BNN framework is launched by calling the +’to implementation()’ function found in test utils.py. +Here, an upper and lower bound is set for the batch sizes, +which are expressed as powers of 2 (e.g. with ’b l = 0’ and +’b u = 4’, the batch sizes used are 20 = 1, 21 = 2, 22 = 4, +and 23 = 8). +The mapping algorithm is then run inside of the nested +for-loops, one for each configuration, and the other one +for each batch size. It consists of two important functions, +’prepare fastinference()’ and ’run experiment()’. The former +generates all the necessary files for the model to compile and +run (more details will be given in section III-E), while the +latter function compiles and runs the generated model, after +which it outputs and stores the results of the inference. +After running all of the generated models, the best con- +figuration for each layer is mapped according to the lowest +inference time (see Alg. 1 – lines 23:26). Finally, the efficient +configuration is generated and inferred, achieving the lowest +inference time out of all other combinations. +The BNN model is stored in ONNX format under the +following path: +fastinference/implementations/ +neuralnet/cuda/automatic/model/ +cifar10/model_cifar10.onnx +Test data is stored under: +fastinference/implementations/ +neuralnet/cuda/automatic/data/ +cifar10/testing.csv +HEP-BNN is launched by running the following command +in the root folder (’fastinference/’): +fastinference/implementations/neuralnet/ +cuda/automatic/test_cuda.py +--outpath tmp/fastinference/cuda_auto +--dataset cifar +5 + +TABLE I +STRUCTURE OF THE CIFAR-10 BNN MODELS. +CIFAR-10 BNN model Structure +In → C64 → S → C64 → MP16 → S → C256 → S → C256 +→ MP8 → S → C512 → S → C512 → MP4 → S +→ FLAT → FC1024 → S → FC1024 → 10 +TABLE II +STRUCTURE OF THE FASHIONMNIST BNN MODELS. +LDB +FashionMNIST BNN model structure +In → C64 → MP14 → S → C64 → MP7 → S +→ FLAT → FC2048 → S → FC2048 → 10 +E. Generated files +In this section, we present the list of generated files for +each configuration, including a brief description. +• ’utils.h’ and ’utils.cuh’: contain utility functions (for C++ +and CUDA code respectively) +• ’cuda kernel.h’ and ’cuda model.h’: are headers contain- +ing functions that link the C++ code to the CUDA code +• ’modelW.hpp’: contains declarations of the output arrays +for every layer, and stores the weights, biases, and thresh- +olds +• ’model.h’ and ’model.cpp’: depending on the configura- +tion, contains either the sequential C++ model for the +CPU, or the calls to parallel CUDA model for the GPU +(both implementations are present, but the unused part is +commented out for debugging and comparison purposes) +• ’model.cu’: CUDA code for the GPU (if applicable) +There are also a few files which are the same, regardless of +configuration, which are already written and copied from the +’../cuda/automatic/’ folder to every generated implementation: +• ’main.cpp’: splits the dataset into batches, calls the model +predictor and calculates the accuracy and latency +• ’CMakeLists.txt’: generates the ’Makefile’ that compiles +the entire project +Note that file changes to any of the Jinja2 templates and +’implement.py’ files require the following command to be run +’python setup.py install’, before calling the ’make’ command. +IV. EXPERIMENT SETUPS AND RESULTS / EVALUATION +In this section, we present our experimental results. Infor- +mation about the hardware, datasets, and models are presented +in Section IV-A. In Section IV-B, the results of our HEP- +BNN framework, i.e., implementing the suitable configurations +for every layer, are presented and compared to the baseline +sequential implementation, as well as other parallel configu- +rations. +A. Profiling Environment +1) Hardware: We execute our experiments on a Linux +based server equipped with an Intel Core i7-8700K CPU +and a GTX1080 GPU with 8GB of video memory each. +Additionally, we run the same experiments on a Windows- +system with a consumer-grade GPU (GTX 1650Ti), as well +as on an embedded Jetson TX2 board. Table III also lists the +amount of available CUDA cores for each GPU. +TABLE III +OVERVIEW OF HARDWARE USED FOR EVALUATION +Name +CPU +GPU +CUDA Cores +Server +i7-8700K +GTX1080 +2560 +Laptop +i7-10750H +GTX1650Ti +1024 +Jetson TX2 +Cortex-A57 +Pascal-based +256 +Kernel +launches +are +wrapped +around +cudaEventRecord() functions, in order to accurately +measure the GPU-time. The communication cost (i.e. memory +allocation and transfer between host and device) in the CUDA +code is executed before the kernel launch, by the CPU, and +is included in the CPU-overhead time. We implement +independent layers for the models, therefore data transfer +between CPU and GPU takes place before and after every +layer’s execution (even if two consecutive layers are executed +on the GPU). The code generator can be adapted to consider +this case in future works/implementations. +2) Datasets: The algorithm is evaluated on two com- +monly used benchmarking datasets, namely FashionMNIST +and CIFAR-10. The FashionMNIST [9] dataset consists of +70, 000 gray-scale images and labels from 10 classes, rep- +resenting different clothing articles. The size of each image is +28 × 28 pixels in 1 channel, with 0 representing the brightest +and 255 the darkest values. Out of the total amount of images, +60, 000 are used for training, while the remaining 10, 000 for +testing. The CIFAR-10 [10] dataset contains 60, 000 colour +images (3 channels), each with a size of 32 × 32 for a +total of 1024 pixels. It is split into 50, 000 training and +10, 000 test images, which are classified in 10 different classes +representing means of transportation (i.e. airplane, ship, truck, +automobile) and animals (i.e. bird, cat, dog, deer, frog, horse). +3) BNN Architecture: The BNN models used for inferring +the datasets are VGG-type architectures [11] adapted for the +binarized variant of NNs. The FashionMNIST network model +contains a total of 10 layers, each of them belonging to one of +the types presented in Section II-A. Specifically, the 1st and +4th layer are convolutional layers, with a size of 28 × 28 × 64 +and 14 × 14 × 64 respectively. The convolutional layers are +down-sampled to half of their input size, in the immediately +following maxpool layers, namely in the 2nd and 5th layer. +Step layers are employed on the 3rd, 6th, and 9th layer, which +apply batch normalization and the activation function. After +convoluting and down-sampling the input image, it is flattened +into a 1-dimensional array in layer 7. Finally, a total of 2048 +neurons are fully-connected in the 8th and 10th layer. +The structures of the networks are listed in Table II and I, +using the notations from Section II-A. The CIFAR-10 model +also contains the standard layer types for BNNs, totalling +to 19 layers. Convolutional layers are placed on the 1st, +3rd, 6th, 8th, 11th, and 13th position. Down-sampling using +maxpooling occurs only three times, namely in the 4th, 9th, +and 14th layer. The 16th layer flattens the image for the fully- +connected layers at position 17 and 19. The rest of the layers +are step layers. +To simulate the inference process, the sets of test images +6 + +TABLE IV +EFFICIENT CONFIGURATIONS FOR THE CIFAR10 MODEL +C64 +S +C64 +MP16 +S +C256 +S +C256 +MP8 +S +C512 +S +C512 +MP4 +S +FLAT +FC1024 +S +FC1024 +Server +CPU +CPU +Z +CPU +CPU +XZ +CPU +XYZ +XY +CPU +XZ +CPU +XZ +X +CPU +CPU +X +CPU +CPU +Laptop +CPU +CPU +Y +CPU +CPU +XYZ +CPU +XYZ +CPU +CPU +XYZ +CPU +XYZ +CPU +CPU +CPU +X +CPU +CPU +TX2 +CPU +CPU +XYZ +CPU +CPU +Z +CPU +Z +CPU +CPU +XZ +CPU +XZ +CPU +CPU +CPU +XY +CPU +CPU +TABLE V +EFFICIENT CONFIGURATIONS FOR FASHION-MNIST MODEL +C64 +MP14 +S +C64 +MP7 +S +FLAT +FC2048 +S +FC2048 +Server +CPU +CPU +CPU +XZ +X +CPU +CPU +CPU +CPU +CPU +Laptop +CPU +CPU +CPU +CPU +CPU +CPU +CPU +CPU +CPU +CPU +TX2 +CPU +CPU +CPU +XZ +CPU +CPU +CPU +CPU +CPU +CPU +from both models respectively are used. This guarantees a +controlled benchmarking environment for the algorithm that +profiles each layer of the BNN models. The weights and biases +were trained over the course of 100 epochs, and achieve an +inference accuracy of 77.24% for the FashionMNIST, and +67.08% for the CIFAR-10 model. +Preliminary observations showed that the batch size has an +impact on runtime for certain configurations. Therefore, the +experiments also apply different batch sizes for the two BNN +models, from {1, 2, ..., 128}, in increments of powers of 2. +B. Results +Table VI presents the inference times measured by running +the BNN models (each with 10000 data images as input), +for every tested target hardware. The latency (displayed in +seconds) is the time required by the BNN to process the +entire test dataset of 10000 images. Recall from Section III-B, +that the suitable configuration is chosen over all the different +batch sizes, and therefore it is important to note the batch size +alongside the runtime. Specifically, a batch size of n means +that n data-images are processed in parallel. +It can be observed that the server, which has the most +CUDA cores out of the three tested hardware, has overall +faster runtimes. On the other hand, the resource-constrained +TX2, having the least amount of CUDA cores, has significantly +higher runtimes. +The suitable configurations mapped for each layer is +presented in Tables IV and V for the CIFAR-10 and +FashionMNIST models respectively. From there, we can notice +the following observations: +• Since the Fashion-MNIST BNN model is smaller out +of the two, almost all of the layers are mapped to the +CPU for sequential execution. A notable exception is the +second convolution layer, which is mapped to the XZ +configuration on the Server and the TX2. +• In the case of the CIFAR10 BNN model, we observe +that the maxpool layer is mapped to the CPU in almost +every case, whereas the convolutional and fully-connected +layers are mapped to the GPU using different parallel +configurations. For example, the second convolutional +layer is mapped to the Z configuration on the server, Y +on the Laptop and XYZ on the TX2. +Finally, Figure 5 compares the purely sequential CPU +implementation and two intuitive ways of parallelizing the +BNN models to the results of the mapping algorithm. The +naive GPU implementation considers the parallelization of +TABLE VI +THE MINIMUM INFERENCE TIMES OF THE EFFICIENT CONFIGURATIONS +Dataset +Fashion-MNIST +CIFAR10 +Hardware +Server +Laptop +TX2 +Server +Laptop +TX2 +Runtime +2.11s +2.84s +9.31s +41.6s +55s +297s +Batch size +16 +2 +64 +16 +128 +16 +every layer suitable for GPU acceleration using only the +Data (X) configuration. The full-parallel GPU implementation +parallelizes every suitable layer as much as possible, i.e., +applying the Data + Window + Neuron (XYZ) configuration. +These are the two most intuitive ways of parallelizing the +workload, while not considering the fact that some layers may +not benefit from GPU acceleration. +The results from running the efficient configurations for +every batch size, show a significant speedup overall. Specifi- +cally, compared to the fully-parallel implementation, running +the HEP-BNN framework on the server leads to at most 2× +speedup, while on the Jetson TX2, it achieves at most 2.6× +speedup, and on the Laptop-system the efficient configuration +results in a 11.8× improvement. +V. CONCLUSION +We propose a framework that generates efficient BNN +layer-to-device mappings for heterogeneous multiprocessor +platforms comprised of CPU and CUDA-capable GPU. Given +a trained BNN model, our proposed HEP-BNN framework +systematically evaluates the execution time of the model on +CPU and on GPU under different parallel configurations. We +evaluate our framework with two BNN architectures on well- +known datasets, running on three different types of hardware +platforms. The results show that, across the tested dataset- +s/BNNs and the different hardware platforms, our proposed +framework generates mappings for BNN inference which +achieve significantly higher speedup compared to a fully- +parallelized approach. Specifically, the efficient parallel con- +figuration from our HEP-BNN framework reduces inference +times by up to 2×, 2.6× and 11.8×, across the tested target +hardware respectively. The generated GPU code from HEP- +BNN containing the efficient configuration can also then be +used for applications using BNN inference in practice. +We believe that our HEP-BNN framework will benefit +researchers and practitioners to find efficient execution con- +figurations for BNN inference systems using heterogeneous +platforms comprised of CPU and GPU. +ACKNOWLEDGMENT +This paper has been supported by Deutsche Forschungs- +gemeinschaft (DFG) project OneMemory (405422836), by +the Collaborative Research Center SFB 876 “Providing In- +formation by Resource-Constrained Analysis” (project number +124020371), subproject A1 (http://sfb876.tu-dortmund.de) and +by the Federal Ministry of Education and Research of Ger- +many and the state of NRW as part of the Lamarr-Institute for +7 + +1 +2 +4 +8 +16 +32 +64 +128 +0 +150 +300 +450 +600 +Latency (s) +CIFAR10 on Server +1 +2 +4 +8 +16 +32 +64 +128 +0 +150 +300 +450 +600 +Latency (s) +CIFAR10 on Laptop +1 +2 +4 +8 +16 +32 +64 +128 +0 +150 +300 +450 +600 +750 +Latency (s) +CIFAR10 on Jetson TX2 +1 +2 +4 +8 +16 +32 +64 +128 +0 +5 +10 +15 +20 +25 +Latency (s) +Fashion-MNIST on Server +1 +2 +4 +8 +16 +32 +64 +128 +0 +25 +50 +75 +100 +Latency (s) +Fashion-MNIST on Laptop +1 +2 +4 +8 +16 +32 +64 +128 +0 +25 +50 +75 +100 +125 +150 +Latency (s) +Fashion-MNIST on Jetson TX2 +CPU only +Naive GPU +Full parallel GPU +Efficient (HEP-BNN Framework) +Fig. 5. +Execution time over batch size comparison for the entire FashionMNIST test images dataset (upper three figures) and entire CIFAR10 test images +dataset (lower three figures). Each dataset is evalauted with three different hardware configurations, namely: Server, Laptop, and TX2. +ML and AI, LAMARR22B. This work has received funding +by the German Federal Ministry of Education and Research +(BMBF) in the course of the 6GEM research hub under grant +number 16KISK038. +REFERENCES +[1] O. I. Abiodun, A. Jantan, A. E. Omolara, K. V. Dada, N. A. Mohamed, +and H. Arshad, “State-of-the-art in artificial neural network applications: +A survey,” Heliyon, vol. 4, no. 11, p. e00938, 2018. +[2] I. Hubara, M. Courbariaux, D. Soudry, R. El-Yaniv, and Y. Bengio, “Bi- +narized neural networks,” in Advances in Neural Information Processing +Systems (NIPS), 2016. +[3] E. Nurvitadhi, D. Sheffield, J. Sim, A. Mishra, G. Venkatesh, and +D. Marr, “Accelerating binarized neural networks: Comparison of +fpga, cpu, gpu, and asic,” in 2016 International Conference on Field- +Programmable Technology (FPT), pp. 77–84, 2016. +[4] X. Xu and M. Pedersoli, “A computing kernel for network binarization +on pytorch,” CoRR, vol. abs/1911.04477, 2019. +[5] A. Li and S. Su, “Accelerating binarized neural networks via bit-tensor- +cores in turing gpus,” IEEE Transactions on Parallel and Distributed +Systems, vol. 32, no. 7, pp. 1878–1891, 2021. +[6] G. Chen, S. He, H. Meng, and K. Huang, “Phonebit: Efficient gpu- +accelerated binary neural network inference engine for mobile phones,” +in 2020 Design, Automation Test in Europe Conference Exhibition +(DATE), pp. 786–791, 2020. +[7] E. Sari, M. Belbahri, and V. P. Nia, “How does batch normalization help +binary training?,” arXiv:1909.09139, 2019. +[8] S. Buschj¨ager, “fastinference github repository.” https://github.com/ +sbuschjaeger/fastinference. +[9] H. Xiao, K. Rasul, and R. Vollgraf, “Fashion-mnist: a novel image +dataset for benchmarking machine learning algorithms,” 2017. +[10] A. Krizhevsky, “Learning multiple layers of features from tiny images,” +tech. rep., 2009. +[11] K. Simonyan and A. Zisserman, “Very deep convolutional networks for +large-scale image recognition,” in International Conference on Learning +Representations, (ICLR), 2015. +8 + diff --git a/JdE4T4oBgHgl3EQfhg2P/content/tmp_files/load_file.txt b/JdE4T4oBgHgl3EQfhg2P/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..581b4c488e1ba336a4bdc2aed9a64b2af931090e --- /dev/null +++ b/JdE4T4oBgHgl3EQfhg2P/content/tmp_files/load_file.txt @@ -0,0 +1,703 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf,len=702 +page_content='This paper has been accepted for presentation in the 5th Workshop on Accelerated Machine Learning (AccML), co-located with HiPEAC’23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' HEP-BNN: A Framework for Finding Low-Latency Execution Configurations of BNNs on Heterogeneous Multiprocessor Platforms Leonard David Bereholschi, Ching-Chi Lin, Mikail Yayla, Jian-Jia Chen Technical University of Dortmund, Germany {leonard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='bereholschi, chingchi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='lin, mikail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='yayla, jian-jia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='chen}@tu-dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='de Abstract—Binarized Neural Networks (BNNs) significantly reduce the computation and memory demands with binarized weights and activations compared to full-precision NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Execut- ing a layer in a BNN on different devices of a heterogeneous multiprocessor platform consisting of CPU and GPU can affect the inference performance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', accuracy and latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Usually, a heterogeneous HW platform consisting of a CPU and a GPU is available to execute the BNN workloads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' However, to use the heterogeneous HW effectively, it is necessary to find an efficient strategy for BNN workload mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In this work, we propose a framework that generates efficient BNN layer- to-device mappings (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' suitable parallel configuration for each layer of the model) for execution platforms comprised of CPU and CUDA-capable GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We evaluate our proposed framework with two BNN architectures using two well-known datasets, Fashion-MNIST and CIFAR-10, on three hardware platforms with different characteristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The results show that compared to running a fully-parallelized GPU implementation, our framework generates an efficient configuration up to 2×, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='6× and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8× faster on our tested hardware respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Index Terms—Binarized Neural Network, inference, GPU, CUDA I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' INTRODUCTION Neural Networks (NN) have been applied to various prac- tical domains in the last decade, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', image recognition in computer vision, prediction of chemical patterns in chemistry, and cancer detection in medical science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [1] Given a well- trained NN model, inference is the process of using the model to make predictions against previously unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Depending on the structure of the NN model, the inference can be time- and resource-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Binarized Neural Network (BNN) [2] is a resource-efficient variant of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In BNNs, the weights and the activations are binarized into 1-bit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Multiplications and accu- mulations in a BNN can be computed using the xnor operand and popcount(), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Therefore, BNNs are signifi- cantly more resource-efficient compared to full-precision NNs, which makes BNNs excellent candidates for running AI ap- plication on resource-constrained edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For executing BNN workloads, customized hardware accel- erators (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' on FPGAs or ASICs) are still in the early stages of development [3], while CPUs and GPUs are mature and readily available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Several recent studies have evaluated the use of GPUs for BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Hubara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [2] evaluate BNNs using XNOR kernels for GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [4] implement a computation kernel for BNNs as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [5] run BNNs on Turing GPUs, focusing on the bit-level parallelism and strides in memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [6] develop a BNN acceleration engine for mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Although GPU provides high computing capability, exe- cuting every layer in a BNN on GPU does not guarantee good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We reveal in this work that executing all the BNN layers exclusively on the GPU leads to significant increase in latency compared to running the model on the CPU sequentially (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' model is too small and CPU-overhead too significant, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Therefore, a proper layer-to-device mapping is imperative for achieving efficient BNN inference on heterogeneous multiprocessor platforms consisting of CPU and GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Contributions: In this paper, we propose a framework, HEP-BNN, which automatically generates an efficient layer- to-device mapping (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' the suitable parallel configuration for each layer) for a given BNN model running on a heterogeneous multiprocessor platform consisting of CPU and GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Given a trained BNN model, HEP-BNN systematically evaluates the execution time of the model on CPU and on GPU under different parallel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The configuration with the least execution time is highlighted as the efficient CPU/GPU configuration for the BNN on target platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Such a frame- work would lead to a more efficient use of available hard- ware resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Furthermore, automatically generating directly applicable code containing the optimized mapping, would enable highly efficient BNN inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' It would allow both researchers and practitioners to fully exploit the capabilities of their available hardware platforms in applying BNNs effi- ciently on resource-limited edge devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Our contributions are summarized as follows: We present our HEP-BNN framework that automatically generates the efficient layer-to-device mapping for given BNN models and heterogeneous execution platforms con- sisting of a CPU and a CUDA-capable GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The gener- ated code, containing the efficient configuration for each layer of the model, can then be used for applications using BNN inference in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The proposed framework is published on Github1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='com/LeonardDavid/hep-bnn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='05126v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='DC] 12 Jan 2023 0 25 50 75 100 125 150 Latency (s) GPU-only (parallel) CPU-only (sequential) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Fashion-MNIST on Jetson TX2: example from out results for the difference of total latency between the sequential CPU model and the parallel GPU model (with a higher CPU-overhead).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' To demonstrate the capabilities of our framework, we apply our framework on two BNN models with two commonly used datasets, Fashion-MNIST and CIFAR10, on three hardware platforms with different characteristics: Server, Laptop, and Jetson TX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Across the different BNN models, our results show that by applying properly chosen parallel configurations for layers running on GPU, inference times can be reduced by up to 2×, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='6× and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8× for the three execution platforms, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' SYSTEM MODEL In Section II-A, the basics of BNNs the layers used in our models are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' An introduction into GPU computing is given in Section II-B, where the CUDA framework is also described, including its features, limitations, and some implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The different parallel configurations implemented and used in our framework, as well as their nota- tions used throughout this paper, are presented in Section II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Finally, the problem definition is elaborated in Section II-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Basics of Binarized Neural Networks Binarized Neural Networks (BNNs) [2] are a resource- efficient variant of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In a BNN model, the weights and the activations are binarized into 1-bit representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Unlike the full-precision NNs, where one matrix multiplication must be performed for computing the output of each neuron, we can simply apply xnor operands for computing the outputs of neurons in BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifically, the output of a layer can be computed with 2 ∗ popcount(xnor(W l i , Il−1)) − #bits > T, where W l i are the weights of layer l, Il−1 are the inputs to layer l, popcount() is a function which counts the number of 1s in the results of the xnor operand, #bits is the number of bits in the xnor operand, and T is a learnable threshold pa- rameter which can be computed with the batch normalization parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The binary outputs depend on the truth value of the statement, which represents a shifted binarization function [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In this work, we consider convolutional BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' There are four basic types of layers in a convolutional BNN, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', convo- lutional layer, maxpool layer, step layer, and fully-connected layer, as shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We also employ the flattening layer in the BNNs in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The convolutional layer computes a 2D convolution of the input with filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We use “Cχ” to denote a convolutional layer, where χ is a number indicating the amount of neurons in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For example, “C64” is a convolutional layer with 64 neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In this work, the filter size is fixed at 3 × 3 for all the convolutional layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The maxpool layer downsamples the input by selecting the maximum value of the input in a given window size, which is set to 2×2 in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We use “MPχ” to denote a maxpool layer, where χ indicates it’s output size, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', “MP16” for a output of size “16 × 16”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A step layer performs batch normalization followed by a binary activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Batch normalization [7] is used in NN models for faster and more stable training, which helps increase the accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Note that the threshold values in the batch normalization are still signed integers, even in BNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In our models, we apply Hard-Tanh as our binary activation func- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For inference in a BNN, batch normalization followed by activation can be computed with binary thresholding [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We denote the step layer as “S” in the rest of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A fully-connected layer connects every neuron in the current layer with every neuron in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We denote a fully- connected layer as “FCχ”, where χ is the amount of neurons in the layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Note that fully-connected layers also have binary weights as learnable parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We use “FLAT” to denote a flattening layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The layer rearranges a high dimension matrix into a lower dimension matrix, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', from a 3-dimension matrix into a 1-dimension array in our models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The rearrangement can be done with a simple one-line operation on CPU in C++ code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' GPU Due to their architecture, GPUs are suitable for highly parallel use cases, such as repetitive matrix multiplication for graphics-intensive tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Most state-of-the-art GPUs have numerous computation cores, operating in an efficient manner with large and fast memories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' On GPUs, the computations are performed by threads in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' To program the thread behaviours, specialized GPU code, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', frameworks such as CUDA or OpenCL, needs to be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In our work, we use the CUDA programming language to deploy the computation workload of a BNN model on Nvidia graphics cards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In a CUDA program, threads can be arranged into thread blocks, in which up to 1024 threads are executed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Thread blocks can be further organized into a 3D grid struc- ture, allowing for a greater degree of parallelization on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The size of the grid is limited on the different dimension axes as follows: {x, y, z} → {231−1, 65536, 65536}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Further- more, the usage of internal CUDA variables, such as threadIDx and blockIDx, allow each thread to address individual values from arrays related to its specific computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Functions in which computation tasks are executed on GPU are called kernels in the CUDA programming language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Before launching a CUDA kernel, memory is allocated on the GPU memory, after which all the required data for the computations is transferred from the host (CPU) to the device (GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The sizes of the thread blocks and the grid are also specified before the kernel launch, while respecting previously 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Structure of a Convolutional BNN model demonstrating the three major layer types: Convolution, Maxpool, and Step layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Concepts of the parallelism aspects: a) Data-based, b) Window-based, and c) Neuron-based mentioned limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' After the GPU finishes executing every task in the kernel, the results are copied from the device back to the host, and the previously allocated memory on the GPU is freed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Although GPUs provide massive computational power com- pared to CPU, and are often used as accelerators in many use cases, running an application on the GPU does not always lead to performance improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In some cases, running parallel code on GPU can take longer than the sequential CPU code because of, for example, time overheads in communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Therefore, an analysis on the characteristics of an application, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', degree of parallelism, can determine if it is beneficial to run the application on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For applications that can be accelerated by GPU, how to organize the workload to achieve the optimal performance is a crucial issue that needs to be solved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Data Parallelism Aspects The workload of a BNN model consists of multiple data images which are used as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We define batch size as the amount of data images in a batch, that are processed concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' To process the workload of BNN inference on a data set in parallel, we organize the workloads based on three aspects of data parallelism: 1) Data-based: every data image in a batch is inferred concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 2) Window-based: a data image is divided into convolution windows of consecutive pixels, with the windows being processed concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 3) Neuron-based: the outputs of the neurons in the same layer in a NN model are calculated concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In the Data (X) configuration, multiple data images in a batch are inferred on the GPU concurrently, as shown in Figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each data image in a batch is assigned to one 3 Convoluted Convolution Layer Maxpool Layer Step Layer Input Image Image neuroni neuron1 MAC POPCOUNT MAX HTanH KNOR w w neuron2 neuron2 neuron2 MAC BN POPCOUNT MAX HTanH XNOR h h neuronn neuronn neuronn MAC BN POPCOUNT MAX HTanH XNOR n3 (images) BATCH_SIZE 1 neurons) a) image w P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 p >blocks pixels< > threads 2 P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' h Y (windows) b) c)GPU thread block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' If a thread block is assigned with multiple data images, these images are processed one after another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each pixel (and its subsequent operations) in a data image is processed by one thread in the thread block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Figure 3 (b) demonstrates the idea for the Window (Y) configuration, in which a data image is divided into windows of consecutive pixels in a row-wise manner, with each window being assigned to one GPU thread block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The workloads re- lated to the pixels (threads) in the same window are processed on the GPU concurrently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For the Neuron (Z) configuration, the outputs of neurons from the same layer are calculated concurrently, as shown in Figure 3 (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The output of a neuron is the weighted sum from its predecessors after going through an activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each neuron in a layer is assigned to a GPU thread block, with threads in a thread block taking the corresponding outputs from the previous layer as input, and calculating the output for the neuron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Using these aspects, we consider the following seven par- allel configurations and their notations, which will be used throughout this paper: 1) Data (X), 2) Window (Y), 3) Neuron (Z), 4) Data + Window (XY), 5) Data + Neuron (XZ), 6) Window + Neuron (YZ), and 7) Data + Window + Neuron (XYZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The configurations composed of multiple aspects are implemented according to all of the implementations of the individual aspects at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Note that for all the parallel configurations, the threads in thread blocks perform the same operation depending on the layer, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', convolution of pixels in a convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The blockIDx and threadIDx vari- ables determine which pixel(s) and/or neuron(s) each thread is responsible for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Problem Definition Given a well-trained BNN model, we aim to reduce the in- ference time of a BNN model with the help of GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' However, there are multiple aspects for parallelizing the computation workloads on GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each layer in the BNN model can have different suitable parallel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Nevertheless, there might also be layers with workloads that are not beneficial if running on GPU due to overheads, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', which are caused by data migration between host and the GPU device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Therefore, our objective is to generate an efficient layer-to-device map- ping for a given BNN model, so that the inference time is minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For layers that are mapped to GPU for execution, we also determine their suitable parallel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' FRAMEWORK PRESENTATION We introduce the proposed framework in details in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The operational steps of our HEP-BNN framework are outlined in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The mapping algorithm is de- scribed in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Information about the folder structure, important script files, and generated files of the framework, are detailed in Sections III-C, III-D and III-E, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' High-level Overview of Our HEP-BNN Framework The operational steps performed by our framework are represented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' First, the program receives a BNN Input: BNN model L1 L2 L3 Generate and Infer Configurations t11 t1m tn1 tnm L1 Ln batch size1 Optimal(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' t11 t1m tn1 tnm L1 Ln batch sizeb Optimal(b) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Greedy Mapping Algorithm Output: Optimal Configurations Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Operational steps our HEP-BNN framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' model in ONNX format as input, previously trained on a specific dataset (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Fashion-MNIST, CIFAR10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Then, for every batch size in a defined range, the appropriate C++ and CUDA code for the CPU and GPU is generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' After every layer (L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' , Ln) is implemented using different configura- tions, the model is inferred for every configuration applied, which results in different runtimes (with t11, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' , t1m ∈ L1 and tn1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' , tnm ∈ Ln).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' These timing information are used for choosing an efficient configuration in a greedy manner, according to Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1 described in Section III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Mapping Algorithm In order to determine the suitable parallel configura- tion which achieves the lowest inference time, the fol- lowing layer-to-device mapping algorithm is applied (see Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The algorithm profiles each layer of the BNN model using different batch sizes, both on the CPU and the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' On the GPU, every parallel configuration from Section II-C are implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In total, each layer is pro- filed on 8 different implementations: 1) CPU, 2) Data (X), 3) Window (Y), 4) Neuron (Z), 5) Data + Window (XY), 6) Data + Neuron (XZ), 7) Window + Neuron (YZ), and 8) Data + Window + Neuron (XYZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The implementation that achieves the lowest inference time for the profiled layer is mapped to the specific batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Summing up the lowest inference times for every layer, results in the total runtime for executing the BNN model, using the efficient implementations for the specific batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' After profiling every layer, the minimal total runtime, as well as the proper batch size for which it is achieved, is searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Finally, the proper batch size is used to get the mapped implementation of each layer of the BNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' This creates the efficient parallel configuration, which achieves the lowest expected inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Algorithm 1 represents the pseudocode of the described mapping algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 4 Data: BNN model Result: Efficient Configuration 1 properbatch size ← 0 2 result time ← MAX_INT 3 foreach batch size do 4 summin time ← 0 5 foreach layer do 6 min time ← MAX_INT 7 foreach implem do 8 implement layer using implem 9 (CPUtime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' GPUtime) ← ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='profile(implementedlayer(batch size)) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='inference time ← CPUtime + GPUtime ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='if inference time < min time then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='min time ← inference time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='MAP implem(layer) to batch size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='summin time ← summin time + min time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='if summin time < result time then ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='result time ← summin time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='properbatch size ← batch size ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='22 end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='23 foreach layer do ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='get implem(layer) from MAP[properbatch size] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='add layerimplem to Efficient Configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='26 end ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='27 return Efficient Configuration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Algorithm 1: Mapping algorithm that determines the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='efficient configuration of a BNN model that achieves the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='lowest inference time (Note: implem is short for implemen- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='tation) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Folder Structure Our HEP-BNN framework uses Python to run the imple- mentations and optimizations of the input model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' To exemplify our implementation, we use the open source machine learning compiler and code generator Fastinference [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifically, for the generation of the C++ and CUDA code for the CPU and GPU respectively, the templating language Jinja2 is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' An overview of the most important part of the folder-tree structure (’fastinference/’) will be outlined in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each model, optimizer and implementation is defined in a separate folder in ’fastinference/’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' These are further sepa- rated into the supported algorithm types such as ’ensemble/’, ’tree/’, ’neuralnet/’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifically, in ’fastinference/implemen- tations/neuralnet’ there are folders for the different target hardware: ’cpp/’, ’fpga/’, ’iree/’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' This is where the main part of our work is located, namely in the ’cuda/’ folder, which contains a separate directory for each parallel configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The folder names follow the notations introduced in Section II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In every folder there are Jinja2 files containing mainly CUDA code templates for every layer, for parallel execution of the model on the GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' There is also a ’cpu/’ folder, that contains C++ templates for the sequential operation of the model on the CPU, used for the sequential implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each ’implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='py’ file found in every folder, contains the function ’to implementation()’ and is responsible for selecting the appropriate template files for each layer of the model, and to generate the necessary C++ and CUDA files for the implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Additionally, the ’automatic/’ folder contains the code which runs our mapping algorithm, that automatically reads the model and data from the appropriate path, generates and infers all of the selected configurations, and maps the suitable configuration in a greedy manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In the following section, we give a more detailed description of the usage of certain files, by using the CIFAR10 dataset as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Important script files In test cuda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='py, the implementations of the parallel con- figurations which can be used, are specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Their notations are consistent with the ones described in Section II-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' After- wards, the HEP-BNN framework is launched by calling the ’to implementation()’ function found in test utils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='py.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Here, an upper and lower bound is set for the batch sizes, which are expressed as powers of 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' with ’b l = 0’ and ’b u = 4’, the batch sizes used are 20 = 1, 21 = 2, 22 = 4, and 23 = 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The mapping algorithm is then run inside of the nested for-loops, one for each configuration, and the other one for each batch size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' It consists of two important functions, ’prepare fastinference()’ and ’run experiment()’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The former generates all the necessary files for the model to compile and run (more details will be given in section III-E), while the latter function compiles and runs the generated model, after which it outputs and stores the results of the inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' After running all of the generated models, the best con- figuration for each layer is mapped according to the lowest inference time (see Alg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1 – lines 23:26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Finally, the efficient configuration is generated and inferred, achieving the lowest inference time out of all other combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The BNN model is stored in ONNX format under the following path: fastinference/implementations/ neuralnet/cuda/automatic/model/ cifar10/model_cifar10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='onnx Test data is stored under: fastinference/implementations/ neuralnet/cuda/automatic/data/ cifar10/testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='csv HEP-BNN is launched by running the following command in the root folder (’fastinference/’): fastinference/implementations/neuralnet/ cuda/automatic/test_cuda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='py --outpath tmp/fastinference/cuda_auto --dataset cifar 5 TABLE I STRUCTURE OF THE CIFAR-10 BNN MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' CIFAR-10 BNN model Structure In → C64 → S → C64 → MP16 → S → C256 → S → C256 → MP8 → S → C512 → S → C512 → MP4 → S → FLAT → FC1024 → S → FC1024 → 10 TABLE II STRUCTURE OF THE FASHIONMNIST BNN MODELS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' LDB FashionMNIST BNN model structure In → C64 → MP14 → S → C64 → MP7 → S → FLAT → FC2048 → S → FC2048 → 10 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Generated files In this section, we present the list of generated files for each configuration, including a brief description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' ’utils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='h’ and ’utils.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='cuh’: contain utility functions (for C++ and CUDA code respectively) ’cuda kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='h’ and ’cuda model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='h’: are headers contain- ing functions that link the C++ code to the CUDA code ’modelW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='hpp’: contains declarations of the output arrays for every layer, and stores the weights, biases, and thresh- olds ’model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='h’ and ’model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='cpp’: depending on the configura- tion, contains either the sequential C++ model for the CPU, or the calls to parallel CUDA model for the GPU (both implementations are present, but the unused part is commented out for debugging and comparison purposes) ’model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='cu’: CUDA code for the GPU (if applicable) There are also a few files which are the same, regardless of configuration, which are already written and copied from the ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='./cuda/automatic/’ folder to every generated implementation: ’main.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='cpp’: splits the dataset into batches, calls the model predictor and calculates the accuracy and latency ’CMakeLists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='txt’: generates the ’Makefile’ that compiles the entire project Note that file changes to any of the Jinja2 templates and ’implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='py’ files require the following command to be run ’python setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='py install’, before calling the ’make’ command.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' EXPERIMENT SETUPS AND RESULTS / EVALUATION In this section, we present our experimental results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Infor- mation about the hardware, datasets, and models are presented in Section IV-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In Section IV-B, the results of our HEP- BNN framework, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', implementing the suitable configurations for every layer, are presented and compared to the baseline sequential implementation, as well as other parallel configu- rations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Profiling Environment 1) Hardware: We execute our experiments on a Linux based server equipped with an Intel Core i7-8700K CPU and a GTX1080 GPU with 8GB of video memory each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Additionally, we run the same experiments on a Windows- system with a consumer-grade GPU (GTX 1650Ti), as well as on an embedded Jetson TX2 board.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Table III also lists the amount of available CUDA cores for each GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' TABLE III OVERVIEW OF HARDWARE USED FOR EVALUATION Name CPU GPU CUDA Cores Server i7-8700K GTX1080 2560 Laptop i7-10750H GTX1650Ti 1024 Jetson TX2 Cortex-A57 Pascal-based 256 Kernel launches are wrapped around cudaEventRecord() functions, in order to accurately measure the GPU-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The communication cost (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' memory allocation and transfer between host and device) in the CUDA code is executed before the kernel launch, by the CPU, and is included in the CPU-overhead time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We implement independent layers for the models, therefore data transfer between CPU and GPU takes place before and after every layer’s execution (even if two consecutive layers are executed on the GPU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The code generator can be adapted to consider this case in future works/implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 2) Datasets: The algorithm is evaluated on two com- monly used benchmarking datasets, namely FashionMNIST and CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The FashionMNIST [9] dataset consists of 70, 000 gray-scale images and labels from 10 classes, rep- resenting different clothing articles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The size of each image is 28 × 28 pixels in 1 channel, with 0 representing the brightest and 255 the darkest values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Out of the total amount of images, 60, 000 are used for training, while the remaining 10, 000 for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The CIFAR-10 [10] dataset contains 60, 000 colour images (3 channels), each with a size of 32 × 32 for a total of 1024 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' It is split into 50, 000 training and 10, 000 test images, which are classified in 10 different classes representing means of transportation (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' airplane, ship, truck, automobile) and animals (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' bird, cat, dog, deer, frog, horse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 3) BNN Architecture: The BNN models used for inferring the datasets are VGG-type architectures [11] adapted for the binarized variant of NNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The FashionMNIST network model contains a total of 10 layers, each of them belonging to one of the types presented in Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifically, the 1st and 4th layer are convolutional layers, with a size of 28 × 28 × 64 and 14 × 14 × 64 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The convolutional layers are down-sampled to half of their input size, in the immediately following maxpool layers, namely in the 2nd and 5th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Step layers are employed on the 3rd, 6th, and 9th layer, which apply batch normalization and the activation function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' After convoluting and down-sampling the input image, it is flattened into a 1-dimensional array in layer 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Finally, a total of 2048 neurons are fully-connected in the 8th and 10th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The structures of the networks are listed in Table II and I, using the notations from Section II-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The CIFAR-10 model also contains the standard layer types for BNNs, totalling to 19 layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Convolutional layers are placed on the 1st, 3rd, 6th, 8th, 11th, and 13th position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Down-sampling using maxpooling occurs only three times, namely in the 4th, 9th, and 14th layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The 16th layer flattens the image for the fully- connected layers at position 17 and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The rest of the layers are step layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' To simulate the inference process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' the sets of test images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='TABLE IV ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='EFFICIENT CONFIGURATIONS FOR THE CIFAR10 MODEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='MP16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C256 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='MP8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C512 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='MP4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='FLAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='FC1024 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='FC1024 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Server ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XYZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Laptop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XYZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XYZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XYZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XYZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='TX2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XYZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='TABLE V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='EFFICIENT CONFIGURATIONS FOR FASHION-MNIST MODEL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='MP14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='C64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='MP7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='FLAT ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='FC2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='S ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='FC2048 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Server ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Laptop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='TX2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='XZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='from both models respectively are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' This guarantees a controlled benchmarking environment for the algorithm that profiles each layer of the BNN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The weights and biases were trained over the course of 100 epochs, and achieve an inference accuracy of 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='24% for the FashionMNIST, and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='08% for the CIFAR-10 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Preliminary observations showed that the batch size has an impact on runtime for certain configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Therefore, the experiments also apply different batch sizes for the two BNN models, from {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', 128}, in increments of powers of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Results Table VI presents the inference times measured by running the BNN models (each with 10000 data images as input), for every tested target hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The latency (displayed in seconds) is the time required by the BNN to process the entire test dataset of 10000 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Recall from Section III-B, that the suitable configuration is chosen over all the different batch sizes, and therefore it is important to note the batch size alongside the runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifically, a batch size of n means that n data-images are processed in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' It can be observed that the server, which has the most CUDA cores out of the three tested hardware, has overall faster runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' On the other hand, the resource-constrained TX2, having the least amount of CUDA cores, has significantly higher runtimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The suitable configurations mapped for each layer is presented in Tables IV and V for the CIFAR-10 and FashionMNIST models respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' From there, we can notice the following observations: Since the Fashion-MNIST BNN model is smaller out of the two, almost all of the layers are mapped to the CPU for sequential execution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A notable exception is the second convolution layer, which is mapped to the XZ configuration on the Server and the TX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' In the case of the CIFAR10 BNN model, we observe that the maxpool layer is mapped to the CPU in almost every case, whereas the convolutional and fully-connected layers are mapped to the GPU using different parallel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' For example, the second convolutional layer is mapped to the Z configuration on the server, Y on the Laptop and XYZ on the TX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Finally, Figure 5 compares the purely sequential CPU implementation and two intuitive ways of parallelizing the BNN models to the results of the mapping algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The naive GPU implementation considers the parallelization of TABLE VI THE MINIMUM INFERENCE TIMES OF THE EFFICIENT CONFIGURATIONS Dataset Fashion-MNIST CIFAR10 Hardware Server Laptop TX2 Server Laptop TX2 Runtime 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='11s 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='84s 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='31s 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='6s 55s 297s Batch size 16 2 64 16 128 16 every layer suitable for GPU acceleration using only the Data (X) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The full-parallel GPU implementation parallelizes every suitable layer as much as possible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', applying the Data + Window + Neuron (XYZ) configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' These are the two most intuitive ways of parallelizing the workload, while not considering the fact that some layers may not benefit from GPU acceleration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The results from running the efficient configurations for every batch size, show a significant speedup overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifi- cally, compared to the fully-parallel implementation, running the HEP-BNN framework on the server leads to at most 2× speedup, while on the Jetson TX2, it achieves at most 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='6× speedup, and on the Laptop-system the efficient configuration results in a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8× improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' CONCLUSION We propose a framework that generates efficient BNN layer-to-device mappings for heterogeneous multiprocessor platforms comprised of CPU and CUDA-capable GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Given a trained BNN model, our proposed HEP-BNN framework systematically evaluates the execution time of the model on CPU and on GPU under different parallel configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We evaluate our framework with two BNN architectures on well- known datasets, running on three different types of hardware platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The results show that, across the tested dataset- s/BNNs and the different hardware platforms, our proposed framework generates mappings for BNN inference which achieve significantly higher speedup compared to a fully- parallelized approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Specifically, the efficient parallel con- figuration from our HEP-BNN framework reduces inference times by up to 2×, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='6× and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8×, across the tested target hardware respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' The generated GPU code from HEP- BNN containing the efficient configuration can also then be used for applications using BNN inference in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' We believe that our HEP-BNN framework will benefit researchers and practitioners to find efficient execution con- figurations for BNN inference systems using heterogeneous platforms comprised of CPU and GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' ACKNOWLEDGMENT This paper has been supported by Deutsche Forschungs- gemeinschaft (DFG) project OneMemory (405422836), by the Collaborative Research Center SFB 876 “Providing In- formation by Resource-Constrained Analysis” (project number 124020371), subproject A1 (http://sfb876.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='tu-dortmund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='de) and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='by the Federal Ministry of Education and Research of Ger- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='many and the state of NRW as part of the Lamarr-Institute for ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CIFAR10 on Server ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CIFAR10 on Laptop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='450 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='750 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CIFAR10 on Jetson TX2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Fashion-MNIST on Server ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Fashion-MNIST on Laptop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='64 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='128 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='75 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='125 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Latency (s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Fashion-MNIST on Jetson TX2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='CPU only ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Naive GPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Full parallel GPU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Efficient (HEP-BNN Framework) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Execution time over batch size comparison for the entire FashionMNIST test images dataset (upper three figures) and entire CIFAR10 test images dataset (lower three figures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Each dataset is evalauted with three different hardware configurations, namely: Server, Laptop, and TX2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' ML and AI, LAMARR22B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' This work has received funding by the German Federal Ministry of Education and Research (BMBF) in the course of the 6GEM research hub under grant number 16KISK038.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' REFERENCES [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Abiodun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Jantan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Omolara, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Dada, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Mohamed, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Arshad, “State-of-the-art in artificial neural network applications: A survey,” Heliyon, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 4, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 11, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' e00938, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [2] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Hubara, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Courbariaux, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Soudry, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' El-Yaniv, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Bengio, “Bi- narized neural networks,” in Advances in Neural Information Processing Systems (NIPS), 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Nurvitadhi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Sheffield, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Sim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Mishra, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Venkatesh, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Marr, “Accelerating binarized neural networks: Comparison of fpga, cpu, gpu, and asic,” in 2016 International Conference on Field- Programmable Technology (FPT), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 77–84, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [4] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Xu and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Pedersoli, “A computing kernel for network binarization on pytorch,” CoRR, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' abs/1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='04477, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Li and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Su, “Accelerating binarized neural networks via bit-tensor- cores in turing gpus,” IEEE Transactions on Parallel and Distributed Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 1878–1891, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [6] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' He, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Meng, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Huang, “Phonebit: Efficient gpu- accelerated binary neural network inference engine for mobile phones,” in 2020 Design, Automation Test in Europe Conference Exhibition (DATE), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 786–791, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [7] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Sari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Belbahri, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Nia, “How does batch normalization help binary training?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=',” arXiv:1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='09139, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [8] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Buschj¨ager, “fastinference github repository.” https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content='com/ sbuschjaeger/fastinference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Xiao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Rasul, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Vollgraf, “Fashion-mnist: a novel image dataset for benchmarking machine learning algorithms,” 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Krizhevsky, “Learning multiple layers of features from tiny images,” tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=', 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' [11] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Simonyan and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' Zisserman, “Very deep convolutional networks for large-scale image recognition,” in International Conference on Learning Representations, (ICLR), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JdE4T4oBgHgl3EQfhg2P/content/2301.05126v1.pdf'} diff --git a/JtFJT4oBgHgl3EQfwi0E/content/tmp_files/2301.11630v1.pdf.txt b/JtFJT4oBgHgl3EQfwi0E/content/tmp_files/2301.11630v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a4398cda256f9d87ecb2f1502fa444689623df3 --- /dev/null +++ b/JtFJT4oBgHgl3EQfwi0E/content/tmp_files/2301.11630v1.pdf.txt @@ -0,0 +1,1623 @@ +1 +Joint Geometry and Attribute Upsampling of Point +Clouds Using Frequency-Selective Models with +Overlapped Support +Viktoria Heimann, Andreas Spruck, and Andr´e Kaup +Abstract—With the increasing demand of capturing our en- +vironment in three-dimensions for AR/ VR applications and +autonomous driving among others, the importance of high- +resolution point clouds rises. As the capturing process is a +complex task, point cloud upsampling is often desired. We +propose Frequency-Selective Upsampling (FSU), an upsampling +scheme that upsamples geometry and attribute information of +point clouds jointly in a sequential manner with overlapped +support areas. The point cloud is partitioned into blocks with +overlapping support area first. Then, a continuous frequency +model is generated that estimates the point cloud’s surface +locally. The model is sampled at new positions for upsampling. +In a subsequent step, another frequency model is created that +models the attribute signal. Here, knowledge from the geometry +upsampling is exploited for a simplified projection of the points +in two dimensions. The attribute model is evaluated for the +upsampled geometry positions. In our extensive evaluation, we +evaluate geometry and attribute upsampling independently and +show joint results. The geometry results show best performances +for our proposed FSU in terms of point-to-plane error and plane- +to-plane angular similarity. Moreover, FSU outperforms other +color upsampling schemes by 1.9 dB in terms of color PSNR. +In addition, the visual appearance of the point clouds clearly +increases with FSU. +Index Terms—Point Cloud Upsampling, Frequency Model +I. INTRODUCTION AND RELATED WORK +The increasing demand of capturing our environment for +virtual and augmented reality applications [1], [2], in automo- +tive industry [3], [4], in architecture, and archaeology [5], [6] +drives the need for high-resolution point clouds. Point clouds +are a versatile three-dimensional data type. In a point cloud, +single points are captured using, e.g., a Light Detection and +Ranging (LiDAR) sensor or an RGB-D camera such as the +Microsoft Kinect [7]. For each point in a point cloud, the +location in 3D space is stored. Moreover, each point may +have an attribute assigned such as an intensity value or color +information in RGB format. Such a set of many points forms +a point cloud. As both, geometry and attribute, have to be +stored for each point in a point cloud, this data type requires +large storage capacities. However, many applications demand +for high resolution point clouds. Therefore, point clouds often +have to be upsampled artificially after acquisition. +Manuscript created 14 October 2022. +The authors are with the Chair of Multimedia Communications and +Signal +Processing, +Friedrich-Alexander +Universit¨at, +Erlangen-N¨urnberg +(FAU), 91058 Erlangen, Germany (e-mail: viktoria.heimann@fau.de; an- +dreas.spruck@fau.de; andre.kaup@fau.de). +This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG, +German Research Foundation) – SFB 1483 – Project-ID 442419336, Emp- +kinS. +Fig. 1: Mario point cloud is partitioned into blocks. +The upsampling of point clouds applies to both, the geom- +etry and the attributes of a point cloud. As a consequence of +this, point cloud upsampling is generally separated into two +steps, geometry upsampling and attribute upsampling. In the +geometry upsampling part, we focus on retrieving the best +location for the upsampled points whereas in the attribute +upsampling part, we focus on precisely estimating the attribute +at the upsampled positions. In literature, mainly the geometry +upsampling part has been investigated so far. +Alexa et al. were the first to add additional points to a +point cloud’s surface [8]. They initially investigated the prob- +lem of point cloud reconstruction. Point cloud reconstruction +describes the process of estimating the surface of a point +cloud in order to reconstruct missing areas. In [8], point set +surfaces are presented for point cloud reconstruction. With the +point set surfaces, an estimation of the point cloud’s surface +is established. In a subsequent step the estimated surface +is sampled such that another representation of the surface +is created. The sampling step size steers the accuracy and +smoothness of the new surface representation. Thereby, Alexa +et al. were the first to sample a point cloud’s surface and +thus, adding new points to the set of originally available +points. Other approaches to point cloud reconstruction aim +at solving an indicator function in three-dimensional space. +The surface is then generated by isosurfacing the grid [9]. +Usually, these algorithms work on a regular grid or on octree. +If the normal field agrees with the local derivation of the +surface, the indicator function can be found by solving a +Poisson equation [10]. Apart from point cloud reconstruction, +arXiv:2301.11630v1 [cs.CV] 27 Jan 2023 + +0.8 +0.6 +N +0.4 +0.2 +0 +1 +0.5 +0.5 +y +0 +0 +X2 +point cloud upsampling was shown in Lipman et al. [11]. +They introduced a locally optimal projection (LOP). For this +projection, a set of projected points is defined such that it +minimizes the sum of weighted distances to the given point set. +The LOP can also refine noisy data sets. Thus it is also applied +for the removal of noise and outliers of raw scanned input +data. The edge-aware resampling approach (EAR) by Huang +et al. [12] incorporates normal vectors into the upsampling +scheme. In this approach, the assumption is exploited that +normal vectors of points in homogeneous areas that are far +away from edges are more accurate than normal vectors in +edge-like areas. Thus, the upsampling procedure starts within +the homogeneous areas and continues with the upsampling +progressively to the edge areas. Finally, the remaining regions +are upsampled. EAR produces point sets with accompanying +normal vectors. Normal vectors are also incorporated in the +approach from Dinesh et al. [13]. They assume locally smooth +surfaces and thereby assume only small deviations between +normal vectors of neighboring points. However, a major +problem in point cloud processing is the missing knowledge +regarding neighborhood relations. Dinesh et al. overcome this +problem with a k-nearest-neighbor graph that connects the +single points of a point cloud. The Euclidean distances are +incorporated as a measure to determine the nearest neighbors. +In addition, the graph holds weights that are determined based +on the similarity of neighboring nodes, i.e., points of the +point cloud. The upsampled points are inserted based on a +Delaunay triangulation. Their locations are optimized during +a refinement step. Therefore, the problem is reformulated as +a minimization of a graph-total variation. +Since the development of PointNet in 2017 [14], the pro- +cessing of point cloud problems with neural networks gained +much interest. The first network that performed point cloud +upsampling was Point Cloud Upsampling Net (PU-Net) [15]. +It is built upon PointNet++ [16]. PU-Net splits the input point +cloud into smaller patches. These patches are used to train the +multi-level features using hierarchically learning from Point- +Net++ [16]. The features from each level are subsequently +interpolated and concatenated. As a result, embedded point +features are generated. These are expanded and used for the +three-dimensional coordinate reconstruction. For the training +of the network, a joint loss function is incorporated that +balances between a smooth surface and a uniform distribution +of the points. Numerous further networks build upon PU- +Net. Yifan et al. [17] use PU-Net in a multi-step patch-based +network (MPU-Net). The aim is to adapt the receptive field +of the network. Yu et al. [18] introduced the edge-aware +consolidation network (EC-Net). It especially learns to extract +edges as features during the training phase and mainly adds +upsampled points in edge areas. Furthermore, also a generative +adversarial network (GAN) approach was presented for point +cloud upsampling by Li et al. [19] with PU-GAN. Zhang et +al. [20] do not follow a local patch-based approach but use +the point cloud as a whole as input to their network. The +clear disadvantage of this approach is that the point clouds +always must have the same overall number of points in order to +meet the requested input size of the network. PUGeoNet [21] +does not learn the features in a three-dimensional domain. The +three-dimensional surface is projected onto a two-dimensional +plane first. The local parametrization for the transformation is +learnt. Thereafter, the point cloud upsampling is pursued in the +two-dimensional domain. Finally, the points are shifted back +to the three-dimensional domain by a linear transformation. +Meta-PU [22] is the first data-driven method that aims at +upsampling a point cloud by an arbitrary scaling factor. +However, +all +these +approaches +hold +drawbacks. +The +optimization-based approaches mainly rely on normal vectors. +Unfortunately, normal vectors are not available for every +point cloud. As the calculation of normal vectors is highly +sensitive to noise and point clouds are often noisy due to +their acquisition process, the calculated normal vectors are +not accurate. Also the data-driven neural network based ap- +proaches hold drawbacks. They are mainly trained for distinct +use cases such as a specific data set or scaling factor. Thus, +the generalization to new unseen data sets might be a chal- +lenge. Therefore, Frequency-Selective Geometry Upsampling +(FSGU) was introduced in [23]. A model-based approach +that estimates the object’s surface block-based and iteratively +with cosine basis functions. For the block partitioning, the +points are assigned to solely one block and all points in one +block are processed together. The model exploits the frequency +selectivity principle which is further explained in the upcoming +section. The underlying assumption is that the object’s surface +can be represented locally in terms of a finite number of +basis functions. During model generation, the influence of the +underlying frequency parts are estimated. The continuously +estimated surface can then be sampled at new positions such +that any arbitrary scaling factor can be achieved without the +aid of normal vectors. +The presented methods for upsampling the point cloud +geometry produce only the locations of the upsampled points. +Thus, the missing attribute information has to be assigned +to the upsampled points in a subsequent attribute upsam- +pling step. Upsampling is a well-known problem for two- +dimensional images and is often also referred to as single- +image super-resolution. Numerous methods were developed +for image upsampling [24], [25], [26]. A straight-forward ap- +proach is to use interpolation schemes such as bilinear or bicu- +bic interpolation [24], [26]. These interpolation schemes are +commonly implemented incorporating a triangulation scheme. +For three-dimensional applications, this is a significant draw- +back as for some point locations an extrapolation is required. +This occurs for example in cases of a concave object surface. +Extrapolation is not possible for triangulation-based schemes +as the interpolated point has to be surrounded by points +located at the corners of a triangle. For extrapolation such +a triangle cannot be built and thus the interpolation scheme +fails to estimate an attribute for this point. Hence, the inter- +polation schemes from two-dimensional applications cannot +be transferred directly to three-dimensional surfaces. Dinesh +et al. [27] extended their graph-total variation approach also +to color upsampling of point clouds. A first estimation for +the RGB values is conducted as the mean of the surrounding +color values. Thereafter, the estimation is refined. As for +the geometry, they assume the neighborhood to be piecewise +smooth, i.e., they assume a smooth color surface. With this + +3 +Original signal +Calculate residual r(ν)(m, n) +Calculate residual energy decrease +∆E(ν) for every basis function +Selection of best fitting basis function +Stopping +criterium +met? +Obtain signal at new points (m′, n′) +Final signal +No +Generated model g(ν)(m′, n′) +Yes +Fig. 2: Frequency selectivity principle. +assumption, they can once again reformulate the refinement +as a minimization of a graph total variation term. Based on +the frequency selectivity principle, Frequency-Selective Mesh- +to-Mesh Resampling (FSMMR) was introduced in [28]. The +color attribute of a point cloud is represented in terms of a +weighted superposition of basis functions and the frequency- +selectivity principle is applied. For the model estimation, the +three-dimensional surface of the object is projected into two- +dimensional space. For the projection, a minimum spanning +tree is established. It is based on the Euclidean distances be- +tween neighboring points. The projection into two-dimensional +space is then conducted along the minimum spanning tree +in order to cope for the object’s extension in z-dimension. +Next, the frequency model is established for the color attribute +based on the projected coordinates. However, the minimum +spanning tree has to be established for each block separately +which is a complex process. Hence, we propose to simplify +the projection by the incorporation of geometry information +from the geometry upsampling in this work. +However, no point cloud upsampling scheme has been +reported yet that solves both tasks, geometry and attribute +upsampling in a single and joint scheme. Thus, we propose a +joint geometry and attribute point cloud upsampling scheme +in the following. Therefore, the point cloud is partitioned into +blocks with overlapping support area for smoother results and +less blocking artifacts. We establish a frequency model for +surface extraction deployed for the geometry upsampling as +in [23] and a frequency model for upsampling the corre- +sponding attribute as in [28]. Therefore, the surface of the +three-dimensional object is projected into a two-dimensional +domain. For this transformation, the knowledge from the +geometry upsampling about the surface is incorporated. Our +approach uses the location information from the points and +the attributes solely. No additional information such as normal +vectors are required. Furthermore, the proposed algorithm can +Low-resolution +input point cloud +Block Partitioning +with Overlapped +Support, Sec. III-A +Geometry Upsampling, +Sec. III-B +3D-to-2D projection, +Sec. III-C +Attribute Upsampling, +Sec. III-D +Reverse Block +Partitioning +High-resolution +output point cloud +Fig. 3: Joint Frequency-Selective Upsampling. Newly intro- +duced steps are highlighted in blue. +easily be adapted to new data sets and scaling factors. +The frequency selectivity principle is presented in the +upcoming section. Thereafter, we present our proposed +frequency-selective upsampling in Sec. III. In Section IV, it +follows the extensive evaluation of our proposed upsampling +scheme. Finally, in Section V, a conclusion is drawn. +II. FREQUENCY SELECTIVITY PRINCIPLE +The frequency selectivity principle has already been proven +to be superior in several resampling [28], [29] , reconstruc- +tion [23], [30] and extrapolation scenarios [31]. Therefore, the +signal is first partitioned into blocks. The set of points in one +block is denoted as A and computed jointly. The model always +follows the assumption that a signal f that is known at distinct +positions (m, n) with floating accuracy can be represented in +terms of a weighted position of basis functions ϕ, i.e., +f(m, n) = +� +k,l∈K +ck,lϕk,l(m, n), +(1) +where k, l denote the frequency indices of the basis functions +from the set of available basis functions K and c is the +according expansion coefficient. With our model g, we aim +at estimating (1) in an iterative process. Thus, we define the +model to be +g(ν)(m, n) = g(ν−1)(m, n) + ˆcu,vϕu,v(m, n) +(2) +with (u, v) being the selected frequency indices in one iteration +ν and ˆc being the estimated expansion coefficient. In the +beginning, the model is set to zero, i.e., g(0) = 0. Any +arbitrary type of basis functions can be incorporated into +the model estimation process. We mainly incorporate cosine +basis functions. These provide dense energy compaction such +that a precise model can be found with a small number of +iterations. Furthermore, the basis functions are real-valued +which is advantageous for scattered input data. The difference +between the original signal and the modeled signal is referred +to as residual r. Thus, +r(ν)(m, n) = f(m, n) − g(ν)(m, n). +(3) +The task of the iterative model estimation procedure is to +minimize the deviation between the original signal and the + +4 +75 +76 +77 +78 +79 +17 +18 +19 +20 +21 +x +y +Fig. 4: +Scattered input data that is partitioned into a core +block (blue) of size N = 2 with support area (red) of M = +0.5. Together, they build a block (orange). Points on white +background are not used for model estimation. +model, i.e., to minimize the residual as good as possible. For +the minimization of the residual, the residual energy E +E(ν) = +� +(m,n) +w(m, n) +� +r(ν)(m, n) +�2 +(4) +is calculated in every iteration. Formulating the optimization +problem as an energy holds the advantage that the mini- +mization of the residual is independent of the sign of the +residual. Furthermore, a spatial weighting function w(m, n) +is incorporated that steers the influence of every single point. +The spatial weighting function is usually defined as a decaying +isotropic window function. Thus, the center points have higher +weights assigned than the points in the outer part of the block. +Due to the higher weights in the center, the model estimates +the centered points more accurately than the points in the outer +areas as the smaller weights allow for a larger deviation from +the original positions. +In the final step of each iteration, the best fitting basis +function has to be selected. We select the basis function as +the best fitting basis function in one iteration that reduces the +residual energy the most. Thereby, closing the gap between +original and model as good as possible. The best fitting basis +function is defined by its two-dimensional frequency indices +(u, v) and thus, +(u, v) = argmax +(k,l) +� +∆E(ν) +k,l wf[k, l] +� +. +(5) +During the maximization of the residual energy decrease, an +additional spectral weighting function wf is incorporated. The +spectral weighting function remaps the assumption that the +underlying signal is locally smooth. Hence, the signal mainly +consists of low-frequency basis functions. High frequency +functions tend to produce an oscillating signal that appears +to be noisy. This behavior is usually not desired. Hence, the +spectral weighting function is a smoothly decaying function +that assigns higher weights to low-frequency basis functions +and smaller weights to high-frequency functions. It is once +again described as an isotropically decaying window function, +i.e., +wf[k, l] = σ +√ +k2+l2 +(6) +with decaying factor σ. The isotropically decaying weighting +function allows to include high frequencies if they are domi- +nant in the signal while preserving a mainly low frequency +signal. We now choose the best fitting basis function in +iteration ν and add this function to our estimated model from +the previous iteration step (ν − 1). However, the expansion +coefficient of the chosen basis function has to be determined. +Therefore, we follow the approach from [31] and set the +derivative of the residual energy E(ν) to zero. Thus, the +expansion coefficient is given as +ˆc(ν) +(k,l) = +� +(m,n)∈A r(ν−1)(m, n)ϕ(k,l)(m, n) +� +(m,n)∈A w(m, n)(ϕ(k,l)(m, n))2 . +(7) +If the model estimation is finished, the signal f ′ can be +retrieved at new positions (m′, n′), i.e., +f ′(m′, n′) = +� +k,l∈K +ˆck,lϕk,l(m′, n′). +(8) +The frequency-selective model estimation process is summa- +rized in Fig. 2. +III. PROPOSED FREQUENCY-SELECTIVE UPSAMPLING +In this work, we propose Frequency-Selective Upsampling +(FSU), a point cloud upsampling scheme that can process both, +geometry and attribute upsampling. Geometry and attribute +upsampling are conducted jointly in a sequential manner. An +overview of FSU is given in Fig. 3. Newly introduced steps +are highlighted in blue. Our proposed joint sequential point +cloud upsampling scheme first partitions the point cloud into +local blocks with overlapping support area. Moreover, it uses +geometry information from the geometry upsampling of the +point cloud in the upsampling of the attribute in the conducted +projection step. +A. Block Partitioning +The block partitioning is a crucial part for the proposed +frequency models as the quality of the estimated model +highly depends on the underlying original points. In particular, +the block partitioning is only conducted once and used for +geometry and attribute upsampling. We propose to partition +the three-dimensional volume into blocks with an equal length +of side of N × N × N. These blocks form the core block. An +exemplary three-dimensional block partitioning of the Mario +point cloud is shown in Fig. 1. The point cloud is normalized in +all three dimensions. The shown blocks hold a length of side of +N = +8 +100. However, research for two-dimensional applications +has shown that an additionally added support area around the +core block may increase the final quality [29], [31]. Thus, +we propose to add an additional support area for overlapped +support. Therefore, the incorporated block size for model +estimation is expanded to (N +2M)×(N +2M)×(N +2M) +with M as the margin of the support area. Nevertheless, +additional points in the upsampling procedure are only inserted + +5 +75 +76 +77 +78 +79 17 +18 +19 +20 +21 +2 +4 +6 +x′ +y′ +z′ +(a) Input points in 3D. +75 +76 +77 +78 +79 17 +18 +19 +20 +21 +2 +4 +6 +x′ +y′ +z′ +(b) Continuous model and input +points in 3D. +76 +77 +78 +18 +19 +20 +(c) Delaunay triangulation in 2D +for input points. Output points are +generated in core block. +75 +76 +77 +78 +79 17 +18 +19 +20 +21 +2 +4 +6 +x′ +y′ +z′ +(d) Continuous model with input +and output points in 3D for De- +launay triangulation. +Fig. 5: Model generation process for a block of the Mario point cloud. Blue points are the original points, red points are the +upsampled points. The continuous model is depicted as a mesh plot. +in the area of the core block of N × N × N. An example of +the block partitioning is shown in Fig. 4. Here an excerpt +of Mario is shown in two dimensions for better visibility. +The scattered points are located on arbitrary coordinates +with floating accuracy. We conduct a block partitioning for +N = +2 +100. For better demonstration, the axes are expanded by +a factor 100 such that N = 2 here. The core block covers +the interval x = [76, 78] and y = [18, 20]. This area is +highlighted in blue. All points in this area are assigned to +the core block. In addition, a support area with a margin of +M = 0.5 is added. The support area is shown in red. Hence, +all points that are located in the interval x = [75.5, 78.5] and +y = [17.5, 20.5] are considered for model estimation. This +bears the advantage of a smoother model especially in the +border regions and smoother transitions from one block to +the other. The partitioning process is conducted for all blocks. +Thus, one point might be assigned to the support area of none, +one or more blocks and at the same time it must be a member +of exactly one core block. During this assignment, each point +remains on its original location. Thus, this block partitioning +procedure may not be confused with a voxelization that is +often pursued as a preprocessing for point cloud processing +algorithms. If the block partitioning is finished, one block after +the other is handed over to the geometry upsampling step that +estimates the underlying smooth object’s surface in this one +block. In the following, we refer to the joint set of core block +and support area as a block. The points within one block form +the set of points A. +B. Geometry Upsampling +In the geometry upsampling of a point cloud, points have +to be added to the original point cloud. Therefore, the points’ +locations have to be determined first. These have to satisfy +two essential requirements. First, the points should fit well in +the surface of the object. And second, the points’ distribution +should approximately follow a uniform distribution such that +the newly added points are not located directly next to original +points. We follow the approach as described in [23]. We +assume a point cloud’s surface to be locally smooth and +representable in terms of a function within one block. One +exemplary block is depicted in Fig. 5a. At first sight, the +block just seems to contain a set of loose points. However, we +will estimate a smooth underlying surface in the following. +Subsequently, we place the additional points on this surface. +In a first step, it has to be decided in which dimensions +the surface model is estimated. Therefore, the dimension +that yields the smallest variance is selected. This bears two +advantages. First, the probability that the surface is a closed +form in the dimension with smallest extension is low. Second, +if the first assumption is false, the introduced error is rather +small as only one surface is estimated as averaging surface in +between the other surfaces that build a closed form. Thus, the +modeled dimension z′ is +z′ = min {Var{x}, Var{y}, Var{z}} +(9) +with x, y and z being the three-dimensional coordinates of +the point cloud’s points. The main assumption that we follow +for geometry upsampling is that the underlying surface in one +block can be represented in terms of a function. We assume +the function to be a weighted superposition of basis functions +ϕ, i.e., +z′ = f(x′, y′) = +� +k,l∈K +ck,lϕk,l[x′, y′]. +(10) +This equation is closely related to (1), if we select f(m, n) = +f(x′, y′) with m = x′ and n = y′. Thus, the model estimation +is conducted along the explained steps from Sec. II. The +resulting continuous model of the block is depicted in Fig. 5b. +Finally, the upsampled coordinates ˆz′ are determined based +on the upsampled points (ˆx′, ˆy′) according to +ˆz′ = f(ˆx′, ˆy′) = +� +k,l∈K +ˆck,lϕk,l(ˆx′, ˆy′). +(11) +The upsampled points (ˆx′, ˆy′) are gained by a Delaunay +triangulation of the points (x′, y′). This computation can be +conducted in parallel to the model generation procedure. The +new points are inserted in the middle of the triangle edges +as shown in Fig. 5c. Thereby, high upsampling factors can be +achieved with one triangulation. The final result of the block is +given in Fig. 5d. It is clearly visible, that the blue input points +are located from one block border to the other, whereas the +orange output points are only located in the core block. This +is due to the incorporated support area. In this case a margin +of 0.5 is taken as support for a core block of size 2 × 2 × 2. + +6 +0 +0.25 +0.5 +0.75 +1 +6 +6.5 +7 +7.5 +M/N +C2C ×10−1 +N = 2 +N = 3 +N = 4 +N = 8 +Fig. 6: Geometry results in terms of C2C similarity for +different block sizes (N) and support margins (M). Support +margin is given relative to block size (M/N). +C. Proposed Projection +The attribute upsampling step describes the process of +estimating the attribute’s value at the upsampled positions +determined as in Sec. III-B. An attribute can be anything in +point clouds such as color, intensity or normal vectors. +As a starting point, we use the sparse frequency model ap- +proach from [28]. We assume the point cloud object’s surface +to be a two-dimensional plane in three-dimensional space and +thus, follow the approach to project the surface into a two- +dimensional space. In [28], the Euclidean distances between +the points are measured and incorporated as weights of a graph +that is minimized to a minimum spanning tree. Following this +tree, the points are mapped to a two-dimensional space. The +transformation is conducted for each block. As the calculation +of the euclidean distances and the generation of the minimum +spanning tree is a time-consuming and complex process, we +propose to exploit knowledge from our geometry upsampling +which was presented in the section before. Therefore, we +propose to simplify the 3D to 2D conversion. For geometry +upsampling, the geometry is rotated such that the dimension +with smallest variance is modeled. This bears the advantage +that the standard deviation in z′−direction is small. Exploiting +this, we can directly project the three dimensional points into +two dimensions along the axis of the smallest dimension. Thus, +x′ +2D = x′ +(12) +and +y′ +2D = y′. +(13) +Next, the two-dimensional points are used for the estimation +of the attribute’s frequency model. +D. Attribute Upsampling +The attribute information is assumed to be modeled with +frequencies. As explained in [28], the color signal fa can be +modeled as a weighted superposition of basis functions ϕ from +the set of available basis functions K according to +fa(x′ +2D, y′ +2D) = +� +k,l∈K +ck,lϕk,l(x′ +2D, y′ +2D). +(14) +Once again, this assumption is closely related to Eq. (1). +Hence, for attribute upsampling we can formulate f(m, n) = +0 +0.25 +0.5 +0.75 +1 +1 +2 +3 +4 +M/N +Histogram distance ×10−2 +N = 2 +N = 3 +N = 4 +N = 8 +Fig. 7: Color results in terms of histogram distance [32] for +different block sizes (N) and support margins (M). Support +margin is given relative to block size (M/N). +fa(x′ +2D, y′ +2D) with m = x′ +2D and n = y′ +2D. During the +model estimation process, a continuous model of the attribute +of the point cloud for one block is estimated. Hence, the +resulting model gives a continuous estimation of the point +cloud’s attribute. To assign a proper attribute information to the +upsampled points (ˆx′ +2D, ˆy′ +2D), the estimated model is evaluated +for +fa(ˆx′ +2D, ˆy′ +2D) = +� +k,l∈K +ˆck,lϕk,l(ˆx′ +2D, ˆy′ +2D). +(15) +Finally, the additionally computed points with its geometric +location and the associated color attribute are remapped in +the three-dimensional space of the original point cloud. Thus, +the equations (9), (12) and (13) have to be reversed. The 3D +high-resolution point cloud in geometry and color is the final +result. +IV. EVALUATION +For the evaluation of our proposed joint geometry and +attribute upsampling scheme, we show evaluations for both, +geometry and color, separately. In addition, visual examples +of the joint upsampling scheme are shown. For the evaluations +the 3DColorMesh dataset [33] is incorporated. This dataset +contains colored point clouds with 40,000 to 200,000 points. +A. Metrics +The quality of the upsampled point clouds are determined +for both, geometry and color, separately. For each upsampling +part, different metrics are applied. +1) Geometry: The evaluation of the geometric shape of the +upsampled point cloud is usually done with respect to the +original point cloud. Therefore, the original point cloud serves +as a reference. For each point in the upsampled point cloud, the +nearest neighbor in the reference is searched. The deviations +are summed up and normalized for all points such that the +overall point-to-point (P2P) error is determined. Hence, the +P2P error is the normalized sum of the error vectors being the +smallest distance between the points in the reference and the +upsampled point cloud [34]. + +7 +TABLE I: Geometry results for all point clouds from the 3DColorMesh dataset for scaling factor of 4 in terms of P2P and P2C +errors [34] and C2C similarity [35]. Best qualities are given in bold. Arrows indicate higher values are better ↑ and smaller +values are better ↓, respectively. +P2P ×10−3 ↓ +P2C ×10−3 ↓ +C2C×10−1 ↑ +Point Cloud +PU +EC +FSGU +FSU +PU +EC +FSGU +FSU +PU +EC +FSGU +FSU +4armsMonstre +8.9 +4.5 +3.1 +3.3 +7.2 +2.9 +2.0 +1.9 +4.6 +3.7 +5.4 +5.6 +Asterix +10.1 +4.6 +3.4 +3.5 +8.1 +2.8 +1.9 +1.9 +4.5 +3.6 +4.6 +4.9 +CableCar +11.1 +2.4 +1.7 +1.9 +10.4 +1.3 +0.8 +0.6 +5.0 +3.7 +7.3 +7.6 +Dragon +11.7 +2.1 +1.6 +1.9 +11.2 +6.9 +0.6 +0.5 +4.7 +3.8 +7.8 +8.2 +Duck +11.4 +5.3 +2.9 +3.3 +9.3 +2.1 +0.7 +0.7 +5.5 +3.7 +8.2 +8.5 +GreenDinosaur +8.5 +3.8 +2.8 +2.9 +6.8 +2.0 +1.5 +1.5 +4.1 +3.7 +4.8 +4.9 +GreenMonstre +9.3 +2.3 +1.8 +1.9 +8.4 +0.9 +0.9 +0.9 +4.0 +3.6 +4.7 +4.7 +Horse +7.9 +3.0 +2.7 +2.8 +6.4 +1.7 +1.7 +1.7 +4.3 +3.7 +5.4 +5.2 +Jaguar +11.4 +1.7 +1.3 +1.5 +11.1 +0.6 +0.5 +0.5 +5.2 +3.7 +7.7 +8.0 +LongDinosaur +12.5 +1.5 +1.2 +1.4 +12.2 +0.5 +0.5 +0.5 +5.1 +4.2 +7.8 +8.0 +Man +11.2 +6.2 +3.1 +3.3 +8.6 +2.6 +1.1 +1.1 +4.8 +3.8 +6.6 +6.9 +Mario +11.6 +1.7 +1.3 +1.5 +11.1 +0.6 +0.6 +0.6 +5.0 +3.7 +7.7 +7.8 +MarioCar +11.0 +1.9 +1.4 +1.7 +10.6 +0.7 +0.6 +0.5 +4.9 +3.7 +7.7 +8.0 +PokemonBall +9.7 +8.0 +4.6 +4.5 +6.5 +3.5 +2.7 +2.1 +4.4 +3.4 +4.5 +4.9 +Rabbit +8.7 +3.4 +2.7 +2.9 +7.3 +2.1 +1.8 +1.8 +4.5 +3.6 +5.8 +5.6 +RedHorse +10.8 +1.8 +1.5 +1.8 +10.3 +0.7 +0.7 +0.7 +4.7 +3.7 +7.4 +7.7 +Statue +8.5 +3.6 +2.8 +3.0 +7.0 +2.3 +1.9 +1.8 +4.6 +3.8 +5.7 +5.8 +Average +10.2 +3.5 +2.3 +2.5 +8.9 +1.9 +1.2 +1.1 +4.7 +3.7 +6.4 +6.6 +A related method is to determine the point-to-plane (P2C) +error. Therefore, the same procedure as for the point-to- +point error is followed with the difference that not the direct +deviation between the upsampled point and the nearest point +in the reference is taken but the difference along the normal +of the upsampled point is taken. Once again, these differences +are summed up and normalized such that the overall point-to- +plane error is determined [34]. +As a third metric, the plane-to-plane (C2C) angular similar- +ity is determined. In this metric, a plane is estimated in both, +the reference and the upsampled point cloud. Then, the angular +similarity between the two planes is determined. Thereby, +the visual degradations of a processed point cloud can be +predicted more accurately. The plane-to-plane similarity metric +is determined according to the implementation of Alexiou et +al. [35]. +2) Attribute: A great challenge in the joint upsampling of +the geometry and attribute of a point cloud is the determina- +tion of the final attribute quality as it is highly affected by +geometric distortions. Thus, we decided to evaluate geometry +and attribute separately. In order to have ground truth data +available, we first downsample the original point cloud ran- +domly. The downsampled points incorporate both, geometry +and color information and thus, serve as the original points. +From the remaining points, only the geometry information is +kept such that a possible geometrical distortion is not affecting +the attribute quality during the evaluation. The attribute infor- +mation of these points is determined following the proposed +algorithm described in Sec. III. Thus, the overall point cloud +is separated into blocks. Next, each block is rotated according +to the geometrical variances. The geometrical upsampling is +skipped, it follows the projection from three-dimensional space +to the two-dimensional plane for both point sets. Finally, the +proposed attribute upsampling scheme from Sec. III-D follows. +The downsampling and upsampling is conducted three times +for each point cloud. Thus, the shown results are averaged +over all three runs. +As ground truth data is available, we determine the color +peak-signal-to-noise ratio (PSNR) as it is known from image +processing. Therefore, we first determine the PSNR for each +color channel separately and average for Color-PSNR. We +measure the color reconstruction PSNR, i.e., the color PSNR +is determined solely on the upsampled points. +As a second evaluation scheme, we incorporated a his- +togram comparison as proposed by Viola et al. [32]. As +the histograms from reference and upsampled point cloud +are compared, it can also be applied if no direct ground +truth information is available. The histogram difference of +the luminance channel Y is shown in the evaluation with a +euclidean distance measure. +B. Influence of Block Parameters +The block partitioning is a relevant part for the frequency- +selective upsampling procedure as it jointly sets the block +partitioning for geometry and color upsampling. Hence, the +aim is to determine the block parameters such that geometry +and attribute quality are as good as possible. Therefore, the +influence of block size and size of the newly introduced +support area is analyzed in this section. The geometry results +are shown in terms of C2C similarity in Fig. 6 and the color +result is depicted in Fig. 7, respectively. The choice of the +parameters affects geometry and attribute upsampling at the +same time. Hence, the averaged results for the 3DColorMesh +dataset are depicted for all combinations of core block sizes +N = 2 (blue), N = 3 (red), N = 4 (yellow) and N = 8 +(purple) and support margins relative to block sizes from +M/N += 0 to M/N += 1. The geometry in Fig. 6 is +optimized in terms of C2C similarity as a smooth surface is + +8 +desired. For block sizes larger than 3, the curves show a slight +decrease before the angular similarity increases. For block +size N = 2, the maximum is achieved for a border width +of M/N = 0.25, i.e. the support margin size is M = 0.5. The +color differences in Fig. 7 increase with increasing support +margins. Hence, color quality is maximized for a model that +is as local as possible achieved by a small support margin. +In the conjunction of geometry and color quality, the block +size is set to N = 2 and the support margin is chosen to +be M = 0.5 for the remainder of this work as geometry is +maximized while a good color quality is maintained. +C. Geometry Results +Most of the known point cloud upsampling schemes upsam- +ple the geometry of a point cloud. We compare the geometry +upsampling part of our proposed FSU to FSGU [23]. The +main difference here is that the block partitioning is conducted +differently. In FSU a support area is incorporated whereas +FSGU does not take a support area into account. In addition, +there are two data-driven approaches, namely PU-Net (PU) and +EC-Net (EC). PU-Net focuses on adding new points uniformly +and as distant from given points as possible whereas EC-Net +focuses on upsampling edges properly. For all upsampling +techniques, point-to-point (P2P), point-to-plane (P2C) and +plane-to-plane (C2C) errors are measured. The results for the +17 point clouds of the 3DColorMesh dataset are shown in +Tab. I. In terms of P2P, the results produced by PU-Net are +worst, followed by EC-Net. The best performing approaches +are the frequency-model based approaches. Our proposed FSU +approach performs slightly worse than FSGU. Due to the +incorporated support area in FSU, the upsampled points can +be located within the whole block and not just within the +range of the original points. Thereby, the distances in between +the points increase and thus, also P2P errors may increase. +Even though the incorporated support area may not lead to an +improvement in the P2P error metric, it leads to an enhanced +visual appearance as it is shown in Fig. 10. The original of +the 4armsMonstre in Fig. 10a is upsampled with FSGU +in Fig. 10b and our proposed FSU in Fig 10c, respectively. +Clear block artifacts can be observed with the upsampling of +FSGU. These disappear with FSU such that a better visual +quality is achieved. In terms of the metrics, the behavior in +terms of P2P metric changes. Here, the FSU results outperform +the results generated with FSGU. This shows the improved +plane reconstruction in FSU. The results in terms of angular +similarity C2C are given in the last columns of Tab. I. As this is +a similarity metric, higher values denote better results. Here, +the frequency-model based approaches show highest values +and thus, are the best upsampling methods for this metric. +Hence, FSU outperforms FSGU in terms of P2C error and +C2C similarity showing that FSU produces smoother results +than FSGU. Thus, the estimation and sampling improved with +FSU. The data-driven approaches perform worst. +The behavior in terms of the metrics remain similar for +further scaling factors as well. An overview of the averaged +result for the dataset from scaling factor two to four is shown +in Fig. 8 for the angular similarity C2C. The relative behavior +2 +3 +4 +0 +2 +4 +6 +8 +10 +Scaling factor +C2C ×10−1 +PU (C2C) +EC (C2C) +FSGU (C2C) +FSU (C2C) +Fig. 8: +Evolution of the C2C results for the averaged +3DColorMesh dataset for scaling factors from two to four. +Upsampling with PU (orange), EC (yellow), FSGU (purple), +and FSU (green). +as shown in Tab. I of the curve remains constant. FSU is +the best performing method in terms of C2C for all scaling +factors. As a general trend, the C2C angular similarity metric +decreases with increasing scaling factor. +D. Color Results +In a second evaluation, we analyze the quality of the color +attribute of the point clouds. Our proposed upsampling scheme +is denoted as FSU. We compare FSU to the Frequency- +Selective Mesh-to-Mesh Resampling (FSMMR) as proposed +in [28], linear interpolation on block-level (LIN2) and on +the whole point cloud (LIN3). Furthermore, natural neighbor +interpolation is also appplied on both, block level (NAT2) and +on the whole point cloud (NAT3). +The results for the metrics introduced in Sec. IV-A2 are +given in Tab. II. The upsampling techniques are evaluated for +all 17 point clouds from the 3DColorMesh dataset. The color +PSNR is shown in the first six columns. The results of the +histogram based evaluation of Viola et al. [32] is depicted in +the last six columns of the table. As it is shown in terms of +PSNR, the proposed FSU performs best on average with a +gain of 1.9 dB to FSMMR. Especially for the Dragon point +cloud, gains of up to 3.1 dB are achieved. The results for +our proposed approach FSU show that it is advantageous to +include a support area and thereby, take more neighborhood +information into account for the model estimation. Severe +degradations of the point clouds can be observed for the +interpolation approaches as they are based on triangulations. +Hence, color reconstruction may not be possible in concave +areas where an extrapolation is necessary. Thus, not all color +information can be retrieved and thus, color PSNR degrades. A +similar behavior can be observed for the histogram difference +by Viola et al. [32] in the last six columns of the table. +As it is a distance-based measure, smaller values indicate +better results. Severe degradations can be observed for the +interpolation-based approaches especially for the Duck, Man, +and PokemonBall point clouds. As our model-based FSU +can retrieve all color information independently of whether +inter- or extrapolation is required the histogram distances are + +9 +TABLE II: Color results for all point clouds from the 3DColorMesh dataset for scaling factor of 4 in terms of color PSNR in +dB and the histogram distance by Viola et al [32]. Best qualities are given in bold. Arrows indicate higher values are better ↑ +and smaller values are better ↓, respectively. +Color PSNR ↑ +Histogram distance by Viola et al. ×10e − 2 ↓ +3D +2D +3D +2D +Point Cloud +LIN3 +NAT3 +LIN2 +NAT2 +FSMMR +FSU +LIN3 +NAT3 +LIN2 +NAT2 +FSMMR +FSU +4armsMonstre +22.6 +22.6 +13.0 +13.0 +22.7 +25.6 +3.3 +3.3 +31.1 +31.1 +1.6 +0.7 +Asterix +19.9 +20.0 +7.6 +7.6 +23.0 +21.0 +4.3 +4.8 +47.7 +47.7 +4.5 +3.9 +CableCar +22.8 +23.0 +13.1 +13.1 +19.0 +20.7 +1.5 +1.7 +20.0 +20.0 +2.7 +2.1 +Dragon +25.8 +25.9 +16.6 +16.6 +23.1 +26.2 +1.8 +1.9 +18.9 +18.9 +1.8 +1.0 +Duck +12.3 +12.4 +5.0 +5.0 +14.0 +15.5 +1.8 +20.4 +71.2 +71.3 +5.7 +3.0 +GreenDinosaur +24.6 +24.5 +14.0 +14.0 +22.2 +23.4 +2.0 +2.2 +37.6 +37.6 +1.7 +1.1 +GreenMonstre +25.0 +25.3 +15.2 +15.2 +22.2 +25.2 +1.5 +1.8 +16.1 +16.1 +3.0 +2.0 +Horse +22.3 +22.4 +10.8 +10.8 +17.7 +19.6 +1.7 +2.1 +21.0 +20.9 +4.3 +3.2 +Jaguar +19.8 +19.8 +13.1 +13.1 +23.0 +25.6 +3.5 +3.8 +10.4 +10.5 +4.5 +3.2 +LongDinosaur +20.1 +20.2 +16.2 +16.2 +25.1 +27.8 +2.9 +3.0 +6.7 +6.7 +1.9 +1.4 +Man +28.7 +28.9 +17.6 +17.6 +26.8 +26.5 +4.1 +4.2 +62.6 +62.6 +4.1 +1.8 +Mario +22.0 +22.1 +14.8 +14.8 +22.6 +20.0 +3.2 +4.3 +6.4 +6.2 +3.1 +2.5 +MarioCar +22.7 +22.6 +14.4 +14.4 +22.9 +24.4 +1.7 +1.8 +12.9 +12.9 +1.8 +1.4 +PokemonBall +8.7 +8.7 +7.9 +7.9 +16.8 +18.6 +25.6 +25.4 +41.5 +41.5 +8.9 +5.0 +Rabbit +20.0 +20.0 +10.1 +10.1 +21.1 +23.4 +2.8 +3.2 +20.2 +20.2 +5.0 +3.4 +RedHorse +21.4 +21.5 +13.2 +13.2 +18.9 +21.0 +1.6 +1.7 +14.5 +14.5 +2.3 +1.3 +Statue +22.9 +22.9 +14.1 +14.1 +21.8 +24.2 +2.6 +2.7 +21.3 +21.3 +1.5 +0.7 +Average +21.3 +21.3 +12.7 +12.7 +21.0 +22.9 +4.8 +5.2 +27.1 +27.1 +3.4 +2.2 +(a) Original. +(b) FSGU+FSMMR. +(c) FSU (Proposed). +Fig. 9: Mario point cloud. Upsampling factor is 4. +much smaller. On average, FSU performs best in terms of +color PSNR and in terms of histogram distances. +E. Joint Results +Our proposed FSU upsamples geometry and attribute +jointly. Some visual results are given in Figs. 9 and 10. +The original point cloud with original resolution is given in +Subfigs. (a). Subfigs. (b) depict the results if FSGU [23] and +FSMMR [28] , are combined. Subfigs. (c) depict the proposed +FSU. Clear improvements can be observed for our proposed +FSU. Especially the 4armsMonstre shows clear blocking +artifacts in Fig. 10b. The newly proposed FSU as given in +Fig. 10c improves these artifacts notably and appears to be +much smoother. This is mainly due to the added support area +that is used during the model generation. +V. CONCLUSION +In this work, we presented a joint upsampling scheme for +geometry and attribute of point clouds. We therefore incor- +porate frequency-selective models. The models are estimated +locally on block level. In the block partitioning an overlap- +ping support area is included that incorporates neighborhood +information into the block estimation process. For attribute +upsampling, information from the geometry upsampling step +is exploited. Our proposed Frequency-Selective Upsampling +(FSU) improves point cloud upsampling in both, geometry and +color quality. FSU shows best results in terms of point-to-plane +error and plane-to-plane angular similarity. Furthermore, the +color upsampling quality is on average improved by 1.9 dB +in terms of color PSNR. Also the visual appearance of the +upsampled point cloud is improved notably as the included +support area reduces block artifacts clearly. + +10 +(a) Original. +(b) FSGU + FSMMR. +(c) FSU (Proposed). +Fig. 10: 4armsMonstre point cloud. Upsampling factor is 4. +REFERENCES +[1] R. Mekuria, K. Blom, and P. Cesar, “Design, implementation, and +evaluation of a point cloud codec for tele-immersive video,” IEEE +Transactions on Circuits and Systems for Video Technology, vol. 27, +no. 4, pp. 828–842, 2017. +[2] R. Held, A. Gupta, B. Curless, and M. Agrawala, 3D Puppetry: A Kinect- +Based Interface for 3D Animation. +New York, NY, USA: Association +for Computing Machinery, 2012, p. 423–434. +[3] X. Chen, H. Ma, J. Wan, B. Li, and T. Xia, “Multi-view 3D Object +Detection Network for Autonomous Driving,” in Proceedings IEEE +Conference on Computer Vision and Pattern Recognition, 2017, pp. +6526–6534. +[4] A. Geiger, P. Lenz, and R. Urtasun, “Are we ready for autonomous +driving? The KITTI vision benchmark suite,” in IEEE Conference on +Computer Vision and Pattern Recognition, 2012, pp. 3354–3361. +[5] B. Mahmood, S. Han, and D.-E. Lee, “BIM-Based Registration and +Localization of 3D Point Clouds of Indoor Scenes Using Geometric +Features for Augmented Reality,” Remote Sensing, vol. 12, no. 14, pp. +1–29, 2020. +[6] M. Andriasyan, J. Moyano, J. E. Nieto-Juli´an, and D. Ant´on, “From +Point Cloud Data to Building Information Modelling: An Automatic +Parametric Workflow for Heritage,” Remote Sensing, vol. 12, no. 7, pp. +1–22, 2020. +[7] J. Han, L. Shao, D. Xu, and J. Shotton, “Enhanced computer vision with +microsoft kinect sensor: A review,” IEEE Transactions on Cybernetics, +vol. 43, no. 5, pp. 1318–1334, 2013. +[8] M. Alexa, J. Behr, D. Cohen-Or, S. Fleishmann, D. Levin, and C. T. +Silva, “Point Set Surfaces,” in Proceedings Visualization, 2001. VIS ’01., +2001, pp. 21–29, 537. +[9] M. Berger, J. A. Levine, L. Nonato, G. Taubin, and C. T. Silva, +“A Benchmark for Surface Reconstruction,” in ACM Transactions on +Graphics, Apr. 2013. +[10] M. M. Kazhdan, M. Bolitho, and H. Hoppe, “Poisson Surface Recon- +struction,” in Proceedings of the Fourth Eurographics Symposium on +Geometry Processing, vol. 256, 2006, pp. 61–70. +[11] Y. Lipman, D. Cohen-Or, D. Levin, and H. Tal-Ezer, “Parametrization- +free projection for geometry reconstruction,” ACM Transactions on +Graphics, vol. 26, no. 3, 2007. +[12] H. Huang, S. Wu, M. Gong, D. Cohen-Or, U. Ascher, and H. R. Zhang, +“Edge-aware point set resampling,” ACM Transactions on Graphics, +vol. 32, no. 1, Feb. 2013. +[13] C. Dinesh, G. Cheung, and I. V. Baji´c, “3D Point Cloud Super- +Resolution via Graph Total Variation on Surface Normals,” in Proceed- +ings IEEE International Conference on Image Processing, 2019, pp. +4390–4394. +[14] R. Q. Charles, H. Su, M. Kaichun, and L. J. Guibas, “PointNet: Deep +Learning on Point Sets for 3D Classification and Segmentation,” in IEEE +Conference on Computer Vision and Pattern Recognition, 2017, pp. 77– +85. +[15] L. Yu, X. Li, C. Fu, D. Cohen-Or, and P. Heng, “PU-Net: Point Cloud +Upsampling Network,” in Proceedings IEEE Conference on Computer +Vision and Pattern Recognition, 2018, pp. 2790–2799. +[16] C. R. Qi, L. Yi, H. Su, and L. J. Guibas, “PointNet++: Deep Hierarchical +Feature Learning on Point Sets in a Metric Space,” in Advances in Neural +Information Processing Systems, vol. 30, 2017. +[17] W. Yifan, S. Wu, H. Huang, D. Cohen-Or, and O. Sorkine-Hornung, +“Patch-Based Progressive 3D Point Set Upsampling,” in Proceedings +IEEE/CVF Conference on Computer Vision and Pattern Recognition, +2019, pp. 5951–5960. +[18] L. Yu, X. Li, C.-W. Fu, D. Cohen-Or, and P.-A. Heng, “EC-Net: +An Edge-Aware Point Set Consolidation Network,” in Proceedings +Computer Vision – ECCV 2018, 2018, pp. 398–414. +[19] R. Li, X. Li, C.-W. Fu, D. Cohen-Or, and P.-A. Heng, “PU-GAN: A Point +Cloud Upsampling Adversarial Network,” in Proceedings IEEE/CVF +International Conference on Computer Vision, 2019, pp. 7202–7211. +[20] W. Zhang, H. Jiang, and Zhang, “Data-driven Upsampling of Point +Clouds,” Computer-Aided Design, vol. 112, pp. 1–13, 2019. +[21] Y. Qian, J. Hou, S. Kwong, and Y. He, “PUGeo-Net: A Geometry- +Centric Network for 3D Point Cloud Upsampling,” in European Con- +ference on Computer Vision, 2020, pp. 752–769. +[22] S. Ye, D. Chen, S. Han, Z. Wan, and J. Lia, “Meta-PU: An Arbitrary- +Scale Upsampling Network for Point Cloud,” IEEE Transactions on +Visualization and Computer Graphics, 2021. +[23] V. Heimann, A. Spruck, and A. Kaup, “Frequency-Selective Geometry +Upsampling of Point Clouds,” in IEEE International Conference on +Image Processing, 2022, arXiv:2205.01458. +[24] S. C. Park, M. K. Park, and M. G. Kang, “Super-Resolution Image +Reconstruction: A Technical Overview,” in IEEE Signal Processing +Magazine, vol. 20, no. 3, May 2003, pp. 21–36. +[25] C. Dong, C. C. Loy, K. He, and X. Tang, “Image Super-Resolution Using +Deep Convolutional Networks,” IEEE Transactions on Pattern Analysis +and Machine Intelligence, vol. 38, no. 2, pp. 295–307, 2016. +[26] C.-Y. Yang, C. Ma, and M.-H. Yang, “Single-Image Super-Resolution: +A Benchmark,” in European Conference on Computer Vision, 2014, pp. +372–386. +[27] C. Dinesh, G. Cheung, and I. V. Baji´c, “Super-Resolution of 3D Color +Point Clouds Via Fast Graph Total Variation,” in Proceedings IEEE +International Conference on Acoustics, Speech and Signal Processing, +2020, pp. 1983–1987. +[28] V. Heimann, A. Spruck, and A. Kaup, “Frequency-Selective Mesh-to- +Mesh Resampling for Color Upsampling of Point Clouds,” in Pro- +ceedings IEEE 23rd International Workshop on Multimedia Signal +Processing, 2021. +[29] J. Koloda, J. Seiler, and A. Kaup, “Frequency-Selective Mesh-to-Grid +Resampling for Image Communication,” vol. 19, no. 8, pp. 1689–1701, +August 2017. +[30] J. Seiler, M. Jonscher, M. Sch¨oberl, and A. Kaup, “Resampling Images +to a Regular Grid From a Non-Regular Subset of Pixel Positions Using +Frequency Selective Reconstruction,” vol. 24, no. 11, pp. 4540–4555, +November 2015. +[31] A. Kaup, K. Meisinger, and T. Aach, “Frequency Selective Signal +Extrapolation with Applications to Error Concealment in Image ommu- +nication,” in International Journal of Electronics and Communications, +vol. 59, no. 3, June 2005, pp. 147–156. +[32] I. Viola, S. Subramanyam, and P. Cesar, “A color-based objective +quality metric for point cloud contents,” in 2020 Twelfth International +Conference on Quality of Multimedia Experience. +IEEE, 2020. +[33] A. +Nouri, +C. +Charrier, +and +O. +L´ezoray, +“Greyc +3D +Colored +Mesh Database,” Normandie Universit´e, Unicaen, EnsiCaen, CNRS, +GREYC UMR 6072, Technical Report, Jan. 2017. [Online]. Available: +https://hal.archives-ouvertes.fr/hal-01441721 +[34] D. Tian, H. Ochimizu, C. Feng, R. Cohen, and A. Vetro, “Geometric +distortion metrics for point cloud compression,” in Proceedings IEEE +International Conference on Image Processing, 2017, pp. 3460–3464. +[35] E. Alexiou and T. Ebrahimi, “Towards a Point Cloud Structural Similar- +ity Metric,” in IEEE International Conference on Multimedia and Expo, +2020, pp. 1–6. + diff --git a/JtFJT4oBgHgl3EQfwi0E/content/tmp_files/load_file.txt b/JtFJT4oBgHgl3EQfwi0E/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c8bd99d7aefbaf91a6a40d64acb4fd9b298c99df --- /dev/null +++ b/JtFJT4oBgHgl3EQfwi0E/content/tmp_files/load_file.txt @@ -0,0 +1,1244 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf,len=1243 +page_content='1 Joint Geometry and Attribute Upsampling of Point Clouds Using Frequency-Selective Models with Overlapped Support Viktoria Heimann, Andreas Spruck, and Andr´e Kaup Abstract—With the increasing demand of capturing our en- vironment in three-dimensions for AR/ VR applications and autonomous driving among others, the importance of high- resolution point clouds rises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As the capturing process is a complex task, point cloud upsampling is often desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We propose Frequency-Selective Upsampling (FSU), an upsampling scheme that upsamples geometry and attribute information of point clouds jointly in a sequential manner with overlapped support areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The point cloud is partitioned into blocks with overlapping support area first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Then, a continuous frequency model is generated that estimates the point cloud’s surface locally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The model is sampled at new positions for upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In a subsequent step, another frequency model is created that models the attribute signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Here, knowledge from the geometry upsampling is exploited for a simplified projection of the points in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The attribute model is evaluated for the upsampled geometry positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In our extensive evaluation, we evaluate geometry and attribute upsampling independently and show joint results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The geometry results show best performances for our proposed FSU in terms of point-to-plane error and plane- to-plane angular similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Moreover, FSU outperforms other color upsampling schemes by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 dB in terms of color PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In addition, the visual appearance of the point clouds clearly increases with FSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Index Terms—Point Cloud Upsampling, Frequency Model I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' INTRODUCTION AND RELATED WORK The increasing demand of capturing our environment for virtual and augmented reality applications [1], [2], in automo- tive industry [3], [4], in architecture, and archaeology [5], [6] drives the need for high-resolution point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Point clouds are a versatile three-dimensional data type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In a point cloud, single points are captured using, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', a Light Detection and Ranging (LiDAR) sensor or an RGB-D camera such as the Microsoft Kinect [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For each point in a point cloud, the location in 3D space is stored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Moreover, each point may have an attribute assigned such as an intensity value or color information in RGB format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Such a set of many points forms a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As both, geometry and attribute, have to be stored for each point in a point cloud, this data type requires large storage capacities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, many applications demand for high resolution point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, point clouds often have to be upsampled artificially after acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Manuscript created 14 October 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The authors are with the Chair of Multimedia Communications and Signal Processing, Friedrich-Alexander Universit¨at, Erlangen-N¨urnberg (FAU), 91058 Erlangen, Germany (e-mail: viktoria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='heimann@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' an- dreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='spruck@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' andre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='kaup@fau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='de).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This work was partly funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – SFB 1483 – Project-ID 442419336, Emp- kinS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1: Mario point cloud is partitioned into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The upsampling of point clouds applies to both, the geom- etry and the attributes of a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As a consequence of this, point cloud upsampling is generally separated into two steps, geometry upsampling and attribute upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In the geometry upsampling part, we focus on retrieving the best location for the upsampled points whereas in the attribute upsampling part, we focus on precisely estimating the attribute at the upsampled positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In literature, mainly the geometry upsampling part has been investigated so far.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Alexa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' were the first to add additional points to a point cloud’s surface [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' They initially investigated the prob- lem of point cloud reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Point cloud reconstruction describes the process of estimating the surface of a point cloud in order to reconstruct missing areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In [8], point set surfaces are presented for point cloud reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' With the point set surfaces, an estimation of the point cloud’s surface is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In a subsequent step the estimated surface is sampled such that another representation of the surface is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The sampling step size steers the accuracy and smoothness of the new surface representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereby, Alexa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' were the first to sample a point cloud’s surface and thus, adding new points to the set of originally available points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Other approaches to point cloud reconstruction aim at solving an indicator function in three-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The surface is then generated by isosurfacing the grid [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Usually, these algorithms work on a regular grid or on octree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' If the normal field agrees with the local derivation of the surface, the indicator function can be found by solving a Poisson equation [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Apart from point cloud reconstruction, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='11630v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='CV] 27 Jan 2023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 0 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 y 0 0 X2 point cloud upsampling was shown in Lipman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' They introduced a locally optimal projection (LOP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For this projection, a set of projected points is defined such that it minimizes the sum of weighted distances to the given point set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The LOP can also refine noisy data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus it is also applied for the removal of noise and outliers of raw scanned input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The edge-aware resampling approach (EAR) by Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [12] incorporates normal vectors into the upsampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In this approach, the assumption is exploited that normal vectors of points in homogeneous areas that are far away from edges are more accurate than normal vectors in edge-like areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the upsampling procedure starts within the homogeneous areas and continues with the upsampling progressively to the edge areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Finally, the remaining regions are upsampled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' EAR produces point sets with accompanying normal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Normal vectors are also incorporated in the approach from Dinesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' They assume locally smooth surfaces and thereby assume only small deviations between normal vectors of neighboring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, a major problem in point cloud processing is the missing knowledge regarding neighborhood relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Dinesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' overcome this problem with a k-nearest-neighbor graph that connects the single points of a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The Euclidean distances are incorporated as a measure to determine the nearest neighbors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In addition, the graph holds weights that are determined based on the similarity of neighboring nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', points of the point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The upsampled points are inserted based on a Delaunay triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Their locations are optimized during a refinement step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the problem is reformulated as a minimization of a graph-total variation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Since the development of PointNet in 2017 [14], the pro- cessing of point cloud problems with neural networks gained much interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The first network that performed point cloud upsampling was Point Cloud Upsampling Net (PU-Net) [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' It is built upon PointNet++ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' PU-Net splits the input point cloud into smaller patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These patches are used to train the multi-level features using hierarchically learning from Point- Net++ [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The features from each level are subsequently interpolated and concatenated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As a result, embedded point features are generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These are expanded and used for the three-dimensional coordinate reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For the training of the network, a joint loss function is incorporated that balances between a smooth surface and a uniform distribution of the points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Numerous further networks build upon PU- Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yifan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [17] use PU-Net in a multi-step patch-based network (MPU-Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The aim is to adapt the receptive field of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [18] introduced the edge-aware consolidation network (EC-Net).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' It especially learns to extract edges as features during the training phase and mainly adds upsampled points in edge areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Furthermore, also a generative adversarial network (GAN) approach was presented for point cloud upsampling by Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [19] with PU-GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [20] do not follow a local patch-based approach but use the point cloud as a whole as input to their network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The clear disadvantage of this approach is that the point clouds always must have the same overall number of points in order to meet the requested input size of the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' PUGeoNet [21] does not learn the features in a three-dimensional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The three-dimensional surface is projected onto a two-dimensional plane first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The local parametrization for the transformation is learnt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereafter, the point cloud upsampling is pursued in the two-dimensional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Finally, the points are shifted back to the three-dimensional domain by a linear transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Meta-PU [22] is the first data-driven method that aims at upsampling a point cloud by an arbitrary scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, all these approaches hold drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The optimization-based approaches mainly rely on normal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Unfortunately, normal vectors are not available for every point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As the calculation of normal vectors is highly sensitive to noise and point clouds are often noisy due to their acquisition process, the calculated normal vectors are not accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Also the data-driven neural network based ap- proaches hold drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' They are mainly trained for distinct use cases such as a specific data set or scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the generalization to new unseen data sets might be a chal- lenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, Frequency-Selective Geometry Upsampling (FSGU) was introduced in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A model-based approach that estimates the object’s surface block-based and iteratively with cosine basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For the block partitioning, the points are assigned to solely one block and all points in one block are processed together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The model exploits the frequency selectivity principle which is further explained in the upcoming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The underlying assumption is that the object’s surface can be represented locally in terms of a finite number of basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' During model generation, the influence of the underlying frequency parts are estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The continuously estimated surface can then be sampled at new positions such that any arbitrary scaling factor can be achieved without the aid of normal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The presented methods for upsampling the point cloud geometry produce only the locations of the upsampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the missing attribute information has to be assigned to the upsampled points in a subsequent attribute upsam- pling step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Upsampling is a well-known problem for two- dimensional images and is often also referred to as single- image super-resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Numerous methods were developed for image upsampling [24], [25], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A straight-forward ap- proach is to use interpolation schemes such as bilinear or bicu- bic interpolation [24], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These interpolation schemes are commonly implemented incorporating a triangulation scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For three-dimensional applications, this is a significant draw- back as for some point locations an extrapolation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This occurs for example in cases of a concave object surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Extrapolation is not possible for triangulation-based schemes as the interpolated point has to be surrounded by points located at the corners of a triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For extrapolation such a triangle cannot be built and thus the interpolation scheme fails to estimate an attribute for this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the inter- polation schemes from two-dimensional applications cannot be transferred directly to three-dimensional surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Dinesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [27] extended their graph-total variation approach also to color upsampling of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A first estimation for the RGB values is conducted as the mean of the surrounding color values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereafter, the estimation is refined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As for the geometry, they assume the neighborhood to be piecewise smooth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', they assume a smooth color surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' With this 3 Original signal Calculate residual r(ν)(m, n) Calculate residual energy decrease ∆E(ν) for every basis function Selection of best fitting basis function Stopping criterium met?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Obtain signal at new points (m′, n′) Final signal No Generated model g(ν)(m′, n′) Yes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2: Frequency selectivity principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' assumption, they can once again reformulate the refinement as a minimization of a graph total variation term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Based on the frequency selectivity principle, Frequency-Selective Mesh- to-Mesh Resampling (FSMMR) was introduced in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The color attribute of a point cloud is represented in terms of a weighted superposition of basis functions and the frequency- selectivity principle is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For the model estimation, the three-dimensional surface of the object is projected into two- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For the projection, a minimum spanning tree is established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' It is based on the Euclidean distances be- tween neighboring points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The projection into two-dimensional space is then conducted along the minimum spanning tree in order to cope for the object’s extension in z-dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Next, the frequency model is established for the color attribute based on the projected coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, the minimum spanning tree has to be established for each block separately which is a complex process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, we propose to simplify the projection by the incorporation of geometry information from the geometry upsampling in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, no point cloud upsampling scheme has been reported yet that solves both tasks, geometry and attribute upsampling in a single and joint scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, we propose a joint geometry and attribute point cloud upsampling scheme in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the point cloud is partitioned into blocks with overlapping support area for smoother results and less blocking artifacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We establish a frequency model for surface extraction deployed for the geometry upsampling as in [23] and a frequency model for upsampling the corre- sponding attribute as in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the surface of the three-dimensional object is projected into a two-dimensional domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For this transformation, the knowledge from the geometry upsampling about the surface is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Our approach uses the location information from the points and the attributes solely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' No additional information such as normal vectors are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Furthermore, the proposed algorithm can Low-resolution input point cloud Block Partitioning with Overlapped Support, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III-A Geometry Upsampling, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III-B 3D-to-2D projection, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III-C Attribute Upsampling, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III-D Reverse Block Partitioning High-resolution output point cloud Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3: Joint Frequency-Selective Upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Newly intro- duced steps are highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' easily be adapted to new data sets and scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The frequency selectivity principle is presented in the upcoming section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereafter, we present our proposed frequency-selective upsampling in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In Section IV, it follows the extensive evaluation of our proposed upsampling scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Finally, in Section V, a conclusion is drawn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' FREQUENCY SELECTIVITY PRINCIPLE The frequency selectivity principle has already been proven to be superior in several resampling [28], [29] , reconstruc- tion [23], [30] and extrapolation scenarios [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the signal is first partitioned into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The set of points in one block is denoted as A and computed jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The model always follows the assumption that a signal f that is known at distinct positions (m, n) with floating accuracy can be represented in terms of a weighted position of basis functions ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', f(m, n) = � k,l∈K ck,lϕk,l(m, n), (1) where k, l denote the frequency indices of the basis functions from the set of available basis functions K and c is the according expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' With our model g, we aim at estimating (1) in an iterative process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, we define the model to be g(ν)(m, n) = g(ν−1)(m, n) + ˆcu,vϕu,v(m, n) (2) with (u, v) being the selected frequency indices in one iteration ν and ˆc being the estimated expansion coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In the beginning, the model is set to zero, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Any arbitrary type of basis functions can be incorporated into the model estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We mainly incorporate cosine basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These provide dense energy compaction such that a precise model can be found with a small number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Furthermore, the basis functions are real-valued which is advantageous for scattered input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The difference between the original signal and the modeled signal is referred to as residual r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, r(ν)(m, n) = f(m, n) − g(ν)(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (3) The task of the iterative model estimation procedure is to minimize the deviation between the original signal and the 4 75 76 77 78 79 17 18 19 20 21 x y Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 4: Scattered input data that is partitioned into a core block (blue) of size N = 2 with support area (red) of M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Together, they build a block (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Points on white background are not used for model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', to minimize the residual as good as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For the minimization of the residual, the residual energy E E(ν) = � (m,n) w(m, n) � r(ν)(m, n) �2 (4) is calculated in every iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Formulating the optimization problem as an energy holds the advantage that the mini- mization of the residual is independent of the sign of the residual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Furthermore, a spatial weighting function w(m, n) is incorporated that steers the influence of every single point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The spatial weighting function is usually defined as a decaying isotropic window function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the center points have higher weights assigned than the points in the outer part of the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Due to the higher weights in the center, the model estimates the centered points more accurately than the points in the outer areas as the smaller weights allow for a larger deviation from the original positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In the final step of each iteration, the best fitting basis function has to be selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We select the basis function as the best fitting basis function in one iteration that reduces the residual energy the most.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereby, closing the gap between original and model as good as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The best fitting basis function is defined by its two-dimensional frequency indices (u, v) and thus, (u, v) = argmax (k,l) � ∆E(ν) k,l wf[k, l] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (5) During the maximization of the residual energy decrease, an additional spectral weighting function wf is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The spectral weighting function remaps the assumption that the underlying signal is locally smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the signal mainly consists of low-frequency basis functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' High frequency functions tend to produce an oscillating signal that appears to be noisy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This behavior is usually not desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the spectral weighting function is a smoothly decaying function that assigns higher weights to low-frequency basis functions and smaller weights to high-frequency functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' It is once again described as an isotropically decaying window function, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', wf[k, l] = σ √ k2+l2 (6) with decaying factor σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The isotropically decaying weighting function allows to include high frequencies if they are domi- nant in the signal while preserving a mainly low frequency signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We now choose the best fitting basis function in iteration ν and add this function to our estimated model from the previous iteration step (ν − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, the expansion coefficient of the chosen basis function has to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, we follow the approach from [31] and set the derivative of the residual energy E(ν) to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the expansion coefficient is given as ˆc(ν) (k,l) = � (m,n)∈A r(ν−1)(m, n)ϕ(k,l)(m, n) � (m,n)∈A w(m, n)(ϕ(k,l)(m, n))2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (7) If the model estimation is finished, the signal f ′ can be retrieved at new positions (m′, n′), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', f ′(m′, n′) = � k,l∈K ˆck,lϕk,l(m′, n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (8) The frequency-selective model estimation process is summa- rized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' PROPOSED FREQUENCY-SELECTIVE UPSAMPLING In this work, we propose Frequency-Selective Upsampling (FSU), a point cloud upsampling scheme that can process both, geometry and attribute upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Geometry and attribute upsampling are conducted jointly in a sequential manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' An overview of FSU is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Newly introduced steps are highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Our proposed joint sequential point cloud upsampling scheme first partitions the point cloud into local blocks with overlapping support area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Moreover, it uses geometry information from the geometry upsampling of the point cloud in the upsampling of the attribute in the conducted projection step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Block Partitioning The block partitioning is a crucial part for the proposed frequency models as the quality of the estimated model highly depends on the underlying original points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In particular, the block partitioning is only conducted once and used for geometry and attribute upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We propose to partition the three-dimensional volume into blocks with an equal length of side of N × N × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These blocks form the core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' An exemplary three-dimensional block partitioning of the Mario point cloud is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The point cloud is normalized in all three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The shown blocks hold a length of side of N = 8 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, research for two-dimensional applications has shown that an additionally added support area around the core block may increase the final quality [29], [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, we propose to add an additional support area for overlapped support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the incorporated block size for model estimation is expanded to (N +2M)×(N +2M)×(N +2M) with M as the margin of the support area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Nevertheless, additional points in the upsampling procedure are only inserted 5 75 76 77 78 79 17 18 19 20 21 2 4 6 x′ y′ z′ (a) Input points in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 75 76 77 78 79 17 18 19 20 21 2 4 6 x′ y′ z′ (b) Continuous model and input points in 3D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 76 77 78 18 19 20 (c) Delaunay triangulation in 2D for input points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Output points are generated in core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 75 76 77 78 79 17 18 19 20 21 2 4 6 x′ y′ z′ (d) Continuous model with input and output points in 3D for De- launay triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5: Model generation process for a block of the Mario point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Blue points are the original points, red points are the upsampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The continuous model is depicted as a mesh plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' in the area of the core block of N × N × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' An example of the block partitioning is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Here an excerpt of Mario is shown in two dimensions for better visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The scattered points are located on arbitrary coordinates with floating accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We conduct a block partitioning for N = 2 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For better demonstration, the axes are expanded by a factor 100 such that N = 2 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The core block covers the interval x = [76, 78] and y = [18, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This area is highlighted in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' All points in this area are assigned to the core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In addition, a support area with a margin of M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The support area is shown in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, all points that are located in the interval x = [75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5] and y = [17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5, 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5] are considered for model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This bears the advantage of a smoother model especially in the border regions and smoother transitions from one block to the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The partitioning process is conducted for all blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, one point might be assigned to the support area of none, one or more blocks and at the same time it must be a member of exactly one core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' During this assignment, each point remains on its original location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, this block partitioning procedure may not be confused with a voxelization that is often pursued as a preprocessing for point cloud processing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' If the block partitioning is finished, one block after the other is handed over to the geometry upsampling step that estimates the underlying smooth object’s surface in this one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In the following, we refer to the joint set of core block and support area as a block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The points within one block form the set of points A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Geometry Upsampling In the geometry upsampling of a point cloud, points have to be added to the original point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the points’ locations have to be determined first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These have to satisfy two essential requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' First, the points should fit well in the surface of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' And second, the points’ distribution should approximately follow a uniform distribution such that the newly added points are not located directly next to original points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We follow the approach as described in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We assume a point cloud’s surface to be locally smooth and representable in terms of a function within one block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' One exemplary block is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' At first sight, the block just seems to contain a set of loose points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' However, we will estimate a smooth underlying surface in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Subsequently, we place the additional points on this surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In a first step, it has to be decided in which dimensions the surface model is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the dimension that yields the smallest variance is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This bears two advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' First, the probability that the surface is a closed form in the dimension with smallest extension is low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Second, if the first assumption is false, the introduced error is rather small as only one surface is estimated as averaging surface in between the other surfaces that build a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the modeled dimension z′ is z′ = min {Var{x}, Var{y}, Var{z}} (9) with x, y and z being the three-dimensional coordinates of the point cloud’s points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The main assumption that we follow for geometry upsampling is that the underlying surface in one block can be represented in terms of a function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We assume the function to be a weighted superposition of basis functions ϕ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', z′ = f(x′, y′) = � k,l∈K ck,lϕk,l[x′, y′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (10) This equation is closely related to (1), if we select f(m, n) = f(x′, y′) with m = x′ and n = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the model estimation is conducted along the explained steps from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The resulting continuous model of the block is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Finally, the upsampled coordinates ˆz′ are determined based on the upsampled points (ˆx′, ˆy′) according to ˆz′ = f(ˆx′, ˆy′) = � k,l∈K ˆck,lϕk,l(ˆx′, ˆy′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (11) The upsampled points (ˆx′, ˆy′) are gained by a Delaunay triangulation of the points (x′, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This computation can be conducted in parallel to the model generation procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The new points are inserted in the middle of the triangle edges as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereby, high upsampling factors can be achieved with one triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The final result of the block is given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' It is clearly visible, that the blue input points are located from one block border to the other, whereas the orange output points are only located in the core block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This is due to the incorporated support area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In this case a margin of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 is taken as support for a core block of size 2 × 2 × 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='75 1 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 M/N C2C ×10−1 N = 2 N = 3 N = 4 N = 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 6: Geometry results in terms of C2C similarity for different block sizes (N) and support margins (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Support margin is given relative to block size (M/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Proposed Projection The attribute upsampling step describes the process of estimating the attribute’s value at the upsampled positions determined as in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' An attribute can be anything in point clouds such as color, intensity or normal vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As a starting point, we use the sparse frequency model ap- proach from [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We assume the point cloud object’s surface to be a two-dimensional plane in three-dimensional space and thus, follow the approach to project the surface into a two- dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In [28], the Euclidean distances between the points are measured and incorporated as weights of a graph that is minimized to a minimum spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Following this tree, the points are mapped to a two-dimensional space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The transformation is conducted for each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As the calculation of the euclidean distances and the generation of the minimum spanning tree is a time-consuming and complex process, we propose to exploit knowledge from our geometry upsampling which was presented in the section before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, we propose to simplify the 3D to 2D conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For geometry upsampling, the geometry is rotated such that the dimension with smallest variance is modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This bears the advantage that the standard deviation in z′−direction is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Exploiting this, we can directly project the three dimensional points into two dimensions along the axis of the smallest dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, x′ 2D = x′ (12) and y′ 2D = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (13) Next, the two-dimensional points are used for the estimation of the attribute’s frequency model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Attribute Upsampling The attribute information is assumed to be modeled with frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As explained in [28], the color signal fa can be modeled as a weighted superposition of basis functions ϕ from the set of available basis functions K according to fa(x′ 2D, y′ 2D) = � k,l∈K ck,lϕk,l(x′ 2D, y′ 2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (14) Once again, this assumption is closely related to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, for attribute upsampling we can formulate f(m, n) = 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='75 1 1 2 3 4 M/N Histogram distance ×10−2 N = 2 N = 3 N = 4 N = 8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 7: Color results in terms of histogram distance [32] for different block sizes (N) and support margins (M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Support margin is given relative to block size (M/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' fa(x′ 2D, y′ 2D) with m = x′ 2D and n = y′ 2D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' During the model estimation process, a continuous model of the attribute of the point cloud for one block is estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the resulting model gives a continuous estimation of the point cloud’s attribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' To assign a proper attribute information to the upsampled points (ˆx′ 2D, ˆy′ 2D), the estimated model is evaluated for fa(ˆx′ 2D, ˆy′ 2D) = � k,l∈K ˆck,lϕk,l(ˆx′ 2D, ˆy′ 2D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (15) Finally, the additionally computed points with its geometric location and the associated color attribute are remapped in the three-dimensional space of the original point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the equations (9), (12) and (13) have to be reversed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The 3D high-resolution point cloud in geometry and color is the final result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' EVALUATION For the evaluation of our proposed joint geometry and attribute upsampling scheme, we show evaluations for both, geometry and color, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In addition, visual examples of the joint upsampling scheme are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For the evaluations the 3DColorMesh dataset [33] is incorporated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This dataset contains colored point clouds with 40,000 to 200,000 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Metrics The quality of the upsampled point clouds are determined for both, geometry and color, separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For each upsampling part, different metrics are applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1) Geometry: The evaluation of the geometric shape of the upsampled point cloud is usually done with respect to the original point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the original point cloud serves as a reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For each point in the upsampled point cloud, the nearest neighbor in the reference is searched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The deviations are summed up and normalized for all points such that the overall point-to-point (P2P) error is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the P2P error is the normalized sum of the error vectors being the smallest distance between the points in the reference and the upsampled point cloud [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 7 TABLE I: Geometry results for all point clouds from the 3DColorMesh dataset for scaling factor of 4 in terms of P2P and P2C errors [34] and C2C similarity [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Best qualities are given in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Arrows indicate higher values are better ↑ and smaller values are better ↓, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' P2P ×10−3 ↓ P2C ×10−3 ↓ C2C×10−1 ↑ Point Cloud PU EC FSGU FSU PU EC FSGU FSU PU EC FSGU FSU 4armsMonstre 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 Asterix 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 CableCar 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 Dragon 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 Duck 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 GreenDinosaur 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 GreenMonstre 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 Horse 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 Jaguar 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 LongDinosaur 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 Man 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 Mario 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 MarioCar 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 PokemonBall 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 Rabbit 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 RedHorse 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 Statue 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 Average 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 A related method is to determine the point-to-plane (P2C) error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the same procedure as for the point-to- point error is followed with the difference that not the direct deviation between the upsampled point and the nearest point in the reference is taken but the difference along the normal of the upsampled point is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Once again, these differences are summed up and normalized such that the overall point-to- plane error is determined [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As a third metric, the plane-to-plane (C2C) angular similar- ity is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In this metric, a plane is estimated in both, the reference and the upsampled point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Then, the angular similarity between the two planes is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereby, the visual degradations of a processed point cloud can be predicted more accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The plane-to-plane similarity metric is determined according to the implementation of Alexiou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2) Attribute: A great challenge in the joint upsampling of the geometry and attribute of a point cloud is the determina- tion of the final attribute quality as it is highly affected by geometric distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, we decided to evaluate geometry and attribute separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In order to have ground truth data available, we first downsample the original point cloud ran- domly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The downsampled points incorporate both, geometry and color information and thus, serve as the original points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' From the remaining points, only the geometry information is kept such that a possible geometrical distortion is not affecting the attribute quality during the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The attribute infor- mation of these points is determined following the proposed algorithm described in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the overall point cloud is separated into blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Next, each block is rotated according to the geometrical variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The geometrical upsampling is skipped, it follows the projection from three-dimensional space to the two-dimensional plane for both point sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Finally, the proposed attribute upsampling scheme from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' III-D follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The downsampling and upsampling is conducted three times for each point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the shown results are averaged over all three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As ground truth data is available, we determine the color peak-signal-to-noise ratio (PSNR) as it is known from image processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, we first determine the PSNR for each color channel separately and average for Color-PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We measure the color reconstruction PSNR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', the color PSNR is determined solely on the upsampled points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As a second evaluation scheme, we incorporated a his- togram comparison as proposed by Viola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As the histograms from reference and upsampled point cloud are compared, it can also be applied if no direct ground truth information is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The histogram difference of the luminance channel Y is shown in the evaluation with a euclidean distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Influence of Block Parameters The block partitioning is a relevant part for the frequency- selective upsampling procedure as it jointly sets the block partitioning for geometry and color upsampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the aim is to determine the block parameters such that geometry and attribute quality are as good as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Therefore, the influence of block size and size of the newly introduced support area is analyzed in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The geometry results are shown in terms of C2C similarity in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 6 and the color result is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The choice of the parameters affects geometry and attribute upsampling at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, the averaged results for the 3DColorMesh dataset are depicted for all combinations of core block sizes N = 2 (blue), N = 3 (red), N = 4 (yellow) and N = 8 (purple) and support margins relative to block sizes from M/N = 0 to M/N = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The geometry in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 6 is optimized in terms of C2C similarity as a smooth surface is 8 desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For block sizes larger than 3, the curves show a slight decrease before the angular similarity increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For block size N = 2, the maximum is achieved for a border width of M/N = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='25, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' the support margin size is M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The color differences in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 7 increase with increasing support margins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, color quality is maximized for a model that is as local as possible achieved by a small support margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In the conjunction of geometry and color quality, the block size is set to N = 2 and the support margin is chosen to be M = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 for the remainder of this work as geometry is maximized while a good color quality is maintained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Geometry Results Most of the known point cloud upsampling schemes upsam- ple the geometry of a point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We compare the geometry upsampling part of our proposed FSU to FSGU [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The main difference here is that the block partitioning is conducted differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In FSU a support area is incorporated whereas FSGU does not take a support area into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In addition, there are two data-driven approaches, namely PU-Net (PU) and EC-Net (EC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' PU-Net focuses on adding new points uniformly and as distant from given points as possible whereas EC-Net focuses on upsampling edges properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For all upsampling techniques, point-to-point (P2P), point-to-plane (P2C) and plane-to-plane (C2C) errors are measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The results for the 17 point clouds of the 3DColorMesh dataset are shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In terms of P2P, the results produced by PU-Net are worst, followed by EC-Net.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The best performing approaches are the frequency-model based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Our proposed FSU approach performs slightly worse than FSGU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Due to the incorporated support area in FSU, the upsampled points can be located within the whole block and not just within the range of the original points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thereby, the distances in between the points increase and thus, also P2P errors may increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Even though the incorporated support area may not lead to an improvement in the P2P error metric, it leads to an enhanced visual appearance as it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The original of the 4armsMonstre in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10a is upsampled with FSGU in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10b and our proposed FSU in Fig 10c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Clear block artifacts can be observed with the upsampling of FSGU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' These disappear with FSU such that a better visual quality is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In terms of the metrics, the behavior in terms of P2P metric changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Here, the FSU results outperform the results generated with FSGU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This shows the improved plane reconstruction in FSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The results in terms of angular similarity C2C are given in the last columns of Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As this is a similarity metric, higher values denote better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Here, the frequency-model based approaches show highest values and thus, are the best upsampling methods for this metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, FSU outperforms FSGU in terms of P2C error and C2C similarity showing that FSU produces smoother results than FSGU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, the estimation and sampling improved with FSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The data-driven approaches perform worst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The behavior in terms of the metrics remain similar for further scaling factors as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' An overview of the averaged result for the dataset from scaling factor two to four is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 8 for the angular similarity C2C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The relative behavior 2 3 4 0 2 4 6 8 10 Scaling factor C2C ×10−1 PU (C2C) EC (C2C) FSGU (C2C) FSU (C2C) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 8: Evolution of the C2C results for the averaged 3DColorMesh dataset for scaling factors from two to four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Upsampling with PU (orange), EC (yellow), FSGU (purple), and FSU (green).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' as shown in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' I of the curve remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' FSU is the best performing method in terms of C2C for all scaling factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As a general trend, the C2C angular similarity metric decreases with increasing scaling factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Color Results In a second evaluation, we analyze the quality of the color attribute of the point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Our proposed upsampling scheme is denoted as FSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We compare FSU to the Frequency- Selective Mesh-to-Mesh Resampling (FSMMR) as proposed in [28], linear interpolation on block-level (LIN2) and on the whole point cloud (LIN3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Furthermore, natural neighbor interpolation is also appplied on both, block level (NAT2) and on the whole point cloud (NAT3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The results for the metrics introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' IV-A2 are given in Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The upsampling techniques are evaluated for all 17 point clouds from the 3DColorMesh dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The color PSNR is shown in the first six columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The results of the histogram based evaluation of Viola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [32] is depicted in the last six columns of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As it is shown in terms of PSNR, the proposed FSU performs best on average with a gain of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 dB to FSMMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Especially for the Dragon point cloud, gains of up to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 dB are achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The results for our proposed approach FSU show that it is advantageous to include a support area and thereby, take more neighborhood information into account for the model estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Severe degradations of the point clouds can be observed for the interpolation approaches as they are based on triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hence, color reconstruction may not be possible in concave areas where an extrapolation is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Thus, not all color information can be retrieved and thus, color PSNR degrades.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A similar behavior can be observed for the histogram difference by Viola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [32] in the last six columns of the table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As it is a distance-based measure, smaller values indicate better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Severe degradations can be observed for the interpolation-based approaches especially for the Duck, Man, and PokemonBall point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' As our model-based FSU can retrieve all color information independently of whether inter- or extrapolation is required the histogram distances are 9 TABLE II: Color results for all point clouds from the 3DColorMesh dataset for scaling factor of 4 in terms of color PSNR in dB and the histogram distance by Viola et al [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Best qualities are given in bold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Arrows indicate higher values are better ↑ and smaller values are better ↓, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Color PSNR ↑ Histogram distance by Viola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' ×10e − 2 ↓ 3D 2D 3D 2D Point Cloud LIN3 NAT3 LIN2 NAT2 FSMMR FSU LIN3 NAT3 LIN2 NAT2 FSMMR FSU 4armsMonstre 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 Asterix 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 CableCar 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 Dragon 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 Duck 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 GreenDinosaur 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 GreenMonstre 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 Horse 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 Jaguar 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 LongDinosaur 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 Man 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 Mario 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 MarioCar 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 PokemonBall 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 Rabbit 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 RedHorse 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 Statue 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 Average 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='3 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='7 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='0 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='2 (a) Original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (b) FSGU+FSMMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (c) FSU (Proposed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 9: Mario point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Upsampling factor is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' much smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' On average, FSU performs best in terms of color PSNR and in terms of histogram distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Joint Results Our proposed FSU upsamples geometry and attribute jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Some visual results are given in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 9 and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The original point cloud with original resolution is given in Subfigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Subfigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (b) depict the results if FSGU [23] and FSMMR [28] , are combined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Subfigs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (c) depict the proposed FSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Clear improvements can be observed for our proposed FSU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Especially the 4armsMonstre shows clear blocking artifacts in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The newly proposed FSU as given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10c improves these artifacts notably and appears to be much smoother.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' This is mainly due to the added support area that is used during the model generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' CONCLUSION In this work, we presented a joint upsampling scheme for geometry and attribute of point clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' We therefore incor- porate frequency-selective models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The models are estimated locally on block level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' In the block partitioning an overlap- ping support area is included that incorporates neighborhood information into the block estimation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' For attribute upsampling, information from the geometry upsampling step is exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Our proposed Frequency-Selective Upsampling (FSU) improves point cloud upsampling in both, geometry and color quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' FSU shows best results in terms of point-to-plane error and plane-to-plane angular similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Furthermore, the color upsampling quality is on average improved by 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='9 dB in terms of color PSNR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Also the visual appearance of the upsampled point cloud is improved notably as the included support area reduces block artifacts clearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10 (a) Original.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (b) FSGU + FSMMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' (c) FSU (Proposed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 10: 4armsMonstre point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Upsampling factor is 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' REFERENCES [1] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Mekuria, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Blom, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cesar, “Design, implementation, and evaluation of a point cloud codec for tele-immersive video,” IEEE Transactions on Circuits and Systems for Video Technology, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 27, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 828–842, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [2] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Held, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Gupta, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Curless, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Agrawala, 3D Puppetry: A Kinect- Based Interface for 3D Animation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' New York, NY, USA: Association for Computing Machinery, 2012, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 423–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [3] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ma, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Wan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Li, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Xia, “Multi-view 3D Object Detection Network for Autonomous Driving,” in Proceedings IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 6526–6534.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Geiger, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Lenz, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Urtasun, “Are we ready for autonomous driving?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' The KITTI vision benchmark suite,” in IEEE Conference on Computer Vision and Pattern Recognition, 2012, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3354–3361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Mahmood, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Han, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Lee, “BIM-Based Registration and Localization of 3D Point Clouds of Indoor Scenes Using Geometric Features for Augmented Reality,” Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1–29, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Andriasyan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Moyano, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Nieto-Juli´an, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ant´on, “From Point Cloud Data to Building Information Modelling: An Automatic Parametric Workflow for Heritage,” Remote Sensing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 7, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1–22, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [7] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Han, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Shao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Xu, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Shotton, “Enhanced computer vision with microsoft kinect sensor: A review,” IEEE Transactions on Cybernetics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 43, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1318–1334, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Alexa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Behr, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fleishmann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Levin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Silva, “Point Set Surfaces,” in Proceedings Visualization, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' VIS ’01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=', 2001, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 21–29, 537.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Berger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Levine, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Nonato, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Taubin, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Silva, “A Benchmark for Surface Reconstruction,” in ACM Transactions on Graphics, Apr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kazhdan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Bolitho, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hoppe, “Poisson Surface Recon- struction,” in Proceedings of the Fourth Eurographics Symposium on Geometry Processing, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 256, 2006, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 61–70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [11] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Lipman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Levin, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Tal-Ezer, “Parametrization- free projection for geometry reconstruction,” ACM Transactions on Graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 26, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3, 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [12] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Wu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Gong, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ascher, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Zhang, “Edge-aware point set resampling,” ACM Transactions on Graphics, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 32, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1, Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [13] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Dinesh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cheung, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Baji´c, “3D Point Cloud Super- Resolution via Graph Total Variation on Surface Normals,” in Proceed- ings IEEE International Conference on Image Processing, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 4390–4394.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [14] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Charles, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Su, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kaichun, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Guibas, “PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation,” in IEEE Conference on Computer Vision and Pattern Recognition, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 77– 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [15] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Heng, “PU-Net: Point Cloud Upsampling Network,” in Proceedings IEEE Conference on Computer Vision and Pattern Recognition, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2790–2799.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [16] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Qi, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yi, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Su, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Guibas, “PointNet++: Deep Hierarchical Feature Learning on Point Sets in a Metric Space,” in Advances in Neural Information Processing Systems, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [17] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yifan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Wu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Huang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Sorkine-Hornung, “Patch-Based Progressive 3D Point Set Upsampling,” in Proceedings IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 5951–5960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Heng, “EC-Net: An Edge-Aware Point Set Consolidation Network,” in Proceedings Computer Vision – ECCV 2018, 2018, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 398–414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [19] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Fu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen-Or, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Heng, “PU-GAN: A Point Cloud Upsampling Adversarial Network,” in Proceedings IEEE/CVF International Conference on Computer Vision, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 7202–7211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Zhang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Jiang, and Zhang, “Data-driven Upsampling of Point Clouds,” Computer-Aided Design, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 112, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1–13, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [21] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Qian, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Hou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kwong, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' He, “PUGeo-Net: A Geometry- Centric Network for 3D Point Cloud Upsampling,” in European Con- ference on Computer Vision, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 752–769.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ye, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Wan, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Lia, “Meta-PU: An Arbitrary- Scale Upsampling Network for Point Cloud,” IEEE Transactions on Visualization and Computer Graphics, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [23] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Heimann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Spruck, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kaup, “Frequency-Selective Geometry Upsampling of Point Clouds,” in IEEE International Conference on Image Processing, 2022, arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='01458.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [24] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Park, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Park, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kang, “Super-Resolution Image Reconstruction: A Technical Overview,” in IEEE Signal Processing Magazine, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 20, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3, May 2003, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 21–36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [25] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Dong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Loy, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' He, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Tang, “Image Super-Resolution Using Deep Convolutional Networks,” IEEE Transactions on Pattern Analysis and Machine Intelligence, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 38, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 295–307, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ma, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Yang, “Single-Image Super-Resolution: A Benchmark,” in European Conference on Computer Vision, 2014, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 372–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [27] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Dinesh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cheung, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Baji´c, “Super-Resolution of 3D Color Point Clouds Via Fast Graph Total Variation,” in Proceedings IEEE International Conference on Acoustics, Speech and Signal Processing, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1983–1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [28] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Heimann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Spruck, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kaup, “Frequency-Selective Mesh-to- Mesh Resampling for Color Upsampling of Point Clouds,” in Pro- ceedings IEEE 23rd International Workshop on Multimedia Signal Processing, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [29] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Koloda, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Seiler, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kaup, “Frequency-Selective Mesh-to-Grid Resampling for Image Communication,” vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 19, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1689–1701, August 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [30] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Seiler, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Jonscher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Sch¨oberl, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kaup, “Resampling Images to a Regular Grid From a Non-Regular Subset of Pixel Positions Using Frequency Selective Reconstruction,” vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 11, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 4540–4555, November 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [31] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Kaup, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Meisinger, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Aach, “Frequency Selective Signal Extrapolation with Applications to Error Concealment in Image ommu- nication,” in International Journal of Electronics and Communications, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 59, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3, June 2005, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 147–156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [32] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Viola, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Subramanyam, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cesar, “A color-based objective quality metric for point cloud contents,” in 2020 Twelfth International Conference on Quality of Multimedia Experience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' IEEE, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [33] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Nouri, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Charrier, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' L´ezoray, “Greyc 3D Colored Mesh Database,” Normandie Universit´e, Unicaen, EnsiCaen, CNRS, GREYC UMR 6072, Technical Report, Jan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Available: https://hal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='archives-ouvertes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content='fr/hal-01441721 [34] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Tian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ochimizu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Feng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Cohen, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Vetro, “Geometric distortion metrics for point cloud compression,” in Proceedings IEEE International Conference on Image Processing, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 3460–3464.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' [35] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Alexiou and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' Ebrahimi, “Towards a Point Cloud Structural Similar- ity Metric,” in IEEE International Conference on Multimedia and Expo, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} +page_content=' 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/JtFJT4oBgHgl3EQfwi0E/content/2301.11630v1.pdf'} diff --git a/KNA0T4oBgHgl3EQfCv9N/content/tmp_files/2301.01993v1.pdf.txt b/KNA0T4oBgHgl3EQfCv9N/content/tmp_files/2301.01993v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..5d80849f1a1a82c0a3079770ad83ae8a88f2a0e1 --- /dev/null +++ b/KNA0T4oBgHgl3EQfCv9N/content/tmp_files/2301.01993v1.pdf.txt @@ -0,0 +1,1036 @@ + +1 +Improved in-situ characterization of electrochemical interfaces using +metasurface-driven surface-enhanced infrared absorption spectroscopy +Luca M. Berger 1, ‡, Malo Duportal 2, ‡, Leonardo de Souza Menezes 1,3, Emiliano Cortés 1, Stefan A. +Maier 4,5,1, Andreas Tittl 1,*, Katharina Krischer 2,* +1 Faculty of Physics, Ludwig-Maximilian-University Munich, 80539 München, Germany +2 Department of Physics, Technical University of Munich, 85748 Garching, Germany +3 Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil +4 School of Physics and Astronomy, Monash University, Melbourne, Victoria, Australia +5 Department of Physics, Imperial College London, SW7 2AZ London, United Kingdom + +* e-mail: Andreas.Tittl@physik.uni-muenchen.de; krischer@tum.de + +Abstract +Electrocatalysis plays a crucial role in +realizing the transition towards green energy, +driving research directions from hydrogen +generation to carbon dioxide reduction. +Understanding electrochemical reactions is crucial to improve their efficiency and to bridge the +gap toward a sustainable zero-carbon future. Surface-enhanced infrared absorption spectroscopy +(SEIRAS) is a suitable method for investigating these processes because it can monitor with +chemical specificity the mechanisms of the reactions. However, it remains difficult to detect many +relevant aspects of electrochemical reactions such as short-lived intermediates. Here, we develop +and experimentally realize an integrated nanophotonic-electrochemical SEIRAS platform for the +in situ investigation of molecular signal traces emerging during electrochemical experiments. +Specifically, we implement a platinum nano-slot metasurface featuring strongly enhanced +electromagnetic near fields and spectrally target it at the weak vibrational bending mode of + +CO +CO +2 +adsorbed CO at ~2033 cm-1. Crucially, our platinum nano-slot metasurface provides high +molecular sensitivity. The resonances can be tuned over a broad range in the mid-infrared +spectrum. Compared to conventional unstructured platinum layers, our nanophotonic- +electrochemical platform delivers a substantial improvement of the experimentally detected +characteristic absorption signals by a factor of 27, enabling the detection of new species with weak +signals, fast conversions, or low surface concentrations. By providing a deeper understanding of +catalytic reactions, we anticipate our nanophotonic-electrochemical platform to open exciting +perspectives for electrochemical SEIRAS, surface-enhanced Raman spectroscopy, and the study +of reactions in other fields of chemistry such as photoelectrocatalysis. + +Introduction +Electrochemical reactions underpin many technologies ubiquitous for a future carbon-zero world +such as green-hydrogen generation for long-term sustainable energy storage1 and CO2 degradation +to combat the current trends of climate change2. Unfortunately, in general, the monitoring, and +therefore understanding, of many electrochemical reactions remains a challenge. In particular, +resolving the electrochemical CO2 reduction reaction (CO2RR) with high efficiency, selectivity, +and sensitivity remains an issue3 especially due to the competition with the hydrogen evolution +reaction (HER) at high current densities4. During the CO2RR to desired carbon products, a +compulsory step to the key intermediate CO is still not fully understood and requires further +investigation5. +For the detection and characterization of molecules, optical spectroscopy, mass spectrometry, +chromatography, and fluorescence microscopy are often used6. Optical spectroscopy methods in +particular are highly advantageous because they allow for the retrieval of the spectral fingerprint + + +3 +of molecules via the detection of their rotational or vibrational modes. Within optical spectroscopy, +two strong methods are Raman and infrared (IR) spectroscopy. The former relies on the inelastic +scattering of photons and studies the resulting spectral shift. The latter detects the absorption of +light by molecules when the energy of the photons matches the energy of the vibrational modes. +The mutual exclusion rule dictates that any mode can be IR active, Raman inactive, and vice-versa +but not simultaneously7. Therefore, for a given molecular mode either Raman or IR spectroscopy +can be used, but not both. Here, the CO vibrational mode under investigation is IR active8. Surface- +enhanced infrared absorption spectroscopy (SEIRAS) is a derivative technique from conventional +infrared spectroscopy based on the enhancement of the local electromagnetic (EM) near fields 9,10. +To increase the sensitivity of either surface enhanced Raman spectroscopy (SERS) or SEIRAS +during electrochemical reactions typically a rough metal surface has been chosen to enhance the +local electromagnetic (EM) near fields11. Rough and highly disordered metallic nm-sized edges +coming from perforations and extrusions in the metallic film locally confine and enhance the EM +fields. Unfortunately, this approach is random, does not allow for spectral tailoring of plasmonic +hotspots, and consequently generates a relatively weak EM near-field enhancement. Even after +improvements in the sensitivity of SEIRAS using an attenuated total internal reflection (ATR) +geometry9,10, the characterization of CO adsorption on catalysts is still hampered by weak signal +traces12–14. +We overcome the challenge of detecting weak signal traces by taking inspiration from other fields +of nanophotonics. In biomolecular sensing, a plethora of alternatives are used to improve +molecular detection using controlled and tuneable EM near-field enhancement via the excitation +of resonances through tailored system parameters on the nanoscale. Examples are plasmonic +nanoparticles, non-plasmonic nanogap dimers15, metasurfaces based on plasmonics16 or exotic + + +4 +phenomena like quasi-bound states in the continuum17, waveguides18 or 2D-integrated19 platforms, +among others20. Plasmonic-based sensors have become the method of choice in label-free detection +of biomolecules. They can be used either as 1) refractive index (RI) sensors or 2) by coupling the +resonances to the molecular modes and analysing the perturbation of the intensity either in +reflection or transmission21, termed perturbed intensity sensing here. +In fact, some recent progress has been made to integrate plasmonic structures for refractive index +sensing with electrochemistry21–23. There are also recent examples of plasmonic structures for +perturbed intensity sensing for SERS used to monitor electrochemical reactions24 or to study the +mechanism of an electrocatalytic reaction25. Literature of plasmonic imaging provides other +examples of electrochemical reactions of single nanoparticles26, plasmonics-supported and +electrochemical monitoring of molecular interactions focused on fluorescence and confocal +microscopy27,28, and plasmon-accelerated electrochemical reactions29,30. However, to the best of +our knowledge, the integration of nanostructured metasurfaces for perturbed intensity sensing in +SEIRAS has never been shown in combination with electrochemistry. +Here, we detect in situ the CO vibrational bending mode at 2033 cm-1 emerging during the +electrochemical conversion of CO into CO2 using a platinum nano-slot metasurface on a CaF2 +substrate (Figure 1a) by coupling its resonance to the molecular vibrational mode and analysing +the perturbation of the intensity in reflection. We investigated the linear CO vibrational mode +(COlinear) at 2033 cm-1 because it is the most intense vibrational mode of CO on platinum31–35. The +material of choice was platinum as it could fulfill all requirements, namely to function as a working +electrode, support strong metasurface-driven resonances, and adsorb CO on its surface36. +Moreover, Pt is a catalytic material for many reactions, making this platform very useful not only +for the CO oxidation reaction but also for other reactions. The decision on the inverse structure + + +5 +(i.e. the slots), was made to preserve a connected metallic film that can carry electrical current. +Moreover, compared to resonant rod-type antennas, the inverse counterparts have been shown to +feature superior detection of molecular signal traces due to linearly instead of exponentially +decaying EM near-fields37. The slots can only be excited with transverse electric (TE) polarized +light37. We perform SEIRAS in an ATR geometry to further improve the sensing performance +while maintaining free accessibility of the electrode surface for reactants and products9,10. We +confirm the detection of adsorbed CO via the observation of the typical Stark shift and resolve a +so far scarcely studied 38–40 effect due to the decrease of the CO coverage on the surface of platinum +during the electrochemical oxidation. Furthermore, the presence of a second peak at 2086 cm-1 on +the spectral location of the linear vibrational mode could be attributed to the effect of the crystal +orientation. Finally, we establish a methodology for designing similar nanophotonic- +electrochemical platforms. + + +6 + +Figure 1. Numerical design of the catalytic metasurface. (a) Schematic for the Pt-based nano- +slot metasurface for the in-situ integrated nanophotonic-electrochemical study of CO2 oxidation. +As the potential between the working electrode (WE) and the reference electrode (Ref.) is swept +the presence of adsorbed CO is monitored via the detection of the linear vibrational mode of CO +at 2033 cm-1 with a Fourier transform infrared (FTIR) spectrometer. The nano-slot metasurface +enhances the electromagnetic near-fields of TE polarized light in an ATR configuration coming in +at an azimuthal angle 𝜙 = 0∘ and polar angle 𝜃 = 72∘ w.r.t. the Pt film (𝑥𝑦-plane). (b) Sketch of +the Pt on CaF2 nano-slot unit cell. Two CO model layers were included parallel to the long edges +of the slot (magenta) with dimensions 𝑙 × ℎ × 𝑡. A 1 nm thick Ti adhesion layer was used in the +fabrication of the structures but is not considered in the numerical simulations due to its negligible +effect on the resonance position. The geometrical parameters of the unit cell for (c), (d), (e) are ℎ += 30 nm, 𝑤 = 200 nm, 𝑙 = 1380 nm, 𝑝𝑦 = 1400 nm, 𝑝𝑥 = 1600 nm. In (c) no CO model layer was +included. In (d) 𝑡 = 5 nm. (c) Electric near field intensity (taken at ℎ = 30 nm) of the unit cell. The +maximum near field intensity is 570. (d) The simulated reflectance spectrum of the metasurface +with (pink) and without (blue) the CO model layer. The spectrum includes the Rayleigh anomaly +(RA). (e) The differential absorbance of (d) with the thickness of the CO layer 𝑡 2 nm (blue), 5 nm +(pink) and 10 nm (red) showing clearly visible absorption bands. + +a +CO +CO +Ref. +CE +WE +Py +IE/Eol max=570 +RA +0.20 +t (nm): +0.18 +2 +2040 +2030 +5 +10 +7 +Results and discussion +Numerical design of catalytic nano-slot metasurface +We start the implementation of our electrochemical sensing platform with the numerical design of +the chosen nano-slot metasurface geometry. The structure consists of a unit cell composed of a +single slot in an otherwise connected platinum film submerged in water on CaF2 (Figure 1b). +Notably, we model adsorbed CO by including an artificially created material covering the inside +walls parallel to the long axis of the slot. The choice for the parameters of the unit cell was guided +by Huck et al.37 and modified in accordance with fabrication constraints. Huck et al.37 optimized +a gold nano-slot metasurface in the mid-infrared for normal incidence illumination in air for high +quality factors (Q-factors) and electric near fields. The Q-factor relates the initial energy stored in +a resonator to the energy dissipated in one radian of the cycle of oscillation41. +On the basis of our simulations, the nano-slot metasurface achieves a resonance with a modulation +in the absorbance of over 82% and a Q-factor of ca. 6.3 (see “Methods” section for details on the +Q-factor calculation). Furthermore, the metasurface numerically exhibits an electric near-field +enhancement |𝐸/𝐸0|2 of 570. This value can be increased in future experiments by decreasing the +width of the slots37 but was limited here due to fabrication constraints. The maximum electric near- +field enhancement occurs inside the slots close to the faces parallel to its long axis (Figure 1c), +with its electric field pointing orthogonally to it. +Huck et al.37 found that the highest Q-factor and electric near field enhancement occurs when w is +small, 𝑝𝑦 = 𝜆𝑟𝑒𝑠/2, and 𝑔 = 𝑝𝑥 − 𝑙 = 𝜆𝑟𝑒𝑠/2, where 𝜆𝑟𝑒𝑠 is the central wavelength of the +resonance. However, to satisfy the experimental conditions the nano-slot metasurface was +simulated in water instead of air and for an angle of incidence 𝜃 = 72∘. Under these conditions, + + +8 +tuning the resonance to 2033 cm-1 ≈ 4.92 µm leads to the appearance of a Rayleigh anomaly (RA) +such that 𝜆𝑅𝐴 > 𝜆𝑟𝑒𝑠, where 𝜆𝑅𝐴 is the central wavelength of the RA. The RA is a phenomenon +associated with light diffracted parallel to the surface of a periodic structure42. When 𝜆𝑅𝐴 > 𝜆𝑟𝑒𝑠, +the resonance lifetime and electric near-field enhancement is strongly reduced43. Consequently, a +metasurface where 𝜆𝑅𝐴 > 𝜆𝑟𝑒𝑠 will exhibit poor sensing performance. For this reason, 𝑔 was +reduced to 220 nm to push the resonance on the evanescent side of the RA (Figure 1d). +In coupled-resonator systems, the excitation efficiency of a resonator is significantly dependent on +the ratio of its losses to external radiation 𝛾𝑒, i.e. light scattering, and intrinsic material absorption +𝛾𝑖 which strongly depends on the system design and parameters chosen44. When 𝛾𝑒 ∼ 𝛾𝑖 the system +is critically-coupled and the second oscillator will lead to a dip in the absorption cross-section. +SEIRAS performance can be maximized by utilizing a system that is close to the critical coupling +condition44,45. Here, the nano-slot metasurface is near the critical-coupling condition with 𝛾𝑒/𝛾𝑖 ≈ +1.2. Thus, when the resonance overlaps with the vibrational mode of adsorbed CO at 2033 cm-1 +the coupling between the two resonators leads to a small peak in the reflectance spectrum (Figure +1d). +The idea that coverage effects and different analyte concentration can be sensed using our nano- +slot metasurface is shown by changing the thickness of the model molecular layer representing +adsorbed CO from 2, to 5, to 10 nm (Figure 1e). By increasing the thickness of the model +molecular layer, a stronger differential absorbance log(𝐼0/𝐼) can be obtained, where 𝐼 and 𝐼0 are +the reflectance measured with and without a CO model molecular layer, respectively (Figure 1c). +Thus, a decrease in the coverage of an adsorbed material inside the slot can be linked to a decrease +in the differential absorbance traces. + + +9 +Metasurface characterization +First, the effect of the metasurface-driven resonance position on the coupling with COlinear +vibration is studied in ATR mode using a focal plan array detector (Figure 2a). To test our +nanophotonic-electrochemical platform, we first tuned the resonance position to match the COlinear +vibration mode in 0.5M K2CO3 saturated with carbon monoxide. Then, we detuned the resonance +to the blue and red spectral regions by decreasing and increasing the slot length 𝑙 by 200 nm from +1.33 µm, respectively (Figure 2b). There is a good fit between the numerically and experimentally +obtained resonance positions, with a discrepancy of less than 40 cm-1. As predicted by the +simulations, a dip is observed in the resonance attributed to the COlinear vibration mode. Following +these results, slots with a length of ca. 1.33 µm were found to match the COlinear vibrational mode +on platinum. According to the literature31–35,46the COlinear mode should be spectrally located +between 2020 cm-1 and 2080 cm-1. On the basis of our experiments, COlinear is located at 2033 cm- +1. +The differential absorbance highlights a more intense and well defined COlinear signal for the +sample which has the best spectral overlap (Figure 2c). Two peaks can be observed at 2033 and +2086 cm-1. The CO signals have a Fano-type line shape due to the narrow discrete COlinear vibration +interfering with the broad spectral line of the metasurface-driven resonance47. The redshifted +sample yields a highly asymmetric CO signal due to the strongly off-resonance coupling between +the resonance and COlinear48. In addition, the redshifted sample presents a strong peak around 1843 +cm-1, which is attributed to a second configuration of adsorption, the CO bridge (CObridge)31,32,34,46. +The scanning electron microscopy images show good quality of the fabricated nanostructures +(Figure 2d). For the next part of this work, the electrochemical behavior of the sample with +matching spectral overlap of its resonance with the COlinear vibrational mode is studied. + + +10 + +Figure 2. Testing the nanophotonic-electrochemical platform. (a) A schematic showing the +experimental setup used to perform SEIRAS in an ATR geometry. A continuous mid-IR +collimated linearly polarized light source illuminated the nano-slot metasurface from below at an +angle of ca. 72º. A focal plane detector array was used to collect the signal. (b) Experimental +absorbance spectra of three Pt nano-slot metasurface in CO saturated 0.5M K2CO3 aqueous +electrolyte interfaces with resonance positions around the ideal position for COlinear with slot +lengths 𝑙 1.33 µm (black, ideal) and two detuned resonance positions with slots length of 1.13 µm +(blue) and 1.53 µm (red). The other parameters are ℎ = 30 nm, 𝑤 = 200 nm, 𝑝𝑦 = 1440 nm, gap = +𝑝𝑥 − 𝑙 = 220 nm. The resonance position of COlinear vibration is indicated by the black dahsed line. +The numerically modelled resonance positions at 2312 cm-1, 2066 cm-1, and 1876 cm-1 (yellow) +are shown for comparison. (c) The differential absorbance of the CO signal after baseline +correction for the blueshifted, matched, and redshifted resonances. (d) Scanning electron +microscopy images corresponding to the nano-slot metasurfaces used in (b) and (c). + + +Ref. +2 μm +Focal plane +arraydetector +olarizer +-IR +ATRcrystal +ource +Blueshifted +Redshifted +Matched +Simulated +Blueshifted +Redshifted +Matched +11 +CO adsorption at Open Circuit Potential +Here, we follow in situ the CO adsorption during the saturation of an electrolyte at open circuit +potential (OCP) and characterize the CO adsorption by performing SEIRAS concurrently with +electrochemical cyclic voltammetry. The transition from the Ar to the CO saturated electrolyte is +accompanied by a shift of the OCP due to a change of the equilibrium determining redox reaction +(Figure 3a). At the equilibrium potential of the Ar saturated electrolyte (ca. 300 mVSHE) CO is +oxidized and the OCP drops towards negative values where CO adsorbs on the Pt surface. +The SEIRAS measurements were taken in 0.5M K2CO3 with and without CO (Figure 3b). +Looking at the differential absorbance (Figure 3c), a distortion of the base line appears at 2460 +seconds (+25 mVSHE). Then, after ca. 2800 seconds (-270 mVSHE) two clearly distinguishable +COlinear peaks emerge. These peaks become more discernible with time as the coverage of adsorbed +CO increases. As the intensity of the peaks stabilizes the maximum coverage of CO is reached. +The CO signal obtained with the nano-slot metasurface compared to that obtained with a pure +platinum layer (30 nm) at the OCP is increased by an estimated factor of 27 (Figure 3d). Both +samples have been evaporated simultaneously. This gives both systems the same material +properties such as the surface roughness. For this reason, the 27-fold difference between the signals +obtained with the two systems can be directly linked to the metasurfaces-driven enhancement +provided by nanostructuring the surface of the working electrode. +For adsorbed CO to interact with incident light, the orientation of the transition dipole moment of +the CO vibrational mode relative to the electric field component needs to be non-zero49. +Consequently, only the (interior) side walls parallel to the long axis of the slots can be considered + + +12 +active representing a ratio of active to total surface of 3.6% compared to a smooth platinum layer. +This leads to an experimentally determined local signal enhancement of above 700. +The second peak at 2086 cm-1 was only observed using the nano-slot metasurface. The most likely +explanation could be that the higher resolution achieved with the nano-slot metasurface allows for +the deconvolution of this peak from the background, which was not possible in previous +architectures based on a continuous Pt film. According to the literature, several possibilities exist. +The first assumption is that CO could adsorbs on different crystal orientations with different +binding energies46,50. As reported by A. Cuesta et al.50, an adsorption on Pt(111) single crystals +was found at around 2070 cm-1 38,50,51, while CO adsorbed on Pt(100) electrodes was detected +between 2027 cm-1 52,53 and 2050 cm-1 50,54. These two values are in good agreement with the ones +observed here (2086 cm-1 and 2033 cm-1). Another possibility is the adsorption of CO on terraces +(higher frequency band at 2086 cm-1), steps and defects (lower frequency band at 2033 cm-1) 31,32,55. + + +13 + +Figure 3. Electrochemical and spectroscopic response of the nanophotonic platform at the +OCP during a gas transition of from an Ar-saturated electrolyte to a CO-saturated one. (a) +The evolution of the OCP of the platinum nano-slot metasurface during the transition from an Ar +saturated (Arsat around +310 mVSHE) to a CO saturated electrolyte (COsat around -440 mVSHE). (b) +FTIR spectra of the Pt nano-slot metasurface/electrolyte interface in Arsat and COsat. The heat map +represents the integrated area below the resonance between 2600 and 1800 cm-1 collected by an +array of 64 by 64 detectors. (c) The evolution of the differential absorbance COlinear peaks during +the CO bubbling process. (d) Comparison of COlinear signals obtained in COsat after 80 min of CO +bubbling with a pure Pt layer and with the nanophotonic-electrochemical platform. + + + + +OCP +In Ar +COundetected +In CO +cO detected +Time (s) +4800 +Nano-slot +3780 +3480 +metasurface +3180 +Ptfilm(30nm) +2820 +2426 +2100 +1500 +840 +300 +14 +CO oxidation on platinum +The behavior of the nano-slot metasurface was evaluated during the electrochemical oxidation of +carbon monoxide using electrochemical cyclic voltammetry. The anodic scan in CO-saturated +electrolyte (COsat) presents an initial state with a low current (Figure 4a, black line). When an +applied potential of around -150 mVSHE is reached, the current density plateaus at around +25 +µA.cm-2, which is attributed to CO oxidation56,57. At ca. 450 mVSHE the current density starts to +decrease. The origin of this decrease is still debated in the literature. One explanation attributes the +decreasing current density to competing adsorption of CO and OH on the Pt surface at higher +potentials58. Another possibility discussed is that the formation of a thin oxide or hydroxide Pt +layer prevents the oxidation of CO. The latter assumption is supported by the reduction dip (from ++160 to -80 mVSHE) of platinum in Argon saturated electrolyte (Arsat) (Figure 4a-b). The behavior +of the cathodic scan is similar, except that the onset of CO oxidation is shifted to more negative +potentials resulting in a hysteresis. Moreover, a shift in the onset of the hydrogen evolution reaction +(HER) in Arsat and COsat electrolytes is observed, highlighting the poisoning behavior of adsorbed +CO on the platinum surface59. +Similarly to our Fourier-transform infrared (FTIR) measurements in COsat electrolyte under OCP +(Figure 3c), two COlinear peaks were also found during the electrochemical potential sweeps +(Figure 4c-d). There is a spectral shift during the anodic and cathodic scan (between -650 and - +150 mVSHE) which is attributed to either a higher π -back-donation from the metal to CO40,60 and/or +to the Stark effect. The Stark effect results from the interactions between the surface electric field +and the dipole moment of the adsorbates60–62. During the anodic scan (Figure 4e), the most intense +peak shows a blue-shift of 53 cm-1/V in agreement with the literature38–40,61,63. The second peak +shows a blueshift of 33 cm-1/V. Between -50 and +50 mVSHE a redshift is observed which is not + + +15 +well documented in the literature39,55,60,64. The redshift is attributed to a decrease of the CO +coverage due to its oxidation into CO2, decreasing the dipole-dipole interactions55,65. The +observation of the coverage effect was possible here due to the high resolution reached with the +nano-slot metasurface. It was not resolved with a continuous platinum film. During the anodic +scan, there is a slight increase in the area of the first peak (~2033 cm-1), while the area of the second +peak slightly decreases. This behavior could be explained by a surface migration of adsorbed CO +to a more stable position33,39,40,52. Alternatively, the reconstruction or roughening of the Pt surface +with electrical polarization66,67 could lead to a modification of the surface microstructure and CO +adsorption energy68. Looking at spectra obtained during the cathodic scan (Figure 4f), the second +peak (~2086 cm-1) almost disappeared. This supports the assumption that the cause is the platinum +surface modification at high applied potentials. At high cathodic potentials (-550 to -650 mVSHE) +a decrease of the CO peak is observed and attributed to the HER63, indicating that the adsorption +of hydrogen displaces adsorbed CO. + + +16 + +Figure 4. Behavior of the Pt nano-slot metasurface during cyclic voltammetry in 0.5M K2CO3 +saturated with CO at 0.25 mV.s-1. Evolution of the current density with the potential during the +(a) anodic (from OCP to +1000 mVSHE) and (b) cathodic (from +1000 mVSHE to -700 mVSHE) scan +in COsat electrolyte (black line). For comparison, the blue line depicts CVs in an Arsat electrolyte. +Evolution of SEIRAS spectra with, the (c) anodic and (d) cathodic scans acquired every 100 mV. +Evolution of the position and area of the COlinear peak during the (e) anodic and (f) cathodic scans. + +COad + OHad +CO2 + H+ + e +CO+OH +CO2 + H+ + e +Anodic - COsat +CO undetected +CO detected +Arsat +Anodic - COsat +CO undetected +- CO detected +Arsat +E(mVsHE) ++850 ++750 ++650 ++550 ++450 ++350 ++250 ++150 ++50 +-50 +-150 +-250 +-350 +-450 +-550 +-650 +peak 1 anodic +peak 2 anodic +peak 1 cathodic +peak 2 cathodic +17 +Conclusion +To the best of our knowledge, we have developed the first hybrid nanophotonic-electrochemical +platform for SEIRAS based on a platinum nano-slot metasurface. The resonance of the +metasurface was numerically modelled giving a maximum electric near-field intensity +enhancement of 570. The resonance was tuned to couple with and enhance the CO vibrational +mode at 2033 cm-1. The principle behind the sensing improvement due to the electric near-field +enhancement was tested by fabricating on-resonance and detuned metasurfaces and carefully +analyzing the resonance. The numerical simulations and SEIRAS experimental results were in +good agreement. Two peaks were resolved for the COlinear mode which could be attributed to +adsorption of CO on Pt(111) and Pt(100). COlinear was best observed with a spectrally overlapping +resonance leading to an experimental signal improvement of more than 27 over a conventionally +used platinum film. During the electrochemical oxidation of CO, a classic Stark effect was +observed. Moreover, thanks to the high resolution provided by the nano-slot metasurface, a redshift +of COlinear was observed, linked to a decrease of the coverage of adsorbed CO due to its oxidation. +We anticipate our proof-of-concept nanophotonic-electrochemical platform for SEIRAS to guide +new system designs and material combinations suitable to characterize different electrochemical +interfaces, reaction products, and short-lived intermediates. + +Methods +Numerical simulations +The simulations were performed in CST Studio Suite 2021 using the finite-element frequency- +domain Maxwell solver. CaF2 was simulated using a refractive index, n, of 1.4, the surrounding +medium as water with n ≈ 1.33 and platinum using the data given by Rakić et al.69 The inside walls + + +18 +perpendicular to the electric field were covered with a model material to represent the adsorbed +COlinear vibrational mode at ~2033 cm-1 (see Supplementary Information for more details). The +titanium adhesion layer was not simulated as including it did not lead to substantial spectral shifts +in the resonance position. An impedance-matched open port with a perfectly matched layer +introduced linearly polarized light at an angle of 72º across the CaF2 layer towards the nano-slots. +At 72º the light was totally internally reflected at the CaF2-Pt interface. Therefore, the boundary +opposite the open port was set as perfect electric conductor. The unit cell was defined and then +simulated as infinite periodic array via Floquet boundaries. A field monitor was placed at the center +of the slot in the xy-plane. The highest field enhancement is found slightly above the apex of the +slots. The value of the highest field enhancement of the system was evaluated within the volume +of the numerical model. To extract the Q-factor and coupling ratio 𝛾𝑒/𝛾𝑖, the simulated resonance +was fitted in reflectance (Figure 1d, blue curve) using temporal coupled mode theory according +to Hu et al.70 +Metasurface fabrication +CaF2 was selected as the substrate due to its transparent nature in the mid-IR spectral range, low +solubility, and high chemical stability. The measurements shown in Figures 3 and 4 used +metasurface arrays were with at least 2200 by 2700 unit cells resulting in a pattern area of +approximately 13.3 mm2, which ensured that there are more than enough unit cells for the +measured resonance to correspond to the mode of the infinite periodic array used for the numerical +simulations. After sample cleaning (acetone bath in an ultrasonic cleaner followed by oxygen +plasma cleaning) the substrate was spin-coated first with an adhesion promoter (Surpass 4000), +then with a layer of negative tone photoresist (ma-N 2403) which was baked at 100ºC for 60s, and +finally with a conducting layer (ESpacer 300Z). The metasurface patterns were written via + + +19 +electron-beam lithography (Raith Eline Plus) with an acceleration voltage of 30 kV and an aperture +of 20 µm. The exposed resist was developed in ma-D 525 for 70s at room temperature. The +patterned surface was then coated with a titanium adhesion layer (1 nm at 0.4 Å/s) and a platinum +film (30 nm at 2 Å/s) using electron-beam evaporation (PRO Line PVD 75, Lesker). Finally, an +overnight lift-off in mr-REM 700 concluded the top-down fabrication process. A pure 30 nm thick +platinum film on 1 nm titanium on CaF2 functioned as reference for the in-situ SEIRAS +measurements. +In-situ SEIRAS and electrochemical measurements +SEIRAS was performed using a Vertex 80 coupled with an IMAC chamber from Bruker. Each +sample was mounted on a VeeMax III (purged with N2) from PIKE Technologies in attenuated +total internal reflection (ATR) mode with a light polarizer, an electrochemical Jackfish cell and a +CaF2 prism bevelled at 72°. A classical three electrode system was used with a Saturated Calomel +Electrode (E= 0.244 VSHE), a platinum wire as counter electrode and the platinum sample as +working electrode. The IMAC chamber is equipped with a focal plane array detector composed of +64 x 64 MCT-detectors (a total of 4096 detectors), which allows to perform a mapping of the +studied sample. Each detector collects its own spectrum and then, the active slot covered area is +detected by integration of each spectrum between 1600 to 2800 cm-1(Figure 3b, inset). Finally, an +average can be determined using the spectra of the detectors that probed the resonance. A baseline +correction is applied to this average as well as a Savitzky–Golay filter to smoothen the data. +For the characterization of the resonance, its position was determined using three samples +composed of arrays with different nanostructure sizes. For each sample, the resonance was +measured in 0.5M K2CO3 electrolyte saturated with Ar and then saturated with CO. Prior to the + + +20 +first characterization, a cyclic voltammogram (20 mV.s-1) was recorded in order to confirm the +cleanliness of the electrode surface. Then, an initial background was acquired using p-polarized +light and the Fano resonance was characterized using s-polarized light. Each spectrum was +recorded with a resolution of 4 cm-1 and the final mapping results from a collection of 32 scans. +The enhancement of the nano-slot metasurface is obtained by comparison of the COlinear vibrational +mode on a pure Pt layer (30nm) without nanostructures. +The adsorption of CO during the transition from Arsat to COsat electrolyte (0.5M K2CO3) was +studied using the nano-slot metasurface with the best overlapping resonance with the COlinear +vibration mode. Cleaning and background acquisition protocol were the same as described above. +Carbon monoxide was slowly flowed into the electrochemical cell and spectra were acquired +regularly during the transition from Arsat to COsat at the OCP. +The oxidation of CO during potential sweeps was investigated after 2 hours of CO bubbling. A +cyclic voltammogram, with a slow scan rate (0.25 mV.s-1), from the OCP to +1000 mVSHE and +back to -760 mVSHE was performed and a spectrum was acquired every 100 mV. + +References +(1) Lagadec, M. F.; Grimaud, A. Water Electrolysers with Closed and Open Electrochemical +Systems. Nat. Mater. 2020, 19 (11), 1140–1150. https://doi.org/10.1038/s41563-020-0788-3. +(2) Sullivan, I.; Goryachev, A.; Digdaya, I. A.; Li, X.; Atwater, H. A.; Vermaas, D. A.; Xiang, C. +Coupling Electrochemical CO2 Conversion with CO2 Capture. Nat. Catal. 2021, 4 (11), 952– +958. https://doi.org/10.1038/s41929-021-00699-7. +(3) Stephens, I. E. L.; Chan, K.; Bagger, A.; Boettcher, S. W.; Bonin, J.; Boutin, E.; Buckley, A. +K.; Buonsanti, R.; Cave, E. R.; Chang, X.; Chee, S. W.; Silva, A. H. M. da; Luna, P. de; Einsle, +O.; Endrődi, B.; Escudero-Escribano, M.; Araujo, J. V. F. de; Figueiredo, M. C.; Hahn, C.; +Hansen, K. U.; Haussener, S.; Hunegnaw, S.; Huo, Z.; Hwang, Y. J.; Janáky, C.; Jayathilake, +B. S.; Jiao, F.; Jovanov, Z. P.; Karimi, P.; Koper, M. T. M.; Kuhl, K. P.; Lee, W. H.; Liang, +Z.; Liu, X.; Ma, S.; Ma, M.; Oh, H.-S.; Robert, M.; Cuenya, B. R.; Rossmeisl, J.; Roy, C.; +Ryan, M. P.; Sargent, E. H.; Sebastián-Pascual, P.; Seger, B.; Steier, L.; Strasser, P.; Varela, + + +21 +A. S.; Vos, R. E.; Wang, X.; Xu, B.; Yadegari, H.; Zhou, Y. 2022 Roadmap on Low +Temperature Electrochemical CO2 Reduction. J. Phys. Energy 2022, 4 (4), 042003. +https://doi.org/10.1088/2515-7655/ac7823. +(4) Zhao, S.; Jin, R.; Jin, R. Opportunities and Challenges in CO 2 Reduction by Gold- and Silver- +Based Electrocatalysts: From Bulk Metals to Nanoparticles and Atomically Precise +Nanoclusters. +ACS +Energy +Lett. +2018, +3 +(2), +452–462. +https://doi.org/10.1021/acsenergylett.7b01104. +(5) Wuttig, A.; Yaguchi, M.; Motobayashi, K.; Osawa, M.; Surendranath, Y. Inhibited Proton +Transfer Enhances Au-Catalyzed CO 2 -to-Fuels Selectivity. Proc. Natl. Acad. Sci. 2016, 113 +(32), E4585–E4593. https://doi.org/10.1073/pnas.1602984113. +(6) Hammes, G. G. Spectroscopy for the Biological Sciences; John Wiley & Sons, 2005. +(7) Bernath, P. F. Spectra of Atoms and Molecules; Oxford University Press, USA, 2005. +(8) Lotti, D.; Hamm, P.; Kraack, J. P. Surface-Sensitive Spectro-Electrochemistry Using Ultrafast +2D ATR IR Spectroscopy. J. Phys. Chem. C 2016, 120 (5), 2883–2892. +https://doi.org/10.1021/acs.jpcc.6b00395. +(9) Yang, X.; Sun, Z.; Low, T.; Hu, H.; Guo, X.; García de Abajo, F. J.; Avouris, P.; Dai, Q. +Nanomaterial‐Based Plasmon‐Enhanced Infrared Spectroscopy. Adv. Mater. 2018, 30 (20), +1704896. https://doi.org/10.1002/adma.201704896. +(10) +Neubrech, F.; Huck, C.; Weber, K.; Pucci, A.; Giessen, H. Surface-Enhanced Infrared +Spectroscopy Using Resonant Nanoantennas. Chem. Rev. 2017, 117 (7), 5110–5145. +https://doi.org/10.1021/acs.chemrev.6b00743. +(11) +Ribeiro, J. A.; Sales, M. G. F.; Pereira, C. M. Electrochemistry Combined-Surface Plasmon +Resonance Biosensors: A Review. TrAC Trends Anal. Chem. 2022, 157, 116766. +https://doi.org/10.1016/j.trac.2022.116766. +(12) +Vijay, S.; Hogg, T. V.; Ehlers, J.; Kristoffersen, H. H.; Katayama, Y.; Shao Horn, Y.; +Chorkendorff, I.; Chan, K.; Seger, B. Interaction of CO with Gold in an Electrochemical +Environment. +J. +Phys. +Chem. +C +2021, +125 +(32), +17684–17689. +https://doi.org/10.1021/acs.jpcc.1c04013. +(13) +Katayama, Y.; Nattino, F.; Giordano, L.; Hwang, J.; Rao, R. R.; Andreussi, O.; Marzari, +N.; Shao-Horn, Y. An In Situ Surface-Enhanced Infrared Absorption Spectroscopy Study of +Electrochemical CO2 Reduction: Selectivity Dependence on Surface C-Bound and O-Bound +Reaction +Intermediates. +J. +Phys. +Chem. +C +2019, +123 +(10), +5951–5963. +https://doi.org/10.1021/acs.jpcc.8b09598. +(14) +Dunwell, M.; Lu, Q.; Heyes, J. M.; Rosen, J.; Chen, J. G.; Yan, Y.; Jiao, F.; Xu, B. The +Central Role of Bicarbonate in the Electrochemical Reduction of Carbon Dioxide on Gold. J. +Am. Chem. Soc. 2017, 139 (10), 3774–3783. https://doi.org/10.1021/jacs.6b13287. +(15) +Wang, D.; Shi, F.; Jose, J.; Hu, Y.; Zhang, C.; Zhu, A.; Grzeschik, R.; Schlücker, S.; Xie, +W. In Situ Monitoring of Palladium-Catalyzed Chemical Reactions by Nanogap-Enhanced +Raman Scattering Using Single Pd Cube Dimers. J. Am. Chem. Soc. 2022, 144 (11), 5003– +5009. https://doi.org/10.1021/jacs.1c13240. +(16) +Rodrigo, D.; Tittl, A.; Ait-Bouziad, N.; John-Herpin, A.; Limaj, O.; Kelly, C.; Yoo, D.; +Wittenberg, N. J.; Oh, S.-H.; Lashuel, H. A.; Altug, H. Resolving Molecule-Specific +Information in Dynamic Lipid Membrane Processes with Multi-Resonant Infrared +Metasurfaces. Nat. Commun. 2018, 9 (1), 2160. https://doi.org/10.1038/s41467-018-04594-x. + + +22 +(17) +Tittl, A.; Leitis, A.; Liu, M.; Yesilkoy, F.; Choi, D.-Y.; Neshev, D. N.; Kivshar, Y. S.; +Altug, H. Imaging-Based Molecular Barcoding with Pixelated Dielectric Metasurfaces. +Science 2018, 360 (6393), 1105–1109. https://doi.org/10.1126/science.aas9768. +(18) +Hosseini Farahabadi, S. A.; Entezami, M.; Abouali, H.; Amarloo, H.; Poudineh, M.; +Safavi-Naeini, S. Sub-Terahertz Silicon-Based on-Chip Absorption Spectroscopy Using Thin- +Film +Model +for +Biological +Applications. +Sci. +Rep. +2022, +12 +(1), +17747. +https://doi.org/10.1038/s41598-022-21015-8. +(19) +Li, T.; Shang, D.; Gao, S.; Wang, B.; Kong, H.; Yang, G.; Shu, W.; Xu, P.; Wei, G. Two- +Dimensional Material-Based Electrochemical Sensors/Biosensors for Food Safety and +Biomolecular Detection. Biosensors 2022, 12 (5), 314. https://doi.org/10.3390/bios12050314. +(20) +Wang, J.; Maier, S. A.; Tittl, A. Trends in Nanophotonics-Enabled Optofluidic Biosensors. +Adv. Opt. Mater. 2022, 10 (7), 2102366. https://doi.org/10.1002/adom.202102366. +(21) +Jiang, J.; Wang, X.; Li, S.; Ding, F.; Li, N.; Meng, S.; Li, R.; Qi, J.; Liu, Q.; Liu, G. L. +Plasmonic Nano-Arrays for Ultrasensitive Bio-Sensing. Nanophotonics 2018, 7 (9), 1517– +1531. https://doi.org/10.1515/nanoph-2018-0023. +(22) +Li, N.; Lu, Y.; Li, S.; Zhang, Q.; Wu, J.; Jiang, J.; Liu, G. L.; Liu, Q. Monitoring the +Electrochemical Responses of Neurotransmitters through Localized Surface Plasmon +Resonance +Using +Nanohole +Array. +Biosens. +Bioelectron. +2017, +93, +241–249. +https://doi.org/10.1016/j.bios.2016.08.105. +(23) +Nakamoto, K.; Kurita, R.; Niwa, O. Electrochemical Surface Plasmon Resonance +Measurement Based on Gold Nanohole Array Fabricated by Nanoimprinting Technique. Anal. +Chem. 2012, 84 (7), 3187–3191. https://doi.org/10.1021/ac203160r. +(24) +Wang, X.; Huang, S.-C.; Hu, S.; Yan, S.; Ren, B. Fundamental Understanding and +Applications of Plasmon-Enhanced Raman Spectroscopy. Nat. Rev. Phys. 2020, 2 (5), 253– +271. https://doi.org/10.1038/s42254-020-0171-y. +(25) +Wright, D.; Lin, Q.; Berta, D.; Földes, T.; Wagner, A.; Griffiths, J.; Readman, C.; Rosta, +E.; Reisner, E.; Baumberg, J. J. Mechanistic Study of an Immobilized Molecular +Electrocatalyst by in Situ Gap-Plasmon-Assisted Spectro-Electrochemistry. Nat. Catal. 2021, +4 (2), 157–163. https://doi.org/10.1038/s41929-020-00566-x. +(26) +Fang, Y.; Wang, H.; Yu, H.; Liu, X.; Wang, W.; Chen, H.-Y.; Tao, N. J. Plasmonic Imaging +of Electrochemical Reactions of Single Nanoparticles. Acc. Chem. Res. 2016, 49 (11), 2614– +2624. https://doi.org/10.1021/acs.accounts.6b00348. +(27) +Cheng, X. R.; Wallace, G. Q.; Lagugné-Labarthet, F.; Kerman, K. Au Nanostructured +Surfaces for Electrochemical and Localized Surface Plasmon Resonance-Based Monitoring of +α-Synuclein–Small Molecule Interactions. ACS Appl. Mater. Interfaces 2015, 7 (7), 4081– +4088. https://doi.org/10.1021/am507972b. +(28) +Wang, S.-S.; Zhao, X.-P.; Liu, F.-F.; Younis, M. R.; Xia, X.-H.; Wang, C. Direct Plasmon- +Enhanced Electrochemistry for Enabling Ultrasensitive and Label-Free Detection of +Circulating +Tumor Cells +in +Blood. +Anal. Chem. 2019, +91 +(7), 4413–4420. +https://doi.org/10.1021/acs.analchem.8b04908. +(29) +Cortés, E.; Wendisch, F. J.; Sortino, L.; Mancini, A.; Ezendam, S.; Saris, S.; Menezes, L. +de S.; Tittl, A.; Ren, H.; Maier, S. A. Optical Metasurfaces for Energy Conversion. Chem. Rev. +2022, 122 (19), 15082–15176. https://doi.org/10.1021/acs.chemrev.2c00078. +(30) +Stefancu, A.; Nan, L.; Zhu, L.; Chiș, V.; Bald, I.; Liu, M.; Leopold, N.; Maier, S. A.; +Cortes, E. Controlling Plasmonic Chemistry Pathways through Specific Ion Effects. Adv. Opt. +Mater. 2022, 10 (14), 2200397. https://doi.org/10.1002/adom.202200397. + + +23 +(31) +Hossain, M. J.; Rahman, M. M.; Jafar Sharif, Md. Preference for Low-Coordination Sites +by Adsorbed CO on Small Platinum Nanoparticles. Nanoscale Adv. 2020, 2 (3), 1245–1252. +https://doi.org/10.1039/C9NA00499H. +(32) +Silva, C. D.; Cabello, G.; Christinelli, W. A.; Pereira, E. C.; Cuesta, A. Simultaneous Time- +Resolved ATR-SEIRAS and CO-Charge Displacement Experiments: The Dynamics of CO +Adsorption +on +Polycrystalline +Pt. +J. +Electroanal. +Chem. +2017, +800, +25–31. +https://doi.org/10.1016/j.jelechem.2016.10.034. +(33) +McPherson, I. J.; Ash, P. A.; Jones, L.; Varambhia, A.; Jacobs, R. M. J.; Vincent, K. A. +Electrochemical CO Oxidation at Platinum on Carbon Studied through Analysis of Anomalous +in +Situ +IR +Spectra. +J. +Phys. +Chem. +C +2017, +121 +(32), +17176–17187. +https://doi.org/10.1021/acs.jpcc.7b02166. +(34) +Susarrey-Arce, A.; Tiggelaar, R. M.; Gardeniers, J. G. E.; van Houselt, A.; Lefferts, L. CO +Adsorption on Pt Nanoparticles in Low E-Fields Studied by ATR-IR Spectroscopy in a +Microreactor. +J. +Phys. +Chem. +C +2015, +119 +(44), +24887–24894. +https://doi.org/10.1021/acs.jpcc.5b08392. +(35) +Smolinka, T.; Heinen, M.; Chen, Y. X.; Jusys, Z.; Lehnert, W.; Behm, R. J. CO2 Reduction +on Pt Electrocatalysts and Its Impact on H2 Oxidation in CO2 Containing Fuel Cell Feed Gas +– A Combined in Situ Infrared Spectroscopy, Mass Spectrometry and Fuel Cell Performance +Study. +Electrochimica +Acta +2005, +50 +(25–26), +5189–5199. +https://doi.org/10.1016/j.electacta.2005.02.082. +(36) +Katayama, Y.; Giordano, L.; Rao, R. R.; Hwang, J.; Muroyama, H.; Matsui, T.; Eguchi, +K.; Shao-Horn, Y. Surface (Electro)Chemistry of CO2 on Pt Surface: An in Situ Surface- +Enhanced Infrared Absorption Spectroscopy Study. J. Phys. Chem. C 2018, 122 (23), 12341– +12349. https://doi.org/10.1021/acs.jpcc.8b03556. +(37) +Huck, C.; Vogt, J.; Sendner, M.; Hengstler, D.; Neubrech, F.; Pucci, A. Plasmonic +Enhancement of Infrared Vibrational Signals: Nanoslits versus Nanorods. ACS Photonics +2015, 2 (10), 1489–1497. https://doi.org/10.1021/acsphotonics.5b00390. +(38) +Villegas, I.; Weaver, M. J. Carbon Monoxide Adlayer Structures on Platinum (111) +Electrodes: A Synergy between In‐situ Scanning Tunneling Microscopy and Infrared +Spectroscopy. J. Chem. Phys. 1994, 101 (2), 1648–1660. https://doi.org/10.1063/1.467786. +(39) +Watanabe, S.; Inukai, J.; Ito, M. Coverage and Potential Dependent CO Adsorption on Pt( +111 l), (711) and (100) Electrode Surfaces Studied by Infrared Reflection Absorption +Spectroscopy. 2020, 9. +(40) +Nakamura, M.; Ogasawara, H.; Inukai, J.; Ito, M. CO Migration on Pt(100) and Pt(11 1 1) +Surfaces Studied by Time Resolved Infrared Reflection-Absorption Spectroscopy. Surf. Sci. +1993, 283 (1–3), 248–254. https://doi.org/10.1016/0039-6028(93)90989-W. +(41) +Hickman, I. Analog Electronics: Analog Circuitry Explained; Newnes, 2013. +(42) +Gao, H.; McMahon, J. M.; Lee, M. H.; Henzie, J.; Gray, S. K.; Schatz, G. C.; Odom, T. W. +Rayleigh Anomaly-Surface Plasmon Polariton Resonances in Palladium and Gold +Subwavelength +Hole +Arrays. +Opt. +Express +2009, +17 +(4), +2334. +https://doi.org/10.1364/OE.17.002334. +(43) +Adato, R.; Yanik, A. A.; Wu, C.-H.; Shvets, G.; Altug, H. Radiative Engineering of +Plasmon Lifetimes in Embedded Nanoantenna Arrays. Opt. Express 2010, 18 (5), 4526. +https://doi.org/10.1364/OE.18.004526. + + +24 +(44) +Adato, R.; Artar, A.; Erramilli, S.; Altug, H. Engineered Absorption Enhancement and +Induced Transparency in Coupled Molecular and Plasmonic Resonator Systems. Nano Lett. +2013, 13 (6), 2584–2591. https://doi.org/10.1021/nl400689q. +(45) +Aigner, A.; Tittl, A.; Wang, J.; Weber, T.; Kivshar, Y.; Maier, S. A.; Ren, H. Plasmonic +Bound States in the Continuum to Tailor Light-Matter Coupling. Sci. Adv. 2022, 8 (49), +eadd4816. https://doi.org/10.1126/sciadv.add4816. +(46) +Farias, M. J. S.; Busó-Rogero, C.; Tanaka, A. A.; Herrero, E.; Feliu, J. M. Monitoring of +CO Binding Sites on Stepped Pt Single Crystal Electrodes in Alkaline Solutions by in Situ +FTIR +Spectroscopy. +Langmuir +2020, +36 +(3), +704–714. +https://doi.org/10.1021/acs.langmuir.9b02928. +(47) +Luk’yanchuk, B.; Zheludev, N. I.; Maier, S. A.; Halas, N. J.; Nordlander, P.; Giessen, H.; +Chong, C. T. The Fano Resonance in Plasmonic Nanostructures and Metamaterials. Nat. +Mater. 2010, 9 (9), 707–715. https://doi.org/10.1038/nmat2810. +(48) +Fan, S. Sharp Asymmetric Line Shapes in Side-Coupled Waveguide-Cavity Systems. Appl. +Phys. Lett. 2002, 80 (6), 908–910. https://doi.org/10.1063/1.1448174. +(49) +Ras, R. H. A.; Schoonheydt, R. A.; Johnston, C. T. Relation between S-Polarized and p- +Polarized Internal Reflection Spectra: Application for the Spectral Resolution of Perpendicular +Vibrational +Modes. +J. +Phys. +Chem. +A +2007, +111 +(36), +8787–8791. +https://doi.org/10.1021/jp073108a. +(50) +Cuesta, A.; Couto, A.; Rincón, A.; Pérez, M. C.; López-Cudero, A.; Gutiérrez, C. Potential +Dependence of the Saturation CO Coverage of Pt Electrodes: The Origin of the Pre-Peak in +CO-Stripping Voltammograms. Part 3: Pt(Poly). J. Electroanal. Chem. 2006, 586 (2), 184– +195. https://doi.org/10.1016/j.jelechem.2005.10.006. +(51) +López-Cudero, A.; Cuesta, A.; Gutiérrez, C. Potential Dependence of the Saturation CO +Coverage of Pt Electrodes: The Origin of the Pre-Peak in CO-Stripping Voltammograms. Part +1: +Pt(111). +J. +Electroanal. +Chem. +2005, +579 +(1), +1–12. +https://doi.org/10.1016/j.jelechem.2005.01.018. +(52) +Sun, S.-G.; Zhou, Z.-Y. Surface Processes and Kinetics of CO2 Reduction on Pt(100) +Electrodes of Different Surface Structure in Sulfuric Acid Solutions. Phys. Chem. Chem. Phys. +2001, 3 (16), 3277–3283. https://doi.org/10.1039/b100938i. +(53) +Rodes, A.; Pastor, E.; Iwasita, T. Structural Effects on CO2 Reduction at Pt Single-Crystal +Electrodes: Part 3. Pt(100) and Related Surfaces. J. Electroanal. Chem. 1994, 377 (1), 215– +225. https://doi.org/10.1016/0022-0728(94)03424-9. +(54) +López-Cudero, A.; Cuesta, Á.; Gutiérrez, C. Potential Dependence of the Saturation CO +Coverage of Pt Electrodes: The Origin of the Pre-Peak in CO-Stripping Voltammograms. Part +2: +Pt(100). +J. +Electroanal. +Chem. +2006, +586 +(2), +204–216. +https://doi.org/10.1016/j.jelechem.2005.10.003. +(55) +Chang, S.-C.; Weaver, M. J. In-Situ Infrared Spectroscopy of CO Adsorbed at Ordered +Pt(110)-Aqueous Interfaces. Surf. Sci. 1990, 230 (1), 222–236. https://doi.org/10.1016/0039- +6028(90)90030-C. +(56) +Blizanac, B. B.; Lucas, C. A.; Gallagher, M. E.; Arenz, M.; Ross, P. N.; Marković, N. M. +Anion Adsorption, CO Oxidation, and Oxygen Reduction Reaction on a Au(100) Surface: The +PH Effect. J. Phys. Chem. B 2004, 108 (2), 625–634. https://doi.org/10.1021/jp036483l. +(57) +Mayet, N.; Servat, K.; Kokoh, K. B.; Napporn, T. W. Electrochemical Oxidation of Carbon +Monoxide on Unsupported Gold Nanospheres in Alkaline Medium. Electrocatalysis 2021, 12 +(1), 26–35. https://doi.org/10.1007/s12678-020-00626-7. + + +25 +(58) +Spendelow, J. S.; Goodpaster, J. D.; Kenis, P. J. A.; Wieckowski, A. Mechanism of CO +Oxidation on Pt(111) in Alkaline Media. J. Phys. Chem. B 2006, 110 (19), 9545–9555. +https://doi.org/10.1021/jp060100c. +(59) +Varela, A. S.; Schlaup, C.; Jovanov, Z. P.; Malacrida, P.; Horch, S.; Stephens, I. E. L.; +Chorkendorff, I. CO2 Electroreduction on Well-Defined Bimetallic Surfaces: Cu Overlayers +on Pt(111) and Pt(211). J. Phys. Chem. C 2013, 117 (40), 20500–20508. +https://doi.org/10.1021/jp406913f. +(60) +Samjeské, G.; Komatsu, K.; Osawa, M. Dynamics of CO Oxidation on a Polycrystalline +Platinum Electrode: A Time-Resolved Infrared Study. J. Phys. Chem. C 2009, 113 (23), +10222–10228. https://doi.org/10.1021/jp900582c. +(61) +Katayama, Y.; Giordano, L.; Rao, R. R.; Hwang, J.; Muroyama, H.; Matsui, T.; Eguchi, +K.; Shao-Horn, Y. Surface (Electro)Chemistry of CO 2 on Pt Surface: An in Situ Surface- +Enhanced Infrared Absorption Spectroscopy Study. J. Phys. Chem. C 2018, 122 (23), 12341– +12349. https://doi.org/10.1021/acs.jpcc.8b03556. +(62) +Stamenkovic, V.; Chou, K. C.; Somorjai, G. A.; Ross, P. N.; Markovic, N. M. Vibrational +Properties of CO at the Pt(111)−Solution Interface: The Anomalous Stark-Tuning Slope. J. +Phys. Chem. B 2005, 109 (2), 678–680. https://doi.org/10.1021/jp044802i. +(63) +Dunwell, M.; Wang, J.; Yan, Y.; Xu, B. Surface Enhanced Spectroscopic Investigations of +Adsorption of Cations on Electrochemical Interfaces. Phys. Chem. Chem. Phys. 2017, 19 (2), +971–975. https://doi.org/10.1039/C6CP07207K. +(64) +Dunwell, M.; Yan, Y.; Xu, B. In Situ Infrared Spectroscopic Investigations of Pyridine- +Mediated CO 2 Reduction on Pt Electrocatalysts. ACS Catal. 2017, 7 (8), 5410–5419. +https://doi.org/10.1021/acscatal.7b01392. +(65) +Deshlahra, P.; Conway, J.; Wolf, E. E.; Schneider, W. F. Influence of Dipole–Dipole +Interactions on Coverage-Dependent Adsorption: CO and NO on Pt(111). Langmuir 2012, 28 +(22), 8408–8417. https://doi.org/10.1021/la300975s. +(66) +Jacobse, L.; Rost, M. J.; Koper, M. T. M. Atomic-Scale Identification of the +Electrochemical Roughening of Platinum. ACS Cent. Sci. 2019, 5 (12), 1920–1928. +https://doi.org/10.1021/acscentsci.9b00782. +(67) +Deng, X.; Galli, F.; Koper, M. T. M. In Situ AFM Imaging of Platinum Electrode Surface +during Oxidation–Reduction Cycles in Alkaline Electrolyte. ACS Appl. Energy Mater. 2020, +3 (1), 597–602. https://doi.org/10.1021/acsaem.9b01700. +(68) +Hammer, B.; Nielsen, O. H.; Nrskov, J. K. Structure Sensitivity in Adsorption: CO +Interaction with Stepped and Reconstructed Pt Surfaces. Catal. Lett. 1997, 46 (1), 31–35. +https://doi.org/10.1023/A:1019073208575. +(69) +Rakić, A. D.; Djurišić, A. B.; Elazar, J. M.; Majewski, M. L. Optical Properties of Metallic +Films for Vertical-Cavity Optoelectronic Devices. Appl. Opt. 1998, 37 (22), 5271. +https://doi.org/10.1364/AO.37.005271. +(70) +Hu, H.; Weber, T.; Bienek, O.; Wester, A.; Hüttenhofer, L.; Sharp, I. D.; Maier, S. A.; Tittl, +A.; Cortés, E. Catalytic Metasurfaces Empowered by Bound States in the Continuum. ACS +Nano 2022, 16 (8), 13057–13068. https://doi.org/10.1021/acsnano.2c05680. + + + + +26 +Acknowledgments +We thank Thomas Weber for his help with the temporal-coupled mode theory algorithms and +Simon Stork for the platinum and titanium electron beam evaporation. +Corresponding Author +Andreas Tittl - Chair in Hybrid Nanosystems, Nanoinstitute Munich, Faculty of Physics, Ludwig- +Maximilians-Universität München, Königinstraße 10, 80539 München, Germany; Orcid: +https://orcid.org/0000-0003-3191-7164; Email: Andreas.Tittl@physik.uni-muenchen.de +Katharina Krischer - Department of Physics, Technical University of Munich, 85748 Garching, +Germany; Email: Krischer@tum.de +Author Contributions +The manuscript was written through the contributions of all authors. All authors have given +approval to the final version of the manuscript. ‡These authors contributed equally. +Funding Sources +This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research +Foundation) under grant numbers EXC 2089/1–390776260 (Germany′s Excellence Strategy) and +TI 1063/1 (Emmy Noether Program), ERC-STG 802989 Catalight, the Bavarian program Solar +Energies Go Hybrid (SolTech) and the Center for NanoScience (CeNS). S.A.M. additionally +acknowledges the Lee-Lucas Chair in Physics and the EPSRC (EP/W017075/1). +Notes +The authors declare no competing financial interest. + +Supporting information +Improved in-situ characterization of electrochemical interfaces using +metasurface-driven surface-enhanced infrared absorption spectroscopy +Luca M. Berger 1, ‡, Malo Duportal 2, ‡, Leonardo de Souza Menezes 1,3, Emiliano Cortés 1, Stefan A. +Maier 4,5,1, Andreas Tittl 1,*, Katharina Krischer 2,* +1 Faculty of Physics, Ludwig-Maximilian-University Munich, 80539 München, Germany +2 Department of Physics, Technical University of Munich, 85748 Garching, Germany +3 Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil +4 School of Physics and Astronomy, Monash University, Melbourne, Victoria, Australia +5 Department of Physics, Imperial College London, SW7 2AZ London, United Kingdom + +* e-mail: Andreas.Tittl@physik.uni-muenchen.de; krischer@tum.de + + +Figure S1. The real (blue) and imaginary (red) parts of the permittivity of the artificial material +modeled to represent CO. The permittivity of a material suffices to numerically model its +interaction with light. Using the Lorentz oscillator model, the real and imaginary parts of the +permittivity can be written as 𝜀𝑟 = 1 + +𝜔𝑝2(𝜔02−𝜔2) +(𝜔02−𝜔2) +2+𝛾2𝜔2 and 𝜀𝑖 = +𝛾𝜔𝑝2𝜔 +(𝜔02−𝜔2) +2+𝛾2𝜔2, respectively1. +Here, we modified the baseline of 𝜀𝑟 to be around 1.332 which is the refractive index squared +used for the surrounding medium (water) due to a negligible shift in the refractive index due to +CO2. Here, 𝜔𝑝 is analogous to the plasma frequency in the Drude-Sommerfeld model, 𝜔0 is the +frequency of the absorption band, 𝜔 is the frequency, and 𝛾 is the damping constant. To model +the linear vibrational mode of CO at 2033 cm-1 the parameters used were 𝜔𝑝 +2 = 1 × 1024 s-2, +𝜔0 = 60.95 × 1012 s-1 (corresponding to 2033 cm-1), and 𝛾 = 1 × 1011 s-1. + +Re[ε] +2.5 +Im[] +2.0: +1.5- +m +1.0 - +0.5 - +0.0 +2010 +2015 +2020 +2025 +2030 +2035 +2040 +2045 +2050 +2055 +Wavenumber (cm-1)References +(1) +Novotny, L.; Hecht, B. Principles of Nano-Optics; Cambridge University Press, 2012. +(2) +Harvey, A. H.; Kaplan, S. G.; Burnett, J. H. Effect of Dissolved Air on the Density and +Refractive Index of Water. Int. J. Thermophys. 2005, 26 (5), 1495–1514. +https://doi.org/10.1007/s10765-005-8099-0. + + diff --git a/KNA0T4oBgHgl3EQfCv9N/content/tmp_files/load_file.txt b/KNA0T4oBgHgl3EQfCv9N/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0a996f2045b98fcd19ab664cb90c0c846e9c7042 --- /dev/null +++ b/KNA0T4oBgHgl3EQfCv9N/content/tmp_files/load_file.txt @@ -0,0 +1,1793 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf,len=1792 +page_content='1 Improved in-situ characterization of electrochemical interfaces using metasurface-driven surface-enhanced infrared absorption spectroscopy Luca M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Berger 1, ‡, Malo Duportal 2, ‡, Leonardo de Souza Menezes 1,3, Emiliano Cortés 1, Stefan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier 4,5,1, Andreas Tittl 1,*, Katharina Krischer 2,* 1 Faculty of Physics, Ludwig-Maximilian-University Munich, 80539 München, Germany 2 Department of Physics, Technical University of Munich, 85748 Garching, Germany 3 Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil 4 School of Physics and Astronomy, Monash University, Melbourne, Victoria, Australia 5 Department of Physics, Imperial College London, SW7 2AZ London, United Kingdom e-mail: Andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='Tittl@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='uni-muenchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' krischer@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='de Abstract Electrocatalysis plays a crucial role in realizing the transition towards green energy, driving research directions from hydrogen generation to carbon dioxide reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Understanding electrochemical reactions is crucial to improve their efficiency and to bridge the gap toward a sustainable zero-carbon future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface-enhanced infrared absorption spectroscopy (SEIRAS) is a suitable method for investigating these processes because it can monitor with chemical specificity the mechanisms of the reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' However, it remains difficult to detect many relevant aspects of electrochemical reactions such as short-lived intermediates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Here, we develop and experimentally realize an integrated nanophotonic-electrochemical SEIRAS platform for the in situ investigation of molecular signal traces emerging during electrochemical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Specifically, we implement a platinum nano-slot metasurface featuring strongly enhanced electromagnetic near fields and spectrally target it at the weak vibrational bending mode of CO CO 2 adsorbed CO at ~2033 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Crucially, our platinum nano-slot metasurface provides high molecular sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The resonances can be tuned over a broad range in the mid-infrared spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Compared to conventional unstructured platinum layers, our nanophotonic- electrochemical platform delivers a substantial improvement of the experimentally detected characteristic absorption signals by a factor of 27, enabling the detection of new species with weak signals, fast conversions, or low surface concentrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' By providing a deeper understanding of catalytic reactions, we anticipate our nanophotonic-electrochemical platform to open exciting perspectives for electrochemical SEIRAS, surface-enhanced Raman spectroscopy, and the study of reactions in other fields of chemistry such as photoelectrocatalysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Introduction Electrochemical reactions underpin many technologies ubiquitous for a future carbon-zero world such as green-hydrogen generation for long-term sustainable energy storage1 and CO2 degradation to combat the current trends of climate change2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Unfortunately, in general, the monitoring, and therefore understanding, of many electrochemical reactions remains a challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In particular, resolving the electrochemical CO2 reduction reaction (CO2RR) with high efficiency, selectivity, and sensitivity remains an issue3 especially due to the competition with the hydrogen evolution reaction (HER) at high current densities4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' During the CO2RR to desired carbon products, a compulsory step to the key intermediate CO is still not fully understood and requires further investigation5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For the detection and characterization of molecules, optical spectroscopy, mass spectrometry, chromatography, and fluorescence microscopy are often used6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Optical spectroscopy methods in particular are highly advantageous because they allow for the retrieval of the spectral fingerprint 3 of molecules via the detection of their rotational or vibrational modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Within optical spectroscopy, two strong methods are Raman and infrared (IR) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The former relies on the inelastic scattering of photons and studies the resulting spectral shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The latter detects the absorption of light by molecules when the energy of the photons matches the energy of the vibrational modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The mutual exclusion rule dictates that any mode can be IR active, Raman inactive, and vice-versa but not simultaneously7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Therefore, for a given molecular mode either Raman or IR spectroscopy can be used, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Here, the CO vibrational mode under investigation is IR active8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface- enhanced infrared absorption spectroscopy (SEIRAS) is a derivative technique from conventional infrared spectroscopy based on the enhancement of the local electromagnetic (EM) near fields 9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' To increase the sensitivity of either surface enhanced Raman spectroscopy (SERS) or SEIRAS during electrochemical reactions typically a rough metal surface has been chosen to enhance the local electromagnetic (EM) near fields11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rough and highly disordered metallic nm-sized edges coming from perforations and extrusions in the metallic film locally confine and enhance the EM fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Unfortunately, this approach is random, does not allow for spectral tailoring of plasmonic hotspots, and consequently generates a relatively weak EM near-field enhancement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Even after improvements in the sensitivity of SEIRAS using an attenuated total internal reflection (ATR) geometry9,10, the characterization of CO adsorption on catalysts is still hampered by weak signal traces12–14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' We overcome the challenge of detecting weak signal traces by taking inspiration from other fields of nanophotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In biomolecular sensing, a plethora of alternatives are used to improve molecular detection using controlled and tuneable EM near-field enhancement via the excitation of resonances through tailored system parameters on the nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Examples are plasmonic nanoparticles, non-plasmonic nanogap dimers15, metasurfaces based on plasmonics16 or exotic 4 phenomena like quasi-bound states in the continuum17, waveguides18 or 2D-integrated19 platforms, among others20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Plasmonic-based sensors have become the method of choice in label-free detection of biomolecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' They can be used either as 1) refractive index (RI) sensors or 2) by coupling the resonances to the molecular modes and analysing the perturbation of the intensity either in reflection or transmission21, termed perturbed intensity sensing here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In fact, some recent progress has been made to integrate plasmonic structures for refractive index sensing with electrochemistry21–23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' There are also recent examples of plasmonic structures for perturbed intensity sensing for SERS used to monitor electrochemical reactions24 or to study the mechanism of an electrocatalytic reaction25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Literature of plasmonic imaging provides other examples of electrochemical reactions of single nanoparticles26, plasmonics-supported and electrochemical monitoring of molecular interactions focused on fluorescence and confocal microscopy27,28, and plasmon-accelerated electrochemical reactions29,30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' However, to the best of our knowledge, the integration of nanostructured metasurfaces for perturbed intensity sensing in SEIRAS has never been shown in combination with electrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Here, we detect in situ the CO vibrational bending mode at 2033 cm-1 emerging during the electrochemical conversion of CO into CO2 using a platinum nano-slot metasurface on a CaF2 substrate (Figure 1a) by coupling its resonance to the molecular vibrational mode and analysing the perturbation of the intensity in reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' We investigated the linear CO vibrational mode (COlinear) at 2033 cm-1 because it is the most intense vibrational mode of CO on platinum31–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The material of choice was platinum as it could fulfill all requirements, namely to function as a working electrode, support strong metasurface-driven resonances, and adsorb CO on its surface36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Moreover, Pt is a catalytic material for many reactions, making this platform very useful not only for the CO oxidation reaction but also for other reactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The decision on the inverse structure 5 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' the slots), was made to preserve a connected metallic film that can carry electrical current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Moreover, compared to resonant rod-type antennas, the inverse counterparts have been shown to feature superior detection of molecular signal traces due to linearly instead of exponentially decaying EM near-fields37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The slots can only be excited with transverse electric (TE) polarized light37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' We perform SEIRAS in an ATR geometry to further improve the sensing performance while maintaining free accessibility of the electrode surface for reactants and products9,10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' We confirm the detection of adsorbed CO via the observation of the typical Stark shift and resolve a so far scarcely studied 38–40 effect due to the decrease of the CO coverage on the surface of platinum during the electrochemical oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Furthermore, the presence of a second peak at 2086 cm-1 on the spectral location of the linear vibrational mode could be attributed to the effect of the crystal orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Finally, we establish a methodology for designing similar nanophotonic- electrochemical platforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Numerical design of the catalytic metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (a) Schematic for the Pt-based nano- slot metasurface for the in-situ integrated nanophotonic-electrochemical study of CO2 oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' As the potential between the working electrode (WE) and the reference electrode (Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=') is swept the presence of adsorbed CO is monitored via the detection of the linear vibrational mode of CO at 2033 cm-1 with a Fourier transform infrared (FTIR) spectrometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The nano-slot metasurface enhances the electromagnetic near-fields of TE polarized light in an ATR configuration coming in at an azimuthal angle 𝜙 = 0∘ and polar angle 𝜃 = 72∘ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' the Pt film (𝑥𝑦-plane).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (b) Sketch of the Pt on CaF2 nano-slot unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Two CO model layers were included parallel to the long edges of the slot (magenta) with dimensions 𝑙 × ℎ × 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A 1 nm thick Ti adhesion layer was used in the fabrication of the structures but is not considered in the numerical simulations due to its negligible effect on the resonance position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The geometrical parameters of the unit cell for (c), (d), (e) are ℎ = 30 nm, 𝑤 = 200 nm, 𝑙 = 1380 nm, 𝑝𝑦 = 1400 nm, 𝑝𝑥 = 1600 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In (c) no CO model layer was included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In (d) 𝑡 = 5 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (c) Electric near field intensity (taken at ℎ = 30 nm) of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The maximum near field intensity is 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (d) The simulated reflectance spectrum of the metasurface with (pink) and without (blue) the CO model layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The spectrum includes the Rayleigh anomaly (RA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (e) The differential absorbance of (d) with the thickness of the CO layer 𝑡 2 nm (blue), 5 nm (pink) and 10 nm (red) showing clearly visible absorption bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' a CO CO Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' CE WE Py IE/Eol max=570 RA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='20 t (nm): 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='18 2 2040 2030 5 10 7 Results and discussion Numerical design of catalytic nano-slot metasurface We start the implementation of our electrochemical sensing platform with the numerical design of the chosen nano-slot metasurface geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The structure consists of a unit cell composed of a single slot in an otherwise connected platinum film submerged in water on CaF2 (Figure 1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Notably, we model adsorbed CO by including an artificially created material covering the inside walls parallel to the long axis of the slot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The choice for the parameters of the unit cell was guided by Huck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='37 and modified in accordance with fabrication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Huck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='37 optimized a gold nano-slot metasurface in the mid-infrared for normal incidence illumination in air for high quality factors (Q-factors) and electric near fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The Q-factor relates the initial energy stored in a resonator to the energy dissipated in one radian of the cycle of oscillation41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' On the basis of our simulations, the nano-slot metasurface achieves a resonance with a modulation in the absorbance of over 82% and a Q-factor of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='3 (see “Methods” section for details on the Q-factor calculation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Furthermore, the metasurface numerically exhibits an electric near-field enhancement |𝐸/𝐸0|2 of 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' This value can be increased in future experiments by decreasing the width of the slots37 but was limited here due to fabrication constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The maximum electric near- field enhancement occurs inside the slots close to the faces parallel to its long axis (Figure 1c), with its electric field pointing orthogonally to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Huck et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='37 found that the highest Q-factor and electric near field enhancement occurs when w is small, 𝑝𝑦 = 𝜆𝑟𝑒𝑠/2, and 𝑔 = 𝑝𝑥 − 𝑙 = 𝜆𝑟𝑒𝑠/2, where 𝜆𝑟𝑒𝑠 is the central wavelength of the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' However, to satisfy the experimental conditions the nano-slot metasurface was simulated in water instead of air and for an angle of incidence 𝜃 = 72∘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Under these conditions, 8 tuning the resonance to 2033 cm-1 ≈ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='92 µm leads to the appearance of a Rayleigh anomaly (RA) such that 𝜆𝑅𝐴 > 𝜆𝑟𝑒𝑠, where 𝜆𝑅𝐴 is the central wavelength of the RA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The RA is a phenomenon associated with light diffracted parallel to the surface of a periodic structure42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' When 𝜆𝑅𝐴 > 𝜆𝑟𝑒𝑠, the resonance lifetime and electric near-field enhancement is strongly reduced43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Consequently, a metasurface where 𝜆𝑅𝐴 > 𝜆𝑟𝑒𝑠 will exhibit poor sensing performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For this reason, 𝑔 was reduced to 220 nm to push the resonance on the evanescent side of the RA (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In coupled-resonator systems, the excitation efficiency of a resonator is significantly dependent on the ratio of its losses to external radiation 𝛾𝑒, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' light scattering, and intrinsic material absorption 𝛾𝑖 which strongly depends on the system design and parameters chosen44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' When 𝛾𝑒 ∼ 𝛾𝑖 the system is critically-coupled and the second oscillator will lead to a dip in the absorption cross-section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' SEIRAS performance can be maximized by utilizing a system that is close to the critical coupling condition44,45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Here, the nano-slot metasurface is near the critical-coupling condition with 𝛾𝑒/𝛾𝑖 ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Thus, when the resonance overlaps with the vibrational mode of adsorbed CO at 2033 cm-1 the coupling between the two resonators leads to a small peak in the reflectance spectrum (Figure 1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The idea that coverage effects and different analyte concentration can be sensed using our nano- slot metasurface is shown by changing the thickness of the model molecular layer representing adsorbed CO from 2, to 5, to 10 nm (Figure 1e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' By increasing the thickness of the model molecular layer, a stronger differential absorbance log(𝐼0/𝐼) can be obtained, where 𝐼 and 𝐼0 are the reflectance measured with and without a CO model molecular layer, respectively (Figure 1c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Thus, a decrease in the coverage of an adsorbed material inside the slot can be linked to a decrease in the differential absorbance traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 9 Metasurface characterization First, the effect of the metasurface-driven resonance position on the coupling with COlinear vibration is studied in ATR mode using a focal plan array detector (Figure 2a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' To test our nanophotonic-electrochemical platform, we first tuned the resonance position to match the COlinear vibration mode in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5M K2CO3 saturated with carbon monoxide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Then, we detuned the resonance to the blue and red spectral regions by decreasing and increasing the slot length 𝑙 by 200 nm from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='33 µm, respectively (Figure 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' There is a good fit between the numerically and experimentally obtained resonance positions, with a discrepancy of less than 40 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' As predicted by the simulations, a dip is observed in the resonance attributed to the COlinear vibration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Following these results, slots with a length of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='33 µm were found to match the COlinear vibrational mode on platinum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' According to the literature31–35,46the COlinear mode should be spectrally located between 2020 cm-1 and 2080 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' On the basis of our experiments, COlinear is located at 2033 cm- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The differential absorbance highlights a more intense and well defined COlinear signal for the sample which has the best spectral overlap (Figure 2c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Two peaks can be observed at 2033 and 2086 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The CO signals have a Fano-type line shape due to the narrow discrete COlinear vibration interfering with the broad spectral line of the metasurface-driven resonance47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The redshifted sample yields a highly asymmetric CO signal due to the strongly off-resonance coupling between the resonance and COlinear48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In addition, the redshifted sample presents a strong peak around 1843 cm-1, which is attributed to a second configuration of adsorption, the CO bridge (CObridge)31,32,34,46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The scanning electron microscopy images show good quality of the fabricated nanostructures (Figure 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For the next part of this work, the electrochemical behavior of the sample with matching spectral overlap of its resonance with the COlinear vibrational mode is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 10 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Testing the nanophotonic-electrochemical platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (a) A schematic showing the experimental setup used to perform SEIRAS in an ATR geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A continuous mid-IR collimated linearly polarized light source illuminated the nano-slot metasurface from below at an angle of ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 72º.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A focal plane detector array was used to collect the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (b) Experimental absorbance spectra of three Pt nano-slot metasurface in CO saturated 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5M K2CO3 aqueous electrolyte interfaces with resonance positions around the ideal position for COlinear with slot lengths 𝑙 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='33 µm (black, ideal) and two detuned resonance positions with slots length of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='13 µm (blue) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='53 µm (red).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The other parameters are ℎ = 30 nm, 𝑤 = 200 nm, 𝑝𝑦 = 1440 nm, gap = 𝑝𝑥 − 𝑙 = 220 nm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The resonance position of COlinear vibration is indicated by the black dahsed line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The numerically modelled resonance positions at 2312 cm-1, 2066 cm-1, and 1876 cm-1 (yellow) are shown for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (c) The differential absorbance of the CO signal after baseline correction for the blueshifted, matched, and redshifted resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (d) Scanning electron microscopy images corresponding to the nano-slot metasurfaces used in (b) and (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2 μm Focal plane arraydetector olarizer IR ATRcrystal ource Blueshifted Redshifted Matched Simulated Blueshifted Redshifted Matched 11 CO adsorption at Open Circuit Potential Here, we follow in situ the CO adsorption during the saturation of an electrolyte at open circuit potential (OCP) and characterize the CO adsorption by performing SEIRAS concurrently with electrochemical cyclic voltammetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The transition from the Ar to the CO saturated electrolyte is accompanied by a shift of the OCP due to a change of the equilibrium determining redox reaction (Figure 3a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' At the equilibrium potential of the Ar saturated electrolyte (ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 300 mVSHE) CO is oxidized and the OCP drops towards negative values where CO adsorbs on the Pt surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The SEIRAS measurements were taken in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5M K2CO3 with and without CO (Figure 3b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Looking at the differential absorbance (Figure 3c), a distortion of the base line appears at 2460 seconds (+25 mVSHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Then, after ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2800 seconds (-270 mVSHE) two clearly distinguishable COlinear peaks emerge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' These peaks become more discernible with time as the coverage of adsorbed CO increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' As the intensity of the peaks stabilizes the maximum coverage of CO is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The CO signal obtained with the nano-slot metasurface compared to that obtained with a pure platinum layer (30 nm) at the OCP is increased by an estimated factor of 27 (Figure 3d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Both samples have been evaporated simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' This gives both systems the same material properties such as the surface roughness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For this reason, the 27-fold difference between the signals obtained with the two systems can be directly linked to the metasurfaces-driven enhancement provided by nanostructuring the surface of the working electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For adsorbed CO to interact with incident light, the orientation of the transition dipole moment of the CO vibrational mode relative to the electric field component needs to be non-zero49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Consequently, only the (interior) side walls parallel to the long axis of the slots can be considered 12 active representing a ratio of active to total surface of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='6% compared to a smooth platinum layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' This leads to an experimentally determined local signal enhancement of above 700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The second peak at 2086 cm-1 was only observed using the nano-slot metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The most likely explanation could be that the higher resolution achieved with the nano-slot metasurface allows for the deconvolution of this peak from the background, which was not possible in previous architectures based on a continuous Pt film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' According to the literature, several possibilities exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The first assumption is that CO could adsorbs on different crystal orientations with different binding energies46,50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' As reported by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cuesta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='50, an adsorption on Pt(111) single crystals was found at around 2070 cm-1 38,50,51, while CO adsorbed on Pt(100) electrodes was detected between 2027 cm-1 52,53 and 2050 cm-1 50,54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' These two values are in good agreement with the ones observed here (2086 cm-1 and 2033 cm-1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Another possibility is the adsorption of CO on terraces (higher frequency band at 2086 cm-1), steps and defects (lower frequency band at 2033 cm-1) 31,32,55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 13 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrochemical and spectroscopic response of the nanophotonic platform at the OCP during a gas transition of from an Ar-saturated electrolyte to a CO-saturated one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (a) The evolution of the OCP of the platinum nano-slot metasurface during the transition from an Ar saturated (Arsat around +310 mVSHE) to a CO saturated electrolyte (COsat around -440 mVSHE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (b) FTIR spectra of the Pt nano-slot metasurface/electrolyte interface in Arsat and COsat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The heat map represents the integrated area below the resonance between 2600 and 1800 cm-1 collected by an array of 64 by 64 detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (c) The evolution of the differential absorbance COlinear peaks during the CO bubbling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (d) Comparison of COlinear signals obtained in COsat after 80 min of CO bubbling with a pure Pt layer and with the nanophotonic-electrochemical platform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' OCP In Ar COundetected In CO cO detected Time (s) 4800 Nano-slot 3780 3480 metasurface 3180 Ptfilm(30nm) 2820 2426 2100 1500 840 300 14 CO oxidation on platinum The behavior of the nano-slot metasurface was evaluated during the electrochemical oxidation of carbon monoxide using electrochemical cyclic voltammetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The anodic scan in CO-saturated electrolyte (COsat) presents an initial state with a low current (Figure 4a, black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' When an applied potential of around -150 mVSHE is reached, the current density plateaus at around +25 µA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='cm-2, which is attributed to CO oxidation56,57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' At ca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 450 mVSHE the current density starts to decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The origin of this decrease is still debated in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' One explanation attributes the decreasing current density to competing adsorption of CO and OH on the Pt surface at higher potentials58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Another possibility discussed is that the formation of a thin oxide or hydroxide Pt layer prevents the oxidation of CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The latter assumption is supported by the reduction dip (from +160 to -80 mVSHE) of platinum in Argon saturated electrolyte (Arsat) (Figure 4a-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The behavior of the cathodic scan is similar, except that the onset of CO oxidation is shifted to more negative potentials resulting in a hysteresis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Moreover, a shift in the onset of the hydrogen evolution reaction (HER) in Arsat and COsat electrolytes is observed, highlighting the poisoning behavior of adsorbed CO on the platinum surface59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Similarly to our Fourier-transform infrared (FTIR) measurements in COsat electrolyte under OCP (Figure 3c), two COlinear peaks were also found during the electrochemical potential sweeps (Figure 4c-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' There is a spectral shift during the anodic and cathodic scan (between -650 and - 150 mVSHE) which is attributed to either a higher π -back-donation from the metal to CO40,60 and/or to the Stark effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The Stark effect results from the interactions between the surface electric field and the dipole moment of the adsorbates60–62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' During the anodic scan (Figure 4e), the most intense peak shows a blue-shift of 53 cm-1/V in agreement with the literature38–40,61,63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The second peak shows a blueshift of 33 cm-1/V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Between -50 and +50 mVSHE a redshift is observed which is not 15 well documented in the literature39,55,60,64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The redshift is attributed to a decrease of the CO coverage due to its oxidation into CO2, decreasing the dipole-dipole interactions55,65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The observation of the coverage effect was possible here due to the high resolution reached with the nano-slot metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' It was not resolved with a continuous platinum film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' During the anodic scan, there is a slight increase in the area of the first peak (~2033 cm-1), while the area of the second peak slightly decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' This behavior could be explained by a surface migration of adsorbed CO to a more stable position33,39,40,52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Alternatively, the reconstruction or roughening of the Pt surface with electrical polarization66,67 could lead to a modification of the surface microstructure and CO adsorption energy68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Looking at spectra obtained during the cathodic scan (Figure 4f), the second peak (~2086 cm-1) almost disappeared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' This supports the assumption that the cause is the platinum surface modification at high applied potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' At high cathodic potentials (-550 to -650 mVSHE) a decrease of the CO peak is observed and attributed to the HER63, indicating that the adsorption of hydrogen displaces adsorbed CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 16 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Behavior of the Pt nano-slot metasurface during cyclic voltammetry in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5M K2CO3 saturated with CO at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='25 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Evolution of the current density with the potential during the (a) anodic (from OCP to +1000 mVSHE) and (b) cathodic (from +1000 mVSHE to -700 mVSHE) scan in COsat electrolyte (black line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For comparison, the blue line depicts CVs in an Arsat electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Evolution of SEIRAS spectra with, the (c) anodic and (d) cathodic scans acquired every 100 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Evolution of the position and area of the COlinear peak during the (e) anodic and (f) cathodic scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' COad + OHad CO2 + H+ + e CO+OH CO2 + H+ + e Anodic - COsat CO undetected CO detected Arsat Anodic - COsat CO undetected CO detected Arsat E(mVsHE) +850 +750 +650 +550 +450 +350 +250 +150 +50 50 150 250 350 450 550 650 peak 1 anodic peak 2 anodic peak 1 cathodic peak 2 cathodic 17 Conclusion To the best of our knowledge, we have developed the first hybrid nanophotonic-electrochemical platform for SEIRAS based on a platinum nano-slot metasurface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The resonance of the metasurface was numerically modelled giving a maximum electric near-field intensity enhancement of 570.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The resonance was tuned to couple with and enhance the CO vibrational mode at 2033 cm-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The principle behind the sensing improvement due to the electric near-field enhancement was tested by fabricating on-resonance and detuned metasurfaces and carefully analyzing the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The numerical simulations and SEIRAS experimental results were in good agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Two peaks were resolved for the COlinear mode which could be attributed to adsorption of CO on Pt(111) and Pt(100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' COlinear was best observed with a spectrally overlapping resonance leading to an experimental signal improvement of more than 27 over a conventionally used platinum film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' During the electrochemical oxidation of CO, a classic Stark effect was observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Moreover, thanks to the high resolution provided by the nano-slot metasurface, a redshift of COlinear was observed, linked to a decrease of the coverage of adsorbed CO due to its oxidation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' We anticipate our proof-of-concept nanophotonic-electrochemical platform for SEIRAS to guide new system designs and material combinations suitable to characterize different electrochemical interfaces, reaction products, and short-lived intermediates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Methods Numerical simulations The simulations were performed in CST Studio Suite 2021 using the finite-element frequency- domain Maxwell solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' CaF2 was simulated using a refractive index, n, of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='4, the surrounding medium as water with n ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='33 and platinum using the data given by Rakić et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='69 The inside walls 18 perpendicular to the electric field were covered with a model material to represent the adsorbed COlinear vibrational mode at ~2033 cm-1 (see Supplementary Information for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The titanium adhesion layer was not simulated as including it did not lead to substantial spectral shifts in the resonance position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' An impedance-matched open port with a perfectly matched layer introduced linearly polarized light at an angle of 72º across the CaF2 layer towards the nano-slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' At 72º the light was totally internally reflected at the CaF2-Pt interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Therefore, the boundary opposite the open port was set as perfect electric conductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The unit cell was defined and then simulated as infinite periodic array via Floquet boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A field monitor was placed at the center of the slot in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The highest field enhancement is found slightly above the apex of the slots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The value of the highest field enhancement of the system was evaluated within the volume of the numerical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' To extract the Q-factor and coupling ratio 𝛾𝑒/𝛾𝑖, the simulated resonance was fitted in reflectance (Figure 1d, blue curve) using temporal coupled mode theory according to Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='70 Metasurface fabrication CaF2 was selected as the substrate due to its transparent nature in the mid-IR spectral range, low solubility, and high chemical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The measurements shown in Figures 3 and 4 used metasurface arrays were with at least 2200 by 2700 unit cells resulting in a pattern area of approximately 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='3 mm2, which ensured that there are more than enough unit cells for the measured resonance to correspond to the mode of the infinite periodic array used for the numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' After sample cleaning (acetone bath in an ultrasonic cleaner followed by oxygen plasma cleaning) the substrate was spin-coated first with an adhesion promoter (Surpass 4000), then with a layer of negative tone photoresist (ma-N 2403) which was baked at 100ºC for 60s, and finally with a conducting layer (ESpacer 300Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The metasurface patterns were written via 19 electron-beam lithography (Raith Eline Plus) with an acceleration voltage of 30 kV and an aperture of 20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The exposed resist was developed in ma-D 525 for 70s at room temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The patterned surface was then coated with a titanium adhesion layer (1 nm at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='4 Å/s) and a platinum film (30 nm at 2 Å/s) using electron-beam evaporation (PRO Line PVD 75, Lesker).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Finally, an overnight lift-off in mr-REM 700 concluded the top-down fabrication process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A pure 30 nm thick platinum film on 1 nm titanium on CaF2 functioned as reference for the in-situ SEIRAS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In-situ SEIRAS and electrochemical measurements SEIRAS was performed using a Vertex 80 coupled with an IMAC chamber from Bruker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Each sample was mounted on a VeeMax III (purged with N2) from PIKE Technologies in attenuated total internal reflection (ATR) mode with a light polarizer, an electrochemical Jackfish cell and a CaF2 prism bevelled at 72°.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A classical three electrode system was used with a Saturated Calomel Electrode (E= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='244 VSHE), a platinum wire as counter electrode and the platinum sample as working electrode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The IMAC chamber is equipped with a focal plane array detector composed of 64 x 64 MCT-detectors (a total of 4096 detectors), which allows to perform a mapping of the studied sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Each detector collects its own spectrum and then, the active slot covered area is detected by integration of each spectrum between 1600 to 2800 cm-1(Figure 3b, inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Finally, an average can be determined using the spectra of the detectors that probed the resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A baseline correction is applied to this average as well as a Savitzky–Golay filter to smoothen the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For the characterization of the resonance, its position was determined using three samples composed of arrays with different nanostructure sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' For each sample, the resonance was measured in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5M K2CO3 electrolyte saturated with Ar and then saturated with CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Prior to the 20 first characterization, a cyclic voltammogram (20 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='s-1) was recorded in order to confirm the cleanliness of the electrode surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Then, an initial background was acquired using p-polarized light and the Fano resonance was characterized using s-polarized light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Each spectrum was recorded with a resolution of 4 cm-1 and the final mapping results from a collection of 32 scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The enhancement of the nano-slot metasurface is obtained by comparison of the COlinear vibrational mode on a pure Pt layer (30nm) without nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The adsorption of CO during the transition from Arsat to COsat electrolyte (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5M K2CO3) was studied using the nano-slot metasurface with the best overlapping resonance with the COlinear vibration mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cleaning and background acquisition protocol were the same as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Carbon monoxide was slowly flowed into the electrochemical cell and spectra were acquired regularly during the transition from Arsat to COsat at the OCP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The oxidation of CO during potential sweeps was investigated after 2 hours of CO bubbling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A cyclic voltammogram, with a slow scan rate (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='25 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='s-1), from the OCP to +1000 mVSHE and back to -760 mVSHE was performed and a spectrum was acquired every 100 mV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' References (1) Lagadec, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Grimaud, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Water Electrolysers with Closed and Open Electrochemical Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2020, 19 (11), 1140–1150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/s41563-020-0788-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (2) Sullivan, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Goryachev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Digdaya, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Atwater, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Vermaas, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Coupling Electrochemical CO2 Conversion with CO2 Capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2021, 4 (11), 952– 958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/s41929-021-00699-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (3) Stephens, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Bagger, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Boettcher, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Bonin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Boutin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Buckley, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Buonsanti, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cave, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Silva, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' da;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Luna, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Einsle, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Endrődi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Escudero-Escribano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Araujo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Figueiredo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hahn, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hansen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Haussener, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hunegnaw, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Huo, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hwang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Janáky, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jayathilake, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jiao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jovanov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Karimi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Koper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kuhl, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ma, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ma, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Oh, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Robert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cuenya, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rossmeisl, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Roy, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ryan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sargent, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sebastián-Pascual, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Seger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Steier, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Strasser, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Varela, 21 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Vos, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yadegari, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022 Roadmap on Low Temperature Electrochemical CO2 Reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Energy 2022, 4 (4), 042003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1088/2515-7655/ac7823.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (4) Zhao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Opportunities and Challenges in CO 2 Reduction by Gold- and Silver- Based Electrocatalysts: From Bulk Metals to Nanoparticles and Atomically Precise Nanoclusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Energy Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2018, 3 (2), 452–462.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acsenergylett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='7b01104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (5) Wuttig, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yaguchi, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Motobayashi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Osawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surendranath, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Inhibited Proton Transfer Enhances Au-Catalyzed CO 2 -to-Fuels Selectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2016, 113 (32), E4585–E4593.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1073/pnas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1602984113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (6) Hammes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Spectroscopy for the Biological Sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' John Wiley & Sons, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (7) Bernath, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Spectra of Atoms and Molecules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Oxford University Press, USA, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (8) Lotti, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hamm, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kraack, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface-Sensitive Spectro-Electrochemistry Using Ultrafast 2D ATR IR Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2016, 120 (5), 2883–2892.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='6b00395.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (9) Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Low, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' García de Abajo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Avouris, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Dai, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nanomaterial‐Based Plasmon‐Enhanced Infrared Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2018, 30 (20), 1704896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1002/adma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='201704896.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (10) Neubrech, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Huck, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Weber, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pucci, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Giessen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface-Enhanced Infrared Spectroscopy Using Resonant Nanoantennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2017, 117 (7), 5110–5145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='chemrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='6b00743.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (11) Ribeiro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sales, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pereira, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrochemistry Combined-Surface Plasmon Resonance Biosensors: A Review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' TrAC Trends Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 157, 116766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='trac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='116766.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (12) Vijay, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hogg, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ehlers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kristoffersen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Katayama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shao Horn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chorkendorff, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Seger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Interaction of CO with Gold in an Electrochemical Environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2021, 125 (32), 17684–17689.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1c04013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (13) Katayama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nattino, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Giordano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hwang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Andreussi, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Marzari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shao-Horn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' An In Situ Surface-Enhanced Infrared Absorption Spectroscopy Study of Electrochemical CO2 Reduction: Selectivity Dependence on Surface C-Bound and O-Bound Reaction Intermediates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2019, 123 (10), 5951–5963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='8b09598.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (14) Dunwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Heyes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rosen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jiao, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The Central Role of Bicarbonate in the Electrochemical Reduction of Carbon Dioxide on Gold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2017, 139 (10), 3774–3783.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='6b13287.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (15) Wang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jose, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Grzeschik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Schlücker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In Situ Monitoring of Palladium-Catalyzed Chemical Reactions by Nanogap-Enhanced Raman Scattering Using Single Pd Cube Dimers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 144 (11), 5003– 5009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jacs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1c13240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (16) Rodrigo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tittl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ait-Bouziad, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' John-Herpin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Limaj, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kelly, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yoo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wittenberg, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Oh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lashuel, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Altug, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Resolving Molecule-Specific Information in Dynamic Lipid Membrane Processes with Multi-Resonant Infrared Metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2018, 9 (1), 2160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/s41467-018-04594-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 22 (17) Tittl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Leitis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yesilkoy, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Choi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Neshev, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kivshar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Altug, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Imaging-Based Molecular Barcoding with Pixelated Dielectric Metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Science 2018, 360 (6393), 1105–1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1126/science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='aas9768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (18) Hosseini Farahabadi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Entezami, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Abouali, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Amarloo, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Poudineh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Safavi-Naeini, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sub-Terahertz Silicon-Based on-Chip Absorption Spectroscopy Using Thin- Film Model for Biological Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 12 (1), 17747.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/s41598-022-21015-8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (19) Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wei, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Two- Dimensional Material-Based Electrochemical Sensors/Biosensors for Food Safety and Biomolecular Detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Biosensors 2022, 12 (5), 314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='3390/bios12050314.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (20) Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tittl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Trends in Nanophotonics-Enabled Optofluidic Biosensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 10 (7), 2102366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1002/adom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='202102366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (21) Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ding, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Meng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Qi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Plasmonic Nano-Arrays for Ultrasensitive Bio-Sensing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nanophotonics 2018, 7 (9), 1517– 1531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1515/nanoph-2018-0023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (22) Li, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Li, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Monitoring the Electrochemical Responses of Neurotransmitters through Localized Surface Plasmon Resonance Using Nanohole Array.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Biosens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Bioelectron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2017, 93, 241–249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='bios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (23) Nakamoto, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kurita, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Niwa, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrochemical Surface Plasmon Resonance Measurement Based on Gold Nanohole Array Fabricated by Nanoimprinting Technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2012, 84 (7), 3187–3191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/ac203160r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (24) Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ren, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Fundamental Understanding and Applications of Plasmon-Enhanced Raman Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2020, 2 (5), 253– 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/s42254-020-0171-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (25) Wright, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Berta, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Földes, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wagner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Griffiths, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Readman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rosta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Reisner, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Baumberg, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mechanistic Study of an Immobilized Molecular Electrocatalyst by in Situ Gap-Plasmon-Assisted Spectro-Electrochemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2021, 4 (2), 157–163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/s41929-020-00566-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (26) Fang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Plasmonic Imaging of Electrochemical Reactions of Single Nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Acc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2016, 49 (11), 2614– 2624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='accounts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='6b00348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (27) Cheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wallace, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lagugné-Labarthet, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kerman, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Au Nanostructured Surfaces for Electrochemical and Localized Surface Plasmon Resonance-Based Monitoring of α-Synuclein–Small Molecule Interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Interfaces 2015, 7 (7), 4081– 4088.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/am507972b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (28) Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Younis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xia, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Direct Plasmon- Enhanced Electrochemistry for Enabling Ultrasensitive and Label-Free Detection of Circulating Tumor Cells in Blood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2019, 91 (7), 4413–4420.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='analchem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='8b04908.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (29) Cortés, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wendisch, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sortino, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mancini, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ezendam, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Saris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Menezes, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' de S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tittl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Optical Metasurfaces for Energy Conversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 122 (19), 15082–15176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='chemrev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2c00078.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (30) Stefancu, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nan, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chiș, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Bald, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Liu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Leopold, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cortes, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Controlling Plasmonic Chemistry Pathways through Specific Ion Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 10 (14), 2200397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1002/adom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='202200397.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 23 (31) Hossain, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rahman, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jafar Sharif, Md.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Preference for Low-Coordination Sites by Adsorbed CO on Small Platinum Nanoparticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nanoscale Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2020, 2 (3), 1245–1252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1039/C9NA00499H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (32) Silva, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cabello, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Christinelli, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pereira, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cuesta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Simultaneous Time- Resolved ATR-SEIRAS and CO-Charge Displacement Experiments: The Dynamics of CO Adsorption on Polycrystalline Pt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electroanal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2017, 800, 25–31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jelechem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (33) McPherson, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ash, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jones, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Varambhia, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jacobs, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Vincent, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrochemical CO Oxidation at Platinum on Carbon Studied through Analysis of Anomalous in Situ IR Spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2017, 121 (32), 17176–17187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='7b02166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (34) Susarrey-Arce, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tiggelaar, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gardeniers, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' van Houselt, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lefferts, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' CO Adsorption on Pt Nanoparticles in Low E-Fields Studied by ATR-IR Spectroscopy in a Microreactor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2015, 119 (44), 24887–24894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5b08392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (35) Smolinka, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Heinen, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jusys, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lehnert, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Behm, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' CO2 Reduction on Pt Electrocatalysts and Its Impact on H2 Oxidation in CO2 Containing Fuel Cell Feed Gas – A Combined in Situ Infrared Spectroscopy, Mass Spectrometry and Fuel Cell Performance Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrochimica Acta 2005, 50 (25–26), 5189–5199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='electacta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (36) Katayama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Giordano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hwang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Muroyama, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Matsui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Eguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shao-Horn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface (Electro)Chemistry of CO2 on Pt Surface: An in Situ Surface- Enhanced Infrared Absorption Spectroscopy Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2018, 122 (23), 12341– 12349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='8b03556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (37) Huck, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Vogt, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sendner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hengstler, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Neubrech, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pucci, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Plasmonic Enhancement of Infrared Vibrational Signals: Nanoslits versus Nanorods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Photonics 2015, 2 (10), 1489–1497.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acsphotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5b00390.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (38) Villegas, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Weaver, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Carbon Monoxide Adlayer Structures on Platinum (111) Electrodes: A Synergy between In‐situ Scanning Tunneling Microscopy and Infrared Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1994, 101 (2), 1648–1660.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='467786.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (39) Watanabe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Inukai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Coverage and Potential Dependent CO Adsorption on Pt( 111 l), (711) and (100) Electrode Surfaces Studied by Infrared Reflection Absorption Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2020, 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (40) Nakamura, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ogasawara, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Inukai, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ito, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' CO Migration on Pt(100) and Pt(11 1 1) Surfaces Studied by Time Resolved Infrared Reflection-Absorption Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1993, 283 (1–3), 248–254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/0039-6028(93)90989-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (41) Hickman, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Analog Electronics: Analog Circuitry Explained;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Newnes, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (42) Gao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' McMahon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Henzie, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gray, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Schatz, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Odom, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rayleigh Anomaly-Surface Plasmon Polariton Resonances in Palladium and Gold Subwavelength Hole Arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Express 2009, 17 (4), 2334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1364/OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='002334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (43) Adato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yanik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shvets, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Altug, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Radiative Engineering of Plasmon Lifetimes in Embedded Nanoantenna Arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Express 2010, 18 (5), 4526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1364/OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='004526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 24 (44) Adato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Artar, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Erramilli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Altug, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Engineered Absorption Enhancement and Induced Transparency in Coupled Molecular and Plasmonic Resonator Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2013, 13 (6), 2584–2591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/nl400689q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (45) Aigner, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tittl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Weber, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kivshar, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ren, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Plasmonic Bound States in the Continuum to Tailor Light-Matter Coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2022, 8 (49), eadd4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1126/sciadv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='add4816.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (46) Farias, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Busó-Rogero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tanaka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Herrero, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Feliu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Monitoring of CO Binding Sites on Stepped Pt Single Crystal Electrodes in Alkaline Solutions by in Situ FTIR Spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Langmuir 2020, 36 (3), 704–714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='langmuir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='9b02928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (47) Luk’yanchuk, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zheludev, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Halas, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nordlander, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Giessen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chong, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The Fano Resonance in Plasmonic Nanostructures and Metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2010, 9 (9), 707–715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1038/nmat2810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (48) Fan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sharp Asymmetric Line Shapes in Side-Coupled Waveguide-Cavity Systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2002, 80 (6), 908–910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1448174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (49) Ras, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Schoonheydt, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Johnston, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Relation between S-Polarized and p- Polarized Internal Reflection Spectra: Application for the Spectral Resolution of Perpendicular Vibrational Modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A 2007, 111 (36), 8787–8791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jp073108a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (50) Cuesta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Couto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rincón, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pérez, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' López-Cudero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gutiérrez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Potential Dependence of the Saturation CO Coverage of Pt Electrodes: The Origin of the Pre-Peak in CO-Stripping Voltammograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Part 3: Pt(Poly).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electroanal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2006, 586 (2), 184– 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jelechem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (51) López-Cudero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cuesta, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gutiérrez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Potential Dependence of the Saturation CO Coverage of Pt Electrodes: The Origin of the Pre-Peak in CO-Stripping Voltammograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Part 1: Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electroanal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2005, 579 (1), 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jelechem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (52) Sun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Zhou, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface Processes and Kinetics of CO2 Reduction on Pt(100) Electrodes of Different Surface Structure in Sulfuric Acid Solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2001, 3 (16), 3277–3283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1039/b100938i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (53) Rodes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pastor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Iwasita, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Structural Effects on CO2 Reduction at Pt Single-Crystal Electrodes: Part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Pt(100) and Related Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electroanal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1994, 377 (1), 215– 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/0022-0728(94)03424-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (54) López-Cudero, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cuesta, Á.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gutiérrez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Potential Dependence of the Saturation CO Coverage of Pt Electrodes: The Origin of the Pre-Peak in CO-Stripping Voltammograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Part 2: Pt(100).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electroanal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2006, 586 (2), 204–216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jelechem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (55) Chang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Weaver, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In-Situ Infrared Spectroscopy of CO Adsorbed at Ordered Pt(110)-Aqueous Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1990, 230 (1), 222–236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1016/0039- 6028(90)90030-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (56) Blizanac, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lucas, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Gallagher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Arenz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ross, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Marković, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Anion Adsorption, CO Oxidation, and Oxygen Reduction Reaction on a Au(100) Surface: The PH Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' B 2004, 108 (2), 625–634.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jp036483l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (57) Mayet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Servat, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kokoh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Napporn, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrochemical Oxidation of Carbon Monoxide on Unsupported Gold Nanospheres in Alkaline Medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Electrocatalysis 2021, 12 (1), 26–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1007/s12678-020-00626-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 25 (58) Spendelow, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Goodpaster, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kenis, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wieckowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Mechanism of CO Oxidation on Pt(111) in Alkaline Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' B 2006, 110 (19), 9545–9555.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jp060100c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (59) Varela, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Schlaup, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Jovanov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Malacrida, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Horch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Stephens, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chorkendorff, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' CO2 Electroreduction on Well-Defined Bimetallic Surfaces: Cu Overlayers on Pt(111) and Pt(211).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2013, 117 (40), 20500–20508.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jp406913f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (60) Samjeské, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Komatsu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Osawa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Dynamics of CO Oxidation on a Polycrystalline Platinum Electrode: A Time-Resolved Infrared Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2009, 113 (23), 10222–10228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jp900582c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (61) Katayama, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Giordano, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rao, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hwang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Muroyama, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Matsui, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Eguchi, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Shao-Horn, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface (Electro)Chemistry of CO 2 on Pt Surface: An in Situ Surface- Enhanced Infrared Absorption Spectroscopy Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C 2018, 122 (23), 12341– 12349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='jpcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='8b03556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (62) Stamenkovic, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Somorjai, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Ross, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Markovic, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Vibrational Properties of CO at the Pt(111)−Solution Interface: The Anomalous Stark-Tuning Slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' B 2005, 109 (2), 678–680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/jp044802i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (63) Dunwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Surface Enhanced Spectroscopic Investigations of Adsorption of Cations on Electrochemical Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2017, 19 (2), 971–975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1039/C6CP07207K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (64) Dunwell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Yan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Xu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In Situ Infrared Spectroscopic Investigations of Pyridine- Mediated CO 2 Reduction on Pt Electrocatalysts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2017, 7 (8), 5410–5419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acscatal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='7b01392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (65) Deshlahra, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Conway, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wolf, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Schneider, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Influence of Dipole–Dipole Interactions on Coverage-Dependent Adsorption: CO and NO on Pt(111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Langmuir 2012, 28 (22), 8408–8417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/la300975s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (66) Jacobse, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Rost, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Koper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Atomic-Scale Identification of the Electrochemical Roughening of Platinum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Cent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2019, 5 (12), 1920–1928.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acscentsci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='9b00782.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (67) Deng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Galli, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Koper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' In Situ AFM Imaging of Platinum Electrode Surface during Oxidation–Reduction Cycles in Alkaline Electrolyte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Energy Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2020, 3 (1), 597–602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acsaem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='9b01700.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (68) Hammer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nielsen, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Nrskov, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Structure Sensitivity in Adsorption: CO Interaction with Stepped and Reconstructed Pt Surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Catal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1997, 46 (1), 31–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1023/A:1019073208575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (69) Rakić, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Djurišić, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Elazar, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Majewski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Optical Properties of Metallic Films for Vertical-Cavity Optoelectronic Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 1998, 37 (22), 5271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1364/AO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='005271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (70) Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Weber, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Bienek, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Wester, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hüttenhofer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Sharp, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Tittl, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cortés, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Catalytic Metasurfaces Empowered by Bound States in the Continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ACS Nano 2022, 16 (8), 13057–13068.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1021/acsnano.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='2c05680.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 26 Acknowledgments We thank Thomas Weber for his help with the temporal-coupled mode theory algorithms and Simon Stork for the platinum and titanium electron beam evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Corresponding Author Andreas Tittl - Chair in Hybrid Nanosystems, Nanoinstitute Munich, Faculty of Physics, Ludwig- Maximilians-Universität München, Königinstraße 10, 80539 München, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Orcid: https://orcid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/0000-0003-3191-7164;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Email: Andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='Tittl@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='uni-muenchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='de Katharina Krischer - Department of Physics, Technical University of Munich, 85748 Garching, Germany;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Email: Krischer@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='de Author Contributions The manuscript was written through the contributions of all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' All authors have given approval to the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' ‡These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Funding Sources This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under grant numbers EXC 2089/1–390776260 (Germany′s Excellence Strategy) and TI 1063/1 (Emmy Noether Program), ERC-STG 802989 Catalight, the Bavarian program Solar Energies Go Hybrid (SolTech) and the Center for NanoScience (CeNS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' additionally acknowledges the Lee-Lucas Chair in Physics and the EPSRC (EP/W017075/1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Notes The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Supporting information Improved in-situ characterization of electrochemical interfaces using metasurface-driven surface-enhanced infrared absorption spectroscopy Luca M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Berger 1, ‡, Malo Duportal 2, ‡, Leonardo de Souza Menezes 1,3, Emiliano Cortés 1, Stefan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Maier 4,5,1, Andreas Tittl 1,*, Katharina Krischer 2,* 1 Faculty of Physics, Ludwig-Maximilian-University Munich, 80539 München, Germany 2 Department of Physics, Technical University of Munich, 85748 Garching, Germany 3 Departamento de Física, Universidade Federal de Pernambuco, 50670-901 Recife-PE, Brazil 4 School of Physics and Astronomy, Monash University, Melbourne, Victoria, Australia 5 Department of Physics, Imperial College London, SW7 2AZ London, United Kingdom e-mail: Andreas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='Tittl@physik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='uni-muenchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='de;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' krischer@tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='de Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The real (blue) and imaginary (red) parts of the permittivity of the artificial material modeled to represent CO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' The permittivity of a material suffices to numerically model its interaction with light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Using the Lorentz oscillator model, the real and imaginary parts of the permittivity can be written as 𝜀𝑟 = 1 + 𝜔𝑝2(𝜔02−𝜔2) (𝜔02−𝜔2) 2+𝛾2𝜔2 and 𝜀𝑖 = 𝛾𝜔𝑝2𝜔 (𝜔02−𝜔2) 2+𝛾2𝜔2, respectively1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Here, we modified the baseline of 𝜀𝑟 to be around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='332 which is the refractive index squared used for the surrounding medium (water) due to a negligible shift in the refractive index due to CO2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Here, 𝜔𝑝 is analogous to the plasma frequency in the Drude-Sommerfeld model, 𝜔0 is the frequency of the absorption band, 𝜔 is the frequency, and 𝛾 is the damping constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' To model the linear vibrational mode of CO at 2033 cm-1 the parameters used were 𝜔𝑝 2 = 1 × 1024 s-2, 𝜔0 = 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='95 × 1012 s-1 (corresponding to 2033 cm-1), and 𝛾 = 1 × 1011 s-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Re[ε] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5 Im[] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5- m 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='5 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='0 2010 2015 2020 2025 2030 2035 2040 2045 2050 2055 Wavenumber (cm-1)References (1) Novotny, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Hecht, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Principles of Nano-Optics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Cambridge University Press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' (2) Harvey, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Kaplan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Burnett, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Effect of Dissolved Air on the Density and Refractive Index of Water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' Thermophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' 2005, 26 (5), 1495–1514.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} +page_content='1007/s10765-005-8099-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNA0T4oBgHgl3EQfCv9N/content/2301.01993v1.pdf'} diff --git a/KdE0T4oBgHgl3EQfSQCv/content/tmp_files/2301.02220v1.pdf.txt b/KdE0T4oBgHgl3EQfSQCv/content/tmp_files/2301.02220v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..db27c29e6de5da7c0e6f2750ae6a3522f33eca15 --- /dev/null +++ b/KdE0T4oBgHgl3EQfSQCv/content/tmp_files/2301.02220v1.pdf.txt @@ -0,0 +1,4571 @@ +arXiv:2301.02220v1 [stat.ML] 5 Jan 2023 +Value Enhancement of Reinforcement +Learning via Efficient and Robust Trust +Region Optimization +Chengchun Shia∗, Zhengling Qib∗†, Jianing Wangc and Fan Zhouc† +aDepartment of Statistics, London School of Economics and Political Science +bDepartment of Decision Sciences, The George Washington University +cDepartment of Statistics, Shanghai University of Finance and Economics +Abstract +Reinforcement learning (RL) is a powerful machine learning technique that en- +ables an intelligent agent to learn an optimal policy that maximizes the cumulative +rewards in sequential decision making. Most of methods in the existing literature are +developed in online settings where the data are easy to collect or simulate. Motivated +by high stake domains such as mobile health studies with limited and pre-collected +data, in this paper, we study offline reinforcement learning methods. To efficiently +use these datasets for policy optimization, we propose a novel value enhancement +method to improve the performance of a given initial policy computed by existing +state-of-the-art RL algorithms. Specifically, when the initial policy is not consistent, +our method will output a policy whose value is no worse and often better than that of +the initial policy. When the initial policy is consistent, under some mild conditions, +our method will yield a policy whose value converges to the optimal one at a faster +rate than the initial policy, achieving the desired “value enhancement” property. The +proposed method is generally applicable to any parametrized policy that belongs to +certain pre-specified function class (e.g., deep neural networks). Extensive numerical +studies are conducted to demonstrate the superior performance of our method. +Keywords: Offline reinforcement learning; Trust region optimization; Semi-parametric effi- +ciency; Mobile health studies. +∗The first two authors contribute equally to this paper. +†qizhengling@gwu.edu;zhoufan@mail.shufe.edu.cn +1 + +1 +Introduction +Reinforcement learning (RL, see e.g., Sutton and Barto, 2018) is concerned with how agents +take sequential actions in dynamic environments, with the main goal of maximizing the +cumulative rewards they receive. In recent years, we have seen tremendous achievements +of RL in artificial intelligence (AI). For example, AlphaGo (Silver et al., 2016), one of the +most successful applications in AI, makes use of reinforcement learning and deep learn- +ing algorithms for teaching machines to play the board game called Go, and has beaten +many top human players. The appealing performance of RL has also been demonstrated +in many scientific fields. +In medical applications, RL has been used to help clinicians +make better treatment decisions for patients with sepsis (Komorowski et al., 2018). In eco- +nomics, econometricians often study dynamic discrete choice models (Rust, 1987) in order +to understand the behavior of rational agents, which is similar to the inverse RL problem +(Abbeel and Ng, 2004). In operations research, RL has been widely applied to business +operations such as supply chain management, finance and logistics (Hubbs et al., 2020). +For an overview of various applications of RL, we refer to Section 5 of Li (2017). +Our research in this paper is partly motivated by recently emerging mobile health +(mHealth) studies. Advancements in mobile and sensor technologies provides us with a +unique opportunity to deliver health interventions at anytime and anywhere for promoting +healthy behaviors such as regular physical activities and preventing drug abuse, etc. For +example, the OhioT1DM dataset (Marcolino et al., 2018) was developed for promoting +blood glucose level prediction in order to improve the health and wellbeing of people with +type 1 diabetes. It contains data information of 6 people for 8 weeks. For each patient, +their treatment information was collected during insulin pump therapy with continuous +glucose monitoring (CGM). In addition, blood glucose levels and self-reported times of +meals and exercises were also constantly measured and recorded via a custom smartphone +app. Finding an optimal insulin pumping policy for each patient at different scenarios may +potentially improve their health status (Shi, Zhang, Lu and Song, 2020). This matches the +goal of RL algorithms. +A fundamental question we aim to investigate here is how to learn an optimal policy +efficiently from the batch data in high-stake domains such as mHealth. Solving this question +faces at least two major challenges. First, different from the standard clinical trial data, +2 + +mobile health data usually consist of a large number of decision points for each patient but +the number of patients may be limited (e.g., in OhioT1DM dataset, 6 patients with a few +thousands decision points). This posits a unique challenge for searching an optimal policy. +In statistics, there is a rich literature in studying dynamic treatment regimes (DTR, see +e.g. Murphy, 2003; Chakraborty and Moodie, 2013; Qian and Murphy, 2011; Zhao et al., +2015; Shi et al., 2018; Wang et al., 2018). For a review of DTR, see Laber et al. (2014), +Kosorok and Laber (2019) and Tsiatis et al. (2019). However, these methods are mainly +designed for only a few treatment decision points and often require a large number of +patients in the observed data in order to be consistent. +Second, different from online RL domains such as video games, where actively interact- +ing with the environment is feasible and data are easy to generate or simulate, in some high +stake domains, data are often pre-collected according to some experimental design and very +limited. With such limited data, it is essential to study how to efficiently learn the optimal +policy from the batch data. We remark that the main focus of RL in the computer science +literature is for online learning. Among all the methods available, Q-learning is arguably +the most popular model-free RL algorithms (Watkins and Dayan, 1992). It derives the op- +timal policy by learning an optimal Q-function (see the definition of Q-function in Section +2.2). Follow this line of research, variants of Q-learning methods have been proposed in- +cluding the fitted Q-iteration (FQI, Ernst et al., 2005; Fan et al., 2020a), deep Q-network +(DQN, Mnih et al., 2015), among many others. Policy-based learning is another class of RL +algorithms that searches the optimal policy among a parametrized policy class. Some pop- +ular algorithms include REINFORCE and actor-critic methods (see e.g., Sutton and Barto, +2018, Chapter 13). Since these algorithms are primarily motivated by the application of de- +veloping artificial intelligence in online video games, their generalization to offline settings +such as mobile health applications remain largely unexplored. +To address the first challenge, we model the observed data by a time-homogeneous +Markov decision process (MDP Puterman, 1994). This framework is particularly suitable +to model the data collected from mobile health studies where the total number of decision +points are often large (see e.g., Liao et al., 2019, 2020). The assumptions of Markov and +time homogeneity enable a consistent estimation of the optimal policy even with only a +few patients. +To address the second challenge, we develop a novel procedure to derive the optimal +3 + +policy. Recently, a few algorithms have been developed in the statistics literature for pol- +icy optimization in mHealth applications (Ertefaie and Strawderman, 2018; Luckett et al., +2020; Liao et al., 2020; Hu et al., 2021). In particular, Liao et al. (2020) proposed a statis- +tically efficient batch policy learning method under the average reward MDP. However, due +to the policy dependent structure of nuisance functions such as Q-function and the marginal +density ratio, their proposed algorithm is computational inefficient as it requires updating +the nuisance functions estimation in each iteration of their policy gradient decent algo- +rithm. Instead of proposing a specific algorithm for policy optimization, we devise a “value +enhancement” method that is generally applicable to any given initial policy computed by +some state-of-the-art RL algorithm to improve their performance. Basically, after employ- +ing some computational efficient RL algorithm and obtaining an initial policy, we take a +one-step update of this policy via efficiently estimating the value enhancement component +(defined in Section 2.3) and solving a constrained optimization problem, thus taking advan- +tage of computational efficiency from existing state-of-art RL algorithms without requiring +iteratively updating the nuisance functions. More importantly, the proposed procedure +guarantees that when the initial policy is not consistent, the output policy by the proposed +algorithm is no worse and often better than the initial policy. If consistent, our method +will yield a policy whose value converges to the optimal one at a faster rate, achieving +the desired “value enhancement” property. Recently, in the computer science literature, +Kallus and Uehara (2020) developed an offline policy gradient algorithm that considered +statistically efficient estimation of the policy gradient. Our proposal differs from theirs in +that we focus on developing a general value enhancement tool that is applicable to any +existing RL algorithms to improve their performances. +Our method is inspired by Lemma 6.1 in Kakade and Langford (2002) and the trust +region policy optimization algorithm by Schulman et al. (2015), which was originally de- +signed for the online setting. A key observation is that, the value difference between any +two policies can be decomposed into a first-order component and a higher-order remainder +term. The higher-order term can be lower bounded, based on which a minorization func- +tion can be constructed for the value function of any policy. One big advantage of working +with this minorization function is that it intrinsically disentangles the policy-dependent +structure of nuisance functions. This ensures the computational efficiency of the proposed +algorithm. +4 + +The key “value enhancement” property relies crucially on statistically efficient estima- +tion of the first-order term in the decomposition. In online settings, Schulman et al. (2015) +proposed to simulate data trajectories to estimate this quantity. However, in offline set- +tings, it remains unknown how to effectively evaluate this quantity based on the observed +data. By leveraging semi-parametric statistics, we develop a triply robust estimator for the +first-order term that is shown to achieve the efficiency bound when compared to the initial +policy. By optimizing the proposed estimator, we are able to improve the value of the initial +policy. The triply robustness property guarantees that the “value enhancement” property +holds even when some nuisance function models are misspecified. The semi-parametric ef- +ficiency guarantees that the value can be enhanced at a sufficiently fast rate. This ensures +the statistical efficiency of the proposed algorithm, which is necessary in the offline setting. +In theory, we establish the value enhancement property under mild conditions on the +nuisance function estimators. In particular, we only require them to converge at a non- +parametric rate. See Section 4 for details. This nice property is achieved mainly due to +the innovative way we put together these nuisance function estimators, which leads to the +triply-robust estimator with a parametric convergence rate for the first-order term. In ad- +dition, we remark that all our theoretical results related to estimation are established in +terms of total decision points, thus showing the proposed method is generally applicable +even when the number of trajectories is small but the length of each trajectory is large, +which is commonly seen in the mobile health applications. +The rest of this paper is organized as follows. In Section 2, we introduce the offline +RL problem in the framework of a time-homogeneous MDP and review the trust region +algorithm. In Section 3, we present our value enhanced policy optimization method and +the related estimation. In Section 4, we study statistical properties of our algorithm. In +Section 5, extensive numerical studies including a toy example demonstrating the value +enhancement property, a real-data driven simulation study and a real data application are +conducted to demonstrate the superior performance of the proposed method. Finally, we +conclude our paper in Section 6. All technical proofs and details can be found in the online +supplementary material. +5 + +2 +Preliminaries +This section is organised as follows. +We first introduce the offline data structure and +describe the model setup in Section 2.1. +In Section 2.2, we introduce some notations +needed to derive our method. In Section 2.3, we review the trust region policy optimization +(TRPO) method proposed by Schulman et al. (2015) for online learning, as it is closely +related to our approach. +2.1 +Value function and the optimal policy +Consider a single trajectory {(St, At, Rt)}t≥0 where (St, At, Rt) denotes the state-action- +reward triplet collected at time t. We use S and A to denote the state and action space, +respectively. We assume S and A are discrete, and rewards Rt are uniformly bounded. +The discrete state space assumption is imposed only to simplify the presentation and the +theoretical analysis. Our proposed method is equally applicable to settings with continuous +state space as well. +The observed data consist of N trajectories, corresponding to N +independent and identically distributed copies of {(St, At, Rt)}t≥0. For any i = 1, · · · , N, +data collected from the ith trajectory can be summarized by {(Si,t, Ai,t, Ri,t, Si,t+1)}0≤t 0. The first inequality in (7) implies that |η2(πnew, πold)| is +indeed a second-order term. Based on this observation, we consider a policy optimization +procedure by maximizing a lower bound of (5), given by +πnew ∈ argmaxπ∈Π [η1(π, πold) − c∗ES∗∼dπold,νDKL (πold(•|S∗), π(•|S∗))]. +(8) +Iteratively solving the above optimization yields a type of minorization-maximization (MM) +algorithm (Hunter and Lange, 2004) as we can see when π = πold, the objective function +in (8) becomes 0 and (1 − γ)V(π) becomes (1 − γ)V(πold). So one can guarantee that (5) is +always nonnegative after optimization. Therefore this type of algorithm can greatly reduce +the computational cost by circumventing computing dπnew,ν and meanwhile monotonically +improve the integrated value function. However, in practice, it may be hard to robustly +10 + +choose the penalty coefficients c∗ in (8). +To resolve this issue, one can consider itera- +tively solving the following equivalent optimization problem with a so-called trust region +constraint: +πnew ∈ argmaxπ∈Π η1(π, πold) +subject to ES∗∼dπold,νDKL (πold(•|S∗), π(•|S∗)) ≤ δ, +(9) +for some constant δ > 0. This yields the TRPO algorithm. +3 +Value Enhanced Policy Optimization +In this section, we first present the motivation of our method. To implement TRPO, we +need an estimate for η1(π, πold). In online settings, Schulman et al. (2015) proposed to +simulate trajectories following the policy πold to estimate η1(π, πold). In offline settings, +it remains unknown how to effectively evaluate this quantity based on the observed data. +By its definition, we note that η1 depends on the nuisance functions Aπold and dπold,ν. A +naive method is to first estimate these quantities (denote by �A and �dν) and then use the +corresponding plug-in estimators �η1 = ES∗∼ �dν +� +a∈A π(a|S∗) �A(a, S∗) to estimate η1(π, πold). +However, such a procedure suffers from the following three main drawbacks: +(I) Iteratively computing the optimization problem (9) can still be computationally ex- +pensive as the policy-dependent nuisance functions need to be updated at each it- +eration, especially when we do not have the closed-form expression for estimating +these nuisance functions such as �dν. Therefore it may not be desirable to directly +implement TRPO method. +(II) When either �A or �dν is not consistent, �η1 might not be consistent. Consequently, +there is no guarantee that the resulting new policy πnew can outperform πold. +(III) To ensure both �A and �dν are both consistent, one might consider estimating these +functions nonparametrically. Even when both of them are consistent, the plug-in +estimator �η1 might not be rate-optimal, i.e., (NT)−1/2, due to that the nonparametric +estimators �A and �dν usually converge much slower than (NT)−1/2. Consequently, +compared with πold, the improvement by πnew may be marginal, resulting in a slow +convergence rate to the optimal policy. +11 + +To address the first concern, we propose to first apply some existing state-of-the-art offline +RL method to obtain a good initial policy. Several methods can be applied here, including +the conservative Q-learning (CQL, Kumar et al., 2020), FQI, V-learning, among others. +CQL and neural FQI use neural networks to index the policy class and V-learning considers +a parametrized policy class indexed by a finite-dimensional vector. Note that these methods +basically rely on the estimation of value or Q-functions. They are more computationally +efficient than the iterative procedure described in (I). The value functions under initial +policies obtained by these algorithms, if consistent, may converge at a slow rate as a trade- +off for fast computation. In the second step, we propose to solve (9) to improve their +performances. This corresponds to a one-step update of the initial policy. One may also +update this new policy for a few times to ensure the final estimated policy achieves a fast +convergence rate. To remove the dependence between the initial policy and our policy +optimization, we incorporate a data-splitting strategy, which is commonly seen in statistics +and machine learning, e.g., Chernozhukov et al. (2018); Kallus and Uehara (2019). The +detailed procedure is described in Section 3.2. +To address the second and the third concerns, we develop an efficient and robust estimat- +ing procedure for η1, which is described in Section 3.1. Specifically, when the input policy +πold is consistent, we can guarantee that the output policy πnew by solving (9) achieves the +desired “value enhancement” property. We call this set of methods “value enhanced policy +optimization (VEPO)”. An overview of our algorithm is given in Section 3.2, which inte- +grates Q-learning, discounted stationary probability ratio estimation, transition dynamics +estimation and policy search. We then discuss each component in the rest of the section. +3.1 +An efficient and multiply robust estimator for η1 +For a given πold ∈ Π, we first outline three potential approaches (see (i)-(iii) below) to +estimating η1(π, πold) from the observed data. Each of these methods requires some nuisance +functions to be consistently estimated. We then present our proposal that combines these +three methods to achieve efficient and triply robust estimation. +(i) Plug-in estimator: �η(1) +1 += ES∗∼ �dν +� +a∈A π(a|S∗) �A(a, S∗). This is the plug-in method +discussed earlier. The validity of �η(1) +1 +requires the consistent estimation of dπold,ν and Aπold. +(ii) Importance sampling (IS) estimator I: +12 + +�η(2) +1 += +1 +� +i Ti +N +� +i=1 +Ti−1 +� +t=0 +ES∗∼ �dν +� +a∈A +{π(a|S∗) − πold(a|S∗)}�ω(Ai,t, Si,t; a, S∗)Ri,t, +(10) +where �ω denotes some estimator for the conditional discounted probability ratio ωπold. See +(4) for a detailed definition. The validity of �η(2) +1 +requires consistent estimation of both +dπold,ν and ωπold. Such an IS estimator is motivated by the work of Liu et al. (2018) on the +off-policy value evaluation. A key observation is that, under (A2) and (A3), Qπ(a, s) can +be represented by +� +a′,s′ +{I(s′ = s, a′ = a) + +� +t≥1 +γtπ(a′|s′)pπ +t (s′|a, s)}r(s′, a′) = +1 +1 − γ Eωπ(At, St; a, s)Rt, +for any t, s and a. This yields the following IS estimator for Aπ(a, s): +1 +� +i Ti +N +� +i=1 +Ti−1 +� +t=0 +� +a′∈A +{I(a′ = a) − π(a′|s)}�ω(Ai,t, Si,t; a′, s)Ri,t. +Plugging in the above estimator for Aπold(a, s) and �dν for dπold,ν yields (10). +(iii) IS estimator II: +�η(3) +1 += +1 +� +i Ti +N +� +i=1 +Ti−1 +� +t=0 +� +a∈A +π(a|Si,t) �A(a, Si,t)�ων(Ai,t, Si,t), +(11) +where �ων denotes some estimator for the integrated probability ratio ωπold,ν. The validity +of �η(3) +1 +requires consistent estimation of dπold,ν and the integrated probability ratio ωπold,ν. +To motivate this estimator, we observe that the expectation ES∼dπ,νf(S) can be rewritten +as Ef(St)ωπ,ν(At, St), for any function f, policy π and decision point t. Consequently, we +can represent η1 by +E +� +a∈A +π(a|St)Aπold(a, St)ωπold,ν(At, St). +This yields the IS estimator in (11). +We note that each of the above estimator may be severely biased when the corresponding +estimated nuisance functions fail to be consistent. Toward that end, we develop a multiply +robust estimator by carefully combining the estimating strategies used in (i)-(iii). Mean- +while, the resulting estimator requires much weaker assumptions to achieve consistency. +Let o be a shorthand for a data tuple (s, a, r, s′). The key to constructing our estimator is +13 + +the following estimating function, +ψ(o; π, πold, �V , �A, �ω, �d) = ψ1(π, πold, �A, �d) + ψ2(o; π, πold, �V , �A, �ω, �d) + ψ3(o; π, πold, �A, �ω, �d), +for some given nuisance functions �V , �A, �ω and �d, where, +ψ1(π, πold, �A, �d) = ES∗∼ �dν +� +a∈A +π(a|S∗) �A(a, S∗), +ψ2(o; π, πold, �V , �A, �ω, �d) = +1 +1 − γ ES∗∼ �dν +� +a∗∈A +{π(a∗|S∗) − πold(a∗|S∗)}�ω(a, s; a∗, S∗) +×{r + γ �V (s′) − �V (s) − �A(a, s)} +ψ3(o; π, πold, �A, �ω, �d) = +� +a∗∈A +�ων(a, s) +1 − γ +� +γEa′∼πold(•|s′) +S∗∼ �d(•|a′,s′) +�A(a∗, S∗)π(a∗|S∗) +−ES∗∼ �d(•|a,s) �A(a∗, S∗)π(a∗|S∗) + (1 − γ)π(a∗|s) �A(a∗, s) +� +, +where the nuisance functions �dν and �ων are determined by �d and �ω, given by �dν(•) = +� +a,s πold(a|s)ν(s)�d(•|a, s) and �ων(•, •) = � +a,s πold(a|s)ν(s)�ω(•, •; a, s). +By definition, ψ consists of three terms. The first term ψ1 is essentially the plug-in esti- +mator that depends only on �A and �d. The second and third terms, i.e., ψ2 and ψ3, are the +augmentation terms. Let Ot = (St, At, Rt, St+1) for any t, we have Eψ2(Ot; π, πold, �V , �A, �ω, �d) = +0 when �A = Aπold, �V = V πold and Eψ3(Ot; π, πold, �A, �ω, �d) = 0 when �d = dπold. See Appendix +A.2 for details. The purpose of adding these two terms is to offer an additional protection +against the potential bias of ψ1 resulting from the biases of �A and �d. Therefore we have +the following proposition. +Proposition 1 Suppose � +a πold(a|s) �A(a, s) = 0 for any s. Then ψ(Ot; π, πold, �V , �A, �ω, �d) +is unbiased to η1 as long as one of the following three assumptions are satisfied: (B1) +�A = Aπold, �V = V πold and �d = dπold; (B2) �ω = ωπold and �d = dπold; (B3) �A = Aπold and +�ω = ωπold. +The condition � +a πold(a|s) �A(a, s) is automatically satisfied if we set �A(a, s) = �A∗(a, s) − +� +a πold(a|s) �A∗(a, s) = 0 for any initial advantage estimator �A∗. We remark that if Qπold is +correctly specified, so do Aπold and V πold. Based on this estimating function, a triply-robust +estimator for η1 is given by +1 +� +i Ti +N +� +i=1 +Ti−1 +� +t=0 +ψ(Oi,t; π, πold, �V , �A, �ω, �d), +(12) +14 + +where Oi,t = (Si,t, Ai,t, Ri,t, Si,t+1). It remains to specify the estimation of nuisance func- +tions. We present the details in the next section. In Section 4.2, we show the resulting +estimator is efficient. +3.2 +The complete algorithm +Our main idea is to construct an efficient and robust estimator for η1 to improve the perfor- +mance of an initial policy πold. To achieve this goal, we need to estimate four key nuisance +functions: (a) An initial policy πold; (b) The value and advantage function V πold and Aπold; +(c) The conditional discounted stationary probability ratio ωπold; (d) The conditional dis- +counted visitation probability function dπold. +Correspondingly, our estimating procedure involves four key steps, described in Sec- +tions 3.2.1-3.2.4 respectively. In addition to these four main estimating components, we +also propose to couple the estimator in (12) with a data-splitting and cross-fitting strategy. +Specifically, without loss of generality, we randomly divide the indices of all trajectories +{1, 2, · · · , N} into L subsets ∪L +ℓ=1{Oi,t}i∈Iℓ,0≤t 0. This implies that π(ℓ) +old is consistent to πopt as either +N or T diverges to infinity. We further assume V(π∗) = V(πopt)+o(1), as NT → ∞. Recall +that π∗ is defined as the optimal in-class policy that maximizes the value among Π. In other +words, the value under the optimal in-class policy approaches to the optimal value function +as the sample size increases (because the size of policy class also increases). To simplify the +theoretical analysis, we assume πopt ∈ Π such that π∗ = πopt. Meanwhile, our theories are +equally applied to settings where πopt /∈ Π but the value difference V(π∗)−V(πopt) converges +at a sufficiently fast rate. This assumption is reasonable in practice when we either have +domain knowledge on the parametric form of πopt or use function classes with the universal +approximation capabilities (e.g., neural networks) to parametrize Π. To establish the value +enhancement property, we need the following set of conditions. +(C1) Suppose E(A,S)∼p∞| �Q(ℓ)(A, S)−Qπ(ℓ) +old(A, S)|2 = Op{(NT)−2κ1} for some constant κ1 ≥ +0. In addition, �Q(ℓ) is uniformly bounded almost surely. +(C2) Suppose E(A,S),( � +A, �S)∼p∞|�ω(ℓ)( �A, �S; A, S) − ωπ(ℓ) +old( �A, �S; A, S)| = Op{(NT)−2κ2} for some +constant κ2 ≥ 0, where ( �A, �S) and (A, S) denote two independent state-action pairs gener- +ated according to p∞. In addition, �ω(ℓ) is uniformly bounded almost surely. +(C3) Suppose E(A,S)∼p∞|DTV(dπ(ℓ) +old(•|A, S), �d(ℓ)(•|A, S))|2 = Op{(NT)−2κ3} for some con- +stant κ3 ≥ 0. +22 + +(C4) Suppose Π corresponds to certain VC type function class (Chernozhukov et al., 2014) +with VC indices upper bounded by O{(NT)κ4} for some constant 0 ≤ κ4 < +α +α+1, where α +is defined below. +(C5) The optimal policy is unique. In addition, there exist some positive constants α, ¯c, ¯ǫ +such that Pr(−ǫ ≤ Aπopt(a, S∗) < 0) ≤ ¯cǫα for any a ∈ A and 0 < ǫ ≤ ¯ǫ, where the random +variable S∗ is distributed according to dπopt,ν. +(C6) The process {(St, At, Rt)}t≥0 is exponentially β-mixing. +Conditions (C1)-(C3) characterize the theoretical requirements on the learners in (a)- +(c), respectively. +In particular, (C1)-(C2) require the squared prediction losses of the +estimated Q-function and the conditional probability ratio to satisfy certain convergence +rates, whereas Condition (C3) assumes the squared total variation norm between the tran- +sition function and its estimator to satisfy a certain convergence rate. +If some para- +metric models are imposed to learn Qπold, ωπold and the transition matrix p, we have +κ1 = κ2 = κ3 = 1/2. +In our setup, we only require κi1 + κi2 > 1/(2 + 2α) for any +disjoint i1, i2 ∈ {1, 2, 3}. See the statement of Theorem 2 below. This condition holds +when mini∈{1,2,3} κi > 1/(4 + 4α) and thus is achievable for many nonparametric estima- +tors. It is also strictly weaker than those imposed in the recent literature that require the +nuisance function to converge at a rate faster than (NT)−1/4 for off-policy value evalua- +tion (e.g., Kallus and Uehara, 2019). For example, when the kernel smoother (Feng et al., +2020), sieve method (Shi, Zhang, Lu and Song, 2020; Chen and Qi, 2022) or deep neural +networks (Fan et al., 2020b) are used to approximate the Q-function, it can be shown that +under some technical conditions, (C2) holds with κ1 = β1/(2β1 + dS) and β1 > dS/2 where +dS denotes the dimension of the state space and β1 denotes the H¨older exponent that char- +acterizes the smoothness of the Q-function. Similar result (i.e., optimal non-parametric +convergence rate) can be obtained for the conditional probability ratio function. As dis- +cussed in Section 3.2.4, when T ′ and M are sufficiently large, (C3) essentially requires +the estimated mean and covariance functions in the conditional Gaussian model to con- +verge at a rate of (NT)−κ3. Under some regularity conditions, an optimal non-parametric +convergence rate can also be achieved. +Condition (C4) is mild as the policy class Π is pre-specified. +When a linear policy +class is employed, we have κ4 = #s where #s denotes the number of parameters used to +index the policy class. When Π is set to some deep neural networks, the corresponding +23 + +VC-dimension is also available in the literature (see e.g., Harvey et al., 2017). +The uniqueness of the optimal policy (C5) is commonly assumed in the literature +(Ertefaie and Strawderman, 2018; Luckett et al., 2020). The second part of (C5) is closely +related to margin-type conditions commonly used to bound the excess misclassification er- +ror (Tsybakov et al., 2004; Audibert et al., 2007) and the regret of individualized treatment +regimes in point treatment studies (Qian and Murphy, 2011; Luedtke and Van Der Laan, +2016; Shi, Lu and Song, 2020). +To better understand the margin condition in (C5), we first observe that Aπopt(a, s) ≤ 0 +for any a and s. To elaborate this, we note that πopt maximizes V π(s) for any π. Consider +the following history-dependent policy πopt(a) that assigns a at the initial decision point and +follows πopt in the subsequent steps. The value under such a policy is given by Qπopt(a, s). +It follows that Qπopt(a, s) ≤ V πopt(s). or equivalently, Aπopt(a, s) ≤ 0 for any a and s. +The equality holds only when a = argmaxa′Qπopt(a′, s). The argmax is well-defined by +the uniqueness of the optimal policy. For a ̸= argmaxa′Qπopt(a′, s), the advantage function +corresponds to the value difference between πopt(a) and πopt. The smaller the difference, +the harder it is to identify the optimal policy. To ensure πopt can be consistently identified, +it is thus reasonable to assume Pr(0 < |Aopt(a, S∗)| ≤ ǫ) decays to zero with ǫ as well. (C5) +explicitly characterizes such dependence. For example, when the action space is binary, let +τ(s) denote the contrast function, i.e., τ(s) = Qπopt(1, s) − Qπopt(0, s). It is immediate to +see that Aπopt(0, s) = min(−τ(s), 0) and Aπopt(1, s) = min(τ(s), 0). Thus, the second part +of (C5) essentially requires Pr(0 < |τ(S∗)| ≤ ǫ) ≤ ¯cǫα, which is automatically satisfied with +α = 1 when τ(S∗) has a bounded probability density function. More generally, it holds +when |τ(S∗)|α has a bounded probability density function. For instance, suppose both the +initial reference distribution ν and the Markov transition function have bounded density +functions on (0, +∞). Then, the distribution of S∗, i.e., the mixture distribution of {St}t≥0 +with weights {(1 − γ)γt}t≥0 has a bounded probability density function as well. Suppose +τ(S∗) = (S∗)1/α. Then it is immediate to see that |τ(S∗)|α has a bounded probability +density function. Finally, when |τ(S∗)| is uniformly bounded away from zero, then (C5) +holds with α = +∞. We will see in Theorem 2 below that the convergence rate of πnew +depends crucially on the margin parameter α. +Assumption (C6) characterizes the dependence of the data observations over time. It +essentially requires the β-mixing coefficient (see e.g., Bradley, 2005, for a detailed def- +24 + +inition) of at lag q, which measures the time dependence between the set of variables +{(Sj, Aj, Rj)}j≤t and {(Sj, Aj, Rj)}j≥t+q, to decay to zero at an exponential rate with re- +spect to q. This assumption automatically holds when {(St, At, Rt)}t≥0 forms a geometri- +cally ergodic Markov chain. Geometric ergodicity is less restrictive than those imposed in +the existing reinforcement learning literature that requires observations to be independent +(see e.g., Degris et al., 2012; Farahmand et al., 2016) or to follow a uniform-ergodic Markov +chain (see e.g., Bhandari et al., 2018). +Theorem 2 (Value Enhancement Property) Suppose (C1)-(C6) hold. If the constants +κ1, κ2, κ3 satisfy κi1 + κi2 > 1/(2 + 2α) for any disjoint i1, i2 ∈ {1, 2, 3}, and that V(πopt) − +V(π(ℓ) +old) = Op{(NT)−κ0} for any ℓ, we have V(πopt) − V(πnew) = E1 + E2 where E1 = +Op{(NT)− κ0(2α+1) +α+1 +}, E2 = op{(NT)−1/2}. +Theorem 2 states that the value difference V(πopt) − V(πnew) can be decomposed into +two terms. Here, the first term E1 describes how the input policy πold takes effect. It is +due to the presence of the higher-order remainder term η2(π, πold) resulting from the first +order approximation of the value difference V(π) − V(πold). The second term E2 is due +to the estimation error of the η1(π, πold). In the typical multiply robust setting, it often +requires that κi1 + κi2 > 1/2 so that the bias of estimating η1(π, πold) is op{(NT)−1/2}. +In Theorem 2, since we require slower rates for nuisance parameters, the proposed value +difference estimator for η1(π, πold) may converge slower than the 1/2-root. However, the +value enhancement property can still be established under such a slower rate requirement. +This is due to that Theorem 2 is concerned with the convergence rate of the estimated +optimal policy πnew in terms of the value instead of the rate of the proposed value difference +estimator (denoted by �η1(πnew)). In particular, the convergence rate of πnew in terms of +the value is primarily determined by the difference between �η1(πnew) and �η1(πopt) (see Page +12 of the supplementary material), which converges at a faster rate than �η1(πnew) itself. +This is because πnew is consistent to the optimal policy implied by the condition that +V(πopt) − V(π(ℓ) +old) = Op{(NT)−κ0}. +When κ0 ≤ 1/2, it can be seen that the value under the output policy converges at +a faster rate than the input policy, leading to the desired “value enhancement property”. +One can repeat the one-step update multiple times to guarantee that the value of the +estimated optimal policy converges at a rate of op{(NT)−1/2}. When the initial policy +25 + +already converges faster than the parametric rate (e.g., κ0 > 1/2), then our proposal is not +guaranteed to yield a better policy in theory. However, as shown in our empirical studies +(see Section 5), the values of the proposed policies are often larger than those computed +via state-of-the-art RL algorithms. This suggests that although these initial policies are +consistent, they might converge at a suboptimal rate and have room for improvement. +4.2 +Efficiency of the value difference estimator +In this subsection, we show that conditional on π(ℓ) +old, the proposed estimator for η1(π, π(ℓ) +old), +i.e., +�η1(π, π(ℓ) +old) = +1 +T|Iℓ| +�� +i∈Iℓ +T−1 +� +t=0 +ψ(Oi,t; π, π(ℓ) +old, �V (ℓ), �A(ℓ), �ω(ℓ), �d(ℓ)) +� +. +is nearly unbiased to η1(π, π(ℓ) +old) and its asymptotic variance matches this efficiency bound. +The notion of efficiency bound can be found in Section A.4 of Supplementary Material. +Consequently, �η1 is efficient. +Let ¯p denote the conditional distribution of (Rt, St+1) given (At, St). For any given π +and πold, we note that η1(π, πold) is completely determined by the transition function ¯p. +Let {¯pθ1 : θ1 ∈ Θ1} be a regular parametric submodel for ¯p. This requires ¯pθ1 to be a +transition matrix for any θ1 and ¯p = ¯pθ∗ +1 for some θ∗ +1 ∈ Θ1. Similarly, let {¯bθ2 : θ2 ∈ Θ2} +and {¯νθ3 : θ3 ∈ Θ3} be regular parametric submodels for the behavior policy and the initial +state distribution, respectively. Let θ = (θ1, θ2, θ3) and θ∗ = (θ∗ +1, θ∗ +2, θ∗ +3) where θ∗ +2 and θ∗ +3 +correspond to the true parameters in Θ2 and Θ3. Under a given submodel indexed by θ, +the log-likelihood function of a single data trajectory can be written as +ℓT({Ot}t; θ) = log +� +¯νθ3(S0) +T +� +t=0 +{¯pθ1(Rt, St+1|At, St)¯bθ2(At|St)} +� +. +Note that η1 can be defined as function of θ as well. We define the efficiency bound as +EB(|Iℓ|, T) = |Iℓ| sup ∇θη1(θ∗) +� +E∇θℓT({Ot}t; θ∗)∇⊤ +θ ℓT({Ot}t; θ∗) +�−1 ∇⊤ +θ η1(θ∗), +where the supremum is taken over all regular parametric submodels, and ∇θg(θ′) denotes +the derivative of a function g with respect to θ, evaluated at θ = θ′. As discussed before, +η1 depends on θ only through θ1. +26 + +Theorem 3 Suppose the conditions in Theorem 2 holds with κi1+κi2 > 1/2 for any disjoint +i1, i2 ∈ {1, 2, 3}. Then conditional on π(ℓ) +old, we have for any π that +�η1(π, π(ℓ) +old) − η1(π, π(ℓ) +old) +� +EB(|Iℓ|, T) +d→ N(0, 1). +Theorem 3 implies that �η1 is asymptotically unbiased with asymptotic variance EB(|Iℓ|, T). +This demonstrates the efficiency of the proposed estimator. +5 +Numerical examples +In this section, we use one toy example and real data related studies to demonstrate the +superior performance of our method. Specifically, in Section 5.1, we use a toy example to +demonstrate the multiple robustness of our estimator and the value enhancement property. +We then demonstrate the performance of the proposed method on OhioT1DM related +datasets in Section 5.2. In Appendix D, we conduct another simulation study to illustrate +the finite-sample performance of our algorithm compared with several existing methods. +5.1 +A Toy Example +We design a toy example to illustrate the multiple robustness of our estimator and the +desired value enhancement property. Consider a binary state space S = {0, 1}, where S0 +takes value 0 with probability 0.4 and otherwise. The action space A takes values in {0, 1}. +The reward function is defined as r(a, s) = I(s = a), and then the reward is generated +according to Rt = r(At, St) + et where {et}0≤t 0 such that |η2(πnew, πold)| is bounded from above by +c∗γ +1 − γ +� +Edπold,νDTV{πold(•|S), πnew(•|S)} +�2 ≤ +c∗γ +1 − γ Edπold,νDKL{πold(•|S), πnew(•|S)}. (21) +We remark that the upper bound on the RHS of (21) is tighter than that in Theorem 1 of +Schulman et al. (2015). +Proof : Note that +|η2(πnew, πold)| ≤ +� +a∈A,s∈S +|πnew(a|s) − πold(a|s)||Aπold(a, s)||dπold,ν(s) − dπnew,ν(s)|. +(22) +In the following, we provide an upper bound on |dπold,ν(s) − dπnew,ν(s)|. By definition, we +have +|dπold,ν(s) − dπnew,ν(s)| ≤ (1 − γ)γ ++∞ +� +t=0 +γt|pπold +t+1(s) − pπnew +t+1 (s)|, +(23) +For any t, we can define a time-varying policy π(t) such that the agent follows πold at the +initial t time points and πnew subsequently. It follows that ++∞ +� +t=0 +γt|pπold +t+1(s) − pπnew +t+1 (s)| ≤ ++∞ +� +t=0 +t +� +j=0 +γt|pπ(j+1) +t+1 +(s) − pπ(j) +t+1 (s)| +≤ ++∞ +� +t=0 +t +� +j=0 +γt +������ +� +s−∈S +pπold +j +(s−) +� +a +p(s|a, s−){πold(a|s−) − πnew(a|s−)}pπnew +t−j (s) +������ +. +33 + +By the definition of total variation distance, we have ++∞ +� +t=0 +γt|pπold +t+1(s) − pπnew +t+1 (s)| ≤ 2 +� +s−∈S +∥πnew(•|s−) − πold(•|s−)∥TV ++∞ +� +t=0 +t +� +j=0 +γtpπnew +t−j (s)pπold +j +(s−) += 2 +� +s−∈S +∥πnew(•|s−) − πold(•|s−)∥TV ++∞ +� +j=0 ++∞ +� +t=j +γtpπnew +t−j (s)pπold +j +(s−) +≤ +2 +(1 − γ)2 +� +s−∈S +∥πnew(•|s−) − πold(•|s−)∥TVdπold,ν(s−)dπnew,ν(s). +Under the given conditions, we have ν(s) ≥ C for some constant C > 0 and any s ∈ S. +Consequently, we have dπold,ν(s) ≥ C(1 − γ) and hence dπnew,ν(s)/dπold,ν(s) ≤ 1/dπold,ν(s) ≤ +C−1(1 − γ)−1. It follows that ++∞ +� +t=0 +γt|pπold +t+1(s) − pπnew +t+1 (s)| ≤ +2 +C(1 − γ)2 +� +s−∈S +∥πold(•|s−) − πnew(•|s−)∥TVdπold,ν(s−)dπold,ν(s), +for some constant C > 0. Combining this together with (22) and the above inequality +yields that |η2(πnew, πold)| is upper bounded by +2Cγ +1 − γ +� +a∈A +� +s,s−∈S +|πnew(a|s) − πold(a|s)||Aπold(a, s)|∥πold(•|s−) − πnew(•|s−)∥TV +×dπold,ν(s−)dπold,ν(s). +(24) +Note that the Q-function and value function correspond to some expected discount cumula- +tive rewards. Under the assumption that the immediate reward is bounded, both functions +are bounded. Consequently, we have supa,s |Aπold(a, s)| ≤ c for some positive constant O(1). +It follows that +|η2(πnew, πold)| ≤ +4cγ +C(1 − γ) +�� +s∈S +∥πold(•|s) − πnew(•|s)∥TVdπold,ν(s) +�2 +. +This yields the upper bound on the left-hand-side (LHS) of (21). +By Cauchy-Schwarz inequality, we obtain +|η2(πnew, πold)| ≤ +4cγ +C(1 − γ) +� +s∈S +∥πold(•|s) − πnew(•|s)∥2 +TVdπold,ν(s). +The upper bound on the RHS of (21) thus follows by Pinsker’s inequality. +34 + +A.2 +Some additional details regarding Proposition 1 +We first give some intuition of this proposition: Specifically, suppose �A satisfies � +a πold(a|s) �A(a, s) = +0 for any s. We have the following results: +• When �A = Aπold, �V = V πold and �d = dπold, the expectations Eψ2(Ot; π, πold, �V , �A, �ω, �d) +and Eψ3(Ot; π, πold, �A, �ω, �d) are equal to zero. Consequently, Eψ(Ot; π, πold, �V , �A, �ω, �d) +is equal to the expectation of the plug-in estimator, and thus ψ(Ot; π, πold, �V , �A, �ω, �d) +is unbiased to η1. +• When �ω = ωπold and �d = dπold, the expectation Eψ3(Ot; π, πold, �A, �ω, �d) is equal to +zero. The presence of �ω in ψ2 guarantees the estimating function is robust to the +misspecification of �A and �V . Under the assumption that � +a πold(a|s) �A(a, s) = 0 for +any s, the expectation of ψ1(π, πold, �A, �d) + ψ2(Ot; π, πold, �V , �A, �ω, �d) is equal to that +of the IS estimator (10). Since the IS estimator is unbiased when �ω = ωπold and +�d = dπold, so is ψ(Ot; π, πold, �V , �A, �ω, �d). +• When �A = Aπold and �ω = ωπold, the expectation Eψ2(Ot; π, πold, �V , �A, �ω, �d) is equal to +zero. The presence of �ων (or �ω, as it is completely determined by �ω) in ψ3 guarantees +that the estimating equation is robust to the misspecification of �d. More specifically, +the expectation of ψ1(π, πold, �A, �d) + ψ3(Ot; π, πold, �A, �ω, �d) is equal to that of the IS +estimator (11) when �ων = ωπold,ν. Since the IS estimator is unbiased when �A = Aπold +and �ω = ωπold, so is ψ(Ot; π, πold, �V , �A, �ω, �d). +Now we formally show the expectations of the two argumentation terms ψ2(Ot; �V , �A, �ω, �d) +and ψ3(Ot; �A, �ω, �d) are equal to zero when some of the nuisance functions are correctly +specified. Consider ψ2 first. When �A = Aπold and �V = V πold, it follows from the Bellman’s +equation that +E{Rt + γ �V (St+1) − �V (St) − �A(At, St)|At, St} = 0. +(25) +Consequently,E{ψ2(Ot; π, πold, �V , �A, �ω, �d)|At, St} = 0 and hence Eψ2(Ot; π, πold, �V , �A, �ω, �d) = +0. As for ψ3, note that when �d = dπold, we have +35 + +E +� +Ea∗∼πold(•|St+1) +S∗∼ �d(•|a∗,St+1) +πnew(a′|S∗) �A(a′, S∗) +����� At, St +� += +� +s,a +πnew(a′|s) �A(a′, s)E{dπold(s|a, St+1)πold(a|St+1)|At, St} += +� +s,a +πnew(a′|s) �A(a′, s)(1 − γ) +� +t≥1 +γt−1pπold +t +(s; At, St) += +1 +γ +� +ES∗∼ �d(•|At,St)πnew(a′|S∗) �A(a′, S∗) − (1 − γ)πnew(a′|St) �A(a′, St) +� +, +where the last equality follows from the definition of �d. This yields E{ψ3(Ot; π, πold, �V , �A, �ω, �d)|At, St} += 0 and hence Eψ3(Ot; π, πold, �V , �A, �ω, �d) = 0. +Next, suppose � +a πold(a|s) �A(a, s) = 0 for any s. We show +�A(a, s) − +1 +1 − γ Eωπold(At, St; a, s) �A(At, St) = 0, +(26) +and +� +a +{πnew(a|s) − πold(a|s)}Eωπold(At, St; a, s){γ �V (St+1) − �V (St)} = 0. +(27) +Combining these two equations yields that the expectation E{ψ1(π, πold, �A, �ω)+ψ2(Ot; �V , �A, �ω, �d)} +is equal to the expectation of the IS estimator in (10) when �ω is correctly specified. Con- +sequently, Eψ(Ot; �V , �A, �ω, �d) is unbiased to η1 under the assumption in (B2). +We first show (26). We observe that +1 +1 − γ Eωπold(At, St; a, s) �A(At, St) = �A(a, s) + +� +t≥1 +γt � +a′,s′ +pt(s′|a, s)πold(a′|s′) �A(a′, s′). +The second term on the RHS is equal to zero under the condition that � +a πold(a|s) �A(a, s) = +0 for any s. This yields (26). We next show (27). With some calculations, +Eωπold(At, St; a, s){γ �V (St+1) − �V (St)} = (1 − γ) +� +t≥1 +γt � +s′ +pt(s′|a, s)�V (s′) +−(1 − γ) +� +t≥0 +γt � +s′ +pt(s′|a, s)�V (s′) = −(1 − γ)�V (s). +Note that the RHS is independent of a. Consequently, we have � +a{π(a|s)−πold(a|s)}�V (s) = +0. This yields (27). +Equation (27) implies that Eψ2(Ot; �V , �A, �ω, �d) = Eψ2(Ot; V πold, �A, �ω, �d) when �ω = ωπold. +If further �A is correctly specified, we obtain that Eψ2(Ot; �V , �A, �ω, �d) = 0. +Finally, we show +36 + +E +� +a∗∈A +ωπold,ν(At, St) +1 − γ +� +γEa′∼πold(•|St+1) +S∗∼ �d(•|a′,St+1) +�A(a∗, S∗)π(a∗|S∗) − ES∗∼ �d(•|At,St) �A(a∗, S∗)π(a∗|S∗) +� += −ψ1(π, πold, �A, �d). +This further yields that the expectation of ψ1(π, πold, �A, �d)+ψ3(Ot; �A, �ω, �d) is equal to that +of the IS estimator (11) when �ων = ωπold,ν. Consequently, Eψ(Ot; �V , �A, �ω, �d) is unbiased +when �A and �ω are correctly specified. +With some calculations, the LHS is equal to +E +� +a∗,s∗ +ωπold,ν(At, St) +1 − γ +�� +γ +� +a′ +πold(a′|St+1)�d(s∗|a′, St+1) − �d(s∗|At, St) +� +�A(a∗, s∗)πnew(a∗|s∗) +� += +1 +1 − γ +� +a∗,s∗,a′,s′ +πnew(a∗|s∗)πold(a′|s′)�d(s∗|a′, s′) �A(a∗, s∗){dπold(s′) − (1 − γ)ν(s′)} +− +1 +1 − γ +� +a∗,s∗,a′,s′ +πnew(a∗|s∗)πold(a′|s′)�d(s∗|a′, s′) �A(a∗, s∗)dπold(s′) += − +� +a∗,s∗,a′,s′ +πnew(a∗|s∗)πold(a′|s′)�d(s∗|a′, s′) �A(a∗, s∗)ν(s′) = − +� +a∗,s∗ +πnew(a∗|s∗)�dν(s∗) �A(a∗, s∗), +where the last equation follows from the definition of �dν. The proof is hence completed. +A.3 +VC type class +We introduce the notion of the VC type class in this section. Specifically, let F denote a +class of measurable functions, with a measurable envelope function F such that supf∈F |f| ≤ +F. For any probability measure Q, let eQ denote a semi-metric on F such that eQ(f1, f2) = +∥f1 − f2∥Q,2 = +� +Q|f1 − f2|2. An ǫ-net of the space (F, eQ) is a subset Fǫ of F, such +that for every f ∈ F, there exists some fǫ ∈ Fǫ satisfying eQ(f, fǫ) < ǫ. We say that +F is a VC type class with envelope F, if there exist constants c0 > 0, c1 ≥ 1, such that +supQ N (F, eQ, ǫ∥F∥Q,2) ≤ (c0/ǫ)c1, for all 0 < ǫ ≤ 1, where the supremum is taken over +all finitely discrete probability measures on the support of F, and N (F, eQ, ǫ∥F∥Q,2) is the +infimum of the cardinality of ǫ∥F∥Q,2-nets of F. We refer to c1 as the VC index of F. +A.4 +Semiparametric Efficiency +In the i.i.d. case, for parametric models, the variance of any unbiased estimator must be +greater than or equal to the Cr´amer-Rao lower bound (Casella and Berger, 2002, Section +7.3) and the maximum likelihood estimator is known to be efficiency under certain regular- +37 + +ity conditions. In semiparametric theory, the efficiency bound is defined as the supremum +of Cr´amer-Rao lower bounds over all regular parametric submodels to move beyond para- +metric setup (see e.g., Tsiatis, 2007). In our setup, the observations are time-dependent. +We adopt the definition in Komunjer and Vuong (2010) and Kallus and Uehara (2019) that +corresponds to a generalization of the classical semiparametric efficiency bound to the non +i.i.d. setting. +B +More on the algorithm +B.1 +More on conditional discounted stationary probability ratio +Consider the following optimization problem +argminω∈Ω sup +f∈F +������ +� +(i,t)̸=(i′,t′) +∆(ω, f, πold; i, t, i′, t′) +������ +2 +. +(28) +We set F to a unit ball of a reproducing kernel Hilbert space (RFHS), i.e., F = {f ∈ H : +∥f∥H = 1}, where +H = + + +f(·) = +� +(i,t)̸=(i′,t′) +bi,t,i′,t′κ(Xi′,t′, Xi,t; ·) : bi,t,i′,t′ ∈ R + + + , +for some positive definite kernel κ(·; ·), where Xi,t is a shorthand for the state-action pair +(Si,t, Ai,t). Similar to Theorem 2 of Liu et al. (2018), we can show the optimization problem +in (28) is then reduced to +argminω∈Ω +� +(i1,t1)̸=(i′ +1,t′ +1) +� +(i2,t2)̸=(i′ +2,t′ +2) +D(ω, πold; i1, t1, i′ +1, t′ +1, i2, t2, i′ +2, t′ +2), +where D(ω, π; i1, t1, i′ +1, t′ +1, i2, t2, i′ +2, t′ +2) is given by +ω(Xi′ +1,t′ +1; Xi1,t1) +(1 − γ)−1 +� +γEa∼π(•|Si′ +1,t′ +1+1)κ(Si′ +1,t′ +1+1, a, Xi1,t1; Xi2,t2, Xi2,t2) − κ(Xi′ +1,t′ +1, Xi1,t1; Xi2,t2, Xi2,t2) +� ++ω(Xi′ +2,t′ +2; Xi2,t2) +(1 − γ)−1 +� +γEa∼π(•|Si′ +2,t′ +2+1)κ(Si′ +2,t′ +2+1, a, Xi2,t2; Xi1,t1, Xi1,t1) − κ(Xi′ +2,t′ +2, Xi2,t2; Xi1,t1, Xi1,t1) +� ++ω(Xi′ +1,t′ +1; Xi1,t1)ω(Xi′ +2,t′ +2; Xi2,t2) +� +γ2Ea1∼π(•|Si′ +1,t′ +1+1) +a2∼π(•|Si′ +2,t′ +2+1) +κ(Si′ +2,t′ +2+1, a, Xi2,t2; Si′ +1,t′ +1+1, a1, Xi1,t1) +−γEa1∼π(•|Si′ +1,t′ +1+1)κ(Si′ +1,t′ +1+1, a, Xi1,t1; Xi′ +2,t′ +2, Xi2,t2) − γEa2∼π(•|Si′ +2,t′ +2+1)κ(Si′ +2,t′ +2+1, a, Xi2,t2; Xi′ +1,t′ +1, Xi1,t1) ++κ(Xi′ +2,t′ +2, Xi2,t2; Xi′ +1,t′ +1, Xi1,t1) +� ++ (1 − γ)2κ(Xi1,t1, Xi1,t1; Xi2,t2, Xi2,t2). +38 + +Algorithm 4 Estimation of the density ratio. +Input: The data subset in Iℓ. +Initial: Initial the density ratio ω = ωβ to be a neural network parameterized by β. +for iteration = 1, 2, · · · do +a Randomly sample batches M, M∗ from the data transitions. +b Update the parameter β by +β ← β−ǫ +�|M| +2 +�−2 +� +(i1,t1),(i′ +1,t′ +1)∈M +(i1,t1)̸=(i′ +1,t′ +1) +� +(i2,t2),(i′ +2,t′ +2)∈M +(i2,t2)̸=(i′ +2,t′ +2) +∇βD( ωβ +zωβ +, πold; i1, t1, i′ +1, t′ +1, i2, t2, i′ +2, t′ +2), +where zωβ is a normalization constant +zωβ(·; Si,t, Ai,t) = +1 +|M∗| +� +(i′,t′)∈M∗ +ωβ(Xi′,t′; Xi,t). +Output ωβ. +In our implementation, we set Ω to the class of neural networks. The detailed estimating +procedure is given in Algorithm 4. +B.2 +More on conditional discounted visitation probability +We first provide an upper bound for DTV(�d(ℓ)(•|a, s), dπ(ℓ) +old(•|a, s)) for a given pair (a, s). +Notice that the total variation distance corresponds to a special case of f-divergence. This +allows us to represent DTV(�d(ℓ)(•|a, s), dπ(ℓ) +old(•|a, s)) as +sup +∥f∥∞≤1/2 +|E +S∗∼dπ(ℓ) +old(•|a,s)f(S∗) − ES∗∼ �d(ℓ)(•|a,s)f(S∗)|, +or equivalently, +sup +∥f∥∞≤1/2 +���E +S∗∼dπ(ℓ) +old(•|a,s)f(S∗) − 1 − γ +M +M +� +m=1 +T ′ +� +t′=0 +γtf(�S(m) +t′ +) +���. +By definition, for a given f, we can decompose the difference into +39 + +�����E +S∗∼dπ(ℓ) +old(•|a,s)f(S∗) − 1 − γ +M +M +� +m=1 +T ′ +� +t′=0 +γtf(�S(m) +t′ +) +����� +≤ +T ′ +� +t′=0 +γt′ +�����(1 − γ)E +S∗∼p +π(ℓ) +old +t′ +(•|a,s) +f(S∗) − 1 − γ +M +M +� +m=1 +f(�S(m) +t′ +) +����� ++(1 − γ) ++∞ +� +t′=T ′+1 +γt′|E +S∗∼dπ(ℓ) +old(•|a,s)f(S∗)|. +Since f is uniformly bounded by 1/2, the second term on the RHS is bounded by γT ′ that +converges to zero as T ′ → ∞. +As for the first term, it can be further bounded from above by +T ′ +� +t′=0 +(1 − γ)γt′ +�����E +S∗∼p +π(ℓ) +old +t′ +(•|a,s) +f(S∗) − Ef(�St′) +����� + +T ′ +� +t′=0 +(1 − γ)γt′ +����� +1 +M +M +� +m=1 +f(�S(m) +t′ +) − Ef(�St′) +����� .(29) +The expectation of the second term in (29) can be upper bounded by +T ′ +� +t′=0 +(1 − γ)γt′ +� +� +� +�Var +� +1 +M +M +� +m=1 +f(�S(m) +t′ +) − Ef(�St′) +� +≤ M−1/2, +by Cauchy-Schwarz inequality. Consequently, the second term in (29) decays to zero as +M → ∞. +Finally, consider the first term in (29). We use (S(j) +t , A(j) +t ) to denote the state-action +pair measured at time t that follows π(ℓ) +old and the transition function p at the first jth +steps conditional on (A0, S0) = (a, s), and then follows π(ℓ) +old and the transition function +N (�µ(ℓ), �Σ(ℓ)) in the subsequent steps. For each t′, we have +�����E +S∗∼p +π(ℓ) +old +t′ +(•|a,s) +f(S∗) − Ef(�St′) +����� ≤ +t′ +� +j=1 +���Ef(S(j−1) +t′ +) − Ef(S(j) +t′ ) +��� . +Let �p +π(ℓ) +old +t +(•|a, s) denote the distribution function of the state vector St at time t that follows +πold and the estimated transition function N (�µ(ℓ), �Σ(ℓ)) conditional on (A0 = a, S0 = s). We +omit the subscript t and the superscript π(ℓ) +old when t = 1. Suppose f is uniformly bounded +by some constant c > 0. Then |E +S∗∼�p +π(ℓ) +old +t′−j(•|a∗,s),a∗∼π(ℓ) +old(•|s) +f(S∗)| is uniformly bounded by c +for any s as well. Consequently, +40 + +���Ef(S(j−1) +t′ +) − Ef(S(j) +t′ ) +��� ≤ E +�������� +ES∗∼p(•|A(j−1) +j−1 ,,S(j−1) +j−1 +) + + + + + + + +E +S∗∗∼�p +π(ℓ) +old +t′−j(•|a∗∗,S∗) +a∗∗∼π(ℓ) +old(•|S∗) +f(S∗∗) + + + + + + + +− ES∗∼�p(•|A(j−1) +j−1 ,S(j−1) +j−1 +) + + + + + + + +E +S∗∗∼p +π(ℓ) +old +t′−j(•|a∗∗,S∗) +a∗∗∼π(ℓ) +old(•|S∗) +f(S∗∗) + + + + + + + +�������� +≤ cEDTV{p(•|A(j−1) +j−1 , S(j−1) +j−1 ), �p(•|A(j−1) +j−1 , S(j−1) +j−1 )}. +It follows that the first term in (29) can be upper bounded by ++∞ +� +t′=0 +(1 − γ)γt′ +�����E +S∗∼p +π(ℓ) +old +t′ +(•|a,s) +f(S∗) − Ef(�St′) +����� +≤ +c ++∞ +� +t′=1 +(1 − γ)γt′ +t′ +� +j=1 +EDTV{p(•|A(j−1) +j−1 , S(j−1) +j−1 ), �p(•|A(j−1) +j−1 , S(j−1) +j−1 )} += +c ++∞ +� +j=1 +γℓDTV{p(•|A(j−1) +j−1 , S(j−1) +j−1 ), �p(•|A(j−1) +j−1 , S(j−1) +j−1 )} += +cγ +1 − γ E +(A∗,S∗)∼qπ(ℓ) +old(•,•;a,s)DTV{p(•|A∗, S∗), �p(•|A∗, S∗)}, +(30) +where qπ(ℓ) +old(•, •; a, s) denotes the conditional discounted visitation probability of the state- +action pair, i.e., +qπ(ℓ) +old(a′, s′; a, s) = (1 − γ)I(a′ = a, s′ = s) + (1 − γ) ++∞ +� +t=1 +γtπ(ℓ) +old(a′|s′)p +π(ℓ) +old +t +(s′|a, s). +Let �p denote the normal density function with mean �µ(ℓ) and covariance matrix Σ. +It +follows from the triangle inequality that +DTV(p, �p) ≤ DTV(p, �p) + DTV(�p, �p). +According to Proposition 2.1 of Devroye et al. (2018), the total variation distance between +p and �p is upper bounded by 0.5 +� +(µ − �µ(ℓ))⊤Σ−1(µ − �µ(ℓ)) ≤ c∥µ−�µ(ℓ)∥2 for some constant +c > 0 under the condition that the minimum eigenvalue of Σ is bounded away from zero. +Meanwhile, it follows from Theorem 1.1 of Devroye et al. (2018) that the total variation +distance between �p and �p is upper bounded by 1.5 +�� +i λ2 +i where {λi}i denote the eigenval- +ues of Σ−1(�Σ(ℓ) − Σ), under the conditions that both �Σ(ℓ) and Σ are positive definite. The +sum of squared eigenvalues equals the squared Frobenious norm of Σ−1(�Σ(ℓ) − Σ), which +41 + +can be further upper bounded by ∥Σ−1∥2 +2∥�Σ(ℓ) − Σ∥2 +F ≤ c2∥�Σ(ℓ) − Σ∥2 +F. Therefore, +DTV(p, �p) ≤ c +2∥µ − �µ(ℓ)∥2 + 3c +2 ∥Σ − �Σ(ℓ)∥F. +To summarize, we have shown that +DTV(�d(ℓ)(•|a, s), dπ(ℓ) +old(a, s)) ≤ γT ′ + M−1/2 ++ +O(1)E +(A∗,S∗)∼qπ(ℓ) +old(•,•;a,s)[∥µ(S∗, A∗) − �µ(ℓ)(S∗, A∗)∥2 + ∥Σ(S∗, A∗) − �Σ(ℓ)(S∗, A∗)∥F], +for some positive constant O(1). +According to the Cauchy-Schwarz inequality, the aggregated squared total variation +distance is upper bounded by +O(1)E(a,s)∼p∞{E +(A∗,S∗)∼qπ(ℓ) +old(•,•;a,s)[∥µ(S∗, A∗) − �µ(ℓ)(S∗, A∗)∥2 + ∥Σ(S∗, A∗) − �Σ(ℓ)(S∗, A∗)∥F]}2 ++3γ2T ′ + 3 +M , +for some positive constant O(1). Using the Cauchy Schwarz inequality again and notice +that p∞ is bounded away from zero, the first term can be further upper bounded by +O(1)E(a,s)∼p∞E +(A∗,S∗)∼qπ(ℓ) +old(•,•;a,s)[∥µ(S∗, A∗) − �µ(ℓ)(S∗, A∗)∥2 + ∥Σ(S∗, A∗) − �Σ(ℓ)(S∗, A∗)∥F]2 +≤ O(1)E(A∗,S∗)∼p∞[∥µ(S∗, A∗) − �µ(ℓ)(S∗, A∗)∥2 + ∥Σ(S∗, A∗) − �Σ(ℓ)(S∗, A∗)∥F]2. +This completes the proof. +C +Proofs +Throughout this section, we use C and c to denote some generic constant whose value is +allowed to change from place to place. +C.1 +Proof of Theorem 1 +The proof of Theorem 1 is straightforward. We first note that, due to the trust-region +condition in (20), we have +(1 − γ) 1 +L +L +� +ℓ=1 +ES∗∼νDKL(π(ℓ) +old(•|S∗), πnew(•|S∗)) ≤ δ, +(31) +as �d(ℓ),ν(s) ≥ (1 − γ)ν(s) for any s. +42 + +Next, using similar arguments in the proof of Lemma 1, we can show +|V(πnew) − V(πold)| = +�����ES∗∼dπnew +� +a +{πnew(a|S∗) − πold(a|S∗)}Aπold(a, S∗) +����� +≤ O(1)ES∗∼dπnew∥πnew(•|S∗) − πold(•|S∗)∥TV ≤ O(1) +� +ES∗∼dπnewDKL(πold(•|S∗), πnew(•|S∗)), +where O(1) denotes some positive constant. +The first equality is due to (5), the first +inequality is due to the condition that the immediate reward is uniformly bounded (and so +is the advantage function), the second inequality follows from Pinsker’s inequality. Under +the condition that ν(·) is bounded uniformly away from zero, we obtain that +ES∗∼dπnewDKL(πold(•|S∗), πnew(•|S∗)) ≤ CES∗∼νDKL(πold(•|S∗), πnew(•|S∗)), +for some constant C > 0. It follows that +�����V(πnew) − 1 +L +L +� +ℓ=1 +V(π(ℓ) +old) +����� ≤ 1 +L +L +� +ℓ=1 +|V(πnew) − V(π(ℓ) +old)| +≤ c +L +� +ℓ=1 +� +ES∗∼νDKL(π(ℓ) +old(•|S∗), πnew(•|S∗)) ≤ cL +� +� +� +� +L +� +ℓ=1 +ES∗∼νDKL(π(ℓ) +old(•|S∗), πnew(•|S∗)), +for some constant c > 0. Theorem 1 thus follows from (31). +C.2 +Proof of Theorem 2 +We begin with some auxiliary lemmas. +Lemma 2 Under (C5), there exists some constant c0 > 0 such that for any π, +V(πopt) − V(π) ≥ c0{ES∗∼dπopt,ν∥πopt(•|S∗) − π(•|S∗)∥TV}1+1/α, +where α is the exponent in (C5). +Lemma 3 Let �η∗ +1(π, π(ℓ) +old) = (|Iℓ|T)−1 � +i∈Iℓ +� +t 0 be a positive constant, such +43 + +that supf∈F E{f 2(Z0)} ≤ σ2. Suppose the envelop function is uniformly bounded by some +constant C > 0. In addition, suppose F belongs to the class of VC-type functions such that +supQ N(F, eQ, ε∥F∥Q,2) ≤ (A/ε)v for some A ≥ e, v ≥ 1. Then +sup +f∈F +����� +T−1 +� +t=0 +f(Zt) +����� = Op +�� +vqσ2T log +�AC +σ +� ++ vC log +�AC +σ +� ++ q +� +, +for any 1 ≤ q < T/2 such that Tβ(q)/q = o(1). +We next sketch an outline of the proof. We aim to provide an upper bound for the +value difference V(πopt) − V(πnew). It can be represented by +1 +(1 − γ)L +L +� +ℓ=1 +� +η1(πopt, π(ℓ) +old) + η2(πopt, π(ℓ) +old) − η1(πnew, π(ℓ) +old) − η2(πnew, π(ℓ) +old) +� +. +We break the rest of the proof into several steps. In the first step, we provide upper bounds +for the high-order remainder terms η2(πopt, π(ℓ) +old) and η2(πnew, π(ℓ) +old). +We first show that when δ is set to a sufficiently small constant, the high-order remainder +terms |η2(πopt, π(ℓ) +old)| + |η2(πnew, π(ℓ) +old)| can be upper bounded from above by ǫ{V(πopt) − +V(πnew)}+c{V(πopt)−V(π(ℓ) +old)}(α+2)/(α+1) for some constants c > 0, 0 < ǫ ≤ (1−γ)/2, with +probability approaching 1 (w.p.a.1). It follows that +V(πopt) − V(πnew) − +ǫ +1 − γ {V(πopt) − V(πnew)} ≤ c +L +L +� +ℓ=1 +{V(πopt) − V(π(ℓ) +old)}(α+2)/(α+1) ++ +1 +(1 − γ)L +L +� +ℓ=1 +{η1(πopt, π(ℓ) +old) − η1(πnew, π(ℓ) +old)}. +Under the given conditions on the initial policy, we have that +V(πopt) − V(πnew) ≤ Op{(NT)− α+2 +α+1κ0} + +2 +(1 − γ)L +L +� +ℓ=1 +{η1(πopt, π(ℓ) +old) − η1(πnew, π(ℓ) +old)}, +w.p.a.1. +It suffices to upper bound L−1 �L +ℓ=1{η1(πopt, π(ℓ) +old)−η1(πnew, π(ℓ) +old)}. Note that πnew is ob- +tained by maximizing �η1(π), we have �η1(πnew) ≥ �η1(πopt). Consequently, L−1 �L +ℓ=1{η1(πopt, πold)− +η1(πnew, πold)} ≤ L−1 �L +ℓ=1{η1(πopt, πold)−η1(πnew, πold)−�η1(πnew)+�η1(πopt)}. Thus, it suf- +fices to provide an upper bound for +����� +1 +L +L +� +ℓ=1 +{η1(πopt, π(ℓ) +old) − η1(πnew, π(ℓ) +old) − �η1(πnew) + �η1(πopt)} +����� . +44 + +By Lemma 3, the above quantity can be upper bounded by +����� +1 +L +L +� +ℓ=1 +{η1(πopt, π(ℓ) +old) − η1(πnew, π(ℓ) +old) − �η∗ +1(πnew) + �η∗ +1(πopt)} +����� ++ +[op{(NT)−1/(2+2α)} + Op{(NT)}κ4/2−1/2]ES∗∼dπopt,ν∥πopt(•|S∗) − πnew(•|S∗)∥TV. +In the second step, we show the first term can be upper bounded by +ES∗∼dπopt,νDTV{πnew(•|S∗), πopt(•|S∗)}Op{(NT)κ4/2−1/2 log(NT)}. +By Lemma 2, we obtain +V(πopt) − V(πnew) ≤ Op{(NT)− α+2 +α+1 κ0} + {V(πopt) − V(πnew)}α/(1+α)(I1 + I2), +where I1 = Op{(NT)κ4/2−1/2 log(NT)} and I2 = op{(NT)−1/(2+2α)}. Using H¨older’s in- +equality, the second term on the right hand side can be upper bounded by {V(πopt) − +V(πnew)}/2+I(1+α) +1 ++I(1+α) +2 +. This yields V(πopt)−V(πnew) = O{(NT)−κ0 α+2 +α+1}+op{(NT)−1/2}, +under the given condition on κ4. The proof is hence completed. +In the last three steps, we present the proofs of Lemmas 2, 3 and 4. We next present +the details for each of the step. +Step 1. We aim to bound the high-order remainder term |η2(πopt, π(ℓ) +old)| and |η2(πnew, π(ℓ) +old)|. +We first consider |η2(πopt, π(ℓ) +old)|. By definition, we have η2(πopt, π) = η(1) +2 (πopt, π)+η(2) +2 (πopt, π) +for any π where +η(1) +2 (πopt, π) += +� +a∈A,s∈S +{πopt(a|s) − π(a|s)}Aπopt(a, s){dπopt,ν(s) − dπ,ν(s)}, +η(2) +2 (πopt, π) += +� +a∈A,s∈S +{πopt(a|s) − π(a|s)}{Aπ(a, s) − Aπopt(a, s)}{dπopt,ν(s) − dπ,ν(s)}. +Consequently, it suffices to bound |η(j) +2 (πopt, π(ℓ) +old)| for j = 1, 2. +We first consider |η(1) +2 (πopt, π(ℓ) +old)|. Using similar arguments in the proof of Lemma 1 (see +Equation 24), we can show it is upper bounded by +O(1)ES∗∼dπopt,ν∥πopt(•|S∗) − π(ℓ) +old(•|S∗)∥TVE +S∗∼dπ(ℓ) +old,ν +� +a∈A +|π(ℓ) +old(a|S∗) − πopt(a|S∗)||Aπopt(a, s)|, +where O(1) denotes some positive constant. +In the proof of Lemma 2, we show that +{π(a|s)−πopt(a|s)}Aπopt(a, s) is nonpositive for any a, s and π. Consequently, ES∗∼dπ,ν � +a∈A |π(a|S∗)− +πopt(a|S∗)||Aπopt(a, S∗)| = ES∗∼dπ,νold +� +a∈A{πopt(a|S∗) − π(a|S∗)}Aπopt(a, S∗) = V(πopt) − +V(π) for any π. It follows from Lemma 2 that for any π, +45 + +|η(1) +2 (πopt, π(ℓ) +old)| ≤ c{V(πopt) − V(π(ℓ) +old)}ES∗∼dπopt,ν∥πopt(•|S∗) − π(ℓ) +old(•|S∗)∥TV +≤ c1{V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 , +(32) +for some constant c1 > 0. +We next consider |η(2) +2 (πopt, π(ℓ) +old)|. Note that Aπ(a, s)−Aπopt(a, s) = γES∗∼p(•|a,s){V π(S∗)− +V πopt(S∗)}+V πopt(s)−V π(s) for any π. Under the given conditions, there exists some con- +stant C > 0 such that ν(s) ≥ C for any s. Using the change of measure, it follows that +ES∗∼p(•|a,s){V πopt(S∗) − V π(S∗)} ≤ ES∗∼ν{V πopt(S∗) − V π(S∗)}p(S∗|a, s) +ν(S∗) +≤ C−1ES∗∼ν{V πopt(S∗) − V π(S∗)} = C−1{V(πopt) − V(π)}, +(33) +and V πopt(s) − V π(s) ≤ C−1{V(πopt) − V(π)}. +Using similar arguments in (24), we obtain +|η(2) +2 (πopt, π)| ≤ O(1)ES∗∼dπopt,ν +� +a∈A +|π(a|S∗) − πopt(a|S∗)| max +a,s |Aπopt(a, s) − Aπ(a, s)| +≤ O(1){V(πopt) − V(π)}ES∗∼dπopt,ν ∥π(•|S∗) − πopt(•|S∗)∥TV, +where O(1) denotes some positive constant. Similar to (32), we obtain |η(2) +2 (πopt, π(ℓ) +old)| ≤ +c2{V(πopt) − V(π(ℓ) +old)}(2α+1)/(α+1) for some constant c2 > 0. This together with (32) yields +|η2(πopt, π(ℓ) +old)| ≤ (c1 + c2){V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 . +We next bound |η2(πnew, π(ℓ) +old)|. We note that |η2(πnew, π(ℓ) +old)| can be upper bounded by +|η2(πnew, π(ℓ) +old)| ≤ |η(3) +2 (πnew, π(ℓ) +old)| + |η(4) +2 (πnew, π(ℓ) +old)| +≡ +� +a,s +|πnew(a|s) − π(ℓ) +old(a|s)||Aπopt(a, s) − Aπ(ℓ) +old(a, s)||dπnew,ν(s) − dπ(ℓ) +old,ν(s)| ++ +� +a,s +|πnew(a|s) − π(ℓ) +old(a|s)||Aπopt(a, s)||dπnew,ν(s) − dπ(ℓ) +old,ν(s)|. +Using similar arguments in bounding |η(2) +2 (πopt, π(ℓ) +old)|, |η(3) +2 (πnew, π(ℓ) +old)| can be upper bounded +by O(1){V(πopt)−V(π(ℓ) +old)}ES∗∼dπopt,ν∥πnew(a|S∗)−π(ℓ) +old(a|S∗)∥TV where O(1) denotes some +positive constant. By triangle inequality, we have +ES∗∼dπopt,ν∥πnew(a|S∗) − π(ℓ) +old(a|S∗)∥TV ≤ ES∗∼dπopt,ν∥πopt(a|S∗) − π(ℓ) +old(a|S∗)∥TV ++ES∗∼dπopt,ν∥πopt(a|S∗) − πnew(a|S∗)∥TV. +By Lemma 2, the two terms on the RHS can be upper bounded by c−1 +0 {V(πopt)−V(π(ℓ) +old)}α/(1+α) +46 + +and c−1 +0 {V(πopt) − V(πnew)}α/(1+α), respectively. Consequently, +|η(3) +2 (πnew, π(ℓ) +old)| ≤ O(1){V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 ++O(1){V(πopt) − V(πnew)} +α +α+1{V(πopt) − V(π(ℓ) +old)}, +for some positive constant O(1). +Similarly, η(4) +2 (πnew, π(ℓ) +old) can be upper bounded by +|η(4) +2 (πnew, π(ℓ) +old)| ≤ +� +a,s +|πnew(a|s) − πopt(a|s)||Aπopt(a, s)||dπnew,ν(s) − dπ(ℓ) +old,ν(s)| ++ +� +a,s +|π(ℓ) +old(a|s) − πopt(a|s)||Aπopt(a, s)||dopt,ν(s) − dπ(ℓ) +old,ν(s)| ++ +� +a,s +|π(ℓ) +old(a|s) − πopt(a|s)||Aπopt(a, s)||dπnew,ν(s) − dopt,ν(s)|. +Using similar arguments in the proofs of Lemma 1 and Theorem 1, the first term on the +RHS can be upper bounded by O( +√ +δ{V(πopt) − V(πnew)}). The second and third terms +can be upper bounded by +O(1){V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 + O(1){V(πopt) − V(πnew)} +α +α+1{V(πopt) − V(π(ℓ) +old)}, +using similar arguments in bounding |η(3) +2 (πnew, π(ℓ) +old)|. +To summarize, we have shown +|η2(πnew, π(ℓ) +old)| ≤ O(1) +√ +δ{V(πopt) − V(πnew)} + O(1){V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 ++O(1){V(πopt) − V(πnew)} +α +α+1{V(πopt) − V(π(ℓ) +old)}, +for some positive constant O(1). By H¨older’s inequality, the last term on the second line +can be upper bounded by +√ +δ{V(πopt) − V(πnew)} + O(1){V(πopt) − V(π(ℓ) +old)}α+1/ +√ +δ. When +V(π(ℓ) +old) is consistent to V(πopt), {V(πopt) − V(π(ℓ) +old)}α+1/ +√ +δ = {V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 / +√ +δ. +It follows that +|η2(πnew, π(ℓ) +old)| ≤ O(1) +√ +δ{V(πopt) − V(πnew)} + O(1){V(πopt) − V(π(ℓ) +old)} +2α+1 +α+1 . +The proof is hence completed. +Step 2. We begin with some notations. For ℓ = 1, · · · , L, we use ψ(o; π, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ)) +to denote ψ(o; π, π(ℓ) +old, �V (ℓ), �A(ℓ), �ω(ℓ), �d(ℓ)) as both �V (ℓ) and �A(ℓ) are derived from �Q(ℓ). Sim- +ilarly, we use the notations ψ1(π, π(ℓ) +old, �Q(ℓ), �d(ℓ)), ψ2(o; π, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ)) and +ψ3(o; π, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ)) to denote ψ1(π, π(ℓ) +old, �A(ℓ), �d(ℓ)), ψ2(o; π, π(ℓ) +old, �V (ℓ), �A(ℓ), �ω(ℓ), �d(ℓ)) +47 + +and ψ3(o; π, π(ℓ) +old, �A(ℓ), �ω(ℓ), �d(ℓ)). Notations ψ1(π, π(ℓ) +old, Qπ(ℓ) +old, dπ(ℓ) +old), ψ2(o; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old), +ψ3(o; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old) and ψ(o; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old) can be similarly defined. +We aim to apply Lemma 4 to show +sup +π∈Π +|η1(πopt, π(ℓ) +old) − η1(π, π(ℓ) +old) + �η∗ +1(π, π(ℓ) +old) − �η∗ +1(πopt, π(ℓ) +old)| +ES∗∼dπopt∥π(•|S∗) − π(ℓ) +old(•|S∗)∥TV += Op{(NT) +κ4−1 +2 +log(NT)}, +where �η∗ +1(π, π(ℓ) +old) = (|Iℓ|T)−1 � +i∈Iℓ +�T +t=0 ψ(Oi,t; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old). Toward that end, +we first note that it follows from (C6) that {(St, At, Rt, St+1)}t≥0 is exponentially β-mixing. +Since the observed data set consists of multiple independent trajectories, it is exponentially +β-mixing as well. Consequently, by setting the integer q in Lemma 4 to c log(NT) for some +sufficiently large constant c, it follows that NTβ(q)/q = o(1). +Under the given conditions, the immediate reward and the conditional probability ratio +are uniformly bounded. By (C4), Π belongs to a VC type function class with bounded +envelop function and VC index bounded by O{(NT)κ4}. +So is the class of functions +{ψ(o; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old) − ψ(o; π, πopt, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old) : π ∈ Π}. In view of Lemma +4, it suffices to show +Var{ψ(O0; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old) − ψ(O0; π, πopt, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old)} +≤ c{ES∗∼dπopt∥π(•|S∗) − π(ℓ) +old(•|S∗)∥TV}2, +and +|ψ(O0; π, π(ℓ) +old, Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old) − ψ(O0; π, πopt(ℓ), Qπ(ℓ) +old, ωπ(ℓ) +old, dπ(ℓ) +old)| +≤ cES∗∼dπopt∥π(•|S∗) − π(ℓ) +old(•|S∗)∥TV, +almost surely for some constant c > 0. This is immediate to see by the definition of ψ. We +omit the details for brevity. +Step 3. We prove Lemma 2 in this step. By (5), we obtain +V(πopt) − V(π) = − +1 +1 − γ ES∗∼dπ,ν +� +a∈A +π(a|S∗)Aπopt(a, S∗) += − +1 +1 − γ ES∗∼dπ,ν +� +a∈A +{π(a|S∗) − πopt(a|S∗)}Aπopt(a, S∗). +(34) +As discussed in Section 4.1, Aπopt(a, s) ≤ 0 for any a and s. We next claim +{π(a|s) − πopt(a|s)}Aπopt(a, s) ≤ 0, +∀a, s. +(35) +48 + +If a = argmaxa′Qπopt(a′, s), then Aπopt(a, s) = 0 and (35) is automatically satisfied. Other- +wise, we have Aπopt(a, s) < 0 and πopt(a|s) = 0. It follows that (35) holds. +Combining (34) with (35) yields that +V(πopt) − V(π) = +1 +1 − γ ES∗∼dπ,ν +� +a∈A +|π(a|S∗) − πopt(a|S∗)||Aπopt(a, S∗)|. +Since dπ,ν(s) ≥ (1 − γ)ν(s) and ν is uniformly bounded away from zero, there exists some +constant C > 0 such that dπ,ν(s) ≥ (1 − γ)C for any s. Similar to (33), we can show +V(πopt) − V(π) = +1 +1 − γ ES∗∼dπ,ν +� +a∈A +|π(a|S∗) − πopt(a|S∗)||Aπopt(a, S∗)| dπ,ν(S∗) +dπopt,ν(S∗) +≥ CES∗∼dπopt,ν +� +a∈A +|π(a|S∗) − πopt(a|S∗)||Aπopt(a, S∗)|. +Denote a∗ = argmaxa′Qπopt(a′, S∗). Then Aπopt(a∗, S∗) = 0. For any ǫ > 0, it follows that +V(πopt) − V(π) ≥ CES∗∼dπopt,ν +� +a∈A,a̸=a∗ +|π(a|S∗) − πopt(a|S∗)||Aπopt(a, S∗)|I(Aπopt(a, S∗) ≤ −ǫ) +≥ ǫCES∗∼dπopt,ν +� +a∈A,a̸=a∗ +|π(a|S∗) − πopt(a|S∗)|I(Aπopt(a, S∗) ≤ −ǫ) +≥ ǫCES∗∼dπopt,ν +� +a∈A,a̸=a∗ +|π(a|S∗) − πopt(a|S∗)| +−ǫCES∗∼dπopt,ν +� +a∈A,a̸=a∗ +|π(a|S∗) − πopt(a|S∗)|I(−ǫ < Aπopt(a, S∗) < 0). +Under the margin-type condition in (C5), the last line is O(ǫα+1). We choose +ǫ = c{ES∗∼dπopt,ν +� +a∈A,a̸=a∗ +|π(a|S∗) − πopt(a|S∗)|}1/α, +for some constant c > 0. With some property choice of c, the last line is lower bounded by +ǫCES∗∼dπopt,ν +� +a∈A,a̸=a∗ |π(a|S∗) − πopt(a|S∗)|/2. Note that +� +a∈A,a̸=a∗ +|π(a|S∗) − πopt(a|S∗)| = |π(a∗|S∗) − πopt(a∗|S∗)| +Consequently, +V(πopt) − V(π) ≥ cC +4 {ES∗∼dπopt,ν +� +a∈A +|π(a|S∗) − πopt(a|S∗)|}1+1/α. +The proof is completed by noting that � +a∈A |π(a|S∗)−πopt(a|S∗)| = 2∥π(•|S∗)−πopt(•|S∗)∥TV. +Step 4. We prove Lemma 3 in this step. We first show the bias term +49 + +∆ψ(π1, π2, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ)) = E[ψ(O0; π1, πold, �Q(ℓ), �ω(ℓ), �d(ℓ))|{Oi,t}i∈Iℓ,0≤t 0. Note that �d(ℓ),ν(s)−dπ(ℓ) +old,ν(s) can be represented by ES∗∼ �d(ℓ),νI(S∗ = +s)−E +S∗∼dπ(ℓ) +old,νI(S∗ = s) = � +a∗,s∗ π(ℓ) +old(a∗|s∗)ν(s∗)(ES∗∼ �d(ℓ)(•|a∗,s∗)I(S∗ = s)−E +S∗∼dπ(ℓ) +old(•|a∗,s∗)I(S∗ = +50 + +s)). As such, it can be upper bounded by +O(1)ES∗∼dπopt,ν|π1(a|S∗) − π2(a|S∗)| +� +a +ES0∼νπ(ℓ) +old(a|S0)DTV{dπ(ℓ) +old(•|a, S0), �d(ℓ)(•|a, S0)}, +where O(1) denotes some positive constant. Since b is uniformly bounded away from zero, +it can be further upper bounded by +O +� +ES∗∼dπopt,ν|π1(a|S∗) − π2(a|S∗)|E(A,S)∼p∞DTV{dπ(ℓ) +old(•|A, S), �d(ℓ)(•|A, S)} +� +. +Consequently, (37) is Op{(NT)−κ3}ES∗∼dπopt,ν|π1(a|S∗) − π2(a|S∗)| under (C3). +Under +the conditions that κ1 + κ3 > 1/(2 + 2α), we obtain ∆ψ(π1, π2, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ)) = +op{(NT)−1/(2+2α)}ES∗∼dπopt,ν|π1(a|S∗) − π2(a|S∗)|. +It suffices to bound ∆ψ(π1, π2, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ)) − ∆ψ(π1, π2, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ)), or +equivalently, +E{ψ2(O0; π1, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ)) − ψ2(O0; π1, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ))} +−E{ψ2(O0; π2, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ)) − ψ2(O0; π2, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ))} +(38) +E{ψ3(O0; π1, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ)) − ψ3(O0; π1, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ))} +−E{ψ3(O0; π2, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ)) − ψ3(O0; π2, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ))}. +(39) +Similarly, we can show (38) and (39) can be upper bounded by +C +� +E(A,S)∼p∞| �Q(ℓ)(A, S) − Qπ(ℓ) +old(A, S)|2 +� +E(A,S)∼p∞ +( � +A, �S∼p∞) +|�ω(ℓ)( �A, �S; A, S) − ωπ(ℓ) +old( �A, �S; A, S)|2 +×ES∗∼dπopt,ν∥π1(•|S∗) − π2(•|S∗)∥TV, +and +C +� +E(A,S)∼p∞D2 +TV{dπ(ℓ) +old(•|A, S), �d(ℓ)(•|A, S)} +� +E(A,S)∼p∞ +( � +A, �S)∼p∞ +|�ω(ℓ)( �A, �S; A, S) − ωπ(ℓ) +old( �A, �S; A, S)|2 +×ES∗∼dπopt,ν∥π1(•|S∗) − π2(•|S∗)∥TV, +for some constant C > 0. Under (C2) and (C3), we obtain ∆ψ(π1, π2, π(ℓ) +old, �Q(ℓ), ωπ(ℓ) +old, �d(ℓ))− +∆ψ(π1, π2, π(ℓ) +old, �Q(ℓ), �ω(ℓ), �d(ℓ)) = op{(NT)−1/(2+2α)}. +It remains to show supπ1,π2∈Π |�η(ℓ)(π1)−�η∗(π1, π(ℓ) +old)−�η(ℓ)(π2)+�η∗(π2, π(ℓ) +old)−∆ψ(π1, π2, π(ℓ) +old)| = +op{(NT)κ4/2−1/2}. This can be proven using similar arguments in Step 2. We omit the de- +tails to save space. +Step 5. +We prove Lemma 4 in the last step. +We first use Berbee’s coupling lemma +51 + +(see Lemma 4.1 in Dedecker and Louhichi, 2002) to approximate supf∈F | �T−1 +t=0 f(Zt)| +by sum of i.i.d. +variables. +Then we apply the maximal inequality in Corollary 5.1 of +Chernozhukov et al. (2014) to bound the expectation of the empirical process. +Following the discussion below Lemma 4.1 of Dedecker and Louhichi (2002), we can +construct a sequence of random variables {Z0 +t : t ≥ 0} such that +sup +f∈F +����� +T−1 +� +t=0 +f(Zt) +����� = sup +f∈F +����� +T−1 +� +t=0 +f(Z0 +t ) +����� , +(40) +with probability at least 1 − Tβ(q)/q, and that the sequences {U0 +2i : i ≥ 0} and {U0 +2i+1 : +i ≥ 0} are i.i.d. where U0 +i = (Z0 +iq, Z0 +iq+1, · · · , Z0 +iq+q−1). +Recall that Ir = {q⌊T/q⌋, q⌊T/q⌋ + 1, · · · , T − 1}, we have +sup +f∈F +����� +T−1 +� +t=0 +f(Z0 +t ) +����� ≤ +q−1 +� +j=0 +sup +f∈F +������ +⌊T/q⌋ +� +t=0 +f(Z0 +tq+j) +������ ++ sup +f∈F +����� +� +t∈Ir +f(Z0 +t ) +����� . +Under the boundedness assumption on F, the second term on the right-hand-side (RHS) +is bounded from above by Cq. Without loss of generality, suppose ⌊T/q⌋ is an even number. +The first term on the RHS can be bounded from above by �2q−1 +j=0 supf∈F | �⌊T/(2q)⌋ +t=0 +f(Z0 +2tq+j)|. +To summarize, we have shown +sup +f∈F +����� +T−1 +� +t=0 +f(Z0 +t ) +����� ≤ +2q−1 +� +j=0 +sup +f∈F +������ +⌊T/(2q)⌋ +� +t=0 +f(Z0 +2tq+j) +������ ++ Cq. +(41) +By construction, {Z0 +2tq : t ≥ 0} are i.i.d. It remains to bound E supf∈F | �⌊T/(2q)⌋ +t=0 +f(Z0 +2tq)|. +It follows from Corollary 5.1 of Chernozhukov et al. (2014) that +E sup +f∈F +������ +⌊T/(2q)⌋ +� +t=0 +f(Z0 +2tq) +������ +⪯ +� +vσ2T +q +log +�AC +σ +� ++ vC log +�AC +σ +� +. +The assertion thus follows from (40), (41) and Markov’s inequality. +C.3 +Proof of Theorem 3 +We omit the subscript ℓ in π(ℓ) +old to easy notation. Similar to Lemma 3, we can show that +�η1(π, πold) = �η∗ +1(π, πold) + op{(NT)−1/2} for any π, πold ∈ Π, when the nuisance functions +converge at rates faster than op((NT)−1/4). Note that �η∗ +1(π, πold) is unbiased to η1(π, πold). +It suffices to show +52 + +�η∗ +1(π, πold) − η1(π, πold) +� +EB(π, πold) +d→ N(0, 1). +(42) +By definition, �η∗ +1(π, πold)−η1(π, πold) = (NT)−1 �N +i=1 +�T−1 +t=0 {ψ2(Oi,t; π, πold, Qπold, ωπold, dπold)+ +ψ3(Oi,t; π, πold, Qπold, ωπold, dπold)}. The sum on the RHS can be represented as +1 +NT +NT +� +g=1 +{ψ2(Oi(g),t(g); π, πold, Qπold, ωπold, dπold) + ψ3(Oi(g),t(g); π, πold, Qπold, ωπold, dπold)},(43) +where the pair i(g) and t(g) are the unique integers that satisfy {i(g) − 1}T + t(g) = g − 1. +Under (A1) and (A2), (43) corresponds to a sum of martingale difference sequence. Suppose +we can show +Var{ψ2(Oi(g),t(g); π, πold, Qπold, ωπold, dπold) + ψ3(Oi(g),t(g); π, πold, Qπold, ωπold, dπold)} += NTEB(π, πold). +(44) +Under the given assumptions, we can show the conditions in Theorem 1 of Brown et al. +(1971) are automatically satisfied. It follows from the martingale central limit theorem +developed by Brown et al. (1971) that (42) holds. Consequently, it suffices to show (44) +holds. +Note that η1 depends only on the transition function ¯p. +For a given tuple O = +(S, A, R, S′) such that (A, S) ∼ p∞, (S′, R|A, S) ∼ ¯p(•, •|A, S), suppose we can show +(45) holds, +∇θ1η1(θ∗ +1) = E +3 +� +j=2 +ψj(O; π, πold, Qπold, ωπold, dπold)∇θ1 log ¯pθ∗ +1(•, •|A, S). +(45) +Then it follows from Cauchy-Schwarz inequality that +NTEB(N, T) = sup ∇θ1η1(θ∗ +1) +� +E∇θ1 log ¯pθ∗ +1(S′, R|A, S)∇θ1 log ¯p⊤ +θ∗ +1(S′, R|A, S) +�−1 +{∇θ1η1(θ∗ +1)}⊤ +≤ E +� +3 +� +j=2 +ψj(O; π, πold, Qπold, ωπold, dπold) +� � +3 +� +j=2 +ψj(O; π, πold, Qπold, ωπold, dπold) +�⊤ +. +This implies that the variance of the proposed estimator is asymptotically greater than +or equal to the efficiency bound. +Moreover, note that for either j = 2 or 3, we have +E{ψj(O; π, πold, Qπold, ωπold, dπold)|A, S} = 0, almost surely. Using similar arguments in the +proof of Theorem 2 of Kallus and Uehara (2019), we can show that there exists a sequence +of regular parametric submodels whose Cr´amer-Rao lower bound approaches to +53 + +E +� +3 +� +j=2 +ψj(O; π, πold, Qπold, ωπold, dπold) +� � +3 +� +j=2 +ψj(O; π, πold, Qπold, ωπold, dπold) +�⊤ +. +This implies that the variance of the proposed estimator is asymptotically equal to the +efficiency bound. The proof is hence completed. +It remains to show (45). We first observe that, the advantage function and the dis- +counted visitation probability are completely determined by the transition function ¯p. By +the chain rule, +∇θ1η1(θ∗ +1) = +� +a,s +π(a|s){∇θ1Aπold(a, s; θ∗ +1)}dπold,ν(s) + +� +a,s +π(a|s)Aπold(a, s)∇θ1dπold,ν(s; θ∗ +1), +where Aπold(•, •; θ1) and dπold,ν(•; θ1) denote the advantage and the discounted visitation +probability under certain parametric submodel for ¯p indexed by θ1. +Using similar arguments in the proof of Theorem 2 in Kallus and Uehara (2019), we +can show that +{∇θ1Aπold(a, s; θ∗ +1)} = +1 +1 − γ E +� +ωπold(A, S; a, s) − +� +a′ +πold(a′|s)ωπold(A, S; a′, s) +� +×{R + γV πold(S′) − Qπold(A, S)}∇θ1 log ¯p⊤ +θ∗ +1(S′, R|A, S). +This in turns yields that +� +a,s +π(a|s){∇θ1Aπold(a, s; θ∗ +1)}dπold,ν(s) = Eψ2(O; π, πold, Qπold, ωπold, dπold)∇θ1 log ¯p⊤ +θ∗ +1(S′, R|A, S). +It remains to show +� +a,s +π(a|s)Aπold(a, s)∇θ1dπold,ν(s; θ∗ +1) = Eψ3(O; π, πold, Qπold, ωπold, dπold)∇θ1 log ¯p⊤ +θ∗ +1(S′, R|A, S). +For a given parametric submodel {¯pθ1 : θ1 ∈ Θ1}, let p(s′|a, s; θ1) = � +r ¯p(s′, r|a, s; θ1). +With some calculations, we have +54 + +1 +γ(1 − γ)∇θ1dπold,ν(s; θ∗ +1) = +� +t≥0 +γt +� +{(aj,sj)}t +j=0 +∇θ1 +� t� +j=0 +πold(aj|sj)p(s|aj, sj; θ∗ +1) +� +ν(s0) += +� +t≥0 +γt +� +{(aj,sj)}t +j=0,st+1 +I(st+1 = s)∇θ1 +� t� +j=0 +πold(aj|sj)p(sj+1|aj, sj; θ∗ +1) +� +ν(s0) += +� +t≥0 +γt +� +{(aj,sj)}t +j=0 +st+1 +I(st+1 = s) +t� +j=0 +πold(aj|sj)p(sj+1|aj, sj) +t +� +k=0 +∇θ1 log p(sk+1|ak, sk; θ∗ +1)ν(s0) += +1 +1 − γ ++∞ +� +k=0 +γk +� +{(aj,sj)}k +j=0 +sk+1,a +πold(a|sk+1)dπold(s; a, sk+1)∇θ1 log p(sk+1|ak, sk; θ∗ +1) +× +� k +� +j=0 +πold(aj|sj)p(sj+1|aj, sj) +� +ν(s0). +By definition of ωπold, the last equation can be rewritten as +1 +(1 − γ)2 +� +a +πold(a|S′)dπold(s; a, S′)ωπold,ν(A, S)∇θ1 log p(S′|A, S; θ∗ +1). +Note that Eh(A, S)∇θ1 log p(S′|A, S; θ∗ +1) = 0 for any function h, we obtain +∇θ1dπold,ν(s; θ∗ +1) = +1 +1 − γ E +� +γ +� +a +πold(a|S′)dπold(s; a, S′) − E +� +γ +� +a +πold(a|S′)dπold(s; a, S′)|A, S +�� +×ωπold,ν(A, S)∇θ1 log p(S′|A, S; θ∗ +1) += +1 +1 − γ E +� +γ +� +a +πold(a|S′)dπold(s; a, S′) − dπold(s; A, S) + (1 − γ)I(s = S) +� +×ωπold,ν(A, S)∇θ1 log p(S′|A, S; θ∗ +1). +Consequently, we obtain +� +a,s +π(a|s)Aπold(a, s)∇θ1dπold,ν(s; θ∗ +1) = Eψ3(O; π, πold, Qπold, ωπold, dπold)∇θ1 log ¯p⊤ +θ∗ +1(S′, R|A, S). +The proof is hence completed. +55 + +D +Some additional numerical details +D.1 +Additional Results on The Toy Example +In this subsection, we first list some details of our toy example. In particular, we consider +the following transition matrix: +p = + + + + + + + +S′ = 0 +S′ = 1 +(S = 0, A = 0) +0.75 +0.25 +(S = 0, A = 1) +0.4 +0.6 +(S = 1, A = 0) +0.1 +0.9 +(S = 1, A = 1) +0.85 +0.15 + + + + + + + +The behavior policy to generate the simulated data is +b = + + +A = 0 +A = 1 +S = 0 +0.7 +0.3 +S = 1 +0.2 +0.8 + +. +Below is the detailed design of each scenario for testing the value enhancement property +and the triple robustness. +(i) origin: all the nuisance functions are set to their oracle values. +(ii) mod1: we inject uniform(0, 2) noises to the marginal density ratio, whereas other +nuisance functions are set to their oracle values. +(iii) mod2: we inject uniform(0, 2) noises to the Q-function, whereas other nuisance func- +tions are set to their oracle values. +(iv) mod3: we multiply the transition matrix p by a random variable following exponential(1) +and clip all values into [0, 1], whereas other nuisance functions are set to their oracle +values. +(vi) mod4: we inject random errors to all the three nuisance functions according to the +procedures described in (ii)-(iv). +We next report values of estimated policies in the toy example (see Section 5.1) where +δ is set to 0.05 and 0.2. These values are depicted in Figure 4 and 5, respectively. +56 + +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +Figure 4: +Values of estimated policies in a toy example. +First row represents results using +(T, N) pair as (30, 30) while the second row using (50, 50). The three columns represents initial +policy factor κ taking values 0.8, 0.5, 0.2 respectively. The horizontal axis represents the number +of iterations used in our value enhancement procedure. +When iteration equals zero, we plot +the evaluation value for the initial policy. The optimal value is 10 and δ is fixed to 0.05. The +confidence band is computed based on 100 replications. +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +0 +1 +2 +3 +ITERATION +0 +2 +4 +6 +8 +10 +Policy Value +origin +mod1 +mod2 +mod3 +mod4 +Figure 5: +Values of estimated policies in a toy example. +First row represents results using +(T, N) pair as (30, 30) while the second row using (50, 50). The three columns represents initial +policy factor κ taking values 0.8, 0.5, 0.2 respectively. The horizontal axis represents the number +of iterations used in our value enhancement procedure. When iteration equals zero, we plot the +evaluation value for the initial policy. The optimal value is 10 and δ is fixed to 0.2. The confidence +band is computed based on 100 replications. +Furthermore, to demonstrate the advantage of the proposed method, we use lookup +tables (e.g., linear models with table lookup features) instead of deep learning models to +parametrize all nuisance functions (including the Q-function, the probability ratio and the +57 + +0 +1 +2 +3 +ITERATION +2 +3 +4 +5 +6 +Policy Value +origin +0 +1 +2 +3 +ITERATION +5.0 +5.5 +6.0 +6.5 +7.0 +7.5 +8.0 +8.5 +9.0 +Policy Value +origin +0 +1 +2 +3 +ITERATION +8.0 +8.5 +9.0 +9.5 +10.0 +Policy Value +origin +Figure 6: Values of estimated policies in the toy example. All nuisance parameters and the +policy are modelled by linear functions. Results are computed using (T, N) pair as (100, 100). +These three figures represents the initial policy factor κ taking values 0.8, 0.5, 0.2 respectively. +The horizontal axis represents the number of iterations used in our value enhancement procedure. +When the iteration equals zero, we plot the evaluation value for the initial policy. The optimal +value is 10 and δ is fixed to 0.05. The confidence band is computed based on 100 replications. +The variances of the values in the first two panels are very small such that the upper and lower +confidence bands are largely overlapped. +transition kernel), and apply the proposed method to the toy example setting in Section +5.1 of the main text. Results are reported in Figure 6. It can be seen that the proposed +method is still able to improve the performance of initial policies. +D.2 +One Additional Simulation Study +In this subsection, we conduct another simulation study to show the finite sample perfor- +mance of our proposed method. Consider a 15-dimensional state vector St = (S(1) +t , · · · , S(15) +t +). +We set initiate state as a standard normal vector. Let the first two state variables evolve +according to: for t ≥ 1 +S(1) +t += (3/4)(2At−1 − 1)S(1) +t−1 + (1/4)S(2) +t−1 + ǫ(1) +t , +S(2) +t += (3/4)(1 − 2At−1)S(2) +t−1 + (1/4)S(1) +t−1 + ǫ(2) +t , +where At takes values in {0, 1} with equal probabilities and {ǫ(1) +t }t, {ǫ(2) +t }t are i.i.d. N(0, 1/4) +random errors. The data generating mechanism for these two variable are similar to those +considered in the simulation settings of Luckett et al. (2020) and Liao et al. (2020). Other +state variables are sampled independently from the standard normal distribution for all +0 ≤ t ≤ T − 1. Define the reward function by Rt = 2S(1) +t+1 + S(2) +t+1 − (1/4)(2At − 1). The +sample size pair (T, N) is set to be (50, 100), (25, 200) or (100, 50) in our experiment. We +consider two choices of γ, corresponding to 0.9 and 0.95. +To implement our method, we set the number of folds L to 2 and the constant δ to +58 + +0.05. In order to obtain πold, as discussed in Section 3.2.1, we apply VL, FQI and CQL to +compute three different initial policies in our experiment. Both FQI and CQL require to +model the optimal Q-function. In our implementation, we use a rectified linear unit (ReLU) +neural network with two hidden layers. To implement V-learning, we use the R-package +developed by Luckett et al. (2020). In particular, we use RBF basis functions to model +the state value function and linear basis functions to model the policy class. We also use +a rectified linear unit (ReLU) neural network with two hidden layers to model πnew. +Results are summarized in Figure 7. It can be seen that all these initial policies have +the potential to be improved based on our procedure. +This demonstrates the superior +performance of our value enhancement method. In addition, the improvement is substantial +when πold is not very close to the optimal policy. This is consistent with our observations +in the toy example. It can also be seen that our method may suffer from some slight value +loss when πold is very close to the optimal one. This is probably due to that we used a +fixed δ in the simulation. As shown in the toy example, we should choose a large δ when +πold is far away from the optimal policy and a small δ otherwise. It will be interesting to +study how to adaptively choose δ. However, this is beyond the scope of the current paper. +We leave it for future work. Finally, we conduct some additional studies to investigate the +performance of the proposed method in settings with a smaller sample size and a higher +noise level. In particular, Figure 8 reported results where N ×T = 3000. Figure 9 reported +results where {ǫ(1) +t }t, {ǫ(2) +t }t are i.i.d. N(0, 1) random errors. Findings are similar to those +in Figure 7. +59 + +0 +1 +2 +3 +ITERATION +6.05 +6.10 +6.15 +6.20 +6.25 +6.30 +Policy Value +VL +CQL +FQI +0 +1 +2 +3 +ITERATION +12.25 +12.30 +12.35 +12.40 +12.45 +12.50 +12.55 +Policy Value +VL +CQL +FQI +0 +1 +2 +3 +ITERATION +6.05 +6.10 +6.15 +6.20 +6.25 +6.30 +Policy Value +VL +CQL +FQI +0 +1 +2 +3 +ITERATION +12.25 +12.30 +12.35 +12.40 +12.45 +12.50 +12.55 +Policy Value +VL +CQL +FQI +0 +1 +2 +3 +ITERATION +6.05 +6.10 +6.15 +6.20 +6.25 +6.30 +Policy Value +VL +CQL +FQI +0 +1 +2 +3 +ITERATION +12.25 +12.30 +12.35 +12.40 +12.45 +12.50 +12.55 +Policy Value +VL +CQL +FQI +Figure 7: Values of our estimated policies where initial ones are computed by VL, CQL, FQI. +The first row represents results using γ = 0.9 while the second row using γ = 0.95. Three columns +represents using (T, N) pair as (50, 100), (25, 200), (100, 50) respectively. The optimal value under +γ = 0.9 is approximately 6.89 and 13.47 under γ = 0.95. The confidence band is computed based +on 100 replications. +0 +1 +2 +ITERATION +6.02 +6.04 +6.06 +6.08 +6.10 +6.12 +6.14 +6.16 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +11.95 +12.00 +12.05 +12.10 +12.15 +12.20 +12.25 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +6.04 +6.06 +6.08 +6.10 +6.12 +6.14 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +11.95 +12.00 +12.05 +12.10 +12.15 +12.20 +12.25 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +6.04 +6.06 +6.08 +6.10 +6.12 +6.14 +6.16 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +11.95 +12.00 +12.05 +12.10 +12.15 +12.20 +12.25 +12.30 +Policy Value +VL +CQL +FQI +Figure 8: Values of our estimated policies where initial ones are computed by VL, CQL, FQI. +The first row represents results using γ = 0.9 while the second row using γ = 0.95. Three columns +represents using (T, N) pair as (30, 100), (15, 200), (100, 30) respectively. The optimal value under +γ = 0.9 is approximately 6.89 and 13.47 under γ = 0.95. The confidence band is computed based +on 100 replications. +60 + +0 +1 +2 +ITERATION +24.8 +24.9 +25.0 +25.1 +25.2 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +49.8 +49.9 +50.0 +50.1 +50.2 +50.3 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +24.85 +24.90 +24.95 +25.00 +25.05 +25.10 +25.15 +25.20 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +49.8 +49.9 +50.0 +50.1 +50.2 +50.3 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +25.0 +25.1 +25.2 +25.3 +Policy Value +VL +CQL +FQI +0 +1 +2 +ITERATION +50.0 +50.1 +50.2 +50.3 +50.4 +Policy Value +VL +CQL +FQI +Figure 9: Values of our estimated policies where initial ones are computed by VL, CQL, FQI. +The first row represents results using γ = 0.9 while the second row using γ = 0.95. Three columns +represents using (T, N) pair as (50, 100), (25, 200), (100, 50) respectively. The confidence band is +computed based on 100 replications. +References +Abbeel, P. and Ng, A. Y. (2004), Apprenticeship learning via inverse reinforcement learning, +in ‘Proceedings of the twenty-first international conference on Machine learning’, p. 1. +Audibert, J.-Y., Tsybakov, A. B. et al. (2007), ‘Fast learning rates for plug-in classifiers’, +The Annals of statistics 35(2), 608–633. +Bertsekas, D. P. and Tsitsiklis, J. N. (1996), Neuro-dynamic programming, Vol. 5, Athena +Scientific Belmont, MA. +Bhandari, J., Russo, D. and Singal, R. (2018), ‘A finite time analysis of temporal difference +learning with linear function approximation’, arXiv preprint arXiv:1806.02450 . +Bishop, C. (1994), ‘Mixture density networks’, Technical Report pp. 1–26. +Bradley, R. C. (2005), ‘Basic properties of strong mixing conditions. a survey and some +open questions’, Probability Surveys 2, 107–144. +Brown, B. M. et al. (1971), ‘Martingale central limit theorems’, The Annals of Mathematical +Statistics 42(1), 59–66. +61 + +Casella, G. and Berger, R. L. (2002), Statistical inference, Vol. 2, Duxbury Pacific Grove, +CA. +Chakraborty, B. and Moodie, E. (2013), Statistical methods for dynamic treatment regimes, +Springer. +Chen, X. and Qi, Z. (2022), ‘On well-posedness and minimax optimal rates of nonparametric +q-function estimation in off-policy evaluation’, arXiv preprint arXiv:2201.06169 . +Chernozhukov, V., Chetverikov, D., Demirer, M., Duflo, E., Hansen, C., Newey, W. and +Robins, J. (2018), ‘Double/debiased machine learning for treatment and structural pa- +rameters’. +Chernozhukov, V., Chetverikov, D., Kato, K. et al. (2014), ‘Gaussian approximation of +suprema of empirical processes’, The Annals of Statistics 42(4), 1564–1597. +Dedecker, J. and Louhichi, S. (2002), Maximal inequalities and empirical central limit the- +orems, in ‘Empirical process techniques for dependent data’, Birkh¨auser Boston, Boston, +MA, pp. 137–159. +Degris, T., White, M. and Sutton, R. S. (2012), Off-policy actor-critic, in ‘Proceedings +of the 29th International Coference on International Conference on Machine Learning’, +pp. 179–186. +Devroye, L., Mehrabian, A. and Reddad, T. (2018), ‘The total variation distance between +high-dimensional gaussians’, arXiv preprint arXiv:1810.08693 . +Ernst, D., Geurts, P., Wehenkel, L. and Littman, L. (2005), ‘Tree-based batch mode rein- +forcement learning’, Journal of Machine Learning Research 6, 503–556. +Ertefaie, A. and Strawderman, R. L. (2018), ‘Constructing dynamic treatment regimes over +indefinite time horizons’, Biometrika 105(4), 963–977. +Fan, J., Wang, Z., Xie, Y. and Yang, Z. (2020a), A theoretical analysis of deep q-learning, +in ‘Learning for Dynamics and Control’, PMLR, pp. 486–489. +Fan, J., Wang, Z., Xie, Y. and Yang, Z. (2020b), A theoretical analysis of deep q-learning, +in ‘Learning for Dynamics and Control’, PMLR, pp. 486–489. +62 + +Farahmand, A.-m., Ghavamzadeh, M., Szepesv´ari, C. and Mannor, S. (2016), ‘Regularized +policy iteration with nonparametric function spaces’, The Journal of Machine Learning +Research 17(1), 4809–4874. +Feng, Y., Ren, T., Tang, Z. and Liu, Q. (2020), Accountable off-policy evaluation with +kernel bellman statistics, in ‘International Conference on Machine Learning’, PMLR, +pp. 3102–3111. +Harvey, N., Liaw, C. and Mehrabian, A. (2017), Nearly-tight vc-dimension bounds for +piecewise linear neural networks, in ‘Conference on learning theory’, PMLR, pp. 1064– +1068. +Hu, X., Qian, M., Cheng, B. and Cheung, Y. K. (2021), ‘Personalized policy learning +using longitudinal mobile health data’, Journal of the American Statistical Association +116(533), 410–420. +Hubbs, C. D., Perez, H. D., Sarwar, O., Sahinidis, N. V., Grossmann, I. E. and Wassick, +J. M. (2020), ‘Or-gym: A reinforcement learning library for operations research problem’, +arXiv preprint arXiv:2008.06319 . +Hunter, D. R. and Lange, K. (2004), ‘A tutorial on mm algorithms’, The American Statis- +tician 58(1), 30–37. +Jiang, N. and Li, L. (2016), Doubly robust off-policy value evaluation for reinforcement +learning, in ‘International Conference on Machine Learning’, pp. 652–661. +Jiang, Z., Yang, S. and Ding, P. (2020), ‘Multiply robust estimation of causal effects under +principal ignorability’, arXiv preprint arXiv:2012.01615 . +Kakade, S. and Langford, J. (2002), Approximately optimal approximate reinforcement +learning, in ‘ICML’, Vol. 2, pp. 267–274. +Kallus, N. and Uehara, M. (2019), ‘Efficiently breaking the curse of horizon: Double rein- +forcement learning in infinite-horizon processes’, arXiv preprint arXiv:1909.05850 . +Kallus, N. and Uehara, M. (2020), Statistically efficient off-policy policy gradients, in +‘International Conference on Machine Learning’, PMLR, pp. 5089–5100. +63 + +Komorowski, M., Celi, L. A., Badawi, O., Gordon, A. C. and Faisal, A. A. (2018), ‘The +artificial intelligence clinician learns optimal treatment strategies for sepsis in intensive +care’, Nature medicine 24(11), 1716–1720. +Komunjer, I. and Vuong, Q. (2010), ‘Semiparametric efficiency bound in time-series models +for conditional quantiles’, Econometric Theory pp. 383–405. +Kosorok, M. R. and Laber, E. B. (2019), ‘Precision medicine’, Annual review of statistics +and its application 6, 263–286. +Kumar, A., Zhou, A., Tucker, G. and Levine, S. (2020), ‘Conservative q-learning for offline +reinforcement learning’, arXiv preprint arXiv:2006.04779 . +Laber, E. B., Lizotte, D. J., Qian, M., Pelham, W. E. and Murphy, S. A. (2014), ‘Dynamic +treatment regimes: Technical challenges and applications’, Electronic journal of statistics +8(1), 1225. +Le, H., Voloshin, C. and Yue, Y. (2019), Batch policy learning under constraints, in ‘Inter- +national Conference on Machine Learning’, pp. 3703–3712. +LeCun, Y., Bengio, Y. and Hinton, G. (2015), ‘Deep learning’, nature 521(7553), 436–444. +Levine, S., Kumar, A., Tucker, G. and Fu, J. (2020), ‘Offline reinforcement learning: Tu- +torial, review, and perspectives on open problems’, arXiv preprint arXiv:2005.01643 . +Li, Y. (2017), ‘Deep reinforcement learning: An overview’, arXiv preprint arXiv:1701.07274 +. +Liao, P., Klasnja, P. and Murphy, S. (2019), ‘Off-policy estimation of long-term average +outcomes with applications to mobile health’, arXiv preprint arXiv:1912.13088 . +Liao, P., Qi, Z. and Murphy, S. (2020), ‘Batch policy learning in average reward markov +decision processes’, arXiv preprint arXiv:2007.11771 . +Liu, Q., Li, L., Tang, Z. and Zhou, D. (2018), Breaking the curse of horizon: Infinite- +horizon off-policy estimation, in ‘Advances in Neural Information Processing Systems’, +pp. 5356–5366. +64 + +Luckett, D. J., Laber, E. B., Kahkoska, A. R., Maahs, D. M., Mayer-Davis, E. and Kosorok, +M. R. (2020), ‘Estimating dynamic treatment regimes in mobile health using v-learning’, +Journal of the American Statistical Association 115(530), 692–706. +Luedtke, A. R. and Van Der Laan, M. J. (2016), ‘Statistical inference for the mean outcome +under a possibly non-unique optimal treatment strategy’, Annals of statistics 44(2), 713. +Marcolino, M. S., Oliveira, J. A. Q., D’Agostino, M., Ribeiro, A. L., Alkmim, M. B. M. +and Novillo-Ortiz, D. (2018), ‘The impact of mhealth interventions: systematic review +of systematic reviews’, JMIR mHealth and uHealth 6(1), e23. +Mnih, V., Kavukcuoglu, K., Silver, D., Rusu, A. A., Veness, J., Bellemare, M. G., Graves, +A., Riedmiller, M., Fidjeland, A. K., Ostrovski, G. et al. (2015), ‘Human-level control +through deep reinforcement learning’, nature 518(7540), 529–533. +Murphy, S. A. (2003), ‘Optimal dynamic treatment regimes’, Journal of the Royal Statistical +Society: Series B (Statistical Methodology) 65(2), 331–355. +Puterman, M. L. (1994), Markov Decision Processes: Discrete Stochastic Dynamic Pro- +gramming, John Wiley & Sons, Inc. +Qian, M. and Murphy, S. A. (2011), ‘Performance guarantees for individualized treatment +rules’, Annals of statistics 39(2), 1180. +Rust, J. (1987), ‘Optimal replacement of gmc bus engines: An empirical model of harold +zurcher’, Econometrica: Journal of the Econometric Society pp. 999–1033. +Schulman, J., Levine, S., Abbeel, P., Jordan, M. and Moritz, P. (2015), Trust region policy +optimization, in ‘International conference on machine learning’, pp. 1889–1897. +Shi, C., Fan, A., Song, R. and Lu, W. (2018), ‘High-dimensional a-learning for optimal +dynamic treatment regimes’, Annals of statistics 46(3), 925. +Shi, C., Lu, W. and Song, R. (2020), ‘Breaking the curse of nonregularity with subagging: +inference of the mean outcome under optimal treatment regimes’, Journal of Machine +Learning Research accepted. +65 + +Shi, C., Wan, R., Chernozhukov, V. and Song, R. (2021), Deeply-debiased off-policy interval +estimation, in ‘Proceedings of the 38th International Conference on Machine Learning’, +Vol. 139 of Proceedings of Machine Learning Research, PMLR, pp. 9580–9591. +Shi, C., Wan, R., Song, R., Lu, W. and Leng, L. (2020), Does the markov decision process fit +the data: Testing for the markov property in sequential decision making, in ‘International +Conference on Machine Learning’, PMLR, pp. 8807–8817. +Shi, C., Zhang, S., Lu, W. and Song, R. (2020), ‘Statistical inference of the value function +for reinforcement learning in infinite horizon settings’, arXiv preprint arXiv:2001.04515 +. +Shi, X., Miao, W., Nelson, J. C. and Tchetgen Tchetgen, E. J. (2020), ‘Multiply robust +causal inference with double-negative control adjustment for categorical unmeasured con- +founding’, Journal of the Royal Statistical Society: Series B (Statistical Methodology) +82(2), 521–540. +Silver, D., Huang, A., Maddison, C. J., Guez, A., Sifre, L., Van Den Driessche, G., Schrit- +twieser, J., Antonoglou, I., Panneershelvam, V., Lanctot, M. et al. (2016), ‘Mastering +the game of go with deep neural networks and tree search’, nature 529(7587), 484–489. +Sutton, R. S. and Barto, A. G. (2018), Reinforcement learning: An introduction, MIT +press. +Tchetgen Tchetgen, E. J. and Shpitser, I. (2012), ‘Semiparametric theory for causal medi- +ation analysis: efficiency bounds, multiple robustness, and sensitivity analysis’, Annals +of statistics 40(3), 1816. +Tsiatis, A. (2007), Semiparametric theory and missing data, Springer Science & Business +Media. +Tsiatis, A. A., Davidian, M., Holloway, S. T. and Laber, E. B. (2019), Dynamic Treatment +Regimes: Statistical Methods for Precision Medicine, CRC press. +Tsybakov, A. B. et al. (2004), ‘Optimal aggregation of classifiers in statistical learning’, +The Annals of Statistics 32(1), 135–166. +66 + +Wang, L. and Tchetgen Tchetgen, E. (2018), ‘Bounded, efficient and multiply robust esti- +mation of average treatment effects using instrumental variables’, Journal of the Royal +Statistical Society: Series B (Statistical Methodology) 80(3), 531–550. +Wang, L., Zhou, Y., Song, R. and Sherwood, B. (2018), ‘Quantile-optimal treatment +regimes’, Journal of the American Statistical Association 113(523), 1243–1254. +Watkins, C. J. and Dayan, P. (1992), ‘Q-learning’, Machine learning 8(3-4), 279–292. +Wu, Y., Tucker, G. and Nachum, O. (2019), ‘Behavior regularized offline reinforcement +learning’, arXiv preprint arXiv:1911.11361 . +Yu, T., Thomas, G., Yu, L., Ermon, S., Zou, J. Y., Levine, S., Finn, C. and Ma, T. +(2020), ‘Mopo: Model-based offline policy optimization’, Advances in Neural Information +Processing Systems 33, 14129–14142. +Zhao, Y.-Q., Zeng, D., Laber, E. B. and Kosorok, M. R. (2015), ‘New statistical learning +methods for estimating optimal dynamic treatment regimes’, Journal of the American +Statistical Association 110(510), 583–598. +67 + +This figure "toy_est.png" is available in "png"� format from: +http://arxiv.org/ps/2301.02220v1 + diff --git a/KdE0T4oBgHgl3EQfSQCv/content/tmp_files/load_file.txt b/KdE0T4oBgHgl3EQfSQCv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..448cc3cad6caf151c0f3b42da4edaa8849e8db80 --- /dev/null +++ b/KdE0T4oBgHgl3EQfSQCv/content/tmp_files/load_file.txt @@ -0,0 +1,2431 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf,len=2430 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='02220v1 [stat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='ML] 5 Jan 2023 Value Enhancement of Reinforcement Learning via Efficient and Robust Trust Region Optimization Chengchun Shia∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Zhengling Qib∗†,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Jianing Wangc and Fan Zhouc† aDepartment of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' London School of Economics and Political Science bDepartment of Decision Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The George Washington University cDepartment of Statistics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Shanghai University of Finance and Economics Abstract Reinforcement learning (RL) is a powerful machine learning technique that en- ables an intelligent agent to learn an optimal policy that maximizes the cumulative rewards in sequential decision making.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Most of methods in the existing literature are developed in online settings where the data are easy to collect or simulate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Motivated by high stake domains such as mobile health studies with limited and pre-collected data, in this paper, we study offline reinforcement learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' To efficiently use these datasets for policy optimization, we propose a novel value enhancement method to improve the performance of a given initial policy computed by existing state-of-the-art RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Specifically, when the initial policy is not consistent, our method will output a policy whose value is no worse and often better than that of the initial policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' When the initial policy is consistent, under some mild conditions, our method will yield a policy whose value converges to the optimal one at a faster rate than the initial policy, achieving the desired “value enhancement” property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The proposed method is generally applicable to any parametrized policy that belongs to certain pre-specified function class (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', deep neural networks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Extensive numerical studies are conducted to demonstrate the superior performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Keywords: Offline reinforcement learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Trust region optimization;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Semi-parametric effi- ciency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Mobile health studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' ∗The first two authors contribute equally to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' †qizhengling@gwu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='edu;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='zhoufan@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='shufe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='cn 1 1 Introduction Reinforcement learning (RL, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', Sutton and Barto, 2018) is concerned with how agents take sequential actions in dynamic environments, with the main goal of maximizing the cumulative rewards they receive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In recent years, we have seen tremendous achievements of RL in artificial intelligence (AI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' For example, AlphaGo (Silver et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2016), one of the most successful applications in AI, makes use of reinforcement learning and deep learn- ing algorithms for teaching machines to play the board game called Go, and has beaten many top human players.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The appealing performance of RL has also been demonstrated in many scientific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In medical applications, RL has been used to help clinicians make better treatment decisions for patients with sepsis (Komorowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In eco- nomics, econometricians often study dynamic discrete choice models (Rust, 1987) in order to understand the behavior of rational agents, which is similar to the inverse RL problem (Abbeel and Ng, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In operations research, RL has been widely applied to business operations such as supply chain management, finance and logistics (Hubbs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' For an overview of various applications of RL, we refer to Section 5 of Li (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Our research in this paper is partly motivated by recently emerging mobile health (mHealth) studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Advancements in mobile and sensor technologies provides us with a unique opportunity to deliver health interventions at anytime and anywhere for promoting healthy behaviors such as regular physical activities and preventing drug abuse, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' For example, the OhioT1DM dataset (Marcolino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2018) was developed for promoting blood glucose level prediction in order to improve the health and wellbeing of people with type 1 diabetes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' It contains data information of 6 people for 8 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' For each patient, their treatment information was collected during insulin pump therapy with continuous glucose monitoring (CGM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In addition, blood glucose levels and self-reported times of meals and exercises were also constantly measured and recorded via a custom smartphone app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Finding an optimal insulin pumping policy for each patient at different scenarios may potentially improve their health status (Shi, Zhang, Lu and Song, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' This matches the goal of RL algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' A fundamental question we aim to investigate here is how to learn an optimal policy efficiently from the batch data in high-stake domains such as mHealth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Solving this question faces at least two major challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' First, different from the standard clinical trial data, 2 mobile health data usually consist of a large number of decision points for each patient but the number of patients may be limited (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', in OhioT1DM dataset, 6 patients with a few thousands decision points).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' This posits a unique challenge for searching an optimal policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In statistics, there is a rich literature in studying dynamic treatment regimes (DTR, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Murphy, 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Chakraborty and Moodie, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Qian and Murphy, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' For a review of DTR, see Laber et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' (2014), Kosorok and Laber (2019) and Tsiatis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' However, these methods are mainly designed for only a few treatment decision points and often require a large number of patients in the observed data in order to be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Second, different from online RL domains such as video games, where actively interact- ing with the environment is feasible and data are easy to generate or simulate, in some high stake domains, data are often pre-collected according to some experimental design and very limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' With such limited data, it is essential to study how to efficiently learn the optimal policy from the batch data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' We remark that the main focus of RL in the computer science literature is for online learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Among all the methods available, Q-learning is arguably the most popular model-free RL algorithms (Watkins and Dayan, 1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' It derives the op- timal policy by learning an optimal Q-function (see the definition of Q-function in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Follow this line of research, variants of Q-learning methods have been proposed in- cluding the fitted Q-iteration (FQI, Ernst et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Fan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2020a), deep Q-network (DQN, Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2015), among many others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Policy-based learning is another class of RL algorithms that searches the optimal policy among a parametrized policy class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Some pop- ular algorithms include REINFORCE and actor-critic methods (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', Sutton and Barto, 2018, Chapter 13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Since these algorithms are primarily motivated by the application of de- veloping artificial intelligence in online video games, their generalization to offline settings such as mobile health applications remain largely unexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' To address the first challenge, we model the observed data by a time-homogeneous Markov decision process (MDP Puterman, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' This framework is particularly suitable to model the data collected from mobile health studies where the total number of decision points are often large (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2019, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The assumptions of Markov and time homogeneity enable a consistent estimation of the optimal policy even with only a few patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' To address the second challenge, we develop a novel procedure to derive the optimal 3 policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Recently, a few algorithms have been developed in the statistics literature for pol- icy optimization in mHealth applications (Ertefaie and Strawderman, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Luckett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In particular, Liao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' (2020) proposed a statis- tically efficient batch policy learning method under the average reward MDP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' However, due to the policy dependent structure of nuisance functions such as Q-function and the marginal density ratio, their proposed algorithm is computational inefficient as it requires updating the nuisance functions estimation in each iteration of their policy gradient decent algo- rithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Instead of proposing a specific algorithm for policy optimization, we devise a “value enhancement” method that is generally applicable to any given initial policy computed by some state-of-the-art RL algorithm to improve their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Basically, after employ- ing some computational efficient RL algorithm and obtaining an initial policy, we take a one-step update of this policy via efficiently estimating the value enhancement component (defined in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='3) and solving a constrained optimization problem, thus taking advan- tage of computational efficiency from existing state-of-art RL algorithms without requiring iteratively updating the nuisance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' More importantly, the proposed procedure guarantees that when the initial policy is not consistent, the output policy by the proposed algorithm is no worse and often better than the initial policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' If consistent, our method will yield a policy whose value converges to the optimal one at a faster rate, achieving the desired “value enhancement” property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Recently, in the computer science literature, Kallus and Uehara (2020) developed an offline policy gradient algorithm that considered statistically efficient estimation of the policy gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Our proposal differs from theirs in that we focus on developing a general value enhancement tool that is applicable to any existing RL algorithms to improve their performances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Our method is inspired by Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='1 in Kakade and Langford (2002) and the trust region policy optimization algorithm by Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' (2015), which was originally de- signed for the online setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' A key observation is that, the value difference between any two policies can be decomposed into a first-order component and a higher-order remainder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The higher-order term can be lower bounded, based on which a minorization func- tion can be constructed for the value function of any policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' One big advantage of working with this minorization function is that it intrinsically disentangles the policy-dependent structure of nuisance functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' This ensures the computational efficiency of the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' 4 The key “value enhancement” property relies crucially on statistically efficient estima- tion of the first-order term in the decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In online settings, Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' (2015) proposed to simulate data trajectories to estimate this quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' However, in offline set- tings, it remains unknown how to effectively evaluate this quantity based on the observed data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' By leveraging semi-parametric statistics, we develop a triply robust estimator for the first-order term that is shown to achieve the efficiency bound when compared to the initial policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' By optimizing the proposed estimator, we are able to improve the value of the initial policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The triply robustness property guarantees that the “value enhancement” property holds even when some nuisance function models are misspecified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The semi-parametric ef- ficiency guarantees that the value can be enhanced at a sufficiently fast rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' This ensures the statistical efficiency of the proposed algorithm, which is necessary in the offline setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In theory, we establish the value enhancement property under mild conditions on the nuisance function estimators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In particular, we only require them to converge at a non- parametric rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' See Section 4 for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' This nice property is achieved mainly due to the innovative way we put together these nuisance function estimators, which leads to the triply-robust estimator with a parametric convergence rate for the first-order term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In ad- dition, we remark that all our theoretical results related to estimation are established in terms of total decision points, thus showing the proposed method is generally applicable even when the number of trajectories is small but the length of each trajectory is large, which is commonly seen in the mobile health applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In Section 2, we introduce the offline RL problem in the framework of a time-homogeneous MDP and review the trust region algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In Section 3, we present our value enhanced policy optimization method and the related estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In Section 4, we study statistical properties of our algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In Section 5, extensive numerical studies including a toy example demonstrating the value enhancement property, a real-data driven simulation study and a real data application are conducted to demonstrate the superior performance of the proposed method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Finally, we conclude our paper in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' All technical proofs and details can be found in the online supplementary material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' 5 2 Preliminaries This section is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' We first introduce the offline data structure and describe the model setup in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='2, we introduce some notations needed to derive our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' In Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='3, we review the trust region policy optimization (TRPO) method proposed by Schulman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' (2015) for online learning, as it is closely related to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content='1 Value function and the optimal policy Consider a single trajectory {(St, At, Rt)}t≥0 where (St, At, Rt) denotes the state-action- reward triplet collected at time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' We use S and A to denote the state and action space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' We assume S and A are discrete, and rewards Rt are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The discrete state space assumption is imposed only to simplify the presentation and the theoretical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' Our proposed method is equally applicable to settings with continuous state space as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' The observed data consist of N trajectories, corresponding to N independent and identically distributed copies of {(St, At, Rt)}t≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KdE0T4oBgHgl3EQfSQCv/content/2301.02220v1.pdf'} +page_content=' For any i = 1, · · · , N, data collected from the ith trajectory can be summarized by {(Si,t, Ai,t, Ri,t, Si,t+1)}0≤t 0, and M∞ is the disjoint union of two 3-spheres. +Theorem A shows that something stronger than width is required for a stability conjecture +related to the rigidity theorem of Marques-Neves. A conjecture in [Sor+21] attributed to +Marques and Neves does hypothesize a stronger condition. In particular, it replaces the +uniform lower bound on width with a uniform lower bound on +MinA(M, g) = inf{|Σ|g : Σ is a closed minimal hypersurface in M}. +Since width is achieved by a minimal surface, we have that width(M3, g) ≥ MinA(M3, g). +Moreover, Marques-Neves show in [MN12] that if (S3, g) contains no stable minimal surfaces, +then we have that MinA(S3, g) = width(S3, g). +Conjecture 1.2. Suppose M3 +j = (S3, gj) are homeomorphic spheres satisfying +Rj ≥ 6 − 1 +j , MinA(M3 +j ) ≥ 4π − 1 +j , diam +� +M3 +j +� +≤ D, and vol +� +M3 +j +� +≤ V +where Rj is the scalar curvature of M3 +j . Then M3 +j converges in the VF-sense to (S3, grd) +the standard unit round sphere. +The sequence of Riemannian manifolds constructed in Theorem A has MinA(Mn +j ) → 0 +and so does not satisfy the hypotheses of Conjecture 1.2. + +EXAMPLES FOR SCALAR SPHERE STABILITY +3 +Figure 1. A sequence of spheres that converge in VF-sense to the disjoint +union of two spheres. +Sormani proposed the MinA condition in [Sor17] to prevent bad limiting behavior, such as +bubbling and pinching, along the sequence. The motivation for such a condition comes from +the sewing construction of Basilio, Dodziuk, and Sormani [BDS18]. This construction shows +the existence of a sequence of manifolds with positive scalar curvature, which has an F-limit +that does not have positive scalar curvature in some generalized sense. Other sequences of +positive scalar curvature manifolds have also been constructed ([BS21], [BKS20]) whose +F-limits have undesirable properties. The key to the construction of these examples is the +ability to glue in tunnels with controlled geometry. In those examples, it is unknown if the +scalar curvature of the tunnel and of the resulting manifold can be kept close to the scalar +curvature of the manifold to which the tunnel is being glued. Therefore, these examples may +not satisfy the curvature condition in Conjecture 1.2. In Section 4, we prove our two main +technical propositions, which are of independent interest. One of which allows us to get +quantitative control over the scalar curvature of the tunnel and of the resulting manifold. +In particular, given a manifold with scalar curvature bounded below by κ, then for small +enough ϵ > 0 there exists a tunnel such that the resulting manifold has scalar curvature +bounded below by κ − ϵ. +We use this new way of attaching tunnels to manifolds that maintains control over the +scalar curvature to construct the sequence in Theorem A. Moreover, we can make a similar +example related to Llarull’s rigidity theorem. First, let us recall Llarull’s theorem [Lla98] +which says that if there is a degree non-zero, smooth, distance non-increasing map from a +smooth, Riemannian, spin, n-manifold, Mn, to the standard unit round n-sphere and the +scalar curvature of Mn is greater than or equal to n(n − 1), then the map is a Riemannian +isometry. +Gromov in [Gro18] proposed studying the stability question related to Llarull’s rigidity +theorem by investigating sequences of Riemannian manifolds Mn +j = (Mn +j , gj) with inf Rj → +n(n − 1) and RadSn(Mj) → 1. RadSn(Mn) is defined as the maximal radius r of the n- +sphere, Sn(r), such that Mn admits a distance non-increasing map from Mn to Sn(r) of +non-zero degree. Based on this, Sormani [Sor22] proposed the following stability conjecture. +Conjecture 1.3. Suppose Mn +j = (Mn +j , gj), n ≥ 3, are closed smooth connected spin Rie- +mannian manifolds such that +Rj ≥ n(n − 1) − 1 +j , MinA(Mn +j ) ≥ 4π − 1 +j , diam +� +Mn +j +� +≤ D, vol +� +Mn +j +� +≤ V + +69 +←(4 +PAUL SWEENEY JR +where Rj is the scalar curvature of Mn +j . Furthermore, suppose there are smooth maps to +the standard unit round n-sphere +fj : Mn +j → Sn +which are 1-Lipschitz and deg fj ̸= 0. Then Mn +j converges in the VF-sense to the standard +unit round n-sphere. +In a similar manner to Theorem A, we are able to construct a sequence of manifolds each +of which is two spheres connected by a thin tunnel, which is related to Conjecture 1.3. This +sequence satisfies all the hypotheses of Conjecture 1.3 except the lower bound on MinA. +Theorem A′. There exists a convergent sequence of Riemannian manifolds Mn +j = (Sn, gj), +n ≥ 3, with Mn +j +VF +−−→ M∞ such that +Rj ≥ n(n − 1) − 1 +j , diam (Mj) ≤ D, and vol (Mj) ≤ V, +for some constants D, V > 0. Furthermore, there are smooth degree one, 1-Lipschitz maps +fj : Mn +j → (Sn, grd) which converge to a 1-Lipschitz map f∞ : M∞ → (Sn, grd), and M∞ is +the disjoint union of two n-spheres. +Theorems A and A′ show the necessity of including a hypothesis like the bound on MinA +to prevent bubbling along the sequence. +When studying a stability conjecture related to scalar curvature, one also often considers +examples similar to the example described by Ilmanen. Ilmanen first described this example +to demonstrate that a sequence of manifolds of positive scalar curvature need not converge in +the Gromov-Hausdorff (GH) sense. The example is a sequence of spheres with increasingly +many arbitrarily thin wells attached to them (see Figure 2). Sormani and Wenger [SW11, +Example A.7] showed, using their intrinsic flat (F) convergence for integral currents, that +the Ilmanen example converges in the F-sense. Over the past decade, Ilmanen-like examples +have been constructed in varying settings to demonstrate that GH-convergence is not the +appropriate convergence in which to ask stability conjectures related to scalar curvature +([LS13], [Lak16], [Per20], [LS14], [LS15], [LS12], [AP20], [APS20]). In these examples, it is +unknown if one can attach a well and only decrease the scalar curvature by a small amount; +consequently, it was unknown if Ilmanen-like examples could exist for Conjecture 1.2 and +Conjecture 1.3. +Our other main technical proposition (see Section 4 below) shows that one can attach +a well to a manifold with scalar curvature bounded below and only decrease the scalar +curvature by an arbitrarily small amount. Therefore, we are able to construct Ilmanen- +like examples related to Conjecture 1.2 and Conjecture 1.3. In particular, we are able to +construct a sequence of spheres M3 +j with scalar curvature larger than 6 − ϵj, width larger +than 4π, and volumes and diameters bounded that does not converge in the GH-sense +but does converge in the volume above distance below (V ADB) sense and the VF-sense. +Likewise, we construct a sequence of spheres with scalar curvature larger than n(n−1)−ϵj, +volumes and diameters bounded, and smooth maps to the unit round n-sphere which are +1-Lipschitz and deg fj ̸= 0. Therefore, we can construct Ilmanen-like examples related to +Conjecture 1.2 and Conjecture 1.3. We, however, cannot verify that MinA stays uniformly +bounded from below even though we expect that it does. + +EXAMPLES FOR SCALAR SPHERE STABILITY +5 +Theorem B. There exists a convergent sequence of Riemannian manifolds M3 +j = (S3, gj), +with M3 +j +VADB +−−−−→ M∞ and M3 +j +VF +−−→ M∞ such that +Rj ≥ 6 − 1 +j , width(M3 +j ) ≥ 4π, diam +� +M3 +j +� +≤ D, and vol +� +M3 +j +� +≤ V, +for some constants D, V > 0, and M∞ is the standard unit round 3-sphere. However, the +sequence has no convergent subsequence in the GH-topology. +In [Sor+21, Remark 9.4], Sormani suggests that it is believable that someone can con- +struct a sequence of spheres with increasingly many increasingly thin wells which satisfy the +hypothesis of Conjecture 1.2. Theorem B partially answers this question by constructing +such a sequence that satisfies all the hypotheses of Conjecture 1.2 except the bound on +MinA. +Theorem B′. There exists a convergent sequence of Riemannian manifolds Mn +j = (Sn, gj), +with Mn +j +VADB +−−−−→ M∞ and Mn +j +VF +−−→ M∞ such that +Rj ≥ n(n − 1) − 1 +j , diam +� +Mn +j +� +≤ D, and vol +� +Mn +j +� +≤ V, +for some constants D, V > 0, and M∞ is the n-sphere. Furthermore, there are smooth +degree non-zero, 1-Lipschitz maps fj : Mn +j → (Sn, grd), and M∞ is the standard unit round +n-sphere. However, the sequence has no subsequence that converges in the GH-sense. +Figure 2. A sequence of spheres with increasingly many thin wells that +converges in the VADB-sense and VF-sense to a sphere but has no convergent +subsequence in the GH-topology +The main tools to prove the above theorems are new construction propositions which are +proved in Section 4. We adapt the bending argument of Gromov and Lawson in [GL80]. +Originally, the construction in [GL80] was used to make tunnels of positive scalar curvature +to show, for example, that the connected sum of two manifolds with positive scalar curvature +carries a metric of positive scalar curvature. For manifolds with constant positive sectional +curvature, Dodziuk, Basilio, and Sormani in [BDS18] refined the construction to give control +over the volume and diameter of the tunnel while maintaining positive scalar curvature. +Dodziuk in [Dod20] further refined the construction by replacing the positive sectional +curvature condition with positive scalar curvature. In this paper, we construct wells and +tunnels such that, if the scalar curvature of a manifold is bounded below, then one can +attach a well or tunnel and only decrease the lower bound by an arbitrarily small amount +while maintaining bounds on the diameter and volume. + +6 +PAUL SWEENEY JR +The new well construction allows us to generalize the construction of Sormani and Wenger +[SW11, Example A.11] of a sequence of manifolds that converge in the F-sense to space +that is not precompact. In particular, we are able to construct a sequence of spheres with +scalar curvatures greater than κ ≥ 0, uniformly bounded diameters, and uniformly bounded +volumes such that the sequence converges in the VF-sense to a limit that is not precompact. +To construct the sequence we attach a sequence of increasingly thin wells to a sphere (see +Figure 3). +Theorem C. There exists a convergent sequence of Riemannian manifolds Mn +j = (Sn, gj), +n ≥ 3, with Mn +j +VF +−−→ M∞ such that +Rj ≥ κ, diam +� +Mn +j +� +≤ D, and vol +� +Mn +j +� +≤ V, +for some nonnegative constants κ, D, V , and M∞ is not precompact. +Figure 3. A sequence of spheres with increasingly many thin wells that +converges in the VF-sense to a limit which is not precompact. +The new tunnel construction allows us to extend the sewing construction in [BDS18] +and [BS21] to a more general setting. Basilio, Dodziuk, and Sormani [BDS18] used sewing +manifolds to investigate the following question of Gromov which asks: What is the weak- +est notion of convergence such that a sequence of Riemannian manifolds, Mn +j with scalar +curvature Rj ≥ κ subconverges to a limit M∞ which may not be a manifold but has scalar +curvature greater than κ in some suitably generalized sense? They were able to show that +when κ = 0 there is a sequence of Riemannian manifolds with non-negative scalar curvature +whose limit fails to have non-negative generalized scalar curvature where generalized scalar +curvature is defined as +wR(p0) := lim +r→0 6(n + 2)volEn B(0, r) − Hn(B(p0, r)) +r2 · volEn B(0, r) +≥ 0 +(1.1) +for the limit space. +Remark 1.4. For a Riemannian manifold (Mn, g) with scalar curvature R, we see for all +p ∈ Mn that wR(p) = R(p). +We are able to provide a similar answer to Gromov’s question for any κ. In particular, +for any κ, there exists a sequence of increasingly tightly sewn manifolds all of which have +scalar curvature greater than κ. Furthermore, this sequence of increasingly tightly sewn +manifolds will converge in the F-sense to a pulled metric space (see [BS21, Section 2] for +discussion of such spaces) which fail to have generalized scalar curvature greater than or +equal to κ at the pulled point. + +EXAMPLES FOR SCALAR SPHERE STABILITY +7 +Theorem D. There exists a sequence of manifolds Mn +j = (Mn, gj) with scalar curvature +Rj ≥ κ − 1 +j which converges in the F-sense to a metric space M∞. Moreover, there is a +point p0 ∈ M∞ such that +wR(p0) := lim +r→0 6(n + 2)volEn B(0, r) − Hn(B(p0, r)) +r2 · volEn B(0, r) += −∞ +(1.2) +Lastly, the new tunnel construction allows us to generalize the construction of Basilio, +Kazaras, and Sormani [BKS20]. They use long thin tunnels with positive scalar curvature +to construct a sequence of manifolds that converges in the F-sense to a space with no +geodesics. Similarly, for any κ > 0, we are able to construct a sequence of manifolds with +scalar curvature bounded below by κ whose limit is not a geodesic space. +Theorem E. There is a sequence of closed, oriented, Riemannian manifolds (Mn +j , gj), +n ≥ 3, with scalar curvature Rj > κ > 0 such that the corresponding integral current spaces +converge in the intrinsic flat sense to +M∞ = +� +N, dEn+1, +� +N +� +, +where N is the round n-sphere of curvature +2κ +n(n−1) and dEn+1 is the Euclidean distance +induced from the standard embedding of N into En+1. +Furthermore, M∞ is not locally +geodesic. +Properties of +Property of the +Type of +Does +Mj +limit, M∞ +convergence +MinAj → 0? +Rj > 6 − 1 +j +Counterexample to +Theorem A +width(Mj) ≥ 4π +Conjecture 1.1. +VF, F +Yes +Shows necessity of +MinA lower bound +Theorem A′ +Rj > n(n − 1) − 1 +j +in Conjecture 1.3. +VF, F +Yes +Rj > 6 − 1 +j +No GH-convergent +Theorem B +width(Mj) ≥ 4π +subsequence. +VADB, VF, F +? +No GH-convergent +Theorem B′ +Rj > n(n − 1) − 1 +j +subsequence. +VADB, VF, F +? +Theorem C +Rj > κ > 0 +Not precompact. +VF, F +? +Generalized Scalar +Rj > κ +curvature is negative +Theorem D +(Rj > 0, [BDS18]) +infinity at a point. +F +Yes +No two points +Rj > κ > 0 +are connected by +Theorem E +(Rj > 0, [BKS20]) +a geodesic. +F +Yes +Table 1. Here we summarize the examples constructed in this paper. +The paper is structured as follows. In Section 3, the background is discussed including +some definitions and theorems related to different notions of convergence for Riemannian +manifolds. +In Section 4, we prove our main construction propositions: Proposition 4.1 +(Constructing Wells) and Proposition 4.2 (Constructing Tunnels). In Section 5, we use + +8 +PAUL SWEENEY JR +Proposition 4.2 to prove Theorems A and A′. +In Section 6, we prove Theorems B, B′, +and C using Proposition 4.1. Finally, in Sections 7 and 8, we discuss how the construc- +tion propositions can be used to generalize the construction of sewing manifolds and the +construction of sequences of smooth manifolds whose limit does not have any geodesics. +2. Acknowledgements +The author would like to thank Marcus Khuri and Raanan Schul for their invaluable +guidance and encouragement throughout the process of producing this result. The author +would also like to thank Christina Sormani for her helpful discussions and the suggestion to +construct examples related to Llarull’s rigidity theorem. The author gratefully acknowledges +support from the Simons Center for Geometry and Physics, Stony Brook University, at +which some of the research for this paper was performed. This work was supported in part +by NSF Grant DMS-2104229 and NSF Grant DMS-2154613. +3. Background +In this section, we will review different types of convergences between two Riemannian +manifolds. +3.1. Gromov-Hausdorff convergence. Here we will review the Gromov-Hausdorff dis- +tance between two metric spaces. Gromov defined this distance between two metric spaces +by generalizing the concept of Hausdorff distance between two subsets of a metric space. +We refer the reader to [Gro99] for further details. +The Gromov-Hausdorff distance between two metric spaces (X1, d1) and (X2, d2) is +dGH((X1, d1), (X2, d2)) = inf +Z {dZ +H(φ1(X1), φ2(X2))} +where the infimum is taken over all complete metric spaces (Z, dZ) and all distance preserv- +ing maps φi : Xi → Z. We say that a metric spaces (Xj, dj) converge in the GH-sense to a +metric space (X∞, d∞) if +dGH((Xj, dj), (X∞, d∞)) → 0 +If, in addition, µj and µ∞ are measures on Xj and X∞, respectively, then Fukaya [Fuk87] +introduced the notion of metric measure convergence for metric measure spaces. We say +(Xj, dj, µj) converges to a metric measure space (X∞, d∞, µ∞) in metric measure (mGH) +sense if we have convergence in the GH-sense and +φj∗µj → φ∞∗µ∞ weakly as measures in Z. +We note that both define a distance between two Riemannian manifolds since there is +a natural distance function and natural measure associated with a Riemannian manifold +(M, g). +Gromov, in the following theorem, characterizes when a sequence of compact metric +spaces contains a subsequence that converges in the GH-sense. +Theorem 3.1. For a sequence of compact metric spaces (Xj, dj) such that diam (Xj) < +D < ∞, the following are equivalent: +(i) There exists a convergent subsequence. +(ii) There is a function N1 : (0, α) → (0, ∞) such that Capj(ϵ) ≤ N1(ϵ) + +EXAMPLES FOR SCALAR SPHERE STABILITY +9 +(iii) There is a function N2 : (0, α) → (0, ∞) such that Covj(ϵ) ≤ N2(ϵ), where +Capj(ϵ) = maximum number of disjoint ϵ +2-balls in Xj, +Covj(ϵ) = minimum number of ϵ-balls it takes to cover Xj. +3.2. Intrinsic Flat Convergence. In this section we will review Sormani-Wenger intrinsic +flat distance between two integral current spaces. Sormani and Wenger [SW11] defined +intrinsic flat distance, which generalizes the notion of flat distance for currents in Euclidean +space. To do so they used Ambrosio and Kirchheim’s generalization of Federer and Fleming’s +integral currents to metric spaces. We refer the reader to [AK00] for further details about +currents in arbitrary metric spaces and to [SW11] for further details about integral current +spaces and intrinsic flat distance. +Let (Z, dZ) be a complete metric space. Denote by Lip(Z) and Lipb(Z) the set of real- +valued Lipschitz functions on Z and the set of bounded real-valued Lipschitz functions on +Z. +Definition 3.2 ([AK00], Definition 3.1). We say a multilinear functional +T : Lipb(Z) × [Lip(Z)]m → R +on a complete metric space (Z, d) is an m-dimensional current if it satisfies the following +properties. +(i) Locality: T(f, π1, . . . , πm) = 0 if there exists and i such that πi is constant on a +neighborhood of {f ̸= 0}. +(ii) Continuity: T is continuous with respect to pointwise convergence of πi such that +Lip(πi) ≤ 1. +(iii) Finite mass: there exists a finite Borel measure µ on X such that +|T(f, π1, . . . , πm)| ≤ +m +� +i=1 +Lip(πi) +� +Z +|f|dµ +(3.1) +for any (f, π1, . . . , πm). +We call the minimal measure satisfying (3.1) the mass measure of T and denote it ||T||. +We can now define many concepts related to a current. M(T) = ||T||(Z) is defined to be +the mass of T and the canonical set of a m-current T on Z is +set(T) = +� +p ∈ Z +��� lim inf +r→0 +||T||(B(p, r)) +rm +> 0 +� +. +The boundary of a current T is defined as ∂T : Lipb(X) × [Lip(X)]m−1 → R, where +∂T(f, π1, . . . , πm−1) = T(1, f, π1, . . . , πm−1). +Given a Lipschitz map φ : Z → Z′, we can pushforward a current T on Z to a current φ#T +on Z′ by defining +φ#T(f, π1, . . . , πm) = T(f ◦ φ, f ◦ π1, . . . , f ◦ πm). +A standard example of an m-current on Z is given by +φ#[[θ]](f, π1, . . . , πm) = +� +A +(θ ◦ φ)(f ◦ φ)d(π1 ◦ φ) ∧ · · · ∧ d(πm ◦ φ), +where φ : Rm → Z is bi-Lipschitz and θ ∈ L1(A, Z). We say that an m-current on Z +is integer rectifiable if there is a countable collection of bi-Lipschitz maps φi : Ai → X + +10 +PAUL SWEENEY JR +where Ai ⊂ Rm is precompact Borel measurable with pairwise disjoint images and weight +functions θi ∈ L1(Ai, Z) such that +T = +∞ +� +i=1 +φi#[[θi]]. +Moreover, we say an integer rectifiable current whose boundary is also integer rectifiable is +an integral current. We denote the space of integral m-currents on Z as Im(Z). The flat +distance between two integral currents T1, T2 ∈ I(Z) is +dZ +F (T1, T2) = inf{M(U) + M(V ) | U ∈ Im(X), V ∈ Im+1(X), T2 − T1 = U + ∂V }. +We say that the triple (X, d, T) is an integral current space if (X, d) is a metric space, +T ∈ Im( ¯X) where ¯X is the completion of X, and set(T) = X. The intrinsic flat (F) distance +between two integral current spaces (X1, d1, T1) and (X2, d2, T2) is +dF((X1, d1, T1), (X2, d2, T2)) = inf +Z {dZ +F (φ1#T1, φ2#T2)} +where the infimum is taken over all complete metric spaces (Z, dZ) and isometric embeddings +φ1 : ( ¯X1, d1) → (Z, dZ) and φ2 : ( ¯X2, d2) → (Z, dZ). +We note that if (X1, d1, T1) and +(X2, d2, T2) are precompact integral current spaces such that +dF((X1, d1, T1), (X2, d2, T2)) = 0 +then there is a current preserving isometry between (X1, d1, T1) and (X2, d2, T2), i.e., there +exists an isometry f : X1 → X2 whose extension ¯f : ¯X1 → ¯X2 pushes forward the current: +¯f#T1 = T2. We say a sequence of (Xj, dj, Tj) precompact integral current spaces converges +to (X∞, d∞, T∞) in the F-sense if +dF((Xj, dj, Tj), (X∞, d∞, T∞)) → 0. +If, in addition, M(Ti) → M(T∞), then we say (Xj, dj, Tj) converges to (X∞, d∞, T∞) in the +voulme preserving intrinsic flat (VF) sense. We note that we can view compact Riemannian +manifolds (Mn, g) as precompact integral current spaces (Mn, dg, +� +Mn dvolg), where dg is the +natural distance function on the Riemannian manifold and integration over the manifold, +� +Mn dvolg, can be viewed as an integral current. Moreover, M(Mn) = vol (Mn). Lakzian +and Sormani in [LS13] were able to estimate the intrinsic distance between two diffeomorphic +manifolds: +Theorem 3.3. Suppose Mn +1 = (Mn, g1) and Mn +2 = (Mn, g2) are oriented precompact Rie- +mannian manifolds with diffeomorphic subregions Uj ⊂ Mn +j and diffeomorphisms ψj : U → +Uj such that for all v ∈ TU we have +1 +(1 + ϵ)2 ψ∗ +1g1(v, v) < ψ∗ +2g2(v, v) < (1 + ϵ)2ψ∗ +1g1(v, v). +We define the following quantities +(i) DUj = sup{diamMj (W) : W is a component of Uj}. +(ii) Define a to be a number such that a > arccos(1+ϵ)−1 +π +max{DU1, DU2}. +(iii) λ = supx,y∈U |dM1 (ψ1(x), ψ1(y)) − dM2 (ψ2(x), ψ2(y)) |. +(iv) h = +� +λ +� +max{DU1, DU2} + λ +4 +� +. +(v) ¯h = max +� +h, +√ +ϵ2 + 2ϵDU1, +√ +ϵ2 + 2ϵDU2 +� +. + +EXAMPLES FOR SCALAR SPHERE STABILITY +11 +Then the intrinsic flat distance between Mn +1 and Mn +2 is bounded: +dF(M1, M2) ≤ +� +2¯h + a +� +(volm(U1) + volm(U2) + volm−1(∂U1) + volm−1(∂U2)) ++ volm(M1 \ U1) + volm(M2 \ U2). +Moreover, Sormani [Sor18] proves the following Arzela-Ascoli theorem in the setting of +F-convergence. +Theorem 3.4. Fix L > 0. Suppose Mj = (Xj, dj, Tj) are integral current spaces for +j ∈ {1, 2, . . . , ∞} and Mj +F−→ M∞ and Fj : Xj → W are L-Lipschitz maps into a compact +metric space W, then a subsequence converges to an L-Lipschitz map F∞ : X∞ → W. +Specifically, there exists isometric embeddings of the subsequence φj : Xj → Z, such that +dZ +F (φj#Tj, φ∞#T∞) → 0 and for any sequence pj ∈ Xj converging to p ∈ X∞, +dZ(φj(pj), φ∞(p)) → 0, +one has converging images +dW (Fj(pj), F∞(p)) → 0. +3.3. Volume above distance below convergence. Allen, Perales, and Sormani in [APS20] +introduced a new notion of convergence of manifolds called volume above distance below +(VADB) convergence. It is based on the volume-distance rigidity theorem which states that +if there is a C1-diffeomorphism F : M → N between two Riemannian manifolds which +is also distance non-increasing then vol (N) ≤ vol (M); moreover, in case of equality the +manifolds are isometric. +Definition 3.5. A sequence of Riemannian manifolds without boundary Mn +j = (Mn, gj) +converge in the VADB-sense to a Riemannian manifold Mn +∞ = (Mn, g∞) if +(i) vol (Mn +j ) → vol (Mn +∞). +(ii) diam (Mn +j ) ≤ D. +(iii) There exists a C1-diffeomorphisms Ψj : Mn +∞ → Mn +j such that for all p, q ∈ Mn +∞ we +have +dj(Ψj(p), Ψj(q)) ≥ d∞(p, q). +We also record the following lemma from [APS20] which says that the above condition +on the distance functions in the definition of VADB-convergence can be converted into a +condition on Riemannian metrics. +Lemma 3.6. Let Mn +1 = (Mn, g1) and Mn +0 = (Mn, g0) be Riemannian manifolds and F : +Mn +1 → Mn +0 be a C1-diffeomorphism. Then +g0(dF(v), dF(v)) ≤ g1(v, v) +for all v ∈ TMn +1 +if and only if +d0(F(p), F(q)) ≤ d1(p, q) +for all p, q ∈ Mn +1 . +Finally, we record the following theorem from [APS20] which describes the relationship +between VADB-convergence and VF-convergence. +Theorem 3.7. If Mn +j = (Mn, gj) and Mn +∞ = (Mn, g∞) are compact oriented Riemannian +manifolds such that Mn +j +VADB +−−−−→ Mn +∞ then Mn +j +VF +−−→ Mn +∞. + +12 +PAUL SWEENEY JR +4. Wells and Tunnels +In this section, we prove the main new technical propositions: Proposition 4.1 (Con- +structing Wells) and Proposition 4.2 (Constructing Tunnels). These are an improvement +of the constructions of Gromov-Lawson [GL80], Basilio, Dodziuk, Sormani [BDS18], and +Dodziuk [Dod20]. We construct wells and tunnels and get control over the volume and +diameter while keeping the scalar curvature close to the scalar curvature of the manifold to +which we are attaching the well. Proposition 4.1 (Constructing Wells) allows us to remove +a ball from a Riemannian manifold M with scalar curvature RM ≥ κ and glue in a well +to create a new Riemannian manifold N; moreover, M and N will be isometric away from +the gluing and the scalar curvature RN of N will satisfy RN ≥ κ − ϵ for arbitrarily small +ϵ. Proposition 4.2 (Constructing Tunnels) allows the analogous construction for connecting +two manifolds with a tunnel. Therefore, given a Riemannian manifold M with RM ≥ κ +we can remove two balls and glue in a tunnel to create a Riemannian manifold P with +RP ≥ κ − ϵ for arbitrarily small ϵ. +Proposition 4.1 (Constructing Wells). Let (Mn, g), n ≥ 3, be a Riemannian manifold +with scalar curvature RM. Let δ > 0 be small enough, j ∈ N, and d > 0. If RM ≥ κ +on Bg(p, 2δ) a ball in (M, g), then we can construct a well Wj = (Bg(p, 2δ), gj) and a new +complete Riemannian manifold (Nn, h), +Nn = Mn, +h|M\Bg(p,2δ) = g|M\Bg(p,2δ), +h|Bg(p,2δ) = gj|Bg(p,2δ). +Furthermore, the following properties are satisfied: +(i) The scalar curvature, Rj, of Wj satisfies Rj > κ − 1 +j . +(ii) gj|E = g|E where E = Bg(p, 2δ) \ Bg(p, δ) is identified with a subset of Wj. +(iii) There exists constant C > 0 independent of j and d such that +diam (Wj) < C(δ + d) +and +vol (Wj) < C(δn + dδn−1). +(iv) N has scalar curvature RN > κ − 1 +j . +Proposition 4.2 (Constructing Tunnels). Let (Mn, g), n ≥ 3, be a Riemannian manifold +with scalar curvature RM. Let δ > 0 be small enough, j ∈ N, and d ≥ 0. If RM ≥ κ on two +balls Bg(p, 2δ) and Bg(p′, 2δ) in (Mn, g), then we can construct a new complete Riemannian +manifold P n, where we remove two balls and glue cylindrical region (Tj, gj) diffeomorphic +to Sn−1 × [0, 1], +P n = Mn \ +� +Bg(p, 2δ) ∪ Bg(p′, 2δ) +� +⊔ Tj. +Furthermore, the following properties are satisfied: +(i) The scalar curvature, Rj, of Tj satisfies Rj > κ − 1 +j . +(ii) gj|E = g|E and gj|E′ = g|E′ where E = Bg(p, 2δ) \ Bg(p, δ) and E′ = Bg(p′, 2δ) \ +Bg(p′, δ) are identified with subsets of P. +(iii) There exists constant C > 0 independent of j and d such that +diam (Tj) < C(δ + d) +and +vol (Tj) < C(δn + dδn−1). +(iv) P has scalar curvature RP > κ − 1 +j . +We adapt the proof from [Dod20]. The well and tunnel will be constructed as a codi- +mension one submanifold. The submanifold will be defined by a curve, and this curve will + +EXAMPLES FOR SCALAR SPHERE STABILITY +13 +control the geometry of the submanifold. First, we show how the curve defines the subman- +ifold and how it affects its geometry. Second, we carefully construct the curve so that the +submanifold will inherit the desired properties. +In particular, the construction will follow the following outline. First, we will describe +how, given a curve, we can define a submanifold and write the scalar curvature in terms +of quantities related to the curve. Second, we carefully construct a C1-curve, γ, which will +be used to define a submanifold that is the precursor to a well or a tunnel. Third, we +adjust the construction of γ so the resulting manifold will be a well. Fourth, we describe +the smoothing procedure to make γ a C∞-curve. Fifth, we construct a well and check it has +the desired properties. Sixth, we perform the analogous steps to construct a tunnel with +the desired properties. +4.1. A Submanifold defined by a curve. Let (Mn, g) be a compact Riemannian man- +ifold with scalar curvature RM ≥ κ. Let δ > 0 and B = B(p, 2δ) be a geodesic ball in +M. Consider the Riemannian product (X, gX) = (R × B, dt2 + dr2 + gr). Let ρ ∈ B be a +geodesic radius from p to ∂B and define S = R × ρ, which is a total geodesic submanifold +of R × B with coordinates (t, r). Let γ be a smooth curve in S to be determined later. +Finally, let Σ = {(y, q) ∈ X : (y, ||q||g) ∈ γ} be a submanifold of (X, gX) with the induced +metric, where || · ||g is the distance from p to q with respect to g. Now we want to calculate +the scalar curvature of Σ. To do so we will need the following lemma from [Dod20]: +Lemma 4.3. The principal curvatures of the hypersurface Sn−1(ϵ) in B are each of the +form +1 +−ϵ + O(ϵ) for small ϵ. Furthermore, let gϵ be the induced metric on Sn−1(ϵ) and let +grd,ϵ be the round metric of curvature +1 +ϵ2 . Then, as ϵ → 0, +1 +ϵ2 gϵ → +1 +ϵ2 grd,ϵ = grd in the C2 +topology, moreover, ||grd − 1 +ϵ2 gϵ|| ≤ ϵ2. +Now to calculate the scalar curvature of Σ, fix q ∈ Σ∩S. Let e1, . . . , en be an orthonormal +basis of of Tq(Σ) where e1 is tangent to γ. Note that the for points in Σ ∩ S the normal ν +to W in X is the same as the normal to γ in S. +From the Gauss equations: +RX(X, Y, Z, U) = RΣ(X, Y, Z, U) − A(X, U)A(Y, Z) + A(X, Z)A(Y, U) +we see +KΣ +ij = KX +ij + λiλj. +where λi are principal curvatures corresponding to ei and KΣ +ij and KX +ij are the respective +sectional curvatures. We note that λ1 = k where k is the geodesic curvature of γ. For +i = 2, . . . , n we see by Lemma 4.3 +λi = ⟨∇∂iν, ∂i⟩ += ⟨∇∂i cos θ∂t + sin θ∂r, ∂i⟩ += cos θ⟨∇∂i∂t, ∂i⟩ + sin θ⟨∇∂i∂r, ∂i⟩ += sin θ⟨∇∂i∂r, ∂i⟩ += +� 1 +−r + O(r) +� +sin θ, +where θ is the angle that between ν and the t-axis. Now note that +KX +1j = RX(ej, e1, e1, ej) = RX(ej, cos θ∂r, cos θ∂r, ej) = cos2 θKM +∂r,j. + +14 +PAUL SWEENEY JR +For i ̸= 1 and j ̸= 1 +KX +ij = RX(ej, ei, ei, ej) = KM +i,j. +Since +RΣ = +� +i̸=j +KΣ +ij +we see +RΣ = RM − 2RicM (∂r, ∂r) sin2 θ ++ (n − 2)(n − 1) +� 1 +r2 + O(1) +� +sin2 θ +− (n − 1) +�1 +r + O(r) +� +k sin θ. +(4.1) +4.2. Constructing the Curve. The construction of the curve that will define the well W +and the construction of the curve that will define the tunnel T are very similar. First, we +will construct a curve that will define a submanifold Σ, which can be thought of as the +precursor to a well or a tunnel. +We want to construct a curve γ so that the resulting manifold Σ has RΣ > κ − 1 +j for any +j ∈ N. We will first construct γ as a piecewise curve of circular arcs and then smooth the +curve. To do this, we will prescribe the geodesic curvature k(s) of γ, and by Theorem 6.7 in +[Gra98], we know that k(s) determines γ. The unit tangent vector to γ and the curvature +are given by +dγ +ds = (sin θ, − cos θ) +and +k = dθ +ds. +Therefore, if γ(s) is defined for s ≤ s′ and k(s) is given for s ≥ s′ we have γ(s) = (t(s), r(s)) +where +θ(s) = θ(s′) + +� s +s′ k(u)du +t(s) = t(s′) + +� s +s′ sin θ(u)du +r(s) = r(s′) − +� s +s′ cos θ(u)du. +(4.2) +Now, we begin the construction of γ. Fix j ∈ N. Let δ0 < δ and let (0, δ0) be a point +in the (t, r)-plane. Next, define the initial segment of γ as the line segment from (0, 2δ) to +(0, δ0) for s ∈ [−2δ, 0]. Define the next segment to be an arc of a circle of curvature k0 = 1 +that is tangent to r-axis at (0, δ0) and let γ run from 0 to s0 ≤ δ0 +2 where s0 is chosen so +that RΣ > κ − 1 +j and that sin θ(s0) +8r(s0) < 1 for all s ≤ s0. We note that s0 exists since θ(0) = 0 +and by the scalar curvature formula (4.1). Next, we prove a lemma that gives a condition +on γ that controls the scalar curvature. +Lemma 4.4. If δ0 is small enough and if +sin θ(s) +4r(s) +> k(s) for s ≥ s0, +(4.3) +then RΣ > κ. + +EXAMPLES FOR SCALAR SPHERE STABILITY +15 +Proof. By (4.1) we see if k ≤ 0 then +RΣ = RM − 2RicM (∂r, ∂r) sin2 θ ++ (n − 2)(n − 1) +� 1 +r2 + O(1) +� +sin2 θ +− (n − 1) +�1 +r + O(r) +� +k sin θ. +(4.4) +and so the third and fourth terms will be nonnegative. By taking δ0 > r small enough, the +third and fourth terms will dominate the second term so RΣ > κ. +Now, if k > 0, then by rewriting the right-hand side of (4.1) we get +RΣ = (n − 2)(n − 1) +2r2 +sin2 θ + +�(n − 2)(n − 1) +2r2 +− 2RicM (∂r, ∂r) + O(1) +� +sin2 θ ++ −2(n − 1)k +r +sin θ + +�(n − 1) +r +− O(r) +� +k sin θ ++ RM, +(4.5) +so second and fourth terms will be positive by taking δ0 > r is small enough and by +assumption we have +sin θ +4r +> k +which implies +(n − 2)(n − 1) +2r2 +sin2 θ + −2(n − 1)k +r +sin θ > 0, +and so RΣ > κ. +⊓⊔ +Thus, as we continue to construct γ, we will ensure that (4.3) is satisfied. We will now +extend γ by a circular arc of curvature k1 = sin θ(s0) +8r(s0) +on [s0, s1] where s1 − s0 = r0 +2 , where +r(s0) = r0. Let θ(s0) = θ0. By (4.2), we have first that sin θ(s) is increasing and r(s) is +decreasing and so on [s0, s1] +sin θ(s) +4r(s) +> sin θ0 +4r0 +> sin θ0 +8r0 += k1. +Second, we see that γ does not cross the t-axis because s1 − s0 = r0 +2 , and third we have +θ(s1) − θ0 = k1(s1 − s0) = sin θ0 +8r0 +r0 +2 = sin θ0 +16 . +Now we proceed inductively. Define: +si = si−1 + ∆si, +∆si = ri−1 +2 , +ri = r(si), +θi = θ(si), +ki = sin θi−1 +8ri−1 +. +As θ(s) is increasing we have that θi − θi−1 = sin θi−1 +16 +> sin θ0 +16 +and so +θi ≥ θ0 + isin θ0 +16 . +Therefore, θi grows without bound so define m to be such that θm−1 < sin−1 � 12 +13 +� +≤ θm. +Redefine sm so that θm := sin−1 � 12 +13 +� += ¯θ. Note that ∆sm ≤ rm−1 +2 +. + +16 +PAUL SWEENEY JR +Now extend again by one circular arc. +To do this we need to define km+1 > 0 and +sm+1 = sm + ∆sm+1. We add a circular arc until θm+1 = π +2 . By the definition of ¯θ, there +exists a km+1 such that 1 − sin ¯θ < km+1 rm +2 < sin ¯θ +8 +and by (4.2) we know +rm+1 = rm − +� sm+1 +sm +cos θ(u)du += rm − +� sm+1 +sm +cos (sm + km+1(u − sm)) du += rm − +1 +km+1 +(sin θm+1 − sin θm) += rm − +1 +km+1 +� +1 − sin ¯θ +� +> rm +2 . +and km+1 < sin ¯θ +4rm . +Remark 4.5. Up until this point the curve γ works for both the construction of a well +and a tunnel. However, from here on the construction of γ differs slightly for the well and +the tunnel. We will continue now with the construction of the well and discuss the tunnel +construction later in Subsection 4.6. +4.3. Adjusting the curve to construct a well. Now we will refine our construction of +γ in order to construct a well. We want to extend by a line with a negative slope of length +d > 0 and not have γ cross the t-axis. By the intermediate value theorem there exists an +ˆs ∈ (sm, sm+1) such that θ(ˆs) = �θ where +� +max +�¯θ, cos−1 � rm+1 +2 +�� +< �θ < π +2 +if d ≤ 1 +max +�¯θ, cos−1 � rm+1 +2d +�� +< �θ < π +2 +if d > 1 +(4.6) +since θm+1 = π +2 . +Redefine sm+1 such that sm+1 = ˆs and θm+1 = �θ. Extend γ to [sm+1, sm+1+d] by setting +k = 0 on [sm+1, sm+1 + d]. Furthermore, note that by (4.2) we have θ(u) ≡ θm+1 on that +interval and +r(sm+1 + d) = rm+1 − +� rm+1+d +rm+1 +cos θ(u)du += rm+1 − d cos θm+1 +≥ rm+1 − rm+1 +2 +> 0. +Let sm+1 + d = sm+2 and θ(sm+1 + d) = θm+2. We now extend on [sm+2, sm+3] by a +small circular arc of negative geodesic curvature such that θ(sm+3) = 0. Take +km+3 < −2 sin θm+2 +rm+2 +. +Since, + +EXAMPLES FOR SCALAR SPHERE STABILITY +17 +θ(s) = θm+2 + +� s +sm+2 +km+3du = θm+2 + km+3(s − sm+2). +we have +r(sm+3) = rm+2 − +� sm+3 +sm+2 +cos θ(u)du +rm+3 = rm+2 − +1 +km+3 +(sin θm+3 − sin θm+2) +rm+3 = rm+2 + +1 +km+3 +sin θm+2 +> 0. +We can extend γ on [sm+3, sm+4] by a vertical straight line by setting km+4 = 0, where +sm+4 is chosen so that r(sm+4) = 0. +Since γ is parameterized by arclength, we note that a bound on sm+4 is a bound on +arclength. In following lemmas, we prove an upper bound for sm+4. +Lemma 4.6. There exists a constant 0 < C1 < 1 independent of j and d such that +ri +ri−1 +≤ C1 +for 1 ≤ i ≤ m − 1 +Proof. By (4.2) and by the mean value theorem we have +ri = ri−1 − +� si +si−1 +cos θ(u)du = ri−1 − ∆si cos ξi = ri−1 +� +1 − cos ξi +2 +� +for some ξi ∈ [si−1, si]. Recalling that ¯θ ≥ ξi for 1 ≤ i ≤ m − 1 we see that +ri +ri−1 +≤ 1 − cos ¯θ +2 += 21 +26. +⊓⊔ +Lemma 4.7. There is a constant C2 independent of j and d such that sm+4 ≤ C2δ0 + d +which implies that the length of γ is bounded by C2δ0 + d. +Proof. We recall for 1 ≤ i ≤ m, ∆si ≤ ri−1 +2 +so +sm = 2δ − δ0 + s0 + ∆s1 + · · · + ∆sm +≤ 2δ + s0 + 1 +2 (r0 + r1 + · · · + rm−1) +≤ 2δ + s0 + r0 +2 +� +1 + C1 + · · · + Cm−1 +1 +� +≤ 3δ + δ0 +2 +� +1 +1 − C1 +� +≤ 28 +5 δ +(4.7) + +18 +PAUL SWEENEY JR +Now, we note that by (4.2) and (4.6): +∆sm+1 ≤ +1 +km+1 +�π +2 − ¯θ +� +< rm +π +2 − ¯θ +1 − sin ¯θ ≤ +π +2 − ¯θ +1 − sin ¯θδ0 = 13 +�π +2 − sin−1 +�12 +13 +�� +δ0. +By (4.2), we have that +θm+3 = θm+2 + km+3∆sm+3. +Therefore, +∆sm+3 = +θm+2 +−km+3 +≤ 2rm+2θm+2 +sin θm+2 +≤ +π +sin ¯θδ0 = 13π +12 δ0 +(4.8) +because θm+2 ≤ π +2 , rm+2 < δ0, and ¯θ < θm+2. +By construction, +∆sm+2 = d +and +∆sm+4 ≤ δ0. +Thus, +sm+4 = s0 + ∆s1 + · · · + ∆sm + ∆sm+1 + ∆sm+2 + ∆sm+3 + ∆sm+4 +≤ C2δ + d. +⊓⊔ +4.4. Smoothing the curve that defines the well. So far we have constructed k(s) as +a piecewise constant function, k +�� +(si,si+1] = ki+1 The resulting curve γ is C1 and piecewise +C∞. +We begin the smoothing of γ by first smoothing out k(s) on [0, sm+3]. Let g ∈ C∞(R) +be a smooth function so that g is 0 if s < 0, 1 if s > 1, and strictly increasing on [0, 1]. Let +h(x) = g(1 − x) and H = +� 1 +0 h(x)dx. Let ˜k(s) be the smooth function defined by +�k(s) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +g +� s +α +� +s ∈ +� +− δ0 +2 , α +� +1 +s ∈ [α, s0 − α] +(1 − k1)h +� s−s0 +α +� ++ k1 +s ∈ [s0 − α, s0] +k1 +s ∈ [s0, s1] +(ki+1 − ki) g +� s−si +α +� ++ ki +s ∈ [si, si + α] +ki+1 +s ∈ [si + α, si+1] +km+1h +� +s−sm+1 +α +� +s ∈ [sm+1, sm+1 + α] +0 +s ∈ [sm+1 + α, sm+2] +−km+3h +� +s−sm+2 +α +� ++ km+3 +s ∈ [sm+2, sm+2 + α] +km+3 +s ∈ [sm+2 + α, sm+3], +where 1 ≤ i ≤ m. +We note that α is the same for each i and that its value will be determined later. Now +let ˜θ be the angle function associated to ˜k. By (4.2) we see that the smooth curve ˜γ(s) = +(˜t(s), ˜r(s)) defined by ˜k(s) will converge uniformly to γ on +� +δ0 +2 , sm+3 +� +as α goes to zero. +Also, ˜θ will converge uniformly to θ as α goes to zero. Therefore, take α small enough such +that ˜θ(sm+1) satisfies (4.6); therefore, we will still extend by a line with a negative slope. + +EXAMPLES FOR SCALAR SPHERE STABILITY +19 +Note +˜θ(sm+2 + α) = ˜θ(sm+2) + +� sm+2+α +sm+2 +−km+3h +�u − sm+2 +α +� ++ km+3du += ˜θ(sm+2) + αkm+3(1 − H) +> 0. +By the smoothing process, ˜θ(sm+3) may no longer be greater than 0. We will now fix that. +If ˜θ(sm+3) ≤ 0 pick a s∗ ∈ (sm+2 + α, sm+3] such that 0 < ˜θ(s∗) < α which exists by the +intermediate value theorem. +If ˜θ(sm+3) > 0, we can redefine sm+3 as sm+3 + +˜θ(sm+2) +−km+3 so that ˜θ(sm+3) = 0. By the +intermediate value theorem, pick a s∗ ∈ (sm+2 + α, sm+3] such that 0 < ˜θ(s∗) < α. +Redefine sm+3 in either case as sm+3 = s∗ and note 0 < ˜θ(s∗) < α. On [sm+3, sm+3 + 2β] +define +˜k(s) = +� +−km+3g +� +s−sm+3 +β +� ++ km+3 +s ∈ [sm+3, sm+3 + β] +0 +s ∈ [sm+3 + β, sm+3 + 2β], +where β = +˜θ(sm+3) +−km+3(1−H) so that +� sm+3+β +sm+3 +˜k(s)ds = −˜θ(sm+3). +This makes ˜θ(sm+3 + 2β) = 0. +By (4.2) we see that the smooth curve ˜γ(s) = (˜t(s), ˜r(s)) defined by ˜k(s) will converge +uniformly to γ on [−2δ, sm+3] as α goes to zero. +Also, ˜θ will converge uniformly to θ as α goes to zero; moreover, as α goes to zero so does +β. Finally, take α small enough so that ˜r(sm+3+2β) > 0. Extend the line segment at the end +of ˜γ on [sm+3+2β, L] where L is defined so that ˜r(L) = 0. Note that |L−(sm+3+2β)| < δ0. +4.5. Attaching the Well. We have constructed a smooth curve ˜γ on [−2δ, L] that begins +and ends as a vertical line segment. Define +˜Wj = {(y, q) ∈ X : (y, ||q||M) ∈ ˜γ} +and let gj be the induced metric, i.e., ˜gj = ˜ι∗ +j(dt2+g) where ˜ιj : ¯Wj → R×B is the inclusion +map. +Lemma 4.8. For small enough α we will show that ˜γ satisfies R ˜ +Wj ≥ κ − 1 +j on [−2δ, L]. +Also the length ˜γ is bounded by C3δ + d. +Proof. By construction ˜γ is parameterized by arclength so by Lemma 4.7 length of ˜γ is +bounded by C3δ0 + d. + +20 +PAUL SWEENEY JR +On [−2δ, 0], we have that +R +˜ +Wj = RM − 2RicM (∂˜r, ∂˜r) sin2 ˜θ + (n − 2)(n − 1) +� 1 +˜r2 + O(1) +� +sin2 ˜θ +− (n − 1) +�1 +˜r + O(r) +� +˜k sin ˜θ += RM +> κ − 1 +j . +On [0, s0], we have that k1 < 1 because of our choice of s0 and the construction. +R +˜ +Wj = RM − 2RicM (∂˜r, ∂˜r) sin2 ˜θ + (n − 2)(n − 1) +� 1 +˜r2 + O(1) +� +sin2 ˜θ +− (n − 1) +�1 +˜r + O(r) +� +˜k sin ˜θ +≥ κ − 2RicM (∂˜r, ∂˜r) sin2 ˜θ + (n − 2)(n − 1) +� 1 +˜r2 + O(1) +� +sin2 ˜θ +− (n − 1) +�1 +˜r + O(r) +� +sin ˜θ +> κ − 1 +j . +since for small enough α we have that ˜θ is uniformly close to θ and ˜r is uniformly close to +r. +On [s0, s1] we have that ˜k(s) = k1 and so +sin ˜θ(s) +4˜r(s) − ˜k(s) > 0 +for small enough α. +On [si, si+1] for 1 ≤ i ≤ m, we have that +sin ˜θ(s) +4˜r(s) − ˜k(s) = +� +sin ˜θ(s) +4˜r(s) − ki+1 +� ++ +� +ki+1 − ˜k(s) +� +and so for small enough α we have the the first term is positive since sin θ(s) +4r(s) > k(s) and the +second term is positive by construction. +On [sm+1, sm+2], we have that +sin ˜θ(s) +4˜r(s) +≥ sin ˜θ(sm+1) +4˜r(sm+1) +> km+1 > ˜k(s). +We have the first inequality since +sin ˜θ(s) +4˜r(s) +is non-decreasing. +The second inequality was +already verified above. The third inequality holds since by construction km+1 > ˜k(s). +On [sm+2, L], we have by construction that ˜k(s) is non-positive so +sin ˜θ(s) +4˜r(s) +≥ ˜k(s). + +EXAMPLES FOR SCALAR SPHERE STABILITY +21 +Therefore, by Lemma 4.4 we have shown R ˜ +Wj > κ − 1 +j on +� +− δ0 +2 , L +� +. +⊓⊔ +Next we will prove the diameter and volume bounds for the well, but before we prove +those bounds, we need to recall the following fact. +Proposition 4.9. Let B(p, r) be a geodesic ball of radius r in a closed Riemannian manifold +(Mn, g). Then there exists constants C, r0 depending on g such that for any p and for all +r ≤ r0 we have +volg(B(p, r)) ≤ Crn +volg(∂B(p, r)) ≤ Crn−1. +(4.9) +Lemma 4.10. There is a constant C(g) independent of j, d such that diameter diam ( ˜Wj) +and volume vol ( ˜Wj) of ˜Wj satisfy +d ≤ diam ( ˜Wj) < C(δ + d) and vol ( ˜Wj) < C(δn + dδn−1). +Proof. Let p, q ∈ ˜Wj be two points and let x be the point at the tip of ˜Wj, i.e., corresponding +to ˜γ(L). By the triangle inequality and Lemma 4.3 we have +dgj(p, q) ≤ dgj(p, x) + dgj(x, q) ≤ length(˜γ) + length(˜γ) ≤ C(δ + d). +By construction we have d ≤ diam (W). Therefore, d ≤ diam (W) < C(δ + d). +By possibly taking δ smaller, we have by Lemma 4.8 and Proposition 4.9 that +vol ( ˜Wj) = +� L +−δ0 +2 +|∂B(p, r(s))|˜gjds ≤ +� L +−δ0 +2 +Cδn−1ds ≤ C(δn + δn−1d). +⊓⊔ +Lemma 4.11. ( ˜Wj, ˜gj) is isometric to Wj = (Bg(p, 2δ), gj = dF 2 +j + g) and Wj attaches +smoothly to M. +Proof. Let B = Bg(p, 2δ) and recall that ||q||g is the distance from q to p in B. Consider +the function Fj : B → R, Fj(q) = ˜t +� +˜r−1(||q||g) +� +. By construction ˜t is smooth and ˜r′(s) < 0 +so ˜r−1 is smooth. Moreover, ||q|| is smooth away from p. Thus, away from p, F is smooth. +In a neighborhood of p we have by construction that (˜t(s), ˜r(s)) is a vertical line segment +so in that neighborhood ˜t ◦ ˜r−1 ≡ const and so Fj is smooth everywhere. Furthermore, by +construction, we have that +˜gj|E = g|E where E = Bg(p, 2δ) \ Bg(p, δ). +Let Γj = {(t, p) ∈ X : Fj(p) = t}. Note that Γj ⊂ X and that Γj = ˜Wj. Let g′ +j = +(ι′ +j)∗(dt2 +g) where ι′ +j : ˜Wj → R×B is the inclusion map ι′ +j(t, p) = (t, p). Let idj : Γj → ˜Wj +be the identity map and conclude that g′ +j = ˜gj. Consider the diffeomorphism Φj : B → Γj +where Φj(q) �→ (Fj(q), q). And so +Φ∗ +jg′ +j = Φ∗((ι′ +j)∗g′ +j) = dF 2 +j + g. +⊓⊔ +And this completes the construction of N from Proposition 4.1 (Constructing Wells). + +22 +PAUL SWEENEY JR +4.6. Constructing a Tunnel. We will pick up the construction of the tunnel from Re- +mark 4.5. Let γ be as it is before Remark 4.5. The same smoothing procedure as above can +be used to smooth γ into a smooth curve. We will abuse notation and call this smoothed-out +curve ˜γ as well. +Let g ∈ C∞(R) be the smooth function so that g is 0 if s < 0, 1 if s > 1, and strictly +increasing on [0, 1]. Let h(x) = g(1 − x) and H = +� 1 +0 h(x)dx. Let ˜k(s) be the smooth +function defined by +�k(s) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +g +� s +α +� +s ∈ +� +− δ0 +2 , α +� +1 +s ∈ [α, s0 − α] +(1 − k1)h +� s−s0 +α +� ++ k1 +s ∈ [s0 − α, s0] +k1 +s ∈ [s0, s1] +(ki+1 − ki) g +� s−si +α +� ++ ki +s ∈ [si, si + α] +ki+1 +s ∈ [si + α, si+1] +where 1 ≤ i ≤ m. +Note that ˜θ(sm+1) could no longer equal π +2 by the smoothing process. We fix that now. +We note that ˜θ(s) converges uniformly to θ(s) as α goes to zero. Take α be small enough +such that ˜θ(sm + α) < π +2 . +We want ˜θ(sm+1) < π +2 . Therefore, if not, then ˜θ(sm+1) ≥ π +2 . Pick a s∗ ∈ (sm + α, sm+1] +such that π +2 − α < ˜θ(s∗) < π +2 which exists by the intermediate value theorem and redefine +sm+1 = s∗. +Let sm+2 = sm+1 + 2β. On [sm+1, sm+2], define +˜k(s) = +� +−km+1g +� +s−sm+1 +β +� ++ km+1 +s ∈ [sm+1, sm+1 + β] +0 +s ∈ [sm+1 + β, sm+2], +where β = +π +2 −˜θ(sm+1) +−km+1(1−H) so that +� sm+1+β +sm+1 +˜k(s)ds = π +2 − ˜θ(sm+1). +Thus, ˜θ(s) = π +2 for all s ∈ [sm+1 + β, sm+2]. Moreover, we have finished smoothing γ to ˜γ. +Define a half tunnel Aj = {(y, q) ∈ X : (y, ||q||g) ∈ ˜γ} with the induced metric. Later, we +will glue two half tunnels together to make a tunnel Tj. In the following lemma, we record +properties of Aj whose proofs are analogous to the ones above. +Lemma 4.12. There is a constant C independent of j such that (Aj, hj) satisfies the +following +(i) The scalar curvature Rj of Aj satisfies Rj > κ − 1 +j . +(ii) diam (Aj) < C(δ). +(iii) vol (Aj) < C(δn). +(iv) Aj smoothly attaches to M \ Bg(p, 2δ) +(v) The new manifold (M \ Bg(p, 2δ)) ⊔ Aj is a manifold with boundary. +We have constructed half of a tunnel, Aj. We now wish to modify the metric at the end +of Aj so that it is a product metric of a round sphere and an interval. We follow the same +procedure as [Dod20]. Let a = t(sm+1 + β), b = t(sm+2), and c = r(sm+2). We note that, + +EXAMPLES FOR SCALAR SPHERE STABILITY +23 +by construction, the induced metric on {(q, y) ∈ X : a ≤ t ≤ b} is h0 = gc + dt2, where +gc is the induced metric on Sn−1(c). Let h1 = c2grd + dt2 where grd is the round metric +on the unit round sphere. Let φ(t) = ψ +� +t−a +η +� +where ψ(u) is a smooth function on [0, 1] +vanishing near zero, increasing to 1 at u = 3 +4 and equal to 1 for u > 3 +4. Define the metric h +for t ∈ [a, b] as +h(q, y) = gc(q, y) + φ(t) +� +c2grd − gc +� ++ dt2. +This metric transitions smoothly between h0 and h1. Note +h − h0 = φ(t) +� +c2grd − gc +� += φ(t)c2 +� +grd − 1 +c2 gc +� +and that the first and second derivatives of φ(t) are O(η−1) and O(η−2), respectively. So +by Lemma 4.3, we have that the second derivatives of h − h0 are O(η2). Therefore, for η +small enough, the scalar curvature of h is close to the scalar curvature of h0 which, again by +Lemma 4.3, has scalar curvature larger than κ − 1 +j for small enough η. Therefore, we have +changed the metric at the end of Aj so that it looks like c2grd + dt2. Thus, given another +ball Bg(p′, 2δ) on M we can construct A′ +j with a metric at the one end that it looks like +c2grd + dt2 with the same c by making the same choices in the construction as we did for +Aj. Now we can immediately glue a cylinder, ([0, d] × Sn−1, dt2 + c2grd), connecting A′ +j to +Aj and so construct the tunnel Tj between ∂Bg(p′, 2δ) and ∂Bg(p, 2δ). +We note that the diameter and volume of the cylinder ([0, d] × Sn−1, dt2 + gSn−1) are +bounded by d and C(n)dδn−1, respectively, where C(n) is a constant that only depends on +the dimension. Therefore, we can conclude that diam(Tj) and vol(Tj) satisfy the bounds in +Proposition 4.2. Therefore, this completes the construction for Proposition 4.2. +5. Manifolds with shrinking tunnels +In this section, we will use Proposition 4.2 (Constructing Tunnels) to construct sequences +of manifolds with thinner and thinner long tunnels. Furthermore, we will prove Theorems +A and A′. +We will need first the following preliminary results. +Proposition 5.1. There exists a sequence of rotationally symmetric manifolds Mj = +(Sn, gj), n ≥ 3, such that Mj satisfies +Rj ≥ n(n − 1) − 1 +j , diam (Mj) ≤ D, and vol (Mj) ≤ V, +for some constants 0 < D, V and converges to M∞ which is the disjoint union of two +n-spheres. +Proof. We will construct the Mj as the connected sum of two standard unit round n- +spheres for which the tunnel that connects the two spheres gets skinnier as j increases. By +Proposition 4.2, we can remove a geodesic ball from both of the spheres and then construct +a tunnel Tj connecting the two spheres. Let (N, h) = (N′, h′) = (Sn, grd). Let j ∈ N, +j ≥ 10, d = 30. Define +Bj := Bh +� +p, 2 +j +� +⊂ N, and B′ := Bh′ +� +p′, 2 +j +� +⊂ N′ + +24 +PAUL SWEENEY JR +where Bj and B′ +j are geodesic balls in N, N′ respectively. +By Proposition 4.2, we can +construct a tunnel Tj connecting ∂Bj to ∂B′ +j and the resulting manifold Mj will have the +following properties: +(i) Mj = +� +(N ⊔ N′) \ +� +Bj ∪ B′ +j +�� +⊔ Tj +(ii) Rj ≥ n(n − 1) − 1 +j . +(iii) Mj \ Tj is isometric to (N \ Bj) ⊔ (N′ \ B′ +j). +(iv) diam (Mj) ≤ 4π + 30, +2 volgrd (Sn) − volh (Bj) − volh′ (B′ +j) ≤ vol (Mj) ≤ 2 volgrd (Sn) + volgj (Tj), +and +lim +j→∞ volh (Bj) = lim +j→∞ volh′ (B′ +j) = lim +j→∞ volgj (Tj) = 0. +In particular, limj→∞ vol (Mj) = 2 volgrd (Sn). +By Theorem 3.3, we have the intrinsic flat distance between Mj and N ⊔ N′ is +dF(Mj, N1 ⊔ N2) ≲ 1 +j +� +volgrd(Sn) + volgrd(Sn−1) +� ++ volh (Bj) + volh′ (B′ +j) + volgj (Tj). +As j → ∞0, we that volh (Bj), volh′ (B′ +j), and volgj (Tj) go to zero. Therefore, we conclude +that Mj converges to N ⊔ N′ in the VF sense. +⊓⊔ +Remark 5.2. From the construction in Proposition 4.2 (Constructing Tunnels) we see that +Mn +j = ([0, Dj]×Sn−1, gj) defined above is rotationally symmetric. Moreover, near {0}×Sn−1 +and {Dj} × Sn−1, we have that Mn +j is isometric to the standard unit round n-sphere. In +particular, the metric takes the form gj = dt2 +sin2(ρj(t))gSn−1 where Dj is the diameter of +Mj and for ρj : [0, Dj] → [0, ∞) is a smooth function with the following properties. Recall +˜γj = (˜tj(s), ˜rj(s)) to be the curve define in Lemma 4.12 that defines the half tunnel Aj. +Then +ρ(t) = +� +ˆr(s), +s ∈ +� +0, 1 +2Dj +� +ˆr(D − s), +s ∈ +� 1 +2Dj, Dj +� and ˆr(t) = +� +� +� +π − s, +s ∈ +� +0, π − 2 +j +� +˜r(s + (δ − π)), +s ∈ +� +π − 2 +j , 1 +2D +� +. +We will now construct smooth 1-Lipschitz maps Fj : Mn +j → (Sn, grd). But first, we need +the following result based on the mollification in [Mia02, Section 3]. Since our lemma varies +slightly from what is stated in [Mia02] we provide an analogous proof. +Lemma 5.3. Let h : R → R be an L-Lipschitz continuous function such that +h(t) = +� +h+(t), +t ∈ (0, ∞) +h−(t), +t ∈ (−∞, 0), +where h+ and h− are smooth functions. Then for small enough ϵ > 0 there exists a function +hϵ : R → R such that +||hϵ(t) − h(t)||C2 ≲ ϵ2, h′ +ϵ(t) ≤ sup{h′(t) : t ∈ R \ {0}}, and |h′ +ϵ(t)| ≤ L. +Proof. Let 0 < ϵ0 < 1. We will restrict our attention to (−ϵ0, ϵ0). Let ϕ ∈ C∞ +c ([−1, 1]) be +the standard mollifier in R such that +0 ≤ ϕ ≤ 1 +and +� 1 +−1 +ϕ(t)dt = 1. + +EXAMPLES FOR SCALAR SPHERE STABILITY +25 +Let σ(t) ∈ C∞ +c +�� +− 1 +2, 1 +2 +�� +be another bump function such that +0 ≤ σ(t) ≤ +1 +100 for t ∈ R, +σ(t) = +1 +100 for |t| < 1 +4, +0 < σ(t) ≤ +1 +100 for 1 +4 < |t| < 1 +2. +Let 0 < ϵ < 1 +10ϵ0. Define σϵ(t) = ϵ3σ +� t +ϵ +� +. Moreover, define +hδ(t) = +� +R +h(t − σδ(t)s)ϕ(s)ds, +t ∈ (−ϵ0, ϵ0) += +�� +R h(s) · +1 +σδ(t)ϕ +� +t−s +σδ(t) +� +ds, +σδ(t) > 0 +h(t), +σδ(t) = 0. +(5.1) +Now we want to compute h′ +δ(s). For |t| > +ϵ3 +100, +h′ +ϵ(t) = d +dt +� +R +h(t − σϵ(t)s)ϕ(s)ds += +� +R +h′(t − σϵ(t)s) +� +1 − sϵ2σ′ +�t +ϵ +�� +ϕ(s)ds. +For |t| < ϵ +4, +h′ +δ(t) = d +dt +� +R +h(s) · +1 +σϵ(t)ϕ +�t − s +σϵ(t) +� +ds += +� +R +h(s) · d +dt +� +1 +σϵ(t)ϕ +�t − s +σϵ(t) +�� +ds += +� +R +h(s) · d +dt +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds += (−1) · +� +R +h(s) · d +ds +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds += (−1) · +� 0 +−∞ +h−(s) · d +ds +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds ++ (−1) · +� ∞ +0 +h+(s) · d +ds +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds += +� 0 +−∞ +h′ +−(s) · +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds ++ +� ∞ +0 +h′ ++(s) · +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds += +� +R +h′(s) · +�100 +ϵ3 ϕ +�100(t − s) +ϵ3 +�� +ds += +� +R +h′(t − σϵ(t)s)ϕ(s)ds. + +26 +PAUL SWEENEY JR +Now note for |t| < ϵ +4 that σϵ is a constant function; therefore, for all t ∈ (−ϵ0, ϵ0) +h′ +ϵ(t) = +� +R +h′(t − σϵ(t)s) +� +1 − sϵ2σ′ +�t +ϵ +�� +ϕ(s)ds. +(5.2) +By (5.1) and (5.2) we have +||hϵ(t) − h(t)||u ≤ +� +R +||h(t − σϵ(t)s) − h(t)||uϕ(s)ds +≲ ϵ3. +and +||h′ +ϵ(t) − h′(t)||u ≤ +� +R +||h′(t − σϵ(t)s) − h′(t)||uϕ(s)ds ++ +� +R +���� +����h′(t − σϵ(t)s)sϵ2σ′ +�t +ϵ +����� +���� +u +ϕ(s)ds +≲ ϵ3 + ϵ2 +� +R +���� +����h′(t − σϵ(t)s)σ′ +�t +ϵ +����� +���� +u +ϕ(s)ds +≲ ϵ2. +Lastly, note that +|h′ +ϵ(t)| = +� +R +|h′(t − σϵ(t)s)| +���� +� +1 − sϵ2σ′ +�t +ϵ +������ |ϕ(s)|ds +≤ L +� +R +� +1 − sϵ2σ′ +�t +ϵ +�� +(ϕ(s))ds += L +� +1 − ϵ2σ′ +�t +ϵ +� � +R +sϕ(s)ds +� +≤ L, +where the first inequality follows if ϵ is small enough and the last inequality follows since +sϕ(s) is an odd function. Moreover, redoing this computation without the absolute values +shows that h′ +ϵ(t) ≤ sup{h′(t) : t ∈ R \ {0}}. +⊓⊔ +Now we are ready to construct smooth 1-Lipschitz maps Fj : Mn +j → (Sn, grd). +Lemma 5.4. There exists a function Fj : Mn +j → Sn that is a 1-Lipschitz diffeomorphism +with deg Fj ̸= 0 +Proof. First define a decreasing 1-Lipschitz function fj : [0, Dj] → [0, π]. +fj(t) = +� +π − t, +t ∈ [0, tj] +aj(t − tj) + bj, +t ∈ [tj, Dj], +where aj = −π+tj +Dj−tj , bj = π − tj, and tj is chosen so that fj(tj) = 1 +10ρ +� 1 +2Dj +� +. Note ρ +� 1 +2Dj +� +is +the radius of the cylindrical part of the tunnel which is also the minimum that ρj(t) attains +on +� π +2 , Dj − π +2 +� +. +By Lemma 5.3, we can smooth fj to fj,ϵ by choosing ϵ0 and ϵ small enough. And so +define Fj,ϵ(t, θ) = (fj,ϵ(t), θ). Since f′ +j,ϵ(t) < 0 and fj,ϵ is a bijection, we have that Fj,ϵ is a +diffeomorphism. We want to show that for all v ∈ TMj +F ∗ +j,ϵgrd(v, v) ≤ gj(v, v). + +EXAMPLES FOR SCALAR SPHERE STABILITY +27 +Note that +F ∗ +j,ϵgrd = +� +f′ +j,ϵ(t) +�2 dt2 + sin2(fj,ϵ(t))gSn−1. +and +gj = dt2 + sin2(ρj(t))gSn−1. +First by (4.2) and Lemma 5.3 we know that |f′ +j,ϵ(t)| ≤ 1 for all t. Now we will show that +sin2(fj,ϵ(t)) ≤ sin2(ρj(t)). +On [0, π − tj − 20ϵ] we have by (4.2) that +ρj(t) = π − +� t +0 +cos (θj(u)) du ≥ π − t = fj(t) = fj,ϵ(t). +On [π − tj − 20ϵ, π − tj] we have +ρj(t) = π − +� t +0 +cos (θj(u)) du > π − t = fj(t) +and so for small enough ϵ, we have that fj,ϵ(t) will also satisfy this inequality. +On +� +π − tj, Dj − π +2 +� +, we have that fj(t) ≤ +1 +10ρ +� 1 +2Dj +� +and that +1 +10ρ +� 1 +2Dj +� +< ρj(t) ≤ π +2 . +Therefore, sin2(fj(t)) ≤ sin2(ρj(t)) on +� +0, Dj − π +2 +� +. +Lastly on +� +Dj − π +2 , Dj +� +we have the following: ρj(t) = π − Dj + t and fj,ϵ(t) = fj(t) = +aj(t − tj) + bj by the construction. Moreover, +−fj(t) + π ≥ ρj(t) +since if we define ψj(t) = ρj(t) + fj(t) − π, then we see that ψ′(t) ≥ 0 and ψ(Dj) = 0. +We also note on +� +Dj − π +2 , Dj +� +that π +2 ≤ −fj(t) + π ≤ π and π +2 ≤ ρj(t) ≤ π. Therefore, we +conclude that sin2(fj,ϵ(t)) = sin2(−fj,ϵ(t) + π) ≤ sin2(ρj(t)) on +� +Dj − π +2 , Dj +� +. +Thus, for all v ∈ TMj we have +F ∗ +j,ϵgrd(v, v) ≤ gj(v, v), +which implies +ℓSn (Fj,ϵ ◦ c) ≤ ℓMj(c) +where c : [0, 1] → (Sn, grd) is a path connecting p and q. This implies that +dSn (Fj,ϵ(p), Fj,ϵ(q)) ≤ dMj(p, q). +Thus, we have that Fj,ϵ is 1-Lipschitz. Moreover, deg Fj,ϵ ̸= 0 since Fj,ϵ is a diffeomorphism. +⊓⊔ +Lemma 5.5. Let (S3, g1), (S3, g2) be 3-spheres such that there exists a diffeomorphism F : +(S3, g1) → (S3, g2) that is 1-Lipschitz and is isotopic to the identity then +width(S3, g2) ≤ width(S3, g1). +Proof. By the definition of width for any δ > 0 there exists {Σt} such that +sup +t +|Σt|1 < width(S3, g1) + δ; +therefore, +width(S3, g2) ≤ sup +t +|F(Σt)|2 ≤ sup +t +|Σt|1 ≤ width(S3, g1) + δ +where the first inequality follows since F(Σt) ∈ Λ′ and the second inequality follows since +F is 1-Lipschitz. +⊓⊔ + +28 +PAUL SWEENEY JR +Proof of Theorem A. Let Mn +j be as in Proposition 5.1; therefore, Mn +j → M∞ in VF-sense +where M∞ is the disjoint union of two spheres. Let Fj : Mn +j → Sn be as in Lemma 5.4 and +define ˜Fj(r, θ) = Fj(Dj − r, θ). Consider the diffeomorphism +Φ : [0, Dj] × Sn−1 → [0, π] × Sn−1, +Φ(r, θ) = +� π +Dj +r, θ +� +Note that Φ is an isometry between ([0, Dj] × Sn−1, Φ∗(dr2 + sin2(r)gSn−1)) and +([0, π] × Sn−1, dr2 + sin2(r)gSn−1). And now consider +(Φ−1 ◦ ˜Fj)(r, θ) = +�Dj +π fj(Dj − r), θ +� +. +This map is a 1-Lipschitz orientation preserving diffeomorphism from Mn +j to the round +n-sphere and Φ−1 ◦ Fj is isotopic to the identity. Therefore, by Lemma 5.5 we have that +width(Mn +j ) ≥ 4π. +⊓⊔ +Proof of Theorem A′. Let Mj be as in Proposition 5.1; therefore, Mn +j → M∞ where M∞ +is the disjoint union of two spheres. Let Fj : Mn +j → Sn be as in Lemma 5.4. Then by +Arzela-Ascoli Theorem 3.4 there is a subsequence Fjk that converges to a 1-Lipschitz map +F∞ : M∞ → Sn. +This map is not a Riemannian isometry since Sn is connected and N ⊔ N′ is not. +⊓⊔ +6. Manifolds with many wells +In this section, we will use Proposition 4.1 (Constructing Wells) to construct sequences +of manifolds with many wells. Furthermore, we will prove Theorems B and B′. +Theorem 6.1. Let (Mn, g) be a closed Riemannian manifold of dimension n ≥ 3 with scalar +curvature R ≥ κ. Then there exists a sequence of Riemannian manifolds Mn +j = (Mn, gj) +such that Rj ≥ κ − 1 +j and Mn +j converge in the VADB-sense and VF-sense to Mn but has +no convergent subsequence in the GH-topology. +Proof. Define +Xj = +� +(B(pj +i, δj), g) +�j +i=1 +to be a collection of disjoint geodesic balls in Mn where 0 < δj < 1 +j is chosen small enough +so that by Proposition 4.1 we replace each B(pj +i, δj) with the well Wi,j = (B(pj +i, δj), gj) such +that the scalar curvature of each of the wells satisfies Rj > κ − 1 +j . Moreover, choose d = 1 +2 +in Proposition 4.1 so that diam(Wi,j) ≥ 1 +2. Call the resulting manifold Mn +j = (Mn, gj). +Now we note that +lim +j→∞ volj (Mn +j ) = lim +j→∞ volg (Mn) − +j +� +i=1 +volg (B(pj +i, δj)) + +j +� +i=1 +volj (Wi,j). +Thus, by Proposition 4.1 and Proposition 4.9 +lim +j→∞ volg (Mn) − jCδn +j ≤ lim +j→∞ volj (Mn +j ) ≤ lim +j→∞ volg (Mn) + Cj +� +δn +j + +δn−1 +j +2 +� +. + +EXAMPLES FOR SCALAR SPHERE STABILITY +29 +and so +lim +j→∞ volj (Mn +j ) = volg (Mn). +Also by Proposition 4.1 and the triangle inequality, we have that +diam +� +Mn +j +� +≤ diam ((Mn, g)) + 2 diam (Wj) ≤ diam ((Mn, g)) + 2 +� +C + 1 +2 +� +. +so the diameters are uniformly bounded. +Consider the identity map id : (Mn, gj) → (Mn, g). Denote id∗gj = gj. Now by con- +struction and Lemma 4.11 we have for any p ∈ Wi,j that +g(v, v) ≤ gj(v, v) for all v ∈ TpM +because gj = dF 2 +j + g and if p /∈ Wi,j then g(v, v) = gj(v, v) for all v ∈ TpM. +Therefore, Mn +j converges to (Mn, g) in the VADB-sense and by Theorem 3.7 we have +that Mn +j converges to (Mn, g) in the VF-sense. +Fix ϵ0 < 1 +4. Note that ϵ0 < d and so B(pj +i, ϵ0) ⊂ Mn +j are disjoint. Therefore, +j < Covj(ϵ0) +and so as j → ∞ we have Covj(ϵ0) → ∞ so by Theorem 3.1 that Mj does not converge in +the GH-sense. +⊓⊔ +Proof of Theorem B. Consider the round 3-sphere (S3, grd). By Theorem 6.1 we see that +there exists a sequence (S3, gj) with scalar curvature Rj ≥ 6 − 1 +j such that (S3, gj) → +(S3, grd) in the V ADB and VF-sense but has no convergent subsequence in the GH-topology. +Moreover, the identity map id : (S3, gj) → (S3, grd) is 1-Lipschitz and by Lemma 5.5 we +have that width(S3, gj) ≥ 4π. +⊓⊔ +Proof of Theorem B′. Consider the round n-sphere (Sn, grd). By Theorem 6.1 we see that +there exists a sequence Mj = (Sn, gj) with scalar curvature Rj ≥ n(n − 1) − 1 +j such that +Mj → (S3, grd) in the V ADB and VF-sense but has no convergent subsequence in the +GH-topology. Furthermore, the identity map id : (Sn, gj) → (Sn, grd) is smooth 1-Lipschitz +diffeomorphism. +⊓⊔ +Proof of Theorem C. Let κ > 0 and let (Mn, g) be the round sphere of curvature +2κ +n(n−1). +Let {pj}∞ +j=1 ⊂ Mn be a sequence of points on a geodesic converging to a point p∞. Define +{B(pj, δj)}∞ +j=1 +to be a collection of disjoint geodesic balls in Mn where 0 < δj < 1 +2j is chosen small enough +so that by Proposition 4.1 there exists a well Wj = (B(pj, δj), gj) such that the scalar +curvature of each of the wells satisfies Rj > 2κ +� +1 − +1 +10j +� +> κ. Let {dj}∞ +j=1 ⊂ [2, 10] be a +strictly increasing sequence of positive numbers, and choose d = dj in Proposition 4.1 so +that diam(Wj) ≥ dj. Now define Mn +i to be the Riemannian obtained by replacing the first +i balls with the corresponding first i wells, i.e., +Mn +i = +� +�Mn \ +i� +j=1 +B(pj, δj) +� +� ⊔ +i� +j=1 +Wj. + +30 +PAUL SWEENEY JR +We note that Mn +i has scalar curvature strictly larger than κ. We also have by Proposition 4.1 +that +diam(Mn +i ) ≤ 25C +and +vol(Mn +i ) ≤ vol(Mn) + +∞ +� +j=1 +vol(Wj) +≤ vol(Mn) + C +� +� +∞ +� +j=1 +1 +2nj + 10 +∞ +� +j=1 +1 +2(n−1)j +� +� +≤ vol(Mn) + 11C. +Now we will define M∞ to be +M∞ = +� +�Mn \ +∞ +� +j=1 +B(pj, δj) +� +� ⊔ +∞ +� +j=1 +Wj +with its induced length metric and natural current structure T∞. Therefore, we have that +vol(Mn +i ) → vol(M∞). Let Ej ⊂ Wj be a ball centered at pj of radius 1 and so M∞ is +noncompact since it contains infinitely many disjoint balls of radius 1. +We will show that Mn +i +converges to M∞ in an analogous many to [SW11, Example +A.11]. Let ϵi = dMn(pi, p∞) and note that if ˜Bi = B(p∞, ϵi − δi), then there is an isometry, +ϕ : Vi → V ′ +i where Ui = Mn +i \ ˜Bi ⊂ Mi and U ′ +i ⊂ M∞. By [SW11, Lemma A.2], there exists +a metric space Z such that +dZ +F (Mn +i , M∞) ≤ vol(Mn +i \ Ui) + vol(M∞ \ U ′ +i) ++ vol(Ui) +�� +2 diamMn +i (∂Ui) diamMn +i (Ui) + diamMn +i (∂Ui) +� ++ vol(U ′ +i) +�� +2 diamMn +i (∂U′ +i) diamMn +i (U ′ +i) + diamMn +i (∂U′ +i) +� +. +We note that +vol(Mn +i \ Ui) ≤ π(ϵi − δi)n, +vol(M∞ \ U ′ +i) ≤ C +� +� +∞ +� +j=i +1 +2nj + 10 +∞ +� +j=i +1 +2(n−1)j +� +� . +Also, diam(∂Ui) and diam(∂U′ +i) converge to zero. Therefore, the right-hand side of the +inequality above goes to zero as i → ∞. We conclude then that Mn +i converges to M∞ in +the VF-sense. +⊓⊔ +7. Sewing Manifolds +We are able to generalize the sewing examples of Basilio, Dodziuk, and Sormani found +in [BDS18] and [BS21]. There are two methods of sewing developed in [BS21]. Method I +generalizes the curve sewing construction of [BDS18]. Here we will extend the construction +using Proposition 4.2 (Constructing Tunnels). We start with Method I which says that +given a fixed manifold one can tightly sew a compact region to a point. +Proposition 7.1. Let (Mn, g) be a complete Riemannian manifold, and A0 ⊂ M a compact +subset with an even number of points pi ∈ A0, i = 1, . . . , n with pairwise disjoint balls + +EXAMPLES FOR SCALAR SPHERE STABILITY +31 +B(pi, 2δ) with scalar curvature greater than κ. For small enough δ > 0, define Aδ := Tδ(A0) +and +A′ +δ = Aδ \ +� n� +i=1 +B(pi, δ) +� +⊔ +n +2� +i=1 +Ti +where Ti are tunnels as in Proposition 4.2 (Constructing Tunnels) connecting ∂B(p2j+1, δ) +and ∂B(p2j+2, δ) for j = 0, 1, . . . , n +2 −1. Then given any ϵ, shrinking δ further, if necessary, +we may create a new complete Riemannian manifold, (Nn, h), +Nn = (Mn \ Aδ) ⊔ A′ +δ +satisfying +vol (Aδ) − ϵ ≤ vol (A′ +δ) ≤ vol (Aδ) + ϵ +and +vol (M) − ϵ ≤ vol (N) ≤ vol (M) + ϵ +If, in addition, M has scalar curvature, RM ≥ κ, then N has scalar curvature, RN ≥ κ − ϵ. +If ∂M ̸= ∅, the balls avoid the boundary and ∂M is isometric to ∂N. +Proof. The proof follows from the proof of [BDS18, Proposition 3.1] while using Proposi- +tion 4.2 (Constructing Tunnels) and Proposition 4.9. +⊓⊔ +Proposition 7.2. Let (Mn, g) be a complete Riemannian manifold and A0 ⊂ M. +Let +Aa = Ta(A0) be a tubular neighborhood of A0. Assume that there is an a > 0 such that +Aa has scalar curvature greater than κ. Let r ∈ (0, a). Given ϵ > 0, there exists δ = +δ(A0, κ, r, ϵ) ∈ (0, r) and there exists even n = ¯n(¯n − 1) depending on A0, κ, ϵ, and r and +points p1, . . . , pn ∈ A0 with B(pi, δ) pairwise disjoint such that we can “sew the region +tightly” to create a new complete Riemannian manifold (Nn, h), +N = (M \ Ar) ⊔ A′ +r, +as in Proposition 7.1, with +A′ +δ = Aδ \ +� 2n +� +i=1 +B(pi, δ) +� +⊔ +n−1 +� +j=0 +T2j+1. +Moreover, +vol (A′ +r) ≤ vol (Ar) + ϵ +and +vol (N) ≤ vol (M) + ϵ +and there is a constant c > 0 such that +diam (A′ +r) ≤ cr. +we say we have sewn the region A0 arbitrarily tight. If M has scalar curvature RM ≥ κ, +then N has scalar curvature RN ≥ κ − ϵ. If ∂M ̸= ∅, the balls avoid the boundary, and ∂M +is isometric to ∂N. +Proof. The proof follows from the proof of [BS21, Proposition 3.6] while using Propositions +4.2, 7.1, and Lemma 4.3. +⊓⊔ + +32 +PAUL SWEENEY JR +These statements allow us to construct sequences of manifolds with scalar curvature +greater than κ which converge to a pulled metric space in a similar manner as in [BS21]. +We recall the following definition from [BS21]. +Definition 7.3. Let (Mn, g) be a Riemannian manifold with a compact set A0 ⊂ M +with tubular neighborhood Aa = Ta(A0) satisfying the hypotheses of Proposition 7.2. We +can construct its sequence of increasingly tightly sewn manifolds, (Nn +j , gj), by applying +Proposition 7.2 taking ϵ = ϵj → 0, n = nj → ∞, and δ = δj → 0 to create each sewn +manifold Nn = Nn +j and the edited regions A′ +δ = A′ +δj which we simply denote A′ +j. Since +these sequences Nj are created using Proposition 7.2, they have scalar curvature greater +than κ−ϵj when M has scalar curvature greater than κ and ∂Nj = ∂M whenever ∂M ̸= ∅. +Theorem 7.4. The sequence Nj, as in Definition 7.3 assuming Mn is compact and A0 is +a compact embedded submanifold of dimension 1 to n, converges in the Gromov-Hausdorff +sense and the intrinsic flat sense to N∞, which is a metric space created by pulling the +region A0 to a point. If, in addition, Hn−1(A0) = 0 then Nj also converges in the metric +measure sense to N∞. +Proof. The proof follows from the proof of [BS21, Theorem 3.8] while using Proposition 7.2. +⊓⊔ +Now we can prove Theorem D. +Proof of Theorem D. Let S be a simply connected space form of dimension n and constant +curvature +κ +n(n−1) and Σm be a constant curvature m-dimensional sphere, 1 ≤ m ≤ n−1. We +note that there exists an embedding of Σm into S. Let (Nn +j , gj) be a sequence of manifolds +constructed from S sewn along an embedded Σm with δ = δj → 0 as in Proposition 7.2 and +the scalar curvature Rj ≥ κ − 1 +j . Then by Theorem 7.4 we have +Nj +mGH +−−−→ N∞ and Nj +F−→ N∞ +where N∞ is the metric space created by taking S and pulling a Σm to a point. Moreover, +at the pulled point p0 ∈ N∞ we have +wR(p0) = lim +r→0 6(n + 2)volEn B(0, r) − Hn(B(p0, r)) +r2 · volEn B(0, r) += −∞. +We can see this because +volN∞ (B(p0, r)) = Hn +N∞(B(p0, r)) = Hn +N∞(B(p0, r) \ {p0}) = Hn +Snκ(Tr(Sm)). +Moreover, there is a constant C(n, m, κ) such that +lim +r→0 +Hn +Snκ(Tr(Σm)) +Crn−m += 1. +We conclude that +wR(p0) = lim +r→0 6(n + 2)ωnrn − Crn−m +ωnrn+2 += −∞. +⊓⊔ +Moreover, using Proposition 4.2 we are to extend Method II for sewing manifolds in +[BS21] to the setting where scalar curvature is bounded below. +In Method II, given a +sequence of Riemannian manifolds whose limit is a Riemannian, then one can create a new +sequence where the sewing occurs along the sequence. + +REFERENCES +33 +Theorem 7.5. Let Mn +j be a sequence of compact Riemannian manifolds each with a com- +pact region Aj,0 ⊂ M3 +j with tubular neighborhood, Aj, with scalar curvature greater than +κ satisfying the hypotheses of Proposition 7.2. We assume Mn +j converge in the biLipschitz +sense to Mn +∞ and the regions Aj,0 converge to a compact set A∞,0 ⊂ Mn +∞ in the sense that +there exists biLipschitz maps +ψj : Mn +j → Mn +∞ +such that +Lj = log (Lip(ψj)) + log +� +Lip +� +ψ−1 +j +�� +→ 0 +and ψj(Aj,0) = A∞,0. Then there exists δj → 0 and applying Proposition 7.2 to Mn = Mn +j +to sew the regions A0 = Aj,0 with δ = δj, to obtain sewn manifolds Nn = Nn +j , we obtain a +sequence Nn +j such that +Nn +j +GH +−−→ N∞ and Nn +j +F−→ N∞,0, +where ¯N∞,0 = N∞ and N∞ is the metric space created by taking Mn +∞ and pulling the region +A∞,0 to a point. +Moreover, if the regions Aj,0 satisfy Hn(Aj,0) = 0, the the sequence Nn +j also converges in +the metric measure sense +Nn +j +mGH +−−−→ N∞. +Proof. The proof follows from the proof of [BS21, Theorem 5.1] while using Proposition 7.2 +and Theorem 7.4 +⊓⊔ +8. Intrinsic Flat limit with no geodesics +We are able to generalize the result of Basilio, Kazaras, and Sormani from [BKS20] which +shows the intrinsic flat limit of Riemannian manifolds need not be geodesically complete. +This follows from Proposition 4.2 (Constructing Tunnels) and the pipe-filling technique +[BKS20, Theorem 3.1]. In particular: +Theorem 8.1. There is a sequence of closed, oriented, Riemannian manifolds (Mn +j , gj), +n ≥ 3, such that the corresponding integral current spaces converge in the intrinsic flat sense +to +M∞ = +� +N, dEn+1, +� +N +� +, +where N is the round n-sphere of curvature +2κ +n(n−1) and dEn+1 is the Euclidean distance +induced from the standard embedding of N into En+1. Moreover, Mj may be chosen so that +Rj, the scalar curvature of Mj, satisfies Rj ≥ 2κ +� +1 − +1 +10j +� +> κ. Moreover, M∞ is not a +length space and is not locally geodesically complete. +References +[AK00] +Luigi Ambrosio and Bernd Kirchheim. “Currents in metric spaces”. In: Acta +Math. 185.1 (2000), pp. 1–80. issn: 0001-5962. doi: 10.1007/BF02392711. url: +https://doi.org/10.1007/BF02392711. +[AP20] +Brian Allen and Raquel Perales. Intrinsic Flat Stability of Manifolds with Bound- +ary where Volume Converges and Distance is Bounded Below. 2020. doi: 10. +48550/ARXIV.2006.13030. url: https://arxiv.org/abs/2006.13030. + +34 +REFERENCES +[APS20] +Brian Allen, Raquel Perales, and Christina Sormani. Volume Above Distance +Below. 2020. doi: 10.48550/ARXIV.2003.01172. url: https://arxiv.org/ +abs/2003.01172. +[BDS18] +J. Basilio, J. Dodziuk, and C. Sormani. “Sewing Riemannian manifolds with +positive scalar curvature”. In: J. Geom. Anal. 28.4 (2018), pp. 3553–3602. issn: +1050-6926. doi: 10.1007/s12220-017-9969-y. url: https://doi.org/10. +1007/s12220-017-9969-y. +[BKS20] +J. Basilio, D. Kazaras, and C. Sormani. “An intrinsic flat limit of Riemannian +manifolds with no geodesics”. In: Geom. Dedicata 204 (2020), pp. 265–284. issn: +0046-5755. doi: 10.1007/s10711-019-00453-1. url: https://doi.org/10. +1007/s10711-019-00453-1. +[BS21] +J. Basilio and C. Sormani. “Sequences of three dimensional manifolds with posi- +tive scalar curvature”. In: Differential Geom. Appl. 77 (2021), Paper No. 101776, +27. issn: 0926-2245. doi: 10.1016/j.difgeo.2021.101776. url: https://doi. +org/10.1016/j.difgeo.2021.101776. +[Dod20] +J´ozef Dodziuk. “Gromov-Lawson tunnels with estimates”. In: Analysis and ge- +ometry on graphs and manifolds. Vol. 461. London Math. Soc. Lecture Note Ser. +Cambridge Univ. Press, Cambridge, 2020, pp. 55–65. +[Fuk87] +Kenji Fukaya. “Collapsing of Riemannian manifolds and eigenvalues of Laplace +operator”. In: Invent. Math. 87.3 (1987), pp. 517–547. issn: 0020-9910. doi: 10. +1007/BF01389241. url: https://doi.org/10.1007/BF01389241. +[GL80] +Mikhael Gromov and H. Blaine Lawson Jr. “The classification of simply con- +nected manifolds of positive scalar curvature”. In: Ann. of Math. (2) 111.3 (1980), +pp. 423–434. issn: 0003-486X. doi: 10.2307/1971103. url: https://doi.org/ +10.2307/1971103. +[Gra98] +Alfred Gray. Modern differential geometry of curves and surfaces with Mathe- +matica. Second. CRC Press, Boca Raton, FL, 1998, pp. xxiv+1053. isbn: 0-8493- +7164-3. +[Gro18] +Misha Gromov. Scalar Curvature of Manifolds with Boundaries: Natural Ques- +tions and Artificial Constructions. 2018. doi: 10.48550/ARXIV.1811.04311. +url: https://arxiv.org/abs/1811.04311. +[Gro99] +Misha Gromov. Metric structures for Riemannian and non-Riemannian spaces. +Vol. 152. Progress in Mathematics. Based on the 1981 French original [ MR0682063 +(85e:53051)], With appendices by M. Katz, P. Pansu and S. Semmes, Translated +from the French by Sean Michael Bates. Birkh¨auser Boston, Inc., Boston, MA, +1999, pp. xx+585. isbn: 0-8176-3898-9. +[Lak16] +Sajjad Lakzian. “On diameter controls and smooth convergence away from sin- +gularities”. In: Differential Geom. Appl. 47 (2016), pp. 99–129. issn: 0926-2245. +doi: 10.1016/j.difgeo.2016.01.003. url: https://doi.org/10.1016/j. +difgeo.2016.01.003. +[Lla98] +Marcelo Llarull. “Sharp estimates and the Dirac operator”. In: Math. Ann. 310.1 +(1998), pp. 55–71. issn: 0025-5831. doi: 10.1007/s002080050136. url: https: +//doi.org/10.1007/s002080050136. +[LS12] +Dan A. Lee and Christina Sormani. “Near-equality of the Penrose inequality for +rotationally symmetric Riemannian manifolds”. In: Ann. Henri Poincar´e 13.7 +(2012), pp. 1537–1556. issn: 1424-0637. doi: 10.1007/s00023- 012- 0172- 1. +url: https://doi.org/10.1007/s00023-012-0172-1. + +REFERENCES +35 +[LS13] +Sajjad Lakzian and Christina Sormani. “Smooth convergence away from singular +sets”. In: Comm. Anal. Geom. 21.1 (2013), pp. 39–104. issn: 1019-8385. doi: +10.4310/CAG.2013.v21.n1.a2. url: https://doi.org/10.4310/CAG.2013. +v21.n1.a2. +[LS14] +Dan A. Lee and Christina Sormani. “Stability of the positive mass theorem for +rotationally symmetric Riemannian manifolds”. In: J. Reine Angew. Math. 686 +(2014), pp. 187–220. issn: 0075-4102. doi: 10.1515/crelle-2012-0094. url: +https://doi.org/10.1515/crelle-2012-0094. +[LS15] +Philippe G. LeFloch and Christina Sormani. “The nonlinear stability of rota- +tionally symmetric spaces with low regularity”. In: J. Funct. Anal. 268.7 (2015), +pp. 2005–2065. issn: 0022-1236. doi: 10 . 1016 / j . jfa . 2014 . 12 . 012. url: +https://doi.org/10.1016/j.jfa.2014.12.012. +[Mia02] +Pengzi Miao. “Positive mass theorem on manifolds admitting corners along a +hypersurface”. In: Adv. Theor. Math. Phys. 6.6 (2002), 1163–1182 (2003). issn: +1095-0761. doi: 10.4310/ATMP.2002.v6.n6.a4. url: https://doi-org.proxy. +library.stonybrook.edu/10.4310/ATMP.2002.v6.n6.a4. +[MN12] +Fernando C. Marques and Andr´e Neves. “Rigidity of min-max minimal spheres +in three-manifolds”. In: Duke Math. J. 161.14 (2012), pp. 2725–2752. issn: 0012- +7094. doi: 10.1215/00127094-1813410. url: https://doi.org/10.1215/ +00127094-1813410. +[Per20] +Raquel Perales. “Convergence of manifolds and metric spaces with boundary”. +In: J. Topol. Anal. 12.3 (2020), pp. 735–774. issn: 1793-5253. doi: 10.1142/ +S1793525319500638. url: https://doi.org/10.1142/S1793525319500638. +[Sor+21] +Christina Sormani, Participants at the IAS Emerging Topics Workshop on Scalar +Curvature, and Convergence. Conjectures on Convergence and Scalar Curvature. +2021. doi: 10.48550/ARXIV.2103.10093. url: https://arxiv.org/abs/2103. +10093. +[Sor17] +Christina Sormani. “Scalar curvature and intrinsic flat convergence”. In: Mea- +sure theory in non-smooth spaces. Partial Differ. Equ. Meas. Theory. De Gruyter +Open, Warsaw, 2017, pp. 288–338. +[Sor18] +Christina Sormani. “Intrinsic flat Arzela-Ascoli theorems”. In: Comm. Anal. +Geom. 26.6 (2018), pp. 1317–1373. issn: 1019-8385. doi: 10.4310/CAG.2018. +v26.n6.a3. url: https://doi.org/10.4310/CAG.2018.v26.n6.a3. +[Sor22] +Christina Sormani. Private communication. 2022. +[SW11] +Christina Sormani and Stefan Wenger. “The intrinsic flat distance between Rie- +mannian manifolds and other integral current spaces”. In: J. Differential Geom. +87.1 (2011), pp. 117–199. issn: 0022-040X. url: http://projecteuclid.org/ +euclid.jdg/1303219774. +[SY79] +R. Schoen and S. T. Yau. “On the structure of manifolds with positive scalar +curvature”. In: Manuscripta Math. 28.1-3 (1979), pp. 159–183. issn: 0025-2611. +doi: 10.1007/BF01647970. url: https://doi.org/10.1007/BF01647970. +Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA +Email address: paul.sweeney@stonybrook.edu + diff --git a/L9AzT4oBgHgl3EQfV_wg/content/tmp_files/load_file.txt b/L9AzT4oBgHgl3EQfV_wg/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..44c514a84d228b6ca1fd734c88945ef3d3679125 --- /dev/null +++ b/L9AzT4oBgHgl3EQfV_wg/content/tmp_files/load_file.txt @@ -0,0 +1,1404 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf,len=1403 +page_content='EXAMPLES FOR SCALAR SPHERE STABILITY PAUL SWEENEY JR Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The rigidity theorems of Marques-Neves and of Llarull, which show two differ- ent ways scalar curvature can characterize the sphere, have associated stability conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Here we produce the first examples related to these stability conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The first set of examples demonstrates the necessity of including a condition on the minimum area of all minimal surfaces to prevent bubbling along the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The second set of examples constructs sequences that do not converge in the Gromov-Hausdorff sense but do converge in the volume preserving intrinsic flat sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In order to construct such sequences, we improve the Gromov-Lawson tunnel construction so that one can attach wells and tunnels to a manifold with scalar curvature bounded below and only decrease the scalar curvature by an arbitrarily small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, we are able to generalize both the sewing con- struction of Basilio, Dodziuk, and Sormani, and the construction due to Basilio, Kazaras, and Sormani of an intrinsic flat limit with no geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Introduction Rigidity theorems are often used to characterize manifolds in Riemannian geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A typical rigidity theorem says that if a Riemannian manifold satisfies some conditions, usually including a bound on curvature, then it must be isometric to a specific model geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' One can naturally formulate a stability theorem from a rigidity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A stability theorem says if the hypotheses of a rigidity theorem are perturbed, then the manifolds that satisfy these hypotheses are quantitatively close to the manifold characterized by the rigidity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In this paper, we are concerned with stability theorems which are associated with rigidity theorems that characterize the sphere using a curvature bound on the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The rigidity theorem of Marques and Neves [MN12] and the rigidity theorem due to Llarull [Lla98] are two results that show how scalar curvature can characterize the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' These two rigidity theorems naturally give rise to stability conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Below, we construct the first examples related to these stability conjectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We demonstrate why a condition preventing bubbling is required, and we investigate different modes of convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In order to construct these examples, we prove an enhancement of the Gromov-Lawson tunnel construction [GL80] (see also Schoen-Yau [SY79]) which retains control over the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The theorem of Marques-Neves pertains to the three dimensional sphere and the min- max quantity width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let us recall the definition of width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let g be a Riemannian metric on the 3-sphere and x4 : S3 ⊆ R4 → R be the height function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For each t ∈ [−1, 1], let Σ′t = {x ∈ S3 : x4 = t} and Λ′ be the collection of all families {Σt} such that Σt = Ft(Σ′t) for some smooth one-parameter family of diffeomorphisms Ft of the 3-sphere all of which are isotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The width of (S3, g) is the following min-max quantity width(S3, g) = inf {Σt}∈Λ′ sup t∈[−1,1] |Σt|g, where |Σ|g is the Hausdorff two measure of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='01292v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='DG] 3 Jan 2023 2 PAUL SWEENEY JR The theorem of Marques-Neves [MN12] says if there is a Riemannian metric on the 3- sphere with scalar curvature larger than 6 and width(S3, g) ≥ 4π then it is isometric to the standard unit round 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This leads to the following naive stability conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Before stating the conjecture, we note that throughout this paper we will condense notation and set Mn j = (Mn j , gj) when we have a sequence of Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Suppose M3 j = (S3, gj) are homeomorphic spheres satisfying Rj ≥ 6 − 1 j , width(M3 j ) ≥ 4π, diam � M3 j � ≤ D, and vol � M3 j � ≤ V where Rj is the scalar curvature of M3 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then M3 j converges in the VF-sense to (S3, grd) where grd is the Riemannian metric for the standard unit round 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We construct a counterexample that refutes Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The counterexample is a sequence of spheres M3 j = (S3, gj) that converges in the volume preserving intrinsic flat (VF) sense to the disjoint union of two spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The M3 j are two spheres connected by a thin tunnel (see Figure 1), and the tunnel gets increasingly thin along the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem A (Counterexample to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a convergent sequence of Riemannian manifolds M3 j = (S3, gj), with M3 j VF −−→ M∞ such that Rj ≥ 6 − 1 j , width(M3 j ) ≥ 4π, diam � M3 j � ≤ D, and vol � M3 j � ≤ V, for some constants D, V > 0, and M∞ is the disjoint union of two 3-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem A shows that something stronger than width is required for a stability conjecture related to the rigidity theorem of Marques-Neves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A conjecture in [Sor+21] attributed to Marques and Neves does hypothesize a stronger condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, it replaces the uniform lower bound on width with a uniform lower bound on MinA(M, g) = inf{|Σ|g : Σ is a closed minimal hypersurface in M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since width is achieved by a minimal surface, we have that width(M3, g) ≥ MinA(M3, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, Marques-Neves show in [MN12] that if (S3, g) contains no stable minimal surfaces, then we have that MinA(S3, g) = width(S3, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Suppose M3 j = (S3, gj) are homeomorphic spheres satisfying Rj ≥ 6 − 1 j , MinA(M3 j ) ≥ 4π − 1 j , diam � M3 j � ≤ D, and vol � M3 j � ≤ V where Rj is the scalar curvature of M3 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then M3 j converges in the VF-sense to (S3, grd) the standard unit round sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The sequence of Riemannian manifolds constructed in Theorem A has MinA(Mn j ) → 0 and so does not satisfy the hypotheses of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A sequence of spheres that converge in VF-sense to the disjoint union of two spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sormani proposed the MinA condition in [Sor17] to prevent bad limiting behavior, such as bubbling and pinching, along the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The motivation for such a condition comes from the sewing construction of Basilio, Dodziuk, and Sormani [BDS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This construction shows the existence of a sequence of manifolds with positive scalar curvature, which has an F-limit that does not have positive scalar curvature in some generalized sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Other sequences of positive scalar curvature manifolds have also been constructed ([BS21], [BKS20]) whose F-limits have undesirable properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The key to the construction of these examples is the ability to glue in tunnels with controlled geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In those examples, it is unknown if the scalar curvature of the tunnel and of the resulting manifold can be kept close to the scalar curvature of the manifold to which the tunnel is being glued.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, these examples may not satisfy the curvature condition in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In Section 4, we prove our two main technical propositions, which are of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' One of which allows us to get quantitative control over the scalar curvature of the tunnel and of the resulting manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, given a manifold with scalar curvature bounded below by κ, then for small enough ϵ > 0 there exists a tunnel such that the resulting manifold has scalar curvature bounded below by κ − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We use this new way of attaching tunnels to manifolds that maintains control over the scalar curvature to construct the sequence in Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, we can make a similar example related to Llarull’s rigidity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' First, let us recall Llarull’s theorem [Lla98] which says that if there is a degree non-zero, smooth, distance non-increasing map from a smooth, Riemannian, spin, n-manifold, Mn, to the standard unit round n-sphere and the scalar curvature of Mn is greater than or equal to n(n − 1), then the map is a Riemannian isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Gromov in [Gro18] proposed studying the stability question related to Llarull’s rigidity theorem by investigating sequences of Riemannian manifolds Mn j = (Mn j , gj) with inf Rj → n(n − 1) and RadSn(Mj) → 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' RadSn(Mn) is defined as the maximal radius r of the n- sphere, Sn(r), such that Mn admits a distance non-increasing map from Mn to Sn(r) of non-zero degree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Based on this, Sormani [Sor22] proposed the following stability conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Suppose Mn j = (Mn j , gj), n ≥ 3, are closed smooth connected spin Rie- mannian manifolds such that Rj ≥ n(n − 1) − 1 j , MinA(Mn j ) ≥ 4π − 1 j , diam � Mn j � ≤ D, vol � Mn j � ≤ V 69 ←(4 PAUL SWEENEY JR where Rj is the scalar curvature of Mn j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, suppose there are smooth maps to the standard unit round n-sphere fj : Mn j → Sn which are 1-Lipschitz and deg fj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then Mn j converges in the VF-sense to the standard unit round n-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In a similar manner to Theorem A, we are able to construct a sequence of manifolds each of which is two spheres connected by a thin tunnel, which is related to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This sequence satisfies all the hypotheses of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 except the lower bound on MinA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a convergent sequence of Riemannian manifolds Mn j = (Sn, gj), n ≥ 3, with Mn j VF −−→ M∞ such that Rj ≥ n(n − 1) − 1 j , diam (Mj) ≤ D, and vol (Mj) ≤ V, for some constants D, V > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, there are smooth degree one, 1-Lipschitz maps fj : Mn j → (Sn, grd) which converge to a 1-Lipschitz map f∞ : M∞ → (Sn, grd), and M∞ is the disjoint union of two n-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorems A and A′ show the necessity of including a hypothesis like the bound on MinA to prevent bubbling along the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' When studying a stability conjecture related to scalar curvature, one also often considers examples similar to the example described by Ilmanen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Ilmanen first described this example to demonstrate that a sequence of manifolds of positive scalar curvature need not converge in the Gromov-Hausdorff (GH) sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The example is a sequence of spheres with increasingly many arbitrarily thin wells attached to them (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sormani and Wenger [SW11, Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7] showed, using their intrinsic flat (F) convergence for integral currents, that the Ilmanen example converges in the F-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Over the past decade, Ilmanen-like examples have been constructed in varying settings to demonstrate that GH-convergence is not the appropriate convergence in which to ask stability conjectures related to scalar curvature ([LS13], [Lak16], [Per20], [LS14], [LS15], [LS12], [AP20], [APS20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In these examples, it is unknown if one can attach a well and only decrease the scalar curvature by a small amount;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' consequently, it was unknown if Ilmanen-like examples could exist for Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 and Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Our other main technical proposition (see Section 4 below) shows that one can attach a well to a manifold with scalar curvature bounded below and only decrease the scalar curvature by an arbitrarily small amount.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we are able to construct Ilmanen- like examples related to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 and Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, we are able to construct a sequence of spheres M3 j with scalar curvature larger than 6 − ϵj, width larger than 4π, and volumes and diameters bounded that does not converge in the GH-sense but does converge in the volume above distance below (V ADB) sense and the VF-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Likewise, we construct a sequence of spheres with scalar curvature larger than n(n−1)−ϵj, volumes and diameters bounded, and smooth maps to the unit round n-sphere which are 1-Lipschitz and deg fj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we can construct Ilmanen-like examples related to Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 and Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We, however, cannot verify that MinA stays uniformly bounded from below even though we expect that it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 5 Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a convergent sequence of Riemannian manifolds M3 j = (S3, gj), with M3 j VADB −−−−→ M∞ and M3 j VF −−→ M∞ such that Rj ≥ 6 − 1 j , width(M3 j ) ≥ 4π, diam � M3 j � ≤ D, and vol � M3 j � ≤ V, for some constants D, V > 0, and M∞ is the standard unit round 3-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' However, the sequence has no convergent subsequence in the GH-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In [Sor+21, Remark 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4], Sormani suggests that it is believable that someone can con- struct a sequence of spheres with increasingly many increasingly thin wells which satisfy the hypothesis of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem B partially answers this question by constructing such a sequence that satisfies all the hypotheses of Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 except the bound on MinA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a convergent sequence of Riemannian manifolds Mn j = (Sn, gj), with Mn j VADB −−−−→ M∞ and Mn j VF −−→ M∞ such that Rj ≥ n(n − 1) − 1 j , diam � Mn j � ≤ D, and vol � Mn j � ≤ V, for some constants D, V > 0, and M∞ is the n-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, there are smooth degree non-zero, 1-Lipschitz maps fj : Mn j → (Sn, grd), and M∞ is the standard unit round n-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' However, the sequence has no subsequence that converges in the GH-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A sequence of spheres with increasingly many thin wells that converges in the VADB-sense and VF-sense to a sphere but has no convergent subsequence in the GH-topology The main tools to prove the above theorems are new construction propositions which are proved in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We adapt the bending argument of Gromov and Lawson in [GL80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Originally, the construction in [GL80] was used to make tunnels of positive scalar curvature to show, for example, that the connected sum of two manifolds with positive scalar curvature carries a metric of positive scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For manifolds with constant positive sectional curvature, Dodziuk, Basilio, and Sormani in [BDS18] refined the construction to give control over the volume and diameter of the tunnel while maintaining positive scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Dodziuk in [Dod20] further refined the construction by replacing the positive sectional curvature condition with positive scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In this paper, we construct wells and tunnels such that, if the scalar curvature of a manifold is bounded below, then one can attach a well or tunnel and only decrease the lower bound by an arbitrarily small amount while maintaining bounds on the diameter and volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 6 PAUL SWEENEY JR The new well construction allows us to generalize the construction of Sormani and Wenger [SW11, Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='11] of a sequence of manifolds that converge in the F-sense to space that is not precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, we are able to construct a sequence of spheres with scalar curvatures greater than κ ≥ 0, uniformly bounded diameters, and uniformly bounded volumes such that the sequence converges in the VF-sense to a limit that is not precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' To construct the sequence we attach a sequence of increasingly thin wells to a sphere (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a convergent sequence of Riemannian manifolds Mn j = (Sn, gj), n ≥ 3, with Mn j VF −−→ M∞ such that Rj ≥ κ, diam � Mn j � ≤ D, and vol � Mn j � ≤ V, for some nonnegative constants κ, D, V , and M∞ is not precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A sequence of spheres with increasingly many thin wells that converges in the VF-sense to a limit which is not precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The new tunnel construction allows us to extend the sewing construction in [BDS18] and [BS21] to a more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Basilio, Dodziuk, and Sormani [BDS18] used sewing manifolds to investigate the following question of Gromov which asks: What is the weak- est notion of convergence such that a sequence of Riemannian manifolds, Mn j with scalar curvature Rj ≥ κ subconverges to a limit M∞ which may not be a manifold but has scalar curvature greater than κ in some suitably generalized sense?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' They were able to show that when κ = 0 there is a sequence of Riemannian manifolds with non-negative scalar curvature whose limit fails to have non-negative generalized scalar curvature where generalized scalar curvature is defined as wR(p0) := lim r→0 6(n + 2)volEn B(0, r) − Hn(B(p0, r)) r2 · volEn B(0, r) ≥ 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) for the limit space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For a Riemannian manifold (Mn, g) with scalar curvature R, we see for all p ∈ Mn that wR(p) = R(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We are able to provide a similar answer to Gromov’s question for any κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, for any κ, there exists a sequence of increasingly tightly sewn manifolds all of which have scalar curvature greater than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, this sequence of increasingly tightly sewn manifolds will converge in the F-sense to a pulled metric space (see [BS21, Section 2] for discussion of such spaces) which fail to have generalized scalar curvature greater than or equal to κ at the pulled point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 7 Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a sequence of manifolds Mn j = (Mn, gj) with scalar curvature Rj ≥ κ − 1 j which converges in the F-sense to a metric space M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, there is a point p0 ∈ M∞ such that wR(p0) := lim r→0 6(n + 2)volEn B(0, r) − Hn(B(p0, r)) r2 · volEn B(0, r) = −∞ (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) Lastly, the new tunnel construction allows us to generalize the construction of Basilio, Kazaras, and Sormani [BKS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' They use long thin tunnels with positive scalar curvature to construct a sequence of manifolds that converges in the F-sense to a space with no geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Similarly, for any κ > 0, we are able to construct a sequence of manifolds with scalar curvature bounded below by κ whose limit is not a geodesic space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There is a sequence of closed, oriented, Riemannian manifolds (Mn j , gj), n ≥ 3, with scalar curvature Rj > κ > 0 such that the corresponding integral current spaces converge in the intrinsic flat sense to M∞ = � N, dEn+1, � N � , where N is the round n-sphere of curvature 2κ n(n−1) and dEn+1 is the Euclidean distance induced from the standard embedding of N into En+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, M∞ is not locally geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Properties of Property of the Type of Does Mj limit, M∞ convergence MinAj → 0?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Rj > 6 − 1 j Counterexample to Theorem A width(Mj) ≥ 4π Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' VF, F Yes Shows necessity of MinA lower bound Theorem A′ Rj > n(n − 1) − 1 j in Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' VF, F Yes Rj > 6 − 1 j No GH-convergent Theorem B width(Mj) ≥ 4π subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' VADB, VF, F ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' No GH-convergent Theorem B′ Rj > n(n − 1) − 1 j subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' VADB, VF, F ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem C Rj > κ > 0 Not precompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' VF, F ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Generalized Scalar Rj > κ curvature is negative Theorem D (Rj > 0, [BDS18]) infinity at a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' F Yes No two points Rj > κ > 0 are connected by Theorem E (Rj > 0, [BKS20]) a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' F Yes Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Here we summarize the examples constructed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In Section 3, the background is discussed including some definitions and theorems related to different notions of convergence for Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In Section 4, we prove our main construction propositions: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (Constructing Wells) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In Section 5, we use 8 PAUL SWEENEY JR Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 to prove Theorems A and A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In Section 6, we prove Theorems B, B′, and C using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Finally, in Sections 7 and 8, we discuss how the construc- tion propositions can be used to generalize the construction of sewing manifolds and the construction of sequences of smooth manifolds whose limit does not have any geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Acknowledgements The author would like to thank Marcus Khuri and Raanan Schul for their invaluable guidance and encouragement throughout the process of producing this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The author would also like to thank Christina Sormani for her helpful discussions and the suggestion to construct examples related to Llarull’s rigidity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The author gratefully acknowledges support from the Simons Center for Geometry and Physics, Stony Brook University, at which some of the research for this paper was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This work was supported in part by NSF Grant DMS-2104229 and NSF Grant DMS-2154613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Background In this section, we will review different types of convergences between two Riemannian manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Gromov-Hausdorff convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Here we will review the Gromov-Hausdorff dis- tance between two metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Gromov defined this distance between two metric spaces by generalizing the concept of Hausdorff distance between two subsets of a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We refer the reader to [Gro99] for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The Gromov-Hausdorff distance between two metric spaces (X1, d1) and (X2, d2) is dGH((X1, d1), (X2, d2)) = inf Z {dZ H(φ1(X1), φ2(X2))} where the infimum is taken over all complete metric spaces (Z, dZ) and all distance preserv- ing maps φi : Xi → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We say that a metric spaces (Xj, dj) converge in the GH-sense to a metric space (X∞, d∞) if dGH((Xj, dj), (X∞, d∞)) → 0 If, in addition, µj and µ∞ are measures on Xj and X∞, respectively, then Fukaya [Fuk87] introduced the notion of metric measure convergence for metric measure spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We say (Xj, dj, µj) converges to a metric measure space (X∞, d∞, µ∞) in metric measure (mGH) sense if we have convergence in the GH-sense and φj∗µj → φ∞∗µ∞ weakly as measures in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that both define a distance between two Riemannian manifolds since there is a natural distance function and natural measure associated with a Riemannian manifold (M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Gromov, in the following theorem, characterizes when a sequence of compact metric spaces contains a subsequence that converges in the GH-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For a sequence of compact metric spaces (Xj, dj) such that diam (Xj) < D < ∞, the following are equivalent: (i) There exists a convergent subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) There is a function N1 : (0, α) → (0, ∞) such that Capj(ϵ) ≤ N1(ϵ) EXAMPLES FOR SCALAR SPHERE STABILITY 9 (iii) There is a function N2 : (0, α) → (0, ∞) such that Covj(ϵ) ≤ N2(ϵ), where Capj(ϵ) = maximum number of disjoint ϵ 2-balls in Xj, Covj(ϵ) = minimum number of ϵ-balls it takes to cover Xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Intrinsic Flat Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In this section we will review Sormani-Wenger intrinsic flat distance between two integral current spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sormani and Wenger [SW11] defined intrinsic flat distance, which generalizes the notion of flat distance for currents in Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' To do so they used Ambrosio and Kirchheim’s generalization of Federer and Fleming’s integral currents to metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We refer the reader to [AK00] for further details about currents in arbitrary metric spaces and to [SW11] for further details about integral current spaces and intrinsic flat distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Z, dZ) be a complete metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Denote by Lip(Z) and Lipb(Z) the set of real- valued Lipschitz functions on Z and the set of bounded real-valued Lipschitz functions on Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 ([AK00], Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We say a multilinear functional T : Lipb(Z) × [Lip(Z)]m → R on a complete metric space (Z, d) is an m-dimensional current if it satisfies the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (i) Locality: T(f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm) = 0 if there exists and i such that πi is constant on a neighborhood of {f ̸= 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) Continuity: T is continuous with respect to pointwise convergence of πi such that Lip(πi) ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) Finite mass: there exists a finite Borel measure µ on X such that |T(f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm)| ≤ m � i=1 Lip(πi) � Z |f|dµ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) for any (f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We call the minimal measure satisfying (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) the mass measure of T and denote it ||T||.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We can now define many concepts related to a current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' M(T) = ||T||(Z) is defined to be the mass of T and the canonical set of a m-current T on Z is set(T) = � p ∈ Z ��� lim inf r→0 ||T||(B(p, r)) rm > 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The boundary of a current T is defined as ∂T : Lipb(X) × [Lip(X)]m−1 → R, where ∂T(f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm−1) = T(1, f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Given a Lipschitz map φ : Z → Z′, we can pushforward a current T on Z to a current φ#T on Z′ by defining φ#T(f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm) = T(f ◦ φ, f ◦ π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , f ◦ πm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A standard example of an m-current on Z is given by φ#[[θ]](f, π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , πm) = � A (θ ◦ φ)(f ◦ φ)d(π1 ◦ φ) ∧ · · · ∧ d(πm ◦ φ), where φ : Rm → Z is bi-Lipschitz and θ ∈ L1(A, Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We say that an m-current on Z is integer rectifiable if there is a countable collection of bi-Lipschitz maps φi : Ai → X 10 PAUL SWEENEY JR where Ai ⊂ Rm is precompact Borel measurable with pairwise disjoint images and weight functions θi ∈ L1(Ai, Z) such that T = ∞ � i=1 φi#[[θi]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, we say an integer rectifiable current whose boundary is also integer rectifiable is an integral current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We denote the space of integral m-currents on Z as Im(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The flat distance between two integral currents T1, T2 ∈ I(Z) is dZ F (T1, T2) = inf{M(U) + M(V ) | U ∈ Im(X), V ∈ Im+1(X), T2 − T1 = U + ∂V }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We say that the triple (X, d, T) is an integral current space if (X, d) is a metric space, T ∈ Im( ¯X) where ¯X is the completion of X, and set(T) = X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The intrinsic flat (F) distance between two integral current spaces (X1, d1, T1) and (X2, d2, T2) is dF((X1, d1, T1), (X2, d2, T2)) = inf Z {dZ F (φ1#T1, φ2#T2)} where the infimum is taken over all complete metric spaces (Z, dZ) and isometric embeddings φ1 : ( ¯X1, d1) → (Z, dZ) and φ2 : ( ¯X2, d2) → (Z, dZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that if (X1, d1, T1) and (X2, d2, T2) are precompact integral current spaces such that dF((X1, d1, T1), (X2, d2, T2)) = 0 then there is a current preserving isometry between (X1, d1, T1) and (X2, d2, T2), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=', there exists an isometry f : X1 → X2 whose extension ¯f : ¯X1 → ¯X2 pushes forward the current: ¯f#T1 = T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We say a sequence of (Xj, dj, Tj) precompact integral current spaces converges to (X∞, d∞, T∞) in the F-sense if dF((Xj, dj, Tj), (X∞, d∞, T∞)) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If, in addition, M(Ti) → M(T∞), then we say (Xj, dj, Tj) converges to (X∞, d∞, T∞) in the voulme preserving intrinsic flat (VF) sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that we can view compact Riemannian manifolds (Mn, g) as precompact integral current spaces (Mn, dg, � Mn dvolg), where dg is the natural distance function on the Riemannian manifold and integration over the manifold, � Mn dvolg, can be viewed as an integral current.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, M(Mn) = vol (Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lakzian and Sormani in [LS13] were able to estimate the intrinsic distance between two diffeomorphic manifolds: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Suppose Mn 1 = (Mn, g1) and Mn 2 = (Mn, g2) are oriented precompact Rie- mannian manifolds with diffeomorphic subregions Uj ⊂ Mn j and diffeomorphisms ψj : U → Uj such that for all v ∈ TU we have 1 (1 + ϵ)2 ψ∗ 1g1(v, v) < ψ∗ 2g2(v, v) < (1 + ϵ)2ψ∗ 1g1(v, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We define the following quantities (i) DUj = sup{diamMj (W) : W is a component of Uj}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) Define a to be a number such that a > arccos(1+ϵ)−1 π max{DU1, DU2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) λ = supx,y∈U |dM1 (ψ1(x), ψ1(y)) − dM2 (ψ2(x), ψ2(y)) |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iv) h = � λ � max{DU1, DU2} + λ 4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (v) ¯h = max � h, √ ϵ2 + 2ϵDU1, √ ϵ2 + 2ϵDU2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 11 Then the intrinsic flat distance between Mn 1 and Mn 2 is bounded: dF(M1, M2) ≤ � 2¯h + a � (volm(U1) + volm(U2) + volm−1(∂U1) + volm−1(∂U2)) + volm(M1 \\ U1) + volm(M2 \\ U2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, Sormani [Sor18] proves the following Arzela-Ascoli theorem in the setting of F-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Fix L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Suppose Mj = (Xj, dj, Tj) are integral current spaces for j ∈ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , ∞} and Mj F−→ M∞ and Fj : Xj → W are L-Lipschitz maps into a compact metric space W, then a subsequence converges to an L-Lipschitz map F∞ : X∞ → W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Specifically, there exists isometric embeddings of the subsequence φj : Xj → Z, such that dZ F (φj#Tj, φ∞#T∞) → 0 and for any sequence pj ∈ Xj converging to p ∈ X∞, dZ(φj(pj), φ∞(p)) → 0, one has converging images dW (Fj(pj), F∞(p)) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Volume above distance below convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Allen, Perales, and Sormani in [APS20] introduced a new notion of convergence of manifolds called volume above distance below (VADB) convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' It is based on the volume-distance rigidity theorem which states that if there is a C1-diffeomorphism F : M → N between two Riemannian manifolds which is also distance non-increasing then vol (N) ≤ vol (M);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' moreover, in case of equality the manifolds are isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A sequence of Riemannian manifolds without boundary Mn j = (Mn, gj) converge in the VADB-sense to a Riemannian manifold Mn ∞ = (Mn, g∞) if (i) vol (Mn j ) → vol (Mn ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) diam (Mn j ) ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) There exists a C1-diffeomorphisms Ψj : Mn ∞ → Mn j such that for all p, q ∈ Mn ∞ we have dj(Ψj(p), Ψj(q)) ≥ d∞(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We also record the following lemma from [APS20] which says that the above condition on the distance functions in the definition of VADB-convergence can be converted into a condition on Riemannian metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Mn 1 = (Mn, g1) and Mn 0 = (Mn, g0) be Riemannian manifolds and F : Mn 1 → Mn 0 be a C1-diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then g0(dF(v), dF(v)) ≤ g1(v, v) for all v ∈ TMn 1 if and only if d0(F(p), F(q)) ≤ d1(p, q) for all p, q ∈ Mn 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Finally, we record the following theorem from [APS20] which describes the relationship between VADB-convergence and VF-convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If Mn j = (Mn, gj) and Mn ∞ = (Mn, g∞) are compact oriented Riemannian manifolds such that Mn j VADB −−−−→ Mn ∞ then Mn j VF −−→ Mn ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 12 PAUL SWEENEY JR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Wells and Tunnels In this section, we prove the main new technical propositions: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (Con- structing Wells) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' These are an improvement of the constructions of Gromov-Lawson [GL80], Basilio, Dodziuk, Sormani [BDS18], and Dodziuk [Dod20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We construct wells and tunnels and get control over the volume and diameter while keeping the scalar curvature close to the scalar curvature of the manifold to which we are attaching the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (Constructing Wells) allows us to remove a ball from a Riemannian manifold M with scalar curvature RM ≥ κ and glue in a well to create a new Riemannian manifold N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' moreover, M and N will be isometric away from the gluing and the scalar curvature RN of N will satisfy RN ≥ κ − ϵ for arbitrarily small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels) allows the analogous construction for connecting two manifolds with a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, given a Riemannian manifold M with RM ≥ κ we can remove two balls and glue in a tunnel to create a Riemannian manifold P with RP ≥ κ − ϵ for arbitrarily small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (Constructing Wells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g), n ≥ 3, be a Riemannian manifold with scalar curvature RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let δ > 0 be small enough, j ∈ N, and d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If RM ≥ κ on Bg(p, 2δ) a ball in (M, g), then we can construct a well Wj = (Bg(p, 2δ), gj) and a new complete Riemannian manifold (Nn, h), Nn = Mn, h|M\\Bg(p,2δ) = g|M\\Bg(p,2δ), h|Bg(p,2δ) = gj|Bg(p,2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, the following properties are satisfied: (i) The scalar curvature, Rj, of Wj satisfies Rj > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) gj|E = g|E where E = Bg(p, 2δ) \\ Bg(p, δ) is identified with a subset of Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) There exists constant C > 0 independent of j and d such that diam (Wj) < C(δ + d) and vol (Wj) < C(δn + dδn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iv) N has scalar curvature RN > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g), n ≥ 3, be a Riemannian manifold with scalar curvature RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let δ > 0 be small enough, j ∈ N, and d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If RM ≥ κ on two balls Bg(p, 2δ) and Bg(p′, 2δ) in (Mn, g), then we can construct a new complete Riemannian manifold P n, where we remove two balls and glue cylindrical region (Tj, gj) diffeomorphic to Sn−1 × [0, 1], P n = Mn \\ � Bg(p, 2δ) ∪ Bg(p′, 2δ) � ⊔ Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, the following properties are satisfied: (i) The scalar curvature, Rj, of Tj satisfies Rj > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) gj|E = g|E and gj|E′ = g|E′ where E = Bg(p, 2δ) \\ Bg(p, δ) and E′ = Bg(p′, 2δ) \\ Bg(p′, δ) are identified with subsets of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) There exists constant C > 0 independent of j and d such that diam (Tj) < C(δ + d) and vol (Tj) < C(δn + dδn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iv) P has scalar curvature RP > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We adapt the proof from [Dod20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The well and tunnel will be constructed as a codi- mension one submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The submanifold will be defined by a curve, and this curve will EXAMPLES FOR SCALAR SPHERE STABILITY 13 control the geometry of the submanifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' First, we show how the curve defines the subman- ifold and how it affects its geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Second, we carefully construct the curve so that the submanifold will inherit the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, the construction will follow the following outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' First, we will describe how, given a curve, we can define a submanifold and write the scalar curvature in terms of quantities related to the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Second, we carefully construct a C1-curve, γ, which will be used to define a submanifold that is the precursor to a well or a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Third, we adjust the construction of γ so the resulting manifold will be a well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Fourth, we describe the smoothing procedure to make γ a C∞-curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Fifth, we construct a well and check it has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sixth, we perform the analogous steps to construct a tunnel with the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' A Submanifold defined by a curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g) be a compact Riemannian man- ifold with scalar curvature RM ≥ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let δ > 0 and B = B(p, 2δ) be a geodesic ball in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the Riemannian product (X, gX) = (R × B, dt2 + dr2 + gr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let ρ ∈ B be a geodesic radius from p to ∂B and define S = R × ρ, which is a total geodesic submanifold of R × B with coordinates (t, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let γ be a smooth curve in S to be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Finally, let Σ = {(y, q) ∈ X : (y, ||q||g) ∈ γ} be a submanifold of (X, gX) with the induced metric, where || · ||g is the distance from p to q with respect to g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we want to calculate the scalar curvature of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' To do so we will need the following lemma from [Dod20]: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The principal curvatures of the hypersurface Sn−1(ϵ) in B are each of the form 1 −ϵ + O(ϵ) for small ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, let gϵ be the induced metric on Sn−1(ϵ) and let grd,ϵ be the round metric of curvature 1 ϵ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then, as ϵ → 0, 1 ϵ2 gϵ → 1 ϵ2 grd,ϵ = grd in the C2 topology, moreover, ||grd − 1 ϵ2 gϵ|| ≤ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now to calculate the scalar curvature of Σ, fix q ∈ Σ∩S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let e1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , en be an orthonormal basis of of Tq(Σ) where e1 is tangent to γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note that the for points in Σ ∩ S the normal ν to W in X is the same as the normal to γ in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' From the Gauss equations: RX(X, Y, Z, U) = RΣ(X, Y, Z, U) − A(X, U)A(Y, Z) + A(X, Z)A(Y, U) we see KΣ ij = KX ij + λiλj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' where λi are principal curvatures corresponding to ei and KΣ ij and KX ij are the respective sectional curvatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that λ1 = k where k is the geodesic curvature of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , n we see by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 λi = ⟨∇∂iν, ∂i⟩ = ⟨∇∂i cos θ∂t + sin θ∂r, ∂i⟩ = cos θ⟨∇∂i∂t, ∂i⟩ + sin θ⟨∇∂i∂r, ∂i⟩ = sin θ⟨∇∂i∂r, ∂i⟩ = � 1 −r + O(r) � sin θ, where θ is the angle that between ν and the t-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now note that KX 1j = RX(ej, e1, e1, ej) = RX(ej, cos θ∂r, cos θ∂r, ej) = cos2 θKM ∂r,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 14 PAUL SWEENEY JR For i ̸= 1 and j ̸= 1 KX ij = RX(ej, ei, ei, ej) = KM i,j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since RΣ = � i̸=j KΣ ij we see RΣ = RM − 2RicM (∂r, ∂r) sin2 θ + (n − 2)(n − 1) � 1 r2 + O(1) � sin2 θ − (n − 1) �1 r + O(r) � k sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Constructing the Curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The construction of the curve that will define the well W and the construction of the curve that will define the tunnel T are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' First, we will construct a curve that will define a submanifold Σ, which can be thought of as the precursor to a well or a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We want to construct a curve γ so that the resulting manifold Σ has RΣ > κ − 1 j for any j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will first construct γ as a piecewise curve of circular arcs and then smooth the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' To do this, we will prescribe the geodesic curvature k(s) of γ, and by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7 in [Gra98], we know that k(s) determines γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The unit tangent vector to γ and the curvature are given by dγ ds = (sin θ, − cos θ) and k = dθ ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, if γ(s) is defined for s ≤ s′ and k(s) is given for s ≥ s′ we have γ(s) = (t(s), r(s)) where θ(s) = θ(s′) + � s s′ k(u)du t(s) = t(s′) + � s s′ sin θ(u)du r(s) = r(s′) − � s s′ cos θ(u)du.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) Now, we begin the construction of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Fix j ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let δ0 < δ and let (0, δ0) be a point in the (t, r)-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Next, define the initial segment of γ as the line segment from (0, 2δ) to (0, δ0) for s ∈ [−2δ, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define the next segment to be an arc of a circle of curvature k0 = 1 that is tangent to r-axis at (0, δ0) and let γ run from 0 to s0 ≤ δ0 2 where s0 is chosen so that RΣ > κ − 1 j and that sin θ(s0) 8r(s0) < 1 for all s ≤ s0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that s0 exists since θ(0) = 0 and by the scalar curvature formula (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Next, we prove a lemma that gives a condition on γ that controls the scalar curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If δ0 is small enough and if sin θ(s) 4r(s) > k(s) for s ≥ s0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3) then RΣ > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) we see if k ≤ 0 then RΣ = RM − 2RicM (∂r, ∂r) sin2 θ + (n − 2)(n − 1) � 1 r2 + O(1) � sin2 θ − (n − 1) �1 r + O(r) � k sin θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4) and so the third and fourth terms will be nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By taking δ0 > r small enough, the third and fourth terms will dominate the second term so RΣ > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now, if k > 0, then by rewriting the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) we get RΣ = (n − 2)(n − 1) 2r2 sin2 θ + �(n − 2)(n − 1) 2r2 − 2RicM (∂r, ∂r) + O(1) � sin2 θ + −2(n − 1)k r sin θ + �(n − 1) r − O(r) � k sin θ + RM, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5) so second and fourth terms will be positive by taking δ0 > r is small enough and by assumption we have sin θ 4r > k which implies (n − 2)(n − 1) 2r2 sin2 θ + −2(n − 1)k r sin θ > 0, and so RΣ > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Thus, as we continue to construct γ, we will ensure that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will now extend γ by a circular arc of curvature k1 = sin θ(s0) 8r(s0) on [s0, s1] where s1 − s0 = r0 2 , where r(s0) = r0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let θ(s0) = θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2), we have first that sin θ(s) is increasing and r(s) is decreasing and so on [s0, s1] sin θ(s) 4r(s) > sin θ0 4r0 > sin θ0 8r0 = k1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Second, we see that γ does not cross the t-axis because s1 − s0 = r0 2 , and third we have θ(s1) − θ0 = k1(s1 − s0) = sin θ0 8r0 r0 2 = sin θ0 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we proceed inductively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define: si = si−1 + ∆si, ∆si = ri−1 2 , ri = r(si), θi = θ(si), ki = sin θi−1 8ri−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' As θ(s) is increasing we have that θi − θi−1 = sin θi−1 16 > sin θ0 16 and so θi ≥ θ0 + isin θ0 16 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, θi grows without bound so define m to be such that θm−1 < sin−1 � 12 13 � ≤ θm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Redefine sm so that θm := sin−1 � 12 13 � = ¯θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note that ∆sm ≤ rm−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 16 PAUL SWEENEY JR Now extend again by one circular arc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' To do this we need to define km+1 > 0 and sm+1 = sm + ∆sm+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We add a circular arc until θm+1 = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By the definition of ¯θ, there exists a km+1 such that 1 − sin ¯θ < km+1 rm 2 < sin ¯θ 8 and by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) we know rm+1 = rm − � sm+1 sm cos θ(u)du = rm − � sm+1 sm cos (sm + km+1(u − sm)) du = rm − 1 km+1 (sin θm+1 − sin θm) = rm − 1 km+1 � 1 − sin ¯θ � > rm 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' and km+1 < sin ¯θ 4rm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Up until this point the curve γ works for both the construction of a well and a tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' However, from here on the construction of γ differs slightly for the well and the tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will continue now with the construction of the well and discuss the tunnel construction later in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Adjusting the curve to construct a well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we will refine our construction of γ in order to construct a well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We want to extend by a line with a negative slope of length d > 0 and not have γ cross the t-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By the intermediate value theorem there exists an ˆs ∈ (sm, sm+1) such that θ(ˆs) = �θ where � max �¯θ, cos−1 � rm+1 2 �� < �θ < π 2 if d ≤ 1 max �¯θ, cos−1 � rm+1 2d �� < �θ < π 2 if d > 1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6) since θm+1 = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Redefine sm+1 such that sm+1 = ˆs and θm+1 = �θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Extend γ to [sm+1, sm+1+d] by setting k = 0 on [sm+1, sm+1 + d].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, note that by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) we have θ(u) ≡ θm+1 on that interval and r(sm+1 + d) = rm+1 − � rm+1+d rm+1 cos θ(u)du = rm+1 − d cos θm+1 ≥ rm+1 − rm+1 2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let sm+1 + d = sm+2 and θ(sm+1 + d) = θm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We now extend on [sm+2, sm+3] by a small circular arc of negative geodesic curvature such that θ(sm+3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Take km+3 < −2 sin θm+2 rm+2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since, EXAMPLES FOR SCALAR SPHERE STABILITY 17 θ(s) = θm+2 + � s sm+2 km+3du = θm+2 + km+3(s − sm+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' we have r(sm+3) = rm+2 − � sm+3 sm+2 cos θ(u)du rm+3 = rm+2 − 1 km+3 (sin θm+3 − sin θm+2) rm+3 = rm+2 + 1 km+3 sin θm+2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We can extend γ on [sm+3, sm+4] by a vertical straight line by setting km+4 = 0, where sm+4 is chosen so that r(sm+4) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since γ is parameterized by arclength, we note that a bound on sm+4 is a bound on arclength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In following lemmas, we prove an upper bound for sm+4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a constant 0 < C1 < 1 independent of j and d such that ri ri−1 ≤ C1 for 1 ≤ i ≤ m − 1 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) and by the mean value theorem we have ri = ri−1 − � si si−1 cos θ(u)du = ri−1 − ∆si cos ξi = ri−1 � 1 − cos ξi 2 � for some ξi ∈ [si−1, si].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Recalling that ¯θ ≥ ξi for 1 ≤ i ≤ m − 1 we see that ri ri−1 ≤ 1 − cos ¯θ 2 = 21 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There is a constant C2 independent of j and d such that sm+4 ≤ C2δ0 + d which implies that the length of γ is bounded by C2δ0 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We recall for 1 ≤ i ≤ m, ∆si ≤ ri−1 2 so sm = 2δ − δ0 + s0 + ∆s1 + · · · + ∆sm ≤ 2δ + s0 + 1 2 (r0 + r1 + · · · + rm−1) ≤ 2δ + s0 + r0 2 � 1 + C1 + · · · + Cm−1 1 � ≤ 3δ + δ0 2 � 1 1 − C1 � ≤ 28 5 δ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7) 18 PAUL SWEENEY JR Now, we note that by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6): ∆sm+1 ≤ 1 km+1 �π 2 − ¯θ � < rm π 2 − ¯θ 1 − sin ¯θ ≤ π 2 − ¯θ 1 − sin ¯θδ0 = 13 �π 2 − sin−1 �12 13 �� δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2), we have that θm+3 = θm+2 + km+3∆sm+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, ∆sm+3 = θm+2 −km+3 ≤ 2rm+2θm+2 sin θm+2 ≤ π sin ¯θδ0 = 13π 12 δ0 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='8) because θm+2 ≤ π 2 , rm+2 < δ0, and ¯θ < θm+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By construction, ∆sm+2 = d and ∆sm+4 ≤ δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, sm+4 = s0 + ∆s1 + · · · + ∆sm + ∆sm+1 + ∆sm+2 + ∆sm+3 + ∆sm+4 ≤ C2δ + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Smoothing the curve that defines the well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' So far we have constructed k(s) as a piecewise constant function, k �� (si,si+1] = ki+1 The resulting curve γ is C1 and piecewise C∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We begin the smoothing of γ by first smoothing out k(s) on [0, sm+3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let g ∈ C∞(R) be a smooth function so that g is 0 if s < 0, 1 if s > 1, and strictly increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let h(x) = g(1 − x) and H = � 1 0 h(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let ˜k(s) be the smooth function defined by �k(s) = � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � g � s α � s ∈ � − δ0 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' α � 1 s ∈ [α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' s0 − α] (1 − k1)h � s−s0 α � + k1 s ∈ [s0 − α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' s0] k1 s ∈ [s0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' s1] (ki+1 − ki) g � s−si α � + ki s ∈ [si,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' si + α] ki+1 s ∈ [si + α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' si+1] km+1h � s−sm+1 α � s ∈ [sm+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' sm+1 + α] 0 s ∈ [sm+1 + α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' sm+2] −km+3h � s−sm+2 α � + km+3 s ∈ [sm+2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' sm+2 + α] km+3 s ∈ [sm+2 + α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' sm+3],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' where 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that α is the same for each i and that its value will be determined later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now let ˜θ be the angle function associated to ˜k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) we see that the smooth curve ˜γ(s) = (˜t(s), ˜r(s)) defined by ˜k(s) will converge uniformly to γ on � δ0 2 , sm+3 � as α goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Also, ˜θ will converge uniformly to θ as α goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, take α small enough such that ˜θ(sm+1) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' therefore, we will still extend by a line with a negative slope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 19 Note ˜θ(sm+2 + α) = ˜θ(sm+2) + � sm+2+α sm+2 −km+3h �u − sm+2 α � + km+3du = ˜θ(sm+2) + αkm+3(1 − H) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By the smoothing process, ˜θ(sm+3) may no longer be greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will now fix that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If ˜θ(sm+3) ≤ 0 pick a s∗ ∈ (sm+2 + α, sm+3] such that 0 < ˜θ(s∗) < α which exists by the intermediate value theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If ˜θ(sm+3) > 0, we can redefine sm+3 as sm+3 + ˜θ(sm+2) −km+3 so that ˜θ(sm+3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By the intermediate value theorem, pick a s∗ ∈ (sm+2 + α, sm+3] such that 0 < ˜θ(s∗) < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Redefine sm+3 in either case as sm+3 = s∗ and note 0 < ˜θ(s∗) < α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [sm+3, sm+3 + 2β] define ˜k(s) = � −km+3g � s−sm+3 β � + km+3 s ∈ [sm+3, sm+3 + β] 0 s ∈ [sm+3 + β, sm+3 + 2β], where β = ˜θ(sm+3) −km+3(1−H) so that � sm+3+β sm+3 ˜k(s)ds = −˜θ(sm+3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This makes ˜θ(sm+3 + 2β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) we see that the smooth curve ˜γ(s) = (˜t(s), ˜r(s)) defined by ˜k(s) will converge uniformly to γ on [−2δ, sm+3] as α goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Also, ˜θ will converge uniformly to θ as α goes to zero;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' moreover, as α goes to zero so does β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Finally, take α small enough so that ˜r(sm+3+2β) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Extend the line segment at the end of ˜γ on [sm+3+2β, L] where L is defined so that ˜r(L) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note that |L−(sm+3+2β)| < δ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Attaching the Well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We have constructed a smooth curve ˜γ on [−2δ, L] that begins and ends as a vertical line segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define ˜Wj = {(y, q) ∈ X : (y, ||q||M) ∈ ˜γ} and let gj be the induced metric, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=', ˜gj = ˜ι∗ j(dt2+g) where ˜ιj : ¯Wj → R×B is the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For small enough α we will show that ˜γ satisfies R ˜ Wj ≥ κ − 1 j on [−2δ, L].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Also the length ˜γ is bounded by C3δ + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By construction ˜γ is parameterized by arclength so by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7 length of ˜γ is bounded by C3δ0 + d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 20 PAUL SWEENEY JR On [−2δ, 0], we have that R ˜ Wj = RM − 2RicM (∂˜r, ∂˜r) sin2 ˜θ + (n − 2)(n − 1) � 1 ˜r2 + O(1) � sin2 ˜θ − (n − 1) �1 ˜r + O(r) � ˜k sin ˜θ = RM > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [0, s0], we have that k1 < 1 because of our choice of s0 and the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' R ˜ Wj = RM − 2RicM (∂˜r, ∂˜r) sin2 ˜θ + (n − 2)(n − 1) � 1 ˜r2 + O(1) � sin2 ˜θ − (n − 1) �1 ˜r + O(r) � ˜k sin ˜θ ≥ κ − 2RicM (∂˜r, ∂˜r) sin2 ˜θ + (n − 2)(n − 1) � 1 ˜r2 + O(1) � sin2 ˜θ − (n − 1) �1 ˜r + O(r) � sin ˜θ > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' since for small enough α we have that ˜θ is uniformly close to θ and ˜r is uniformly close to r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [s0, s1] we have that ˜k(s) = k1 and so sin ˜θ(s) 4˜r(s) − ˜k(s) > 0 for small enough α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [si, si+1] for 1 ≤ i ≤ m, we have that sin ˜θ(s) 4˜r(s) − ˜k(s) = � sin ˜θ(s) 4˜r(s) − ki+1 � + � ki+1 − ˜k(s) � and so for small enough α we have the the first term is positive since sin θ(s) 4r(s) > k(s) and the second term is positive by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [sm+1, sm+2], we have that sin ˜θ(s) 4˜r(s) ≥ sin ˜θ(sm+1) 4˜r(sm+1) > km+1 > ˜k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We have the first inequality since sin ˜θ(s) 4˜r(s) is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The second inequality was already verified above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The third inequality holds since by construction km+1 > ˜k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [sm+2, L], we have by construction that ˜k(s) is non-positive so sin ˜θ(s) 4˜r(s) ≥ ˜k(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 21 Therefore, by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4 we have shown R ˜ Wj > κ − 1 j on � − δ0 2 , L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Next we will prove the diameter and volume bounds for the well, but before we prove those bounds, we need to recall the following fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let B(p, r) be a geodesic ball of radius r in a closed Riemannian manifold (Mn, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then there exists constants C, r0 depending on g such that for any p and for all r ≤ r0 we have volg(B(p, r)) ≤ Crn volg(∂B(p, r)) ≤ Crn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='9) Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There is a constant C(g) independent of j, d such that diameter diam ( ˜Wj) and volume vol ( ˜Wj) of ˜Wj satisfy d ≤ diam ( ˜Wj) < C(δ + d) and vol ( ˜Wj) < C(δn + dδn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let p, q ∈ ˜Wj be two points and let x be the point at the tip of ˜Wj, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=', corresponding to ˜γ(L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By the triangle inequality and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 we have dgj(p, q) ≤ dgj(p, x) + dgj(x, q) ≤ length(˜γ) + length(˜γ) ≤ C(δ + d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By construction we have d ≤ diam (W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, d ≤ diam (W) < C(δ + d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By possibly taking δ smaller, we have by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='8 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='9 that vol ( ˜Wj) = � L −δ0 2 |∂B(p, r(s))|˜gjds ≤ � L −δ0 2 Cδn−1ds ≤ C(δn + δn−1d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ( ˜Wj, ˜gj) is isometric to Wj = (Bg(p, 2δ), gj = dF 2 j + g) and Wj attaches smoothly to M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let B = Bg(p, 2δ) and recall that ||q||g is the distance from q to p in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the function Fj : B → R, Fj(q) = ˜t � ˜r−1(||q||g) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By construction ˜t is smooth and ˜r′(s) < 0 so ˜r−1 is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, ||q|| is smooth away from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, away from p, F is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In a neighborhood of p we have by construction that (˜t(s), ˜r(s)) is a vertical line segment so in that neighborhood ˜t ◦ ˜r−1 ≡ const and so Fj is smooth everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, by construction, we have that ˜gj|E = g|E where E = Bg(p, 2δ) \\ Bg(p, δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Γj = {(t, p) ∈ X : Fj(p) = t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note that Γj ⊂ X and that Γj = ˜Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let g′ j = (ι′ j)∗(dt2 +g) where ι′ j : ˜Wj → R×B is the inclusion map ι′ j(t, p) = (t, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let idj : Γj → ˜Wj be the identity map and conclude that g′ j = ˜gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the diffeomorphism Φj : B → Γj where Φj(q) �→ (Fj(q), q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' And so Φ∗ jg′ j = Φ∗((ι′ j)∗g′ j) = dF 2 j + g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ And this completes the construction of N from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (Constructing Wells).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 22 PAUL SWEENEY JR 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Constructing a Tunnel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will pick up the construction of the tunnel from Re- mark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let γ be as it is before Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The same smoothing procedure as above can be used to smooth γ into a smooth curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will abuse notation and call this smoothed-out curve ˜γ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let g ∈ C∞(R) be the smooth function so that g is 0 if s < 0, 1 if s > 1, and strictly increasing on [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let h(x) = g(1 − x) and H = � 1 0 h(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let ˜k(s) be the smooth function defined by �k(s) = � � � � � � � � � � � � � � � � � � � � � g � s α � s ∈ � − δ0 2 , α � 1 s ∈ [α, s0 − α] (1 − k1)h � s−s0 α � + k1 s ∈ [s0 − α, s0] k1 s ∈ [s0, s1] (ki+1 − ki) g � s−si α � + ki s ∈ [si, si + α] ki+1 s ∈ [si + α, si+1] where 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note that ˜θ(sm+1) could no longer equal π 2 by the smoothing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We fix that now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that ˜θ(s) converges uniformly to θ(s) as α goes to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Take α be small enough such that ˜θ(sm + α) < π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We want ˜θ(sm+1) < π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, if not, then ˜θ(sm+1) ≥ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Pick a s∗ ∈ (sm + α, sm+1] such that π 2 − α < ˜θ(s∗) < π 2 which exists by the intermediate value theorem and redefine sm+1 = s∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let sm+2 = sm+1 + 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [sm+1, sm+2], define ˜k(s) = � −km+1g � s−sm+1 β � + km+1 s ∈ [sm+1, sm+1 + β] 0 s ∈ [sm+1 + β, sm+2], where β = π 2 −˜θ(sm+1) −km+1(1−H) so that � sm+1+β sm+1 ˜k(s)ds = π 2 − ˜θ(sm+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, ˜θ(s) = π 2 for all s ∈ [sm+1 + β, sm+2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, we have finished smoothing γ to ˜γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define a half tunnel Aj = {(y, q) ∈ X : (y, ||q||g) ∈ ˜γ} with the induced metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Later, we will glue two half tunnels together to make a tunnel Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In the following lemma, we record properties of Aj whose proofs are analogous to the ones above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There is a constant C independent of j such that (Aj, hj) satisfies the following (i) The scalar curvature Rj of Aj satisfies Rj > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (ii) diam (Aj) < C(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) vol (Aj) < C(δn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iv) Aj smoothly attaches to M \\ Bg(p, 2δ) (v) The new manifold (M \\ Bg(p, 2δ)) ⊔ Aj is a manifold with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We have constructed half of a tunnel, Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We now wish to modify the metric at the end of Aj so that it is a product metric of a round sphere and an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We follow the same procedure as [Dod20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let a = t(sm+1 + β), b = t(sm+2), and c = r(sm+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that, EXAMPLES FOR SCALAR SPHERE STABILITY 23 by construction, the induced metric on {(q, y) ∈ X : a ≤ t ≤ b} is h0 = gc + dt2, where gc is the induced metric on Sn−1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let h1 = c2grd + dt2 where grd is the round metric on the unit round sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let φ(t) = ψ � t−a η � where ψ(u) is a smooth function on [0, 1] vanishing near zero, increasing to 1 at u = 3 4 and equal to 1 for u > 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define the metric h for t ∈ [a, b] as h(q, y) = gc(q, y) + φ(t) � c2grd − gc � + dt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This metric transitions smoothly between h0 and h1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note h − h0 = φ(t) � c2grd − gc � = φ(t)c2 � grd − 1 c2 gc � and that the first and second derivatives of φ(t) are O(η−1) and O(η−2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' So by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3, we have that the second derivatives of h − h0 are O(η2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, for η small enough, the scalar curvature of h is close to the scalar curvature of h0 which, again by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3, has scalar curvature larger than κ − 1 j for small enough η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we have changed the metric at the end of Aj so that it looks like c2grd + dt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, given another ball Bg(p′, 2δ) on M we can construct A′ j with a metric at the one end that it looks like c2grd + dt2 with the same c by making the same choices in the construction as we did for Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we can immediately glue a cylinder, ([0, d] × Sn−1, dt2 + c2grd), connecting A′ j to Aj and so construct the tunnel Tj between ∂Bg(p′, 2δ) and ∂Bg(p, 2δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that the diameter and volume of the cylinder ([0, d] × Sn−1, dt2 + gSn−1) are bounded by d and C(n)dδn−1, respectively, where C(n) is a constant that only depends on the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we can conclude that diam(Tj) and vol(Tj) satisfy the bounds in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, this completes the construction for Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Manifolds with shrinking tunnels In this section, we will use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels) to construct sequences of manifolds with thinner and thinner long tunnels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, we will prove Theorems A and A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will need first the following preliminary results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a sequence of rotationally symmetric manifolds Mj = (Sn, gj), n ≥ 3, such that Mj satisfies Rj ≥ n(n − 1) − 1 j , diam (Mj) ≤ D, and vol (Mj) ≤ V, for some constants 0 < D, V and converges to M∞ which is the disjoint union of two n-spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will construct the Mj as the connected sum of two standard unit round n- spheres for which the tunnel that connects the two spheres gets skinnier as j increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2, we can remove a geodesic ball from both of the spheres and then construct a tunnel Tj connecting the two spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (N, h) = (N′, h′) = (Sn, grd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let j ∈ N, j ≥ 10, d = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define Bj := Bh � p, 2 j � ⊂ N, and B′ := Bh′ � p′, 2 j � ⊂ N′ 24 PAUL SWEENEY JR where Bj and B′ j are geodesic balls in N, N′ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2, we can construct a tunnel Tj connecting ∂Bj to ∂B′ j and the resulting manifold Mj will have the following properties: (i) Mj = � (N ⊔ N′) \\ � Bj ∪ B′ j �� ⊔ Tj (ii) Rj ≥ n(n − 1) − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iii) Mj \\ Tj is isometric to (N \\ Bj) ⊔ (N′ \\ B′ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (iv) diam (Mj) ≤ 4π + 30, 2 volgrd (Sn) − volh (Bj) − volh′ (B′ j) ≤ vol (Mj) ≤ 2 volgrd (Sn) + volgj (Tj), and lim j→∞ volh (Bj) = lim j→∞ volh′ (B′ j) = lim j→∞ volgj (Tj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, limj→∞ vol (Mj) = 2 volgrd (Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3, we have the intrinsic flat distance between Mj and N ⊔ N′ is dF(Mj, N1 ⊔ N2) ≲ 1 j � volgrd(Sn) + volgrd(Sn−1) � + volh (Bj) + volh′ (B′ j) + volgj (Tj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' As j → ∞0, we that volh (Bj), volh′ (B′ j), and volgj (Tj) go to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we conclude that Mj converges to N ⊔ N′ in the VF sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' From the construction in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels) we see that Mn j = ([0, Dj]×Sn−1, gj) defined above is rotationally symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, near {0}×Sn−1 and {Dj} × Sn−1, we have that Mn j is isometric to the standard unit round n-sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular, the metric takes the form gj = dt2 +sin2(ρj(t))gSn−1 where Dj is the diameter of Mj and for ρj : [0, Dj] → [0, ∞) is a smooth function with the following properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Recall ˜γj = (˜tj(s), ˜rj(s)) to be the curve define in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='12 that defines the half tunnel Aj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then ρ(t) = � ˆr(s), s ∈ � 0, 1 2Dj � ˆr(D − s), s ∈ � 1 2Dj, Dj � and ˆr(t) = � � � π − s, s ∈ � 0, π − 2 j � ˜r(s + (δ − π)), s ∈ � π − 2 j , 1 2D � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will now construct smooth 1-Lipschitz maps Fj : Mn j → (Sn, grd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' But first, we need the following result based on the mollification in [Mia02, Section 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since our lemma varies slightly from what is stated in [Mia02] we provide an analogous proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let h : R → R be an L-Lipschitz continuous function such that h(t) = � h+(t), t ∈ (0, ∞) h−(t), t ∈ (−∞, 0), where h+ and h− are smooth functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then for small enough ϵ > 0 there exists a function hϵ : R → R such that ||hϵ(t) − h(t)||C2 ≲ ϵ2, h′ ϵ(t) ≤ sup{h′(t) : t ∈ R \\ {0}}, and |h′ ϵ(t)| ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let 0 < ϵ0 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will restrict our attention to (−ϵ0, ϵ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let ϕ ∈ C∞ c ([−1, 1]) be the standard mollifier in R such that 0 ≤ ϕ ≤ 1 and � 1 −1 ϕ(t)dt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 25 Let σ(t) ∈ C∞ c �� − 1 2, 1 2 �� be another bump function such that 0 ≤ σ(t) ≤ 1 100 for t ∈ R, σ(t) = 1 100 for |t| < 1 4, 0 < σ(t) ≤ 1 100 for 1 4 < |t| < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let 0 < ϵ < 1 10ϵ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define σϵ(t) = ϵ3σ � t ϵ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, define hδ(t) = � R h(t − σδ(t)s)ϕ(s)ds, t ∈ (−ϵ0, ϵ0) = �� R h(s) · 1 σδ(t)ϕ � t−s σδ(t) � ds, σδ(t) > 0 h(t), σδ(t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) Now we want to compute h′ δ(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For |t| > ϵ3 100, h′ ϵ(t) = d dt � R h(t − σϵ(t)s)ϕ(s)ds = � R h′(t − σϵ(t)s) � 1 − sϵ2σ′ �t ϵ �� ϕ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For |t| < ϵ 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='δ(t) = d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h(s) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='σϵ(t)ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�t − s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='σϵ(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h(s) · d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='σϵ(t)ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�t − s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='σϵ(t) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h(s) · d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='dt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= (−1) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h(s) · d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= (−1) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h−(s) · d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='+ (−1) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h+(s) · d ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='−∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='−(s) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='+(s) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h′(s) · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ϕ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�100(t − s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ϵ3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='ds ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='R ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='h′(t − σϵ(t)s)ϕ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 26 PAUL SWEENEY JR Now note for |t| < ϵ 4 that σϵ is a constant function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' therefore, for all t ∈ (−ϵ0, ϵ0) h′ ϵ(t) = � R h′(t − σϵ(t)s) � 1 − sϵ2σ′ �t ϵ �� ϕ(s)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) we have ||hϵ(t) − h(t)||u ≤ � R ||h(t − σϵ(t)s) − h(t)||uϕ(s)ds ≲ ϵ3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' and ||h′ ϵ(t) − h′(t)||u ≤ � R ||h′(t − σϵ(t)s) − h′(t)||uϕ(s)ds + � R ���� ����h′(t − σϵ(t)s)sϵ2σ′ �t ϵ ����� ���� u ϕ(s)ds ≲ ϵ3 + ϵ2 � R ���� ����h′(t − σϵ(t)s)σ′ �t ϵ ����� ���� u ϕ(s)ds ≲ ϵ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lastly, note that |h′ ϵ(t)| = � R |h′(t − σϵ(t)s)| ���� � 1 − sϵ2σ′ �t ϵ ������ |ϕ(s)|ds ≤ L � R � 1 − sϵ2σ′ �t ϵ �� (ϕ(s))ds = L � 1 − ϵ2σ′ �t ϵ � � R sϕ(s)ds � ≤ L, where the first inequality follows if ϵ is small enough and the last inequality follows since sϕ(s) is an odd function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, redoing this computation without the absolute values shows that h′ ϵ(t) ≤ sup{h′(t) : t ∈ R \\ {0}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Now we are ready to construct smooth 1-Lipschitz maps Fj : Mn j → (Sn, grd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There exists a function Fj : Mn j → Sn that is a 1-Lipschitz diffeomorphism with deg Fj ̸= 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' First define a decreasing 1-Lipschitz function fj : [0, Dj] → [0, π].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' fj(t) = � π − t, t ∈ [0, tj] aj(t − tj) + bj, t ∈ [tj, Dj], where aj = −π+tj Dj−tj , bj = π − tj, and tj is chosen so that fj(tj) = 1 10ρ � 1 2Dj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note ρ � 1 2Dj � is the radius of the cylindrical part of the tunnel which is also the minimum that ρj(t) attains on � π 2 , Dj − π 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3, we can smooth fj to fj,ϵ by choosing ϵ0 and ϵ small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' And so define Fj,ϵ(t, θ) = (fj,ϵ(t), θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since f′ j,ϵ(t) < 0 and fj,ϵ is a bijection, we have that Fj,ϵ is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We want to show that for all v ∈ TMj F ∗ j,ϵgrd(v, v) ≤ gj(v, v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 27 Note that F ∗ j,ϵgrd = � f′ j,ϵ(t) �2 dt2 + sin2(fj,ϵ(t))gSn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' and gj = dt2 + sin2(ρj(t))gSn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' First by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 we know that |f′ j,ϵ(t)| ≤ 1 for all t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we will show that sin2(fj,ϵ(t)) ≤ sin2(ρj(t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [0, π − tj − 20ϵ] we have by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2) that ρj(t) = π − � t 0 cos (θj(u)) du ≥ π − t = fj(t) = fj,ϵ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On [π − tj − 20ϵ, π − tj] we have ρj(t) = π − � t 0 cos (θj(u)) du > π − t = fj(t) and so for small enough ϵ, we have that fj,ϵ(t) will also satisfy this inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' On � π − tj, Dj − π 2 � , we have that fj(t) ≤ 1 10ρ � 1 2Dj � and that 1 10ρ � 1 2Dj � < ρj(t) ≤ π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, sin2(fj(t)) ≤ sin2(ρj(t)) on � 0, Dj − π 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lastly on � Dj − π 2 , Dj � we have the following: ρj(t) = π − Dj + t and fj,ϵ(t) = fj(t) = aj(t − tj) + bj by the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, −fj(t) + π ≥ ρj(t) since if we define ψj(t) = ρj(t) + fj(t) − π, then we see that ψ′(t) ≥ 0 and ψ(Dj) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We also note on � Dj − π 2 , Dj � that π 2 ≤ −fj(t) + π ≤ π and π 2 ≤ ρj(t) ≤ π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we conclude that sin2(fj,ϵ(t)) = sin2(−fj,ϵ(t) + π) ≤ sin2(ρj(t)) on � Dj − π 2 , Dj � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, for all v ∈ TMj we have F ∗ j,ϵgrd(v, v) ≤ gj(v, v), which implies ℓSn (Fj,ϵ ◦ c) ≤ ℓMj(c) where c : [0, 1] → (Sn, grd) is a path connecting p and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This implies that dSn (Fj,ϵ(p), Fj,ϵ(q)) ≤ dMj(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, we have that Fj,ϵ is 1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, deg Fj,ϵ ̸= 0 since Fj,ϵ is a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (S3, g1), (S3, g2) be 3-spheres such that there exists a diffeomorphism F : (S3, g1) → (S3, g2) that is 1-Lipschitz and is isotopic to the identity then width(S3, g2) ≤ width(S3, g1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By the definition of width for any δ > 0 there exists {Σt} such that sup t |Σt|1 < width(S3, g1) + δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' therefore, width(S3, g2) ≤ sup t |F(Σt)|2 ≤ sup t |Σt|1 ≤ width(S3, g1) + δ where the first inequality follows since F(Σt) ∈ Λ′ and the second inequality follows since F is 1-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ 28 PAUL SWEENEY JR Proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Mn j be as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' therefore, Mn j → M∞ in VF-sense where M∞ is the disjoint union of two spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Fj : Mn j → Sn be as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4 and define ˜Fj(r, θ) = Fj(Dj − r, θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the diffeomorphism Φ : [0, Dj] × Sn−1 → [0, π] × Sn−1, Φ(r, θ) = � π Dj r, θ � Note that Φ is an isometry between ([0, Dj] × Sn−1, Φ∗(dr2 + sin2(r)gSn−1)) and ([0, π] × Sn−1, dr2 + sin2(r)gSn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' And now consider (Φ−1 ◦ ˜Fj)(r, θ) = �Dj π fj(Dj − r), θ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This map is a 1-Lipschitz orientation preserving diffeomorphism from Mn j to the round n-sphere and Φ−1 ◦ Fj is isotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5 we have that width(Mn j ) ≥ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Proof of Theorem A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Mj be as in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' therefore, Mn j → M∞ where M∞ is the disjoint union of two spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Fj : Mn j → Sn be as in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then by Arzela-Ascoli Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4 there is a subsequence Fjk that converges to a 1-Lipschitz map F∞ : M∞ → Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This map is not a Riemannian isometry since Sn is connected and N ⊔ N′ is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Manifolds with many wells In this section, we will use Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (Constructing Wells) to construct sequences of manifolds with many wells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, we will prove Theorems B and B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g) be a closed Riemannian manifold of dimension n ≥ 3 with scalar curvature R ≥ κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then there exists a sequence of Riemannian manifolds Mn j = (Mn, gj) such that Rj ≥ κ − 1 j and Mn j converge in the VADB-sense and VF-sense to Mn but has no convergent subsequence in the GH-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define Xj = � (B(pj i, δj), g) �j i=1 to be a collection of disjoint geodesic balls in Mn where 0 < δj < 1 j is chosen small enough so that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 we replace each B(pj i, δj) with the well Wi,j = (B(pj i, δj), gj) such that the scalar curvature of each of the wells satisfies Rj > κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, choose d = 1 2 in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 so that diam(Wi,j) ≥ 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Call the resulting manifold Mn j = (Mn, gj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we note that lim j→∞ volj (Mn j ) = lim j→∞ volg (Mn) − j � i=1 volg (B(pj i, δj)) + j � i=1 volj (Wi,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Thus, by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='9 lim j→∞ volg (Mn) − jCδn j ≤ lim j→∞ volj (Mn j ) ≤ lim j→∞ volg (Mn) + Cj � δn j + δn−1 j 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' EXAMPLES FOR SCALAR SPHERE STABILITY 29 and so lim j→∞ volj (Mn j ) = volg (Mn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Also by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 and the triangle inequality, we have that diam � Mn j � ≤ diam ((Mn, g)) + 2 diam (Wj) ≤ diam ((Mn, g)) + 2 � C + 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' so the diameters are uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the identity map id : (Mn, gj) → (Mn, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Denote id∗gj = gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now by con- struction and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='11 we have for any p ∈ Wi,j that g(v, v) ≤ gj(v, v) for all v ∈ TpM because gj = dF 2 j + g and if p /∈ Wi,j then g(v, v) = gj(v, v) for all v ∈ TpM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, Mn j converges to (Mn, g) in the VADB-sense and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7 we have that Mn j converges to (Mn, g) in the VF-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Fix ϵ0 < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Note that ϵ0 < d and so B(pj i, ϵ0) ⊂ Mn j are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, j < Covj(ϵ0) and so as j → ∞ we have Covj(ϵ0) → ∞ so by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 that Mj does not converge in the GH-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Proof of Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the round 3-sphere (S3, grd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 we see that there exists a sequence (S3, gj) with scalar curvature Rj ≥ 6 − 1 j such that (S3, gj) → (S3, grd) in the V ADB and VF-sense but has no convergent subsequence in the GH-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, the identity map id : (S3, gj) → (S3, grd) is 1-Lipschitz and by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5 we have that width(S3, gj) ≥ 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Proof of Theorem B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Consider the round n-sphere (Sn, grd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 we see that there exists a sequence Mj = (Sn, gj) with scalar curvature Rj ≥ n(n − 1) − 1 j such that Mj → (S3, grd) in the V ADB and VF-sense but has no convergent subsequence in the GH-topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Furthermore, the identity map id : (Sn, gj) → (Sn, grd) is smooth 1-Lipschitz diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Proof of Theorem C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let κ > 0 and let (Mn, g) be the round sphere of curvature 2κ n(n−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let {pj}∞ j=1 ⊂ Mn be a sequence of points on a geodesic converging to a point p∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Define {B(pj, δj)}∞ j=1 to be a collection of disjoint geodesic balls in Mn where 0 < δj < 1 2j is chosen small enough so that by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 there exists a well Wj = (B(pj, δj), gj) such that the scalar curvature of each of the wells satisfies Rj > 2κ � 1 − 1 10j � > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let {dj}∞ j=1 ⊂ [2, 10] be a strictly increasing sequence of positive numbers, and choose d = dj in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 so that diam(Wj) ≥ dj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now define Mn i to be the Riemannian obtained by replacing the first i balls with the corresponding first i wells, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=', Mn i = � �Mn \\ i� j=1 B(pj, δj) � � ⊔ i� j=1 Wj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 30 PAUL SWEENEY JR We note that Mn i has scalar curvature strictly larger than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We also have by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 that diam(Mn i ) ≤ 25C and vol(Mn i ) ≤ vol(Mn) + ∞ � j=1 vol(Wj) ≤ vol(Mn) + C � � ∞ � j=1 1 2nj + 10 ∞ � j=1 1 2(n−1)j � � ≤ vol(Mn) + 11C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Now we will define M∞ to be M∞ = � �Mn \\ ∞ � j=1 B(pj, δj) � � ⊔ ∞ � j=1 Wj with its induced length metric and natural current structure T∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, we have that vol(Mn i ) → vol(M∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Ej ⊂ Wj be a ball centered at pj of radius 1 and so M∞ is noncompact since it contains infinitely many disjoint balls of radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We will show that Mn i converges to M∞ in an analogous many to [SW11, Example A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let ϵi = dMn(pi, p∞) and note that if ˜Bi = B(p∞, ϵi − δi), then there is an isometry, ϕ : Vi → V ′ i where Ui = Mn i \\ ˜Bi ⊂ Mi and U ′ i ⊂ M∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' By [SW11, Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2], there exists a metric space Z such that dZ F (Mn i , M∞) ≤ vol(Mn i \\ Ui) + vol(M∞ \\ U ′ i) + vol(Ui) �� 2 diamMn i (∂Ui) diamMn i (Ui) + diamMn i (∂Ui) � + vol(U ′ i) �� 2 diamMn i (∂U′ i) diamMn i (U ′ i) + diamMn i (∂U′ i) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that vol(Mn i \\ Ui) ≤ π(ϵi − δi)n, vol(M∞ \\ U ′ i) ≤ C � � ∞ � j=i 1 2nj + 10 ∞ � j=i 1 2(n−1)j � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Also, diam(∂Ui) and diam(∂U′ i) converge to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Therefore, the right-hand side of the inequality above goes to zero as i → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We conclude then that Mn i converges to M∞ in the VF-sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sewing Manifolds We are able to generalize the sewing examples of Basilio, Dodziuk, and Sormani found in [BDS18] and [BS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There are two methods of sewing developed in [BS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Method I generalizes the curve sewing construction of [BDS18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Here we will extend the construction using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We start with Method I which says that given a fixed manifold one can tightly sew a compact region to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g) be a complete Riemannian manifold, and A0 ⊂ M a compact subset with an even number of points pi ∈ A0, i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , n with pairwise disjoint balls EXAMPLES FOR SCALAR SPHERE STABILITY 31 B(pi, 2δ) with scalar curvature greater than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' For small enough δ > 0, define Aδ := Tδ(A0) and A′ δ = Aδ \\ � n� i=1 B(pi, δ) � ⊔ n 2� i=1 Ti where Ti are tunnels as in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels) connecting ∂B(p2j+1, δ) and ∂B(p2j+2, δ) for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , n 2 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then given any ϵ, shrinking δ further, if necessary, we may create a new complete Riemannian manifold, (Nn, h), Nn = (Mn \\ Aδ) ⊔ A′ δ satisfying vol (Aδ) − ϵ ≤ vol (A′ δ) ≤ vol (Aδ) + ϵ and vol (M) − ϵ ≤ vol (N) ≤ vol (M) + ϵ If, in addition, M has scalar curvature, RM ≥ κ, then N has scalar curvature, RN ≥ κ − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If ∂M ̸= ∅, the balls avoid the boundary and ∂M is isometric to ∂N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The proof follows from the proof of [BDS18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1] while using Proposi- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels) and Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g) be a complete Riemannian manifold and A0 ⊂ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Aa = Ta(A0) be a tubular neighborhood of A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Assume that there is an a > 0 such that Aa has scalar curvature greater than κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let r ∈ (0, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Given ϵ > 0, there exists δ = δ(A0, κ, r, ϵ) ∈ (0, r) and there exists even n = ¯n(¯n − 1) depending on A0, κ, ϵ, and r and points p1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' , pn ∈ A0 with B(pi, δ) pairwise disjoint such that we can “sew the region tightly” to create a new complete Riemannian manifold (Nn, h), N = (M \\ Ar) ⊔ A′ r, as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1, with A′ δ = Aδ \\ � 2n � i=1 B(pi, δ) � ⊔ n−1 � j=0 T2j+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, vol (A′ r) ≤ vol (Ar) + ϵ and vol (N) ≤ vol (M) + ϵ and there is a constant c > 0 such that diam (A′ r) ≤ cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' we say we have sewn the region A0 arbitrarily tight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If M has scalar curvature RM ≥ κ, then N has scalar curvature RN ≥ κ − ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If ∂M ̸= ∅, the balls avoid the boundary, and ∂M is isometric to ∂N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The proof follows from the proof of [BS21, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6] while using Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1, and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ 32 PAUL SWEENEY JR These statements allow us to construct sequences of manifolds with scalar curvature greater than κ which converge to a pulled metric space in a similar manner as in [BS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We recall the following definition from [BS21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Mn, g) be a Riemannian manifold with a compact set A0 ⊂ M with tubular neighborhood Aa = Ta(A0) satisfying the hypotheses of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We can construct its sequence of increasingly tightly sewn manifolds, (Nn j , gj), by applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 taking ϵ = ϵj → 0, n = nj → ∞, and δ = δj → 0 to create each sewn manifold Nn = Nn j and the edited regions A′ δ = A′ δj which we simply denote A′ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Since these sequences Nj are created using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2, they have scalar curvature greater than κ−ϵj when M has scalar curvature greater than κ and ∂Nj = ∂M whenever ∂M ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The sequence Nj, as in Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 assuming Mn is compact and A0 is a compact embedded submanifold of dimension 1 to n, converges in the Gromov-Hausdorff sense and the intrinsic flat sense to N∞, which is a metric space created by pulling the region A0 to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' If, in addition, Hn−1(A0) = 0 then Nj also converges in the metric measure sense to N∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The proof follows from the proof of [BS21, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='8] while using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Now we can prove Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let S be a simply connected space form of dimension n and constant curvature κ n(n−1) and Σm be a constant curvature m-dimensional sphere, 1 ≤ m ≤ n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We note that there exists an embedding of Σm into S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let (Nn j , gj) be a sequence of manifolds constructed from S sewn along an embedded Σm with δ = δj → 0 as in Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 and the scalar curvature Rj ≥ κ − 1 j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4 we have Nj mGH −−−→ N∞ and Nj F−→ N∞ where N∞ is the metric space created by taking S and pulling a Σm to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, at the pulled point p0 ∈ N∞ we have wR(p0) = lim r→0 6(n + 2)volEn B(0, r) − Hn(B(p0, r)) r2 · volEn B(0, r) = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We can see this because volN∞ (B(p0, r)) = Hn N∞(B(p0, r)) = Hn N∞(B(p0, r) \\ {p0}) = Hn Snκ(Tr(Sm)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, there is a constant C(n, m, κ) such that lim r→0 Hn Snκ(Tr(Σm)) Crn−m = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We conclude that wR(p0) = lim r→0 6(n + 2)ωnrn − Crn−m ωnrn+2 = −∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' ⊓⊔ Moreover, using Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 we are to extend Method II for sewing manifolds in [BS21] to the setting where scalar curvature is bounded below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In Method II, given a sequence of Riemannian manifolds whose limit is a Riemannian, then one can create a new sequence where the sewing occurs along the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' REFERENCES 33 Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Let Mn j be a sequence of compact Riemannian manifolds each with a com- pact region Aj,0 ⊂ M3 j with tubular neighborhood, Aj, with scalar curvature greater than κ satisfying the hypotheses of Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' We assume Mn j converge in the biLipschitz sense to Mn ∞ and the regions Aj,0 converge to a compact set A∞,0 ⊂ Mn ∞ in the sense that there exists biLipschitz maps ψj : Mn j → Mn ∞ such that Lj = log (Lip(ψj)) + log � Lip � ψ−1 j �� → 0 and ψj(Aj,0) = A∞,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Then there exists δj → 0 and applying Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 to Mn = Mn j to sew the regions A0 = Aj,0 with δ = δj, to obtain sewn manifolds Nn = Nn j , we obtain a sequence Nn j such that Nn j GH −−→ N∞ and Nn j F−→ N∞,0, where ¯N∞,0 = N∞ and N∞ is the metric space created by taking Mn ∞ and pulling the region A∞,0 to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, if the regions Aj,0 satisfy Hn(Aj,0) = 0, the the sequence Nn j also converges in the metric measure sense Nn j mGH −−−→ N∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' The proof follows from the proof of [BS21, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1] while using Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 and Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4 ⊓⊔ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Intrinsic Flat limit with no geodesics We are able to generalize the result of Basilio, Kazaras, and Sormani from [BKS20] which shows the intrinsic flat limit of Riemannian manifolds need not be geodesically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' This follows from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2 (Constructing Tunnels) and the pipe-filling technique [BKS20, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In particular: Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' There is a sequence of closed, oriented, Riemannian manifolds (Mn j , gj), n ≥ 3, such that the corresponding integral current spaces converge in the intrinsic flat sense to M∞ = � N, dEn+1, � N � , where N is the round n-sphere of curvature 2κ n(n−1) and dEn+1 is the Euclidean distance induced from the standard embedding of N into En+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, Mj may be chosen so that Rj, the scalar curvature of Mj, satisfies Rj ≥ 2κ � 1 − 1 10j � > κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Moreover, M∞ is not a length space and is not locally geodesically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' References [AK00] Luigi Ambrosio and Bernd Kirchheim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Currents in metric spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (2000), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1–80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0001-5962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/BF02392711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/BF02392711.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [AP20] Brian Allen and Raquel Perales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Intrinsic Flat Stability of Manifolds with Bound- ary where Volume Converges and Distance is Bounded Below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='13030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/abs/2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='13030.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 34 REFERENCES [APS20] Brian Allen, Raquel Perales, and Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Volume Above Distance Below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='01172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/ abs/2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='01172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [BDS18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Basilio, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Dodziuk, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Sewing Riemannian manifolds with positive scalar curvature”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 3553–3602.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 1050-6926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/s12220-017-9969-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1007/s12220-017-9969-y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [BKS20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Basilio, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Kazaras, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “An intrinsic flat limit of Riemannian manifolds with no geodesics”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Dedicata 204 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 265–284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0046-5755.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/s10711-019-00453-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1007/s10711-019-00453-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [BS21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Basilio and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Sequences of three dimensional manifolds with posi- tive scalar curvature”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 77 (2021), Paper No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 101776, 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0926-2245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='difgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='101776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='difgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='101776.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Dod20] J´ozef Dodziuk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Gromov-Lawson tunnels with estimates”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Analysis and ge- ometry on graphs and manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lecture Note Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Cambridge Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Press, Cambridge, 2020, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 55–65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Fuk87] Kenji Fukaya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Collapsing of Riemannian manifolds and eigenvalues of Laplace operator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Invent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 (1987), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 517–547.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0020-9910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1007/BF01389241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/BF01389241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [GL80] Mikhael Gromov and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Blaine Lawson Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “The classification of simply con- nected manifolds of positive scalar curvature”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' (2) 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 (1980), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 423–434.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0003-486X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2307/1971103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/ 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2307/1971103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Gra98] Alfred Gray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Modern differential geometry of curves and surfaces with Mathe- matica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' CRC Press, Boca Raton, FL, 1998, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' xxiv+1053.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' isbn: 0-8493- 7164-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Gro18] Misha Gromov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Scalar Curvature of Manifolds with Boundaries: Natural Ques- tions and Artificial Constructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='04311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/abs/1811.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='04311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Gro99] Misha Gromov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Metric structures for Riemannian and non-Riemannian spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Progress in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Based on the 1981 French original [ MR0682063 (85e:53051)], With appendices by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Katz, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Pansu and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Semmes, Translated from the French by Sean Michael Bates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Birkh¨auser Boston, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=', Boston, MA, 1999, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' xx+585.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' isbn: 0-8176-3898-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Lak16] Sajjad Lakzian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “On diameter controls and smooth convergence away from sin- gularities”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 47 (2016), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 99–129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0926-2245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='difgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' difgeo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Lla98] Marcelo Llarull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Sharp estimates and the Dirac operator”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (1998), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 55–71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0025-5831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/s002080050136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/s002080050136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [LS12] Dan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lee and Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Near-equality of the Penrose inequality for rotationally symmetric Riemannian manifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Henri Poincar´e 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1537–1556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 1424-0637.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/s00023- 012- 0172- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/s00023-012-0172-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' REFERENCES 35 [LS13] Sajjad Lakzian and Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Smooth convergence away from singular sets”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (2013), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 39–104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 1019-8385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4310/CAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='v21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4310/CAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' v21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='n1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [LS14] Dan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Lee and Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Stability of the positive mass theorem for rotationally symmetric Riemannian manifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Reine Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 686 (2014), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 187–220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0075-4102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1515/crelle-2012-0094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1515/crelle-2012-0094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [LS15] Philippe G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' LeFloch and Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “The nonlinear stability of rota- tionally symmetric spaces with low regularity”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='7 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2005–2065.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0022-1236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1016 / j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' jfa .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2014 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 12 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='jfa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Mia02] Pengzi Miao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Positive mass theorem on manifolds admitting corners along a hypersurface”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6 (2002), 1163–1182 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 1095-0761.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4310/ATMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi-org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='edu/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4310/ATMP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='a4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [MN12] Fernando C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Marques and Andr´e Neves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Rigidity of min-max minimal spheres in three-manifolds”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='14 (2012), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2725–2752.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0012- 7094.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1215/00127094-1813410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1215/ 00127094-1813410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Per20] Raquel Perales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Convergence of manifolds and metric spaces with boundary”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Topol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='3 (2020), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 735–774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 1793-5253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1142/ S1793525319500638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1142/S1793525319500638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Sor+21] Christina Sormani, Participants at the IAS Emerging Topics Workshop on Scalar Curvature, and Convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Conjectures on Convergence and Scalar Curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='10093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/abs/2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 10093.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Sor17] Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Scalar curvature and intrinsic flat convergence”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Mea- sure theory in non-smooth spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Partial Differ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Equ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Meas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' De Gruyter Open, Warsaw, 2017, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 288–338.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Sor18] Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “Intrinsic flat Arzela-Ascoli theorems”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='6 (2018), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 1317–1373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 1019-8385.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4310/CAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' v26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='4310/CAG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='v26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='n6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [Sor22] Christina Sormani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Private communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [SW11] Christina Sormani and Stefan Wenger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “The intrinsic flat distance between Rie- mannian manifolds and other integral current spaces”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Differential Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1 (2011), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 117–199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0022-040X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: http://projecteuclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/ euclid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='jdg/1303219774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' [SY79] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Schoen and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' “On the structure of manifolds with positive scalar curvature”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' In: Manuscripta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1-3 (1979), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' 159–183.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' issn: 0025-2611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/BF01647970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' url: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='1007/BF01647970.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content=' Department of Mathematics, Stony Brook University, Stony Brook, NY 11794, USA Email address: paul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='sweeney@stonybrook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} +page_content='edu' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/L9AzT4oBgHgl3EQfV_wg/content/2301.01292v1.pdf'} diff --git a/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf b/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..8b79c4bd51b521ca1330c37c94d5c7293854b092 --- /dev/null +++ b/LdE1T4oBgHgl3EQfGwON/content/2301.02918v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a32c0d439c690de704e9a8a49ab530c86c6826e17d2977c09a688d8ad5e10267 +size 623818 diff --git a/LdE1T4oBgHgl3EQfGwON/vector_store/index.pkl b/LdE1T4oBgHgl3EQfGwON/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..cf12518491ce351aa830d4d67ceae64c1b6223c3 --- /dev/null +++ b/LdE1T4oBgHgl3EQfGwON/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:9452a681cdf614241cf5d8931c9f9620c890c500fdd514ca11fb0655042b0d29 +size 187375 diff --git a/MdE3T4oBgHgl3EQfwQuL/content/tmp_files/2301.04701v1.pdf.txt b/MdE3T4oBgHgl3EQfwQuL/content/tmp_files/2301.04701v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..75c725a37dd2b4ad0219a3098308d85cabc35dc2 --- /dev/null +++ b/MdE3T4oBgHgl3EQfwQuL/content/tmp_files/2301.04701v1.pdf.txt @@ -0,0 +1,1290 @@ +arXiv:2301.04701v1 [math.SP] 11 Jan 2023 +Indices of diagonalizable and universal +realizability of spectra∗ +Charles R. Johnsona, Ana I. Juliob†, Ricardo L. Sotob +aDepartment of Mathematics, College of William and Mary +Williamsburg, VA, USA +bDepartamento de Matem´aticas, Universidad Cat´olica del Norte +Antofagasta, Chile. +Casilla 1280, Antofagasta, Chile. +Abstract +A list Λ = {λ1, . . . , λn} of complex numbers (repeats allowed) is +said to be realizable if it is the spectrum of an entrywise nonnegative +matrix A. Λ is diagonalizably realizable if the realizing matrix A is +diagonalizable. Λ is said to be universally realizable if it is realizable +for each possible Jordan canonical form allowed by Λ. Here, we study +the connection between diagonalizable realizability and universal real- +izability of spectra. In particular, we establish indices of realizability +for diagonalizable and universal realizability. We also define the merge +of two spectra and we prove a result that allow us to easily decide, in +many cases, about the universal realizability of spectra. +AMS classification: 15A18, 15A20, 15A29 +Key words: Nonnegative matrices, Spectra diagonalizably realizable, Spec- +tra universal realizability, Jordan structure, Eigenvalues and eigenvectors +∗Supported by Universidad Cat´olica del Norte-VRIDT 036-2020, NUCLEO UCN +VRIDT-083-2020, Chile. +†Corresponding author, crjohn@wm.edu (Charles R. Johnson), ajulio@ucn.cl, (Ana I. +Julio), rsoto@ucn.cl (Ricardo L. Soto) +1 + +1 +Introduction +A list Λ = {λ1, . . . , λn} of complex numbers (with repeats allowed) is said +to be realizable if there is an n-by-n nonnegative matrix A with spectrum Λ. +In this case A is said to be a realizing matrix. The problem of characterizing +the realizability of lists Λ = {λ1, . . . , λn} of complex numbers is called the +nonnegative inverse eigenvalue problem (NIEP). We say that Λ is diagonal- +izably realizable (DR) if it is the spectrum of a diagonalizable nonnegative +matrix. The problem of determining this kind of realizability is called the +diagonalizable realizability problem. Λ is universally realizable (UR) if it is +realizable for every possible Jordan canonical form (JCF) allowed by Λ. The +problem of determining the universal realizability of spectra is called the uni- +versal realizability problem (URP). The URP seeks to extend the question +of determining the spectral properties allowed by a nonnegative matrix, not +only regarding the eigenvalues themselves, but also from the point of view of +the corresponding JCF. The URP contains the NIEP and both problems are +equivalent if the given complex numbers λ1, . . . , λn are distinct. The NIEP +has attracted the interest of many linear algebraist researchers. The URP, +on the other hand, is a new and even more difficult problem. Both problems +remain unsolved for n ≥ 5. A complete solution, if any, is still far away from +the present state of the art. The URP is studied module the NIEP (see [15]). +This means that the methods applied to the NIEP are not only useful but +also in many cases necessary for the URP. Then, getting answers for some +topics and open questions about the URP means a positive impact on the +progress towards a solution. +The NIEP, as we know it today, begins with the works by Sule˘ımanova [24] +and Perfect [17, 18]. The NIEP has been solved only for the cases n = 3 +by Loewy and London [13], and for n = 4 by Meehan [14] and also inde- +pendently by Torre-Mayo et al. [25]. The first known results on the URP +are due to Minc [15, 16]. In [15] Minc proved that if Λ = {λ1, . . . , λn} is the +spectrum of a diagonalizable positive matrix, then Λ is UR. We want to point +out here that previously the URP was called nonnegative inverse elementary +divisors problem [15], and that in [4] the authors used for the first time the +concept and name universal realizability. There are spectra, not positively +realizable, that are known to be UR [3, 22, 23]. For instance, spectra in the +left half-plane, that is, Λ = {λ1, . . . , λn} with λ1 > 0, Reλi ≤ 0, i = 2, . . . , n, +such as real Sule˘ımanova spectra [22] λ1 > 0 > λ2 ≥ · · · ≥ λn; complex +2 + +Sule˘ımanova spectra [23] +λ1 > 0, λi ∈ {z ∈ C : Rez ≤ 0, |Rez| ≥ |Imz|} , i = 2, . . . , n; +and ˇSmigoc spectra [3] +λ1 > 0, λi ∈ +� +z ∈ C : Rez ≤ 0, +��� +√ +3Rez +��� ≥ |Imz| +� +, i = 2, . . . , n, +which contains the real and complex Suleimanova spectra. These spectra are +realizable if and only if they are UR if and only if �n +i=1 λi ≥ 0. The good +behavior of these kind of spectra led to think that any left half-plane list was +UR. Now, we know that this is not true [9, 10]. +In [12] Laffey and ˇSmigoc proved that a list Λ = {λ1, . . . , λn} in the left +half-plane is realizable if and only if +s1 = +n +� +i=1 +λi ≥ 0, +s2 = +n +� +i=1 +λ2 +i ≥ 0, +s2 +1 ≤ ns2. +The positivity and diagonalizability conditions in the result of Minc [15] are +essential for his proof. Minc says that “it is not known if the the theorem +holds for a general nonnegative matrix”, question that has been open for +almost 40 years. Recently, two extensions for nonnegative realizations have +been obtained: the first one by Collao, Salas, and Soto [2] shows that if +Λ = {λ1, . . . , λn} is the spectrum of a diagonalizable nonnegative matrix +with constant row sums and a positive row or column, then Λ is UR. The +second extension by Johnson, Julio, and Soto [7], shows that if Λ is realiz- +able by a diagonalizable ODP matrix, that is, a diagonalizable nonnegative +matrix having all off-diagonal entries being positive (zeros on diagonal are +permitted), then Λ is also UR. Observe that in both extensions, the set of +realizing matrices contains the set of positive realizing matrices. Moreover, +the extension in [7] allows to decide about the universal realizability of lists +with +n� +i=1 +λi = 0, which it is not possible from the Minc’s result. These two +extensions open a research line that allow to prove the universal realizability +of non-positively realizable spectra, thus significantly extending the class of +spectra that can be shown to be UR. Since DR is a necessary condition for +UR, all lists of complex numbers that are DR, are natural candidates to be +UR. +3 + +Despite the progress that has been made on the URP, there are still nu- +merous open questions. Two of them, until recently open, were: Is any left +half-plane list UR? In [10] the authors show that a realizable left half-plane +list is not necessarily UR. In fact, they prove that the spectrum +Λ = +� +a, −a +4 + +√ +5a +4 i, −a +4 − +√ +5a +4 i, −a +4 + +√ +5a +4 i, −a +4 − +√ +5a +4 i +� +a > 0, +(1) +is realizable, but not DR and therefore not UR The other open question was +whether DR realizable implies UR. In [9] the authors show that the spectra +{λ1, λ1, λ2, λ2} with λ1 > 0 > λ2 ≥ −λ1, λ1 + 2λ2 < 0, have no realiza- +tion with a JCF having a Jordan block J2(λ2) of size 2. Thus, for instance, +Λ = {1, 1, −1, −1} is not UR although it is DR. Since Λ = {1, 1, −1, −1} +has a reducible realization, what may be said about irreducible realizations?. +In [10] it was also shown that irreducible diagonalizable realizations are not +necessarily UR. Then, it remains to know under what conditions a DR left +half-plane list of complex numbers is UR. From extensions in [2, 7] we may +say that DR implies UR if Λ = {λ1, . . . , λn} is the spectrum of a diago- +nalizable nonnegative matrix with constant row sums and a positive row or +column, or it is diagonalizably ODP realizable. The importance of the diag- +onal JCF lies in the fact that we know how to join Jordan blocks to obtain +larger Jordan blocks. Then if we may obtain a diagonalizable realizing ma- +trix for Λ = {λ1, . . . , λn}, we may also obtain a nonnegative matrix with +spectrum Λ for each possible JCF allowed by Λ. What about criteria which +allow to decide about the universal realizability of a list of complex numbers? +In this work we also introduce an operation that we name the merge of two +spectra and a result which allow to easily decide, in many cases, about the +universal realizability of spectra. We also establish indexes of Guo type [5] +for diagonalizable and universal realizability. +A matrix A = [aij] of order n is said to have constant row sums if all its +rows sum to the same constant α. We denote by CSα the set of all n-by-n +real matrices with constant row sums equal to α. It is clear that any matrix +in CSα has an eigenvector e +T = [1, . . . , 1] corresponding to the eigenvalue +α. +The relevance of the real matrices with constant row sums is due to +the fact that, the problem of finding a nonnegative matrix with spectrum +Λ = {λ1, . . . , λn}, λ1 being the Perron eigenvalue, is equivalent to the prob- +4 + +lem of finding a nonnegative matrix in CSλ1, with spectrum Λ (see [6]). We +denote by Ei,j the matrix with 1 in position (i, j) and zeros elsewhere and +we define the matrix +E(K) = +� +i∈K +Ei,i+1, +K ⊂ {1, 2, . . . , n}. +(2) +The paper is organized as follows: In Section 2, we introduce the diag- +onalizable realizability index gd(Λ/λ1) and the universal realizability index +gu(Λ/λ1). Then, we show that a realizable list Λ = {µ, λ2, . . . , λn} of com- +plex numbers is DR for all µ ≥ gd(Λ/λ1) and that Λ is UR for all µ ≥ +gu(Λ/λ1). In Section 3, we define the merge of two spectra and show that if +Γ1 = {λ1, . . . , λn} and Γ2 = {µ1, . . . , µm} have a diagonalizable ODP real- +ization, then Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm} has also a diagonalizable +ODP realization and therefore Γ is UR. This result becomes a useful criterion +to decide, in many cases, the universal realizability of spectra. +2 +Diagonalizable and universal realizability +indices. +In what follows we will use the following results, Theorems 2.1 to 2.3 below. +Theorem 2.1, due to Brauer [1], is a perturbation result that shows how to +change one single eigenvalue of an n-by-n matrix without changing any of +the remaining (n − 1) eigenvalues. Theorem 2.2, by Soto and Ccapa [22], +establishes the JCF of the Brauer perturbation A + eq +T. Theorem 2.3, by +ˇSmigoc [20], shows how to construct a matrix C with a particular JCF from +two given matrices A and B. +Theorem 2.1 [1] Brauer. Let A be an n-by-n matrix with spectrum Λ = +{λ1, . . . , λn}. Let v +T = [v1, . . . , vn] be an eigenvector of A associated with +the eigenvalue λk and let q be any n-dimensional vector. Then the matrix +A + vq +T has eigenvalues λ1, . . . , λk−1, λk + v +Tq, λk+1, . . . , λn. +Theorem 2.2 [22] Soto and Ccapa. Let q +T = [q1, . . . , qn] be an arbitrary +n−dimensional vector. Let A ∈ CSλ1 with JCF +J(A) = S−1AS = diag {J1(λ1), Jn2(λ2), . . . , Jnk(λk)} . +5 + +If λ1 + �n +i=1 qi ̸= λi, i = 2, . . . , n, then the matrix A + eq +T has Jordan +canonical form J(A) + (�n +i=1 qi) E11. In particular, if �n +i=1 qi = 0 then A +and A + eq +T are similar. +Theorem 2.3 [20] ˇSmigoc. Suppose B is an m-by-m matrix with JCF that +contains at least one 1-by-1 Jordan block corresponding to the eigenvalue c: +J(B) = +� c +0 +0 +I(B) +� +. +Let t and s, respectively, be the left and the right eigenvectors of B associated +with the 1-by-1 Jordan block in the above canonical form. Furthermore, we +normalize vectors t and s so that t +Ts = 1. Let J(A) be a JCF for an n-by-n +matrix +A = +� A1 +a +b +T +c +� +, +where A1 is an (n − 1)-by-(n − 1) matrix and a and b are vectors in C +n-1. +Then the matrix +C = +� +A1 +at +T +sb +T +B +� +has JCF +J(C) = +� J(A) +0 +0 +I(B) +� +. +Let Λ = {λ1, . . . , λn} and Λ/λ1 = {λ2, . . . , λn} be self-conjugate lists of com- +plex numbers. Guo [5] proved that there is a minimal nonnegative number +gr(Λ/λ1) such that Λµ = {µ, λ2, . . . , λn} is realizable for all µ ≥ gr(Λ/λ1). +Moreover, Guo established that +max +2≤j≤n |λj| ≤ gr(Λ/λ1) ≤ 2n max +2≤j≤n |λj| . +(3) +In [11] it was shown that the upper bound in (3) may be reduced to (n − +1) max +2≤j≤n |λj| , with n ≥ 5 in the case when λk, k = 2, . . . , n, are conjugates +complex, and that this bound is sharp. In [19] the authors show how to cal- +culate the Guo index gr(Λ/λ1) for circulant nonnegative matrices. However, +to compute this index becomes a prohibitive task for large n. In this section, +we prove that there is also a minimal nonnegative number gd(Λ/λ1), called +6 + +diagonalizable realizability index, such that Λµ = {µ, λ2, . . . , λn} is DR if +µ ≥ gd(Λ/λ1). Of course gd(Λ/λ1) ≥ gr(Λ/λ1). Equality occurs if Λ/λ1 has +distinct elements and may occur otherwise. The proof is similar to the proof +in [11], but it considers all possible cases, which it does not occur in [11]. +Theorem 2.4 If Λ = {λ1, . . . , λn} is realizable, then there is a nonnegative +real number gd(Λ/λ1) such that Λµ = {µ, λ2, . . . , λn} is DR for every µ ≥ +gd(Λ/λ1). Moreover +gr(Λ/λ1) ≤ gd(Λ/λ1) ≤ (n − 1) max +2≤j≤n |λj| . +(4) +Proof. First, we exhibit a value µ such that Λµ is DR. Let A be a realizing +matrix for Λ. Without loss of generality we assume that A ∈ CSλ1, that is, +Ae = λ1e. If A is diagonalizable we are done. If not, we take A = SJS−1, +where J is the JCF of A and Se1 = e. Now let �J be the same as J, except that +any superdiagonal 1′s are replaced with 0´s. So, �J is diagonal with spectrum +Λ. Define �A = S �JS−1. If �A is nonnegative we are done. If not, since �A ∈ +CSλ1 we apply Brauer’s Theorem to produce a nonnegative matrix A′ = +�A+eq +T, where q +T = [q1, . . . , qn] is an appropriate nonnegative vector. From +Theorem 2.2 A′ is diagonalizable with spectrum +� +λ1 + +n� +i=1 +qi, λ2, . . . , λn +� +. +Let gd(Λ/λ1) = λ1 + +n� +i=1 +qi. Thus we have established the existence of a value +gd(Λ/λ1) such that Λµ is DR for every µ ≥ gd(Λ/λ1). +Next, we show that gd(Λ/λ1) satisfies (4). The lower bound is clear. Although +similar to the proof of Theorem 3.2 in [11], this is more complete and consider +all cases. In particular, the case in which µj, j = 2, . . . , n, are all conjugate +complex. For the sake of completeness and since the result is of independent +interest, we set the proof here. +Let m = max +2≤j≤n |λj| and let µj = +λj +(n−1)m, +j = 2, . . . , n. Then, {µ2, . . . , µn} is a list of complex numbers such that +7 + +|µj| ≤ +1 +n−1, j = 2, . . . , n. Consider the initial matrix +B = + + +0 +0 +0 +· · · +· · · +· · · +· · · +· · · +0 +−µ2 +µ2 +... +... +... +... +... +... +... +−µp +... +... +µp +0 +... +−xs +ys +... +xs +−ys +... +−xs +−ys +ys +xs +... +... +... +... +... +... +... +0 +−xt +yt +... +xt +−yt +−xt +−yt +· · · +· · · +· · · +· · · +0 +yt +xt + + +, +where µ2, . . . , µp are real, xj = Reµj, yj = Imµj, p + 1 ≤ j ≤ n+p +2 . Then +B ∈ CS0 has eigenvalues 0, µ2, . . . , µp, µp+1, . . . , µn and is clear that B is +diagonalizable. +• If Reµj ≤ 0, j = 2, . . . , n, then all entries in the first column of B are +nonnegative. Let +q +T = +� +0, +1 +n − 1, . . . , +1 +n − 1 +� +. +From Theorem 2.1 and Theorem 2.2, A′ = B + eq +T is diagonaliz- +able nonnegative with spectrum {1, µ2, . . . , µn} and A = (n − 1)mA′ is +diagonalizable nonnegative with spectrum {(n − 1)m, λ2, . . . , λn}. +• If Reµj > 0 for some j, 3 ≤ j ≤ n, then all the entries in the j +st column +of B (or in the (j − 1) +st column of B if j corresponds to the second +column in the corresponding 2-by-2 complex block) are nonnegative. +Let +q +T = +� +1 +n − 1, . . . , +1 +n − 1, 0, +1 +n − 1, . . . , +1 +n − 1 +� +, +with zero in the j +st position ((j − 1) +st position). Then, again, A′ = +B + eq +T is diagonalizable nonnegative with spectrum {1, µ2, . . . , µn} +and A = (n − 1)mA′ is diagonalizable nonnegative with spectrum +8 + +{(n − 1)m, λ2, . . . , λn}. +Observe that, from the necessary and suffi- +cient conditions by Loewy and London [13], the result still holds for +the special case n = 3 with Λ = {λ1, a + bi, a − bi}. +• If µ2 > 0 with Reµj < 0, j = 3, . . . , n, then we write the −Reµ′ +js, 3 ≤ +j ≤ n, along the second column of B and the ±Imµ′ +js, p + 1 ≤ j ≤ n, +along the first column of B. Then again with +q +T = +� +1 +n − 1, 0, +1 +n − 1, . . . , +1 +n − 1 +� +, +we obtain, as before, the diagonalizable nonnegative matrix A = (n − +1)mA′ with the required spectrum. +• Now we consider the case in which µj, j = 2, . . . , n, are all conjugate +complex numbers, that is, Imµj ̸= 0. Let the 2-by-2 diagonal blocks +� Reµj +−Imµj +Imµj +Reµj +� +, +j = 2k, k = 1, 2, . . . , n − 1 +2 +in the matrix B. In this case we only can use the first column of B to +set −(Reµj + Imµj) in order that B ∈ CS0. Now we have the following +cases: +– If Reµj ≥ 0 for one or more indexes j, then the corresponding j +st +columns in B are nonnegative and the proof follows as before. If +|Reµj + Imµj| > +1 +n−1 for some j, then we set the corresponding +diagonal block +� +Reµj +−Imµj +Imµj +Reµj +� +on the last 2-by-2 diagonal position in the matrix B to distribute +any possible negative amount (from the reciprocal of Reµj +Imµj) +through the last row under some appropriate columns. Observe +that there is at least one negative amount (−Imµj) in each above +2-by-2 block in B. Thus, A′ is nonnegative with spectrum +� n� +i=1 +qi, µ2, . . . , µn +� +, +n� +i=1 +qi ≤ 1 and A = (n − 1)mA′ is nonnegative with spectrum +9 + +� +(n − 1)m +n� +i=1 +qi, λ2, . . . , λn +� +, where (n − 1)m +n� +i=1 +qi ≤ (n − 1)m. +In fact, for n = 5 (the minimum case) with Reµi ≥ 0, we have + + +0 +0 +0 +0 +0 +−Reµi + Imµi +Reµi +−Imµi +0 +0 +−Reµi − Imµi +Imµi +Reµi +0 +0 +−Reµj + Imµj +0 +0 +Reµj +−Imµj +−Reµj − Imµj + Imµi +0 +−Imµi +Imµj +Reµj + + +Then +qT = [Reµj + Imµj, 0, Imµi, 0, Imµj] +and +5� +i=1 +qi = Reµj+2Imµj+ Imµi ≤ +4 +n−1 ≤ 1. As it was said above, +the validity of the case n = 3 is guaranteed from the necessary +and sufficient conditions by Loewy and London [13]. +– If Reµj < 0 with |Reµj| ≥ |Imµj| , j = 2, . . . , n, then the first +column in B is nonnegative and the proof follows as before. +– If Reµj < 0 with |Reµj| < |Imµj|, j = 2, . . . , n, then +|Reµj + Imµj| < | Imµj| < |µj| ≤ +1 +n − 1, +and the proof follows as before again. +The upper bound in (4) is also sharp for gd(Λ/λ1), as it is shown by a diag- +onalizable realization of the list +Λ = {(n − 1), −1, . . . , −1 +� +�� +� +(n−1)times +}. +The spectrum Λ = ∪ +n +2 +i=1{λi, −λi}, with real λi, shows that the lower bound +in (4) is also sharp for gd(Λ/λ1). We may also define, similar to gr(Λ/λ1) and +gd(Λ/λ1), a universal realizability index gu(Λ/λ1). Then, the following result +is clear. +Corollary 2.1 From Theorem 2.4 we have that +gd(Λ/λ1) ≤ gu(Λ/λ1) ≤ (n − 1) max +2≤j≤n |λj| . +(5) +10 + +Proof. The lower bound is clear. To prove the upper bound we consider +the matrix A = (n − 1)mA′, with A′ = (B + eqT) ∈ CS1, in the proof of +Theorem 2.4, which is diagonalizable nonnegative with JCF J(A) = S−1AS, +where Se1 = e. Then, for an appropriate matrix E(K) as defined in (2), +J(A) + E(K) = S−1AS + E(K) = S−1(A + SE(K)S−1)S. +(6) +Then we may reach any JCF allowed by the spectrum of A. In fact, A + +SE(K)S−1 has the desired JCF, J(A) + E(K), although it is no necessar- +ily nonnegative. +Since SE(K)S−1 ∈ CS0, then for a convenient nonneg- +ative vector r +T = [r1, . . . , rn] and ǫ > 0 small enough, the matrix M = +(A + ǫSE(K)S−1)+ er +T is nonnegative with spectrum {(n−1)m, λ2, . . . , λn} +or spectrum {(n − 1)m +n� +i=1 +qi, λ2, . . . , λn}, with +n� +i=1 +qi ≤ 1 in the case µj, +j = 2, . . . , n, are conjugate complex numbers. Then, since (n − 1)m +n� +i=1 +qi ≤ +(n − 1)m, Λ is UR and gu(Λ/λ1) ≤ (n − 1)max |λj| . +2≤j≤n +The following example illustrates Theorem 2.4 and Corollary 2.1. In par- +ticular the case where µj, j = 2, . . . , n, are all conjugates complex numbers. +Example 2.1 Consider the list +{0,1 + i +√ +15, 1 − i +√ +15, − +√ +15 + i, − +√ +15 − i}. +Then n = 5, m = max +2≤j≤5 |λj| = 4 and (n−1)m = 16. In order to have +5� +i=1 +qi ≤ 1 +we need to take the initial matrix B as +B = 1 +16 + + +0 +0 +0 +0 +0 +1 + +√ +15 +− +√ +15 +−1 +0 +0 +−1 + +√ +15 +1 +− +√ +15 +0 +0 +−1 + +√ +15 +0 +0 +1 +− +√ +15 +0 +−1 +− +√ +15 +√ +15 +1 + + +. +Then, A′ = B + eqT with +qT = 1 +16 +� +1, +√ +15, +√ +15, 0, +√ +15 +� +11 + +has the spectrum +1 +16{ +5 +� +i=1 +qi, 1 + i +√ +15, 1 − i +√ +15, − +√ +15 + i, − +√ +15 − i}, +with +5� +i=1 +qi ≤ 1 and (n − 1)m +5� +i=1 +qi ≤ (n − 1)m. +Since the list Λ = {(n − 1), −1, . . . , −1} is UR, then the upper bound in (4) +is also sharp for gu(Λ/λ1). Next, we have: +Theorem 2.5 Let Λ = {λ1, . . . , λn} be a DR list of complex numbers and +let gd(Λ/λ1) be the diagonalizable realizability index of Λ. Then for every +µ > gd(Λ/λ1), Λµ = {µ, λ2, . . . , λn} is UR. +Proof. The result is a straightforward consequence of the Minc’s result, and +the fact that if the Perron eigenvalue is increased for any ǫ > 0, then any +diagonalizably realizable list becomes a diagonalizably positively realizable +list. +Remark 2.1 We have seen that gd(Λ/λ1) ≥ gr(Λ/λ1). Of course, if a re- +alizable list Λ = {λ1, . . . , λn} of complex numbers has all its elements dis- +tinct, then gd(Λ/λ1) = gr(Λ/λ1). In order that gd(Λ/λ1) > gr(Λ/λ1), Λ must +admit repeats. This is necessary, but not sufficient. For instance, the list +Λ = {6, 1, 1, −4, −4} is DR (see [21, Lemma 1]), but gd(Λ/λ1) = gr(Λ/λ1). +It is clear that gd(Λ/λ1) > gr(Λ/λ1) if and only if Λ is not DR. +Example 2.2 The list +Λ = {7, 5, 1, 1, −4, −4, −6}, +is symmetrically realizable by +A = + + +0 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +√ +10 +10 +√ +10 +10 +3+ +√ +5 +2 +0 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +√ +10 +10 +√ +10 +10 +3− +√ +5 +2 +3+ +√ +5 +2 +0 +3+ +√ +5 +2 +3− +√ +5 +2 +√ +10 +10 +√ +10 +10 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +0 +3+ +√ +5 +2 +√ +10 +10 +√ +10 +10 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +0 +√ +10 +10 +√ +10 +10 +√ +10 +10 +√ +10 +10 +√ +10 +10 +√ +10 +10 +√ +10 +10 +0 +6 +√ +10 +10 +√ +10 +10 +√ +10 +10 +√ +10 +10 +√ +10 +10 +6 +0 + + +. +12 + +Then, since A is diagonalizable ODP, Λ is UR and gr(Λ/λ1) = gd(Λ/λ1) = +gu(Λ/λ1) = 7. +A realizable list Λ = {λ1, . . . , λn} of complex numbers is said to be Perron +extreme, if the list Λǫ = {λ1 − ǫ, λ2, . . . , λn} is not realizable for every ǫ > 0. +If Λ is not Perron extreme, there is an ǫ > 0 such that Λǫ is realizable. Then +we have the following result, which is equivalent to Theorem 2.5. +Corollary 2.2 Let Λ = {λ1, . . . , λn} be a list not Perron extreme of complex +numbers with Λǫ = {λ1 − ǫ, λ2, . . . , λn}, ǫ > 0, being DR. Then Λ is UR. +Proof. Let B ∈ CSλ1−ǫ be a diagonalizable nonnegative matrix with spec- +trum Λǫ. Then +A = B + eq +T, with q +T = +� ǫ +n, . . . , ǫ +n +� +is positive with spectrum Λ. Moreover, from Theorem 2.2, A is diagonalizable. +Hence, Λ is UR. +Example 2.3 The list Λ = {8, 2, 2, −3, −4, −4} is not Perron extreme. Since +Λ′ = {7, 2, 2, −3, −4, −4} is DR by +B = + + +0 +4 +0 +2 +0 +1 +4 +0 +0 +2 +0 +1 +0 +2 +0 +4 +0 +1 +0 +2 +4 +0 +0 +1 +0 +2 +0 +2 +0 +3 +0 +2 +0 +2 +3 +0 + + +, +then A = B + eq +T, where q +T = +� 1 +6 +1 +6 +1 +6 +1 +6 +1 +6 +1 +6 +� +is diagonalizable posi- +tive with spectrum Λ. Therefore Λ is UR. +Remark 2.2 We know that DR does not necessarily imply UR. We also +know of two important extensions [2, 7] of Minc’s result [15]. As far as we +know, these extensions are the most general universal realizability criteria for +a solution to URP. Then, for Λ = {λ1, λ2, . . . , λn} diagonalizably realizable +we may say that DR implies UR if: +i) The realizing matrix for Λ is ODP (positive matrices are ODP), or +ii) Λ is the spectrum of a nonnegative matrix A ∈ CSλ1 with a positive row +or column. +13 + +There are spectra, however, which are UR, without to have necessarily a re- +alizing matrix of some above classes. For instance: +iii) Λ = {λ1, λ2, . . . , λ n +2 , −λ n +2 , . . . , −λ2, −λ1}. Is clear that Λ has a diago- +nalizable realization and if it has 2 × 2 blocks repeated, we may obtain any +coarser JCF. +iv) Λ = {λ1, λ2, −λ3, . . . , −λn} with λj > 0, j = 1, 2, . . . , n. λ1 > λ2, +λ1 + λ2 − +n� +j=3 +λj = 0, It was proved in [4] that Λ is UR. +3 +The merge of two spectra +In this section we define the merge of two spectra in the following way: +Definition 3.1 Let Γ1 = {λ1, λ2, . . . , λn} and Γ2 = {µ1, µ2, . . . , µm} be lists +of complex numbers. The merge Γ1 with Γ2 is +Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm}. +Then, we have: +Theorem 3.1 Suppose that A and B are n-by-n and m-by-m diagonalizable +ODP matrices with spectrum Γ1 = {λ1, . . . , λn} and Γ2 = {µ1, . . . , µm}, +respectively. Then, there is an (n + m − 1)-by-(n + m − 1) diagonalizable +ODP matrix C with spectrum Γ = {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm}. Hence, +Γ is UR. +Proof. Since A is diagonalizable ODP with spectrum Γ1, it is irreducible. +Then, there is a positive vector v +T = [v1, . . . , vn] such that Av = λ1v. +Thus, if D = diag{v1, . . . , vn}, then ˜A = D−1AD ∈ CSλ1 is diagonalizable +ODP. If d1, . . . , dn are the diagonal entries of ˜A, then from Theorem 2.1, +A1 = ˜A+e[0, 0, . . . , µ1] = +�A11 +a +b +T +dn + µ1 +� +has spectrum {λ1 +µ1, λ2, . . . , λn} +and diagonal entries d1, d2, . . . , dn + µ1. It is clear, from Theorem 2.2, that +A1 is a diagonalizable ODP matrix. +Since B is diagonalizable ODP with spectrum Γ2 then, as before, there is +a diagonalizable ODP matrix ˜B ∈ CSµ1 with spectrum Γ2. Then, the ma- +trix B1 = ˜B + e[dn, 0 . . . , 0] is diagonalizable ODP with spectrum {µ1 + +14 + +dn, µ2, . . . , µm}. Finally, by applying the ˇSmigoc’s glue technique, Theorem +2.3, there is a matrix +C = +� +A11 +at +T +sb +T +B1 +� +with A11 the (n − 1)-by-(n − 1) matrix in A1 and with s, t such that t +Ts = +1, being the right and left eigenvectors of B1, respectively. +Now, C has +the spectrum {λ1 + µ1, λ2, . . . , λn, µ2, . . . , µm}, with Jordan canonical form +J(C) = J(A1) ⊕ I(B1) with J(B1) = +� +µ1 + dn +I(B1) +� +. Since a, b, s and t +are positive vectors, it is clear that C is a diagonalizable ODP matrix and +therefore Γ is UR. +In many cases, Theorem 3.1 may be a useful tool to decide about the universal +realizability of a list of complex numbers, as the following example shows. +Example 3.1 Is +Λ = {13, 1, 1, −3, −4, −4, 1 + 3i, 1 − 3i}, +universally realizable? To answer the question, consider the lists +Λ1 += +{7, −3, 1 + 3i, 1 − 3i}. +Λ2 += +{6, 1, 1, −4, −4}. +From [8], Λ1 has the normal nonnegative realization +A1 = + + +3 +2 − +√ +3 +√ +6+2 +√ +2 +2 +√ +6+2 +√ +2 +2 +2 + +√ +3 +3 +2 +√ +2− +√ +6 +2 +2 +√ +2− +√ +6 +2 +2 +√ +2− +√ +6 +2 +√ +6+2 +√ +2 +2 +0 +3 +2 +√ +2− +√ +6 +2 +√ +6+2 +√ +2 +2 +3 +0 + + , +which is diagonalizable ODP. From [21, Lemma 1], Λ2 has the symmetric +ODP realization +A2 = + + +0 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +3+ +√ +5 +2 +0 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +0 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +0 +3+ +√ +5 +2 +3+ +√ +5 +2 +3− +√ +5 +2 +3− +√ +5 +2 +3+ +√ +5 +2 +0 + + +. +15 + +Then, from Theorem 3.1, the merge Λ1 with Λ2, that is, +Λ = {13, 1, 1, −3, −4, −4, 1 + 3i, 1 − 3i} +has a diagonalizable ODP realization. Hence, Λ is UR. +Remark 3.1 It is known that lists of Sule˘ımanova type, real or complex, +are UR. Now, this result can be also proved via Theorem 3.1, in the case of +real Sule˘ımanova spectra, and of complex Sule˘ımanova spectra with |Reλi| > +|Imλi| . In fact, in both cases, we may obtain diagonalizable ODP realizations. +Declaration of Competing Interest +There is no competing interest +References +[1] A. Brauer, Limits for the characteristic roots of a matrix IV. Applica- +tions to stochastic matrices, Duke Math. J. 19 (1952) 75-91. +[2] M. Collao, M. Salas, R. L. Soto, Spectra universally realizable by doubly +stochastic matrices, Spec. Matrices 6 (2018) 301-309. +[3] R. C. D´ıaz, R. L. Soto, Nonnegative inverse elementary divisors problem +in the left half plane, Linear and Multilinear Algebra 64 (2016) 258-268. +[4] M. Collao, C. R. Johnson, R. L. Soto, Universal realizability of spectra +with two positive eigenvalues, Linear Algebra Appl. 545 (2018) 226-239. +[5] W. Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl. 266 +(1997) 261-270. +[6] C. R. Johnson, Row stochastic matrices similar to doubly stochastic +matrices, Linear and Multilinear Algebra 10 (1981) 113-130. +[7] C. R. Johnson, A. I. Julio, R. L. Soto, Nonnegative realizability with +Jordan structure, Linear Algebra Appl. 587 (2020) 302-313. +[8] A. I. Julio, C. B. Manzaneda, R. L. Soto, Normal nonnegative realization +of spectra, Linear and Multilinear Algebra 63 (2015) 1204-1215. +16 + +[9] A. I. Julio, C. Mariju´an, M. Pisonero, R. L. Soto, On universal realiz- +ability of spectra, Linear Algebra Appl. 563 (2019) 353-372. +[10] A. I. Julio, C. Mariju´an, M. Pisonero, R. L. Soto, Universal realizability +in low dimension, Linear Algebra Appl. 619 (2021) 107-136. +[11] A. I. Julio, R. L. Soto, The role of certain Brauer and Rado results in the +nonnegative inverse sepectral problems, Electronic J. Linear Algebra 36 +(2020) 484-502. +[12] T. J. Laffey, H. ˇSmigoc, Nonnegative realization of spectra having neg- +ative real parts, Linear Algebra Appl. 416 (2006) 148-159. +[13] R. Loewy, D. London, A note on the inverse problem for nonnegative +matrices, Linear and Multilinear Algebra. 6 (1978) 83-90. +[14] M. E. Meehan, Some results on matrix spectra, Ph.D. thesis, National +University of Ireland, Dublin, (1998). +[15] H. Minc, Inverse elementary divisor problem for nonnegative matrices, +Proc. of the Amer. Math. Society 83 (4) (1981) 665-669. +[16] H. Minc, Inverse elementary divisor problem for doubly stochastic ma- +trices, Linear and Multilinear Algebra. 11 (1982) 121-131. +[17] H. Perfect, Methods of constructing certain stochastic matrices, Duke +Math. J. 20 (1953) 395-404. +[18] H. Perfect, Methods of constructing certain stochastic matrices II, Duke +Math. J. 22 (1955) 305-311. +[19] O. Rojo, R. L. Soto, Existence and construction of nonnegative matrices +with complex spectrum, Linear Algebra Appl. 368 (2003) 53-69. +[20] H. ˇSmigoc, The inverse eigenvalue problem for nonnegative matrices, +Linear Algebra Appl. 393 (2004) 365-374. +[21] R. L. Soto, O. Rojo, Applications of a Brauer Theorem in the non- +negative inverse eigenvalue problem, Linear Algebra Appl. 416 (2006) +844-856. +17 + +[22] R. L. Soto, J. Ccapa, Nonnegative matrices with prescribed elementary +divisors, Electronic Journal of Linear Algebra 17 (2008) 287-303. +[23] R. L. Soto, R. C. D´ıaz, H. Nina, M. Salas, Nonnegative matrices with +prescribed spectrum and elementary divisors, Linear Algebra Appl. 439 +(2013) 3591-3604. +[24] H. R. Suleimanova, Stochastic matrices with real characteristic values, +Dokl. Akad. Nauk SSSR. 66 (1949) 343-345. +[25] J. Torre-Mayo, M. R. Abril-Raymundo, E. Alarc´ıa-Est´evez, C. Mariju´an, +M. Pisonero, The nonnegative inverse eigenvalue problem from the coef- +ficients of the characteristic polynomial. EBL digraphs, Linear Algebra +Appl. 426 (2007) 729-773. +18 + diff --git a/MdE3T4oBgHgl3EQfwQuL/content/tmp_files/load_file.txt b/MdE3T4oBgHgl3EQfwQuL/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..df5baf06cb72acd2c59e1daeec01af9d7ed01dda --- /dev/null +++ b/MdE3T4oBgHgl3EQfwQuL/content/tmp_files/load_file.txt @@ -0,0 +1,910 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf,len=909 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='04701v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='SP] 11 Jan 2023 Indices of diagonalizable and universal realizability of spectra∗ Charles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Johnsona, Ana I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Juliob†, Ricardo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Sotob aDepartment of Mathematics, College of William and Mary Williamsburg, VA, USA bDepartamento de Matem´aticas, Universidad Cat´olica del Norte Antofagasta, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Casilla 1280, Antofagasta, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Abstract A list Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} of complex numbers (repeats allowed) is said to be realizable if it is the spectrum of an entrywise nonnegative matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Λ is diagonalizably realizable if the realizing matrix A is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Λ is said to be universally realizable if it is realizable for each possible Jordan canonical form allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Here, we study the connection between diagonalizable realizability and universal real- izability of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In particular, we establish indices of realizability for diagonalizable and universal realizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We also define the merge of two spectra and we prove a result that allow us to easily decide, in many cases, about the universal realizability of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' AMS classification: 15A18, 15A20, 15A29 Key words: Nonnegative matrices, Spectra diagonalizably realizable, Spec- tra universal realizability, Jordan structure, Eigenvalues and eigenvectors ∗Supported by Universidad Cat´olica del Norte-VRIDT 036-2020, NUCLEO UCN VRIDT-083-2020, Chile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' †Corresponding author, crjohn@wm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='edu (Charles R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Johnson), ajulio@ucn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='cl, (Ana I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Julio), rsoto@ucn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='cl (Ricardo L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto) 1 1 Introduction A list Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} of complex numbers (with repeats allowed) is said to be realizable if there is an n-by-n nonnegative matrix A with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In this case A is said to be a realizing matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The problem of characterizing the realizability of lists Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} of complex numbers is called the nonnegative inverse eigenvalue problem (NIEP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We say that Λ is diagonal- izably realizable (DR) if it is the spectrum of a diagonalizable nonnegative matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The problem of determining this kind of realizability is called the diagonalizable realizability problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Λ is universally realizable (UR) if it is realizable for every possible Jordan canonical form (JCF) allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The problem of determining the universal realizability of spectra is called the uni- versal realizability problem (URP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The URP seeks to extend the question of determining the spectral properties allowed by a nonnegative matrix, not only regarding the eigenvalues themselves, but also from the point of view of the corresponding JCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The URP contains the NIEP and both problems are equivalent if the given complex numbers λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn are distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The NIEP has attracted the interest of many linear algebraist researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The URP, on the other hand, is a new and even more difficult problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Both problems remain unsolved for n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' A complete solution, if any, is still far away from the present state of the art.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The URP is studied module the NIEP (see [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' This means that the methods applied to the NIEP are not only useful but also in many cases necessary for the URP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, getting answers for some topics and open questions about the URP means a positive impact on the progress towards a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The NIEP, as we know it today, begins with the works by Sule˘ımanova [24] and Perfect [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The NIEP has been solved only for the cases n = 3 by Loewy and London [13], and for n = 4 by Meehan [14] and also inde- pendently by Torre-Mayo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The first known results on the URP are due to Minc [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In [15] Minc proved that if Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is the spectrum of a diagonalizable positive matrix, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We want to point out here that previously the URP was called nonnegative inverse elementary divisors problem [15], and that in [4] the authors used for the first time the concept and name universal realizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' There are spectra, not positively realizable, that are known to be UR [3, 22, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' For instance, spectra in the left half-plane, that is, Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} with λ1 > 0, Reλi ≤ 0, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, such as real Sule˘ımanova spectra [22] λ1 > 0 > λ2 ≥ · · · ≥ λn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' complex 2 Sule˘ımanova spectra [23] λ1 > 0, λi ∈ {z ∈ C : Rez ≤ 0, |Rez| ≥ |Imz|} , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' and ˇSmigoc spectra [3] λ1 > 0, λi ∈ � z ∈ C : Rez ≤ 0, ��� √ 3Rez ��� ≥ |Imz| � , i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, which contains the real and complex Suleimanova spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' These spectra are realizable if and only if they are UR if and only if �n i=1 λi ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The good behavior of these kind of spectra led to think that any left half-plane list was UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Now, we know that this is not true [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In [12] Laffey and ˇSmigoc proved that a list Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} in the left half-plane is realizable if and only if s1 = n � i=1 λi ≥ 0, s2 = n � i=1 λ2 i ≥ 0, s2 1 ≤ ns2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The positivity and diagonalizability conditions in the result of Minc [15] are essential for his proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Minc says that “it is not known if the the theorem holds for a general nonnegative matrix”, question that has been open for almost 40 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Recently, two extensions for nonnegative realizations have been obtained: the first one by Collao, Salas, and Soto [2] shows that if Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is the spectrum of a diagonalizable nonnegative matrix with constant row sums and a positive row or column, then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The second extension by Johnson, Julio, and Soto [7], shows that if Λ is realiz- able by a diagonalizable ODP matrix, that is, a diagonalizable nonnegative matrix having all off-diagonal entries being positive (zeros on diagonal are permitted), then Λ is also UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Observe that in both extensions, the set of realizing matrices contains the set of positive realizing matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Moreover, the extension in [7] allows to decide about the universal realizability of lists with n� i=1 λi = 0, which it is not possible from the Minc’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' These two extensions open a research line that allow to prove the universal realizability of non-positively realizable spectra, thus significantly extending the class of spectra that can be shown to be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since DR is a necessary condition for UR, all lists of complex numbers that are DR, are natural candidates to be UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 3 Despite the progress that has been made on the URP, there are still nu- merous open questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Two of them, until recently open, were: Is any left half-plane list UR?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In [10] the authors show that a realizable left half-plane list is not necessarily UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In fact, they prove that the spectrum Λ = � a, −a 4 + √ 5a 4 i, −a 4 − √ 5a 4 i, −a 4 + √ 5a 4 i, −a 4 − √ 5a 4 i � a > 0, (1) is realizable, but not DR and therefore not UR The other open question was whether DR realizable implies UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In [9] the authors show that the spectra {λ1, λ1, λ2, λ2} with λ1 > 0 > λ2 ≥ −λ1, λ1 + 2λ2 < 0, have no realiza- tion with a JCF having a Jordan block J2(λ2) of size 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Thus, for instance, Λ = {1, 1, −1, −1} is not UR although it is DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since Λ = {1, 1, −1, −1} has a reducible realization, what may be said about irreducible realizations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='. In [10] it was also shown that irreducible diagonalizable realizations are not necessarily UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, it remains to know under what conditions a DR left half-plane list of complex numbers is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' From extensions in [2, 7] we may say that DR implies UR if Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is the spectrum of a diago- nalizable nonnegative matrix with constant row sums and a positive row or column, or it is diagonalizably ODP realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The importance of the diag- onal JCF lies in the fact that we know how to join Jordan blocks to obtain larger Jordan blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then if we may obtain a diagonalizable realizing ma- trix for Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}, we may also obtain a nonnegative matrix with spectrum Λ for each possible JCF allowed by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' What about criteria which allow to decide about the universal realizability of a list of complex numbers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In this work we also introduce an operation that we name the merge of two spectra and a result which allow to easily decide, in many cases, about the universal realizability of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We also establish indexes of Guo type [5] for diagonalizable and universal realizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' A matrix A = [aij] of order n is said to have constant row sums if all its rows sum to the same constant α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We denote by CSα the set of all n-by-n real matrices with constant row sums equal to α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' It is clear that any matrix in CSα has an eigenvector e T = [1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , 1] corresponding to the eigenvalue α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The relevance of the real matrices with constant row sums is due to the fact that, the problem of finding a nonnegative matrix with spectrum Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}, λ1 being the Perron eigenvalue, is equivalent to the prob- 4 lem of finding a nonnegative matrix in CSλ1, with spectrum Λ (see [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We denote by Ei,j the matrix with 1 in position (i, j) and zeros elsewhere and we define the matrix E(K) = � i∈K Ei,i+1, K ⊂ {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' (2) The paper is organized as follows: In Section 2, we introduce the diag- onalizable realizability index gd(Λ/λ1) and the universal realizability index gu(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, we show that a realizable list Λ = {µ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} of com- plex numbers is DR for all µ ≥ gd(Λ/λ1) and that Λ is UR for all µ ≥ gu(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In Section 3, we define the merge of two spectra and show that if Γ1 = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} and Γ2 = {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm} have a diagonalizable ODP real- ization, then Γ = {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm} has also a diagonalizable ODP realization and therefore Γ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' This result becomes a useful criterion to decide, in many cases, the universal realizability of spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 2 Diagonalizable and universal realizability indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In what follows we will use the following results, Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1, due to Brauer [1], is a perturbation result that shows how to change one single eigenvalue of an n-by-n matrix without changing any of the remaining (n − 1) eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2, by Soto and Ccapa [22], establishes the JCF of the Brauer perturbation A + eq T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3, by ˇSmigoc [20], shows how to construct a matrix C with a particular JCF from two given matrices A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 [1] Brauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let A be an n-by-n matrix with spectrum Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let v T = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , vn] be an eigenvector of A associated with the eigenvalue λk and let q be any n-dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then the matrix A + vq T has eigenvalues λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λk−1, λk + v Tq, λk+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 [22] Soto and Ccapa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let q T = [q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , qn] be an arbitrary n−dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let A ∈ CSλ1 with JCF J(A) = S−1AS = diag {J1(λ1), Jn2(λ2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , Jnk(λk)} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 5 If λ1 + �n i=1 qi ̸= λi, i = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, then the matrix A + eq T has Jordan canonical form J(A) + (�n i=1 qi) E11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In particular, if �n i=1 qi = 0 then A and A + eq T are similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3 [20] ˇSmigoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Suppose B is an m-by-m matrix with JCF that contains at least one 1-by-1 Jordan block corresponding to the eigenvalue c: J(B) = � c 0 0 I(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let t and s, respectively, be the left and the right eigenvectors of B associated with the 1-by-1 Jordan block in the above canonical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Furthermore, we normalize vectors t and s so that t Ts = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let J(A) be a JCF for an n-by-n matrix A = � A1 a b T c � , where A1 is an (n − 1)-by-(n − 1) matrix and a and b are vectors in C n-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then the matrix C = � A1 at T sb T B � has JCF J(C) = � J(A) 0 0 I(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} and Λ/λ1 = {λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} be self-conjugate lists of com- plex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Guo [5] proved that there is a minimal nonnegative number gr(Λ/λ1) such that Λµ = {µ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is realizable for all µ ≥ gr(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Moreover, Guo established that max 2≤j≤n |λj| ≤ gr(Λ/λ1) ≤ 2n max 2≤j≤n |λj| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' (3) In [11] it was shown that the upper bound in (3) may be reduced to (n − 1) max 2≤j≤n |λj| , with n ≥ 5 in the case when λk, k = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, are conjugates complex, and that this bound is sharp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In [19] the authors show how to cal- culate the Guo index gr(Λ/λ1) for circulant nonnegative matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' However, to compute this index becomes a prohibitive task for large n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In this section, we prove that there is also a minimal nonnegative number gd(Λ/λ1), called 6 diagonalizable realizability index, such that Λµ = {µ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is DR if µ ≥ gd(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Of course gd(Λ/λ1) ≥ gr(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Equality occurs if Λ/λ1 has distinct elements and may occur otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The proof is similar to the proof in [11], but it considers all possible cases, which it does not occur in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='4 If Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is realizable, then there is a nonnegative real number gd(Λ/λ1) such that Λµ = {µ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is DR for every µ ≥ gd(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Moreover gr(Λ/λ1) ≤ gd(Λ/λ1) ≤ (n − 1) max 2≤j≤n |λj| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' (4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' First, we exhibit a value µ such that Λµ is DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let A be a realizing matrix for Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Without loss of generality we assume that A ∈ CSλ1, that is, Ae = λ1e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If A is diagonalizable we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If not, we take A = SJS−1, where J is the JCF of A and Se1 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Now let �J be the same as J, except that any superdiagonal 1′s are replaced with 0´s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' So, �J is diagonal with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Define �A = S �JS−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If �A is nonnegative we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If not, since �A ∈ CSλ1 we apply Brauer’s Theorem to produce a nonnegative matrix A′ = �A+eq T, where q T = [q1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , qn] is an appropriate nonnegative vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 A′ is diagonalizable with spectrum � λ1 + n� i=1 qi, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let gd(Λ/λ1) = λ1 + n� i=1 qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Thus we have established the existence of a value gd(Λ/λ1) such that Λµ is DR for every µ ≥ gd(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Next, we show that gd(Λ/λ1) satisfies (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The lower bound is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Although similar to the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 in [11], this is more complete and consider all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In particular, the case in which µj, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, are all conjugate complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' For the sake of completeness and since the result is of independent interest, we set the proof here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let m = max 2≤j≤n |λj| and let µj = λj (n−1)m, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, {µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µn} is a list of complex numbers such that 7 |µj| ≤ 1 n−1, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Consider the initial matrix B = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 · · · · · · · · · · 0 −µ2 µ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' −µp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' µp 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' −xs ys .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' xs −ys .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' −xs −ys ys xs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 0 −xt yt .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' xt −yt −xt −yt · · · · · · · · 0 yt xt \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , where µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µp are real, xj = Reµj, yj = Imµj, p + 1 ≤ j ≤ n+p 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then B ∈ CS0 has eigenvalues 0, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µp, µp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µn and is clear that B is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If Reµj ≤ 0, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, then all entries in the first column of B are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let q T = � 0, 1 n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , 1 n − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2, A′ = B + eq T is diagonaliz- able nonnegative with spectrum {1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µn} and A = (n − 1)mA′ is diagonalizable nonnegative with spectrum {(n − 1)m, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If Reµj > 0 for some j, 3 ≤ j ≤ n, then all the entries in the j st column of B (or in the (j − 1) st column of B if j corresponds to the second column in the corresponding 2-by-2 complex block) are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let q T = � 1 n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , 1 n − 1, 0, 1 n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , 1 n − 1 � , with zero in the j st position ((j − 1) st position).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, again, A′ = B + eq T is diagonalizable nonnegative with spectrum {1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µn} and A = (n − 1)mA′ is diagonalizable nonnegative with spectrum 8 {(n − 1)m, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Observe that, from the necessary and suffi- cient conditions by Loewy and London [13], the result still holds for the special case n = 3 with Λ = {λ1, a + bi, a − bi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If µ2 > 0 with Reµj < 0, j = 3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, then we write the −Reµ′ js, 3 ≤ j ≤ n, along the second column of B and the ±Imµ′ js, p + 1 ≤ j ≤ n, along the first column of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then again with q T = � 1 n − 1, 0, 1 n − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , 1 n − 1 � , we obtain, as before, the diagonalizable nonnegative matrix A = (n − 1)mA′ with the required spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Now we consider the case in which µj, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, are all conjugate complex numbers, that is, Imµj ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let the 2-by-2 diagonal blocks � Reµj −Imµj Imµj Reµj � , j = 2k, k = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n − 1 2 in the matrix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In this case we only can use the first column of B to set −(Reµj + Imµj) in order that B ∈ CS0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Now we have the following cases: – If Reµj ≥ 0 for one or more indexes j, then the corresponding j st columns in B are nonnegative and the proof follows as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If |Reµj + Imµj| > 1 n−1 for some j, then we set the corresponding diagonal block � Reµj −Imµj Imµj Reµj � on the last 2-by-2 diagonal position in the matrix B to distribute any possible negative amount (from the reciprocal of Reµj +Imµj) through the last row under some appropriate columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Observe that there is at least one negative amount (−Imµj) in each above 2-by-2 block in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Thus, A′ is nonnegative with spectrum � n� i=1 qi, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µn � , n� i=1 qi ≤ 1 and A = (n − 1)mA′ is nonnegative with spectrum 9 � (n − 1)m n� i=1 qi, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn � , where (n − 1)m n� i=1 qi ≤ (n − 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In fact, for n = 5 (the minimum case) with Reµi ≥ 0, we have \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 −Reµi + Imµi Reµi −Imµi 0 0 −Reµi − Imµi Imµi Reµi 0 0 −Reµj + Imµj 0 0 Reµj −Imµj −Reµj − Imµj + Imµi 0 −Imµi Imµj Reµj \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb Then qT = [Reµj + Imµj, 0, Imµi, 0, Imµj] and 5� i=1 qi = Reµj+2Imµj+ Imµi ≤ 4 n−1 ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' As it was said above, the validity of the case n = 3 is guaranteed from the necessary and sufficient conditions by Loewy and London [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' – If Reµj < 0 with |Reµj| ≥ |Imµj| , j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, then the first column in B is nonnegative and the proof follows as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' – If Reµj < 0 with |Reµj| < |Imµj|, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, then |Reµj + Imµj| < | Imµj| < |µj| ≤ 1 n − 1, and the proof follows as before again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The upper bound in (4) is also sharp for gd(Λ/λ1), as it is shown by a diag- onalizable realization of the list Λ = {(n − 1), −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , −1 � �� � (n−1)times }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The spectrum Λ = ∪ n 2 i=1{λi, −λi}, with real λi, shows that the lower bound in (4) is also sharp for gd(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We may also define, similar to gr(Λ/λ1) and gd(Λ/λ1), a universal realizability index gu(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, the following result is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 From Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='4 we have that gd(Λ/λ1) ≤ gu(Λ/λ1) ≤ (n − 1) max 2≤j≤n |λj| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' (5) 10 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The lower bound is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' To prove the upper bound we consider the matrix A = (n − 1)mA′, with A′ = (B + eqT) ∈ CS1, in the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='4, which is diagonalizable nonnegative with JCF J(A) = S−1AS, where Se1 = e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, for an appropriate matrix E(K) as defined in (2), J(A) + E(K) = S−1AS + E(K) = S−1(A + SE(K)S−1)S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' (6) Then we may reach any JCF allowed by the spectrum of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In fact, A + SE(K)S−1 has the desired JCF, J(A) + E(K), although it is no necessar- ily nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since SE(K)S−1 ∈ CS0, then for a convenient nonneg- ative vector r T = [r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , rn] and ǫ > 0 small enough, the matrix M = (A + ǫSE(K)S−1)+ er T is nonnegative with spectrum {(n−1)m, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} or spectrum {(n − 1)m n� i=1 qi, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}, with n� i=1 qi ≤ 1 in the case µj, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, are conjugate complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, since (n − 1)m n� i=1 qi ≤ (n − 1)m, Λ is UR and gu(Λ/λ1) ≤ (n − 1)max |λj| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 2≤j≤n The following example illustrates Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='4 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In par- ticular the case where µj, j = 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n, are all conjugates complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 Consider the list {0,1 + i √ 15, 1 − i √ 15, − √ 15 + i, − √ 15 − i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then n = 5, m = max 2≤j≤5 |λj| = 4 and (n−1)m = 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In order to have 5� i=1 qi ≤ 1 we need to take the initial matrix B as B = 1 16 \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 0 0 0 0 1 + √ 15 − √ 15 −1 0 0 −1 + √ 15 1 − √ 15 0 0 −1 + √ 15 0 0 1 − √ 15 0 −1 − √ 15 √ 15 1 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, A′ = B + eqT with qT = 1 16 � 1, √ 15, √ 15, 0, √ 15 � 11 has the spectrum 1 16{ 5 � i=1 qi, 1 + i √ 15, 1 − i √ 15, − √ 15 + i, − √ 15 − i}, with 5� i=1 qi ≤ 1 and (n − 1)m 5� i=1 qi ≤ (n − 1)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since the list Λ = {(n − 1), −1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , −1} is UR, then the upper bound in (4) is also sharp for gu(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Next, we have: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 Let Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} be a DR list of complex numbers and let gd(Λ/λ1) be the diagonalizable realizability index of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then for every µ > gd(Λ/λ1), Λµ = {µ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The result is a straightforward consequence of the Minc’s result, and the fact that if the Perron eigenvalue is increased for any ǫ > 0, then any diagonalizably realizable list becomes a diagonalizably positively realizable list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 We have seen that gd(Λ/λ1) ≥ gr(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Of course, if a re- alizable list Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} of complex numbers has all its elements dis- tinct, then gd(Λ/λ1) = gr(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In order that gd(Λ/λ1) > gr(Λ/λ1), Λ must admit repeats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' This is necessary, but not sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' For instance, the list Λ = {6, 1, 1, −4, −4} is DR (see [21, Lemma 1]), but gd(Λ/λ1) = gr(Λ/λ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' It is clear that gd(Λ/λ1) > gr(Λ/λ1) if and only if Λ is not DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 The list Λ = {7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' −4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' −6},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='is symmetrically realizable by ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='A = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='\uf8ee ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='\uf8f9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 12 Then, since A is diagonalizable ODP, Λ is UR and gr(Λ/λ1) = gd(Λ/λ1) = gu(Λ/λ1) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' A realizable list Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} of complex numbers is said to be Perron extreme, if the list Λǫ = {λ1 − ǫ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} is not realizable for every ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If Λ is not Perron extreme, there is an ǫ > 0 such that Λǫ is realizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then we have the following result, which is equivalent to Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 Let Λ = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} be a list not Perron extreme of complex numbers with Λǫ = {λ1 − ǫ, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn}, ǫ > 0, being DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Let B ∈ CSλ1−ǫ be a diagonalizable nonnegative matrix with spec- trum Λǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then A = B + eq T, with q T = � ǫ n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , ǫ n � is positive with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Moreover, from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2, A is diagonalizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Hence, Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3 The list Λ = {8, 2, 2, −3, −4, −4} is not Perron extreme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since Λ′ = {7, 2, 2, −3, −4, −4} is DR by B = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 4 0 2 0 1 4 0 0 2 0 1 0 2 0 4 0 1 0 2 4 0 0 1 0 2 0 2 0 3 0 2 0 2 3 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb , then A = B + eq T, where q T = � 1 6 1 6 1 6 1 6 1 6 1 6 � is diagonalizable posi- tive with spectrum Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Therefore Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2 We know that DR does not necessarily imply UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' We also know of two important extensions [2, 7] of Minc’s result [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' As far as we know, these extensions are the most general universal realizability criteria for a solution to URP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, for Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} diagonalizably realizable we may say that DR implies UR if: i) The realizing matrix for Λ is ODP (positive matrices are ODP), or ii) Λ is the spectrum of a nonnegative matrix A ∈ CSλ1 with a positive row or column.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 13 There are spectra, however, which are UR, without to have necessarily a re- alizing matrix of some above classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' For instance: iii) Λ = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λ n 2 , −λ n 2 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , −λ2, −λ1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Is clear that Λ has a diago- nalizable realization and if it has 2 × 2 blocks repeated, we may obtain any coarser JCF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' iv) Λ = {λ1, λ2, −λ3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , −λn} with λj > 0, j = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' λ1 > λ2, λ1 + λ2 − n� j=3 λj = 0, It was proved in [4] that Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 3 The merge of two spectra In this section we define the merge of two spectra in the following way: Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 Let Γ1 = {λ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} and Γ2 = {µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm} be lists of complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' The merge Γ1 with Γ2 is Γ = {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, we have: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 Suppose that A and B are n-by-n and m-by-m diagonalizable ODP matrices with spectrum Γ1 = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} and Γ2 = {µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm}, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, there is an (n + m − 1)-by-(n + m − 1) diagonalizable ODP matrix C with spectrum Γ = {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Hence, Γ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since A is diagonalizable ODP with spectrum Γ1, it is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, there is a positive vector v T = [v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , vn] such that Av = λ1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Thus, if D = diag{v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , vn}, then ˜A = D−1AD ∈ CSλ1 is diagonalizable ODP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' If d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , dn are the diagonal entries of ˜A, then from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1, A1 = ˜A+e[0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µ1] = �A11 a b T dn + µ1 � has spectrum {λ1 +µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn} and diagonal entries d1, d2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , dn + µ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' It is clear, from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='2, that A1 is a diagonalizable ODP matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since B is diagonalizable ODP with spectrum Γ2 then, as before, there is a diagonalizable ODP matrix ˜B ∈ CSµ1 with spectrum Γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Then, the ma- trix B1 = ˜B + e[dn, 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , 0] is diagonalizable ODP with spectrum {µ1 + 14 dn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Finally, by applying the ˇSmigoc’s glue technique, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='3, there is a matrix C = � A11 at T sb T B1 � with A11 the (n − 1)-by-(n − 1) matrix in A1 and with s, t such that t Ts = 1, being the right and left eigenvectors of B1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Now, C has the spectrum {λ1 + µ1, λ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , λn, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' , µm}, with Jordan canonical form J(C) = J(A1) ⊕ I(B1) with J(B1) = � µ1 + dn I(B1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Since a, b, s and t are positive vectors, it is clear that C is a diagonalizable ODP matrix and therefore Γ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In many cases, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 may be a useful tool to decide about the universal realizability of a list of complex numbers, as the following example shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 Is Λ = {13, 1, 1, −3, −4, −4, 1 + 3i, 1 − 3i}, universally realizable?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' To answer the question, consider the lists Λ1 = {7, −3, 1 + 3i, 1 − 3i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Λ2 = {6, 1, 1, −4, −4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' From [8], Λ1 has the normal nonnegative realization A1 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8f0 3 2 − √ 3 √ 6+2 √ 2 2 √ 6+2 √ 2 2 2 + √ 3 3 2 √ 2− √ 6 2 2 √ 2− √ 6 2 2 √ 2− √ 6 2 √ 6+2 √ 2 2 0 3 2 √ 2− √ 6 2 √ 6+2 √ 2 2 3 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fb , which is diagonalizable ODP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' From [21, Lemma 1], Λ2 has the symmetric ODP realization A2 = \uf8ee \uf8ef\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 0 3+ √ 5 2 3− √ 5 2 3− √ 5 2 3+ √ 5 2 3+ √ 5 2 0 3+ √ 5 2 3− √ 5 2 3− √ 5 2 3− √ 5 2 3+ √ 5 2 0 3+ √ 5 2 3− √ 5 2 3− √ 5 2 3− √ 5 2 3+ √ 5 2 0 3+ √ 5 2 3+ √ 5 2 3− √ 5 2 3− √ 5 2 3+ √ 5 2 0 \uf8f9 \uf8fa\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 15 Then, from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1, the merge Λ1 with Λ2, that is, Λ = {13, 1, 1, −3, −4, −4, 1 + 3i, 1 − 3i} has a diagonalizable ODP realization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Hence, Λ is UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1 It is known that lists of Sule˘ımanova type, real or complex, are UR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Now, this result can be also proved via Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='1, in the case of real Sule˘ımanova spectra, and of complex Sule˘ımanova spectra with |Reλi| > |Imλi| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' In fact, in both cases, we may obtain diagonalizable ODP realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Declaration of Competing Interest There is no competing interest References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Brauer, Limits for the characteristic roots of a matrix IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Applica- tions to stochastic matrices, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 19 (1952) 75-91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Collao, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Salas, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Spectra universally realizable by doubly stochastic matrices, Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Matrices 6 (2018) 301-309.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [3] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' D´ıaz, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Nonnegative inverse elementary divisors problem in the left half plane, Linear and Multilinear Algebra 64 (2016) 258-268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Collao, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Johnson, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Universal realizability of spectra with two positive eigenvalues, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 545 (2018) 226-239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [5] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Guo, Eigenvalues of nonnegative matrices, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 266 (1997) 261-270.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [6] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Johnson, Row stochastic matrices similar to doubly stochastic matrices, Linear and Multilinear Algebra 10 (1981) 113-130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [7] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Johnson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Julio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Nonnegative realizability with Jordan structure, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 587 (2020) 302-313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [8] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Julio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Manzaneda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Normal nonnegative realization of spectra, Linear and Multilinear Algebra 63 (2015) 1204-1215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 16 [9] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Julio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Mariju´an, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Pisonero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, On universal realiz- ability of spectra, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 563 (2019) 353-372.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Julio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Mariju´an, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Pisonero, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Universal realizability in low dimension, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 619 (2021) 107-136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [11] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Julio, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, The role of certain Brauer and Rado results in the nonnegative inverse sepectral problems, Electronic J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Linear Algebra 36 (2020) 484-502.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [12] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Laffey, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' ˇSmigoc, Nonnegative realization of spectra having neg- ative real parts, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 416 (2006) 148-159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Loewy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' London, A note on the inverse problem for nonnegative matrices, Linear and Multilinear Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 6 (1978) 83-90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [14] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Meehan, Some results on matrix spectra, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' thesis, National University of Ireland, Dublin, (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Minc, Inverse elementary divisor problem for nonnegative matrices, Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' of the Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Society 83 (4) (1981) 665-669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Minc, Inverse elementary divisor problem for doubly stochastic ma- trices, Linear and Multilinear Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 11 (1982) 121-131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Perfect, Methods of constructing certain stochastic matrices, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 20 (1953) 395-404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Perfect, Methods of constructing certain stochastic matrices II, Duke Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 22 (1955) 305-311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [19] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Rojo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, Existence and construction of nonnegative matrices with complex spectrum, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 368 (2003) 53-69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [20] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' ˇSmigoc, The inverse eigenvalue problem for nonnegative matrices, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 393 (2004) 365-374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [21] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Rojo, Applications of a Brauer Theorem in the non- negative inverse eigenvalue problem, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 416 (2006) 844-856.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 17 [22] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Ccapa, Nonnegative matrices with prescribed elementary divisors, Electronic Journal of Linear Algebra 17 (2008) 287-303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Soto, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' D´ıaz, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Nina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Salas, Nonnegative matrices with prescribed spectrum and elementary divisors, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 439 (2013) 3591-3604.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Suleimanova, Stochastic matrices with real characteristic values, Dokl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Akad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Nauk SSSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 66 (1949) 343-345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' [25] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Torre-Mayo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Abril-Raymundo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Alarc´ıa-Est´evez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Mariju´an, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' Pisonero, The nonnegative inverse eigenvalue problem from the coef- ficients of the characteristic polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' EBL digraphs, Linear Algebra Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 426 (2007) 729-773.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} +page_content=' 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MdE3T4oBgHgl3EQfwQuL/content/2301.04701v1.pdf'} diff --git a/MtE1T4oBgHgl3EQftQWC/content/tmp_files/2301.03375v1.pdf.txt b/MtE1T4oBgHgl3EQftQWC/content/tmp_files/2301.03375v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e628c7f8f506dea0c22b19d24d89ef87c2e94b7e --- /dev/null +++ b/MtE1T4oBgHgl3EQftQWC/content/tmp_files/2301.03375v1.pdf.txt @@ -0,0 +1,1137 @@ +One-Shot Achievable Secrecy Rate Regions for +Quantum Interference Wiretap Channel +Hadi Aghaee +Faculty of Electrical Engineering +K. N. Toosi University of Technology +Tehran, Iran +Email: Aghaee_Hadi@email.kntu.ac.ir + +Bahareh Akhbari +Faculty of Electrical Engineering +K. N. Toosi University of Technology +Tehran, Iran +Email: akhbari@eetd.kntu.ac.ir + +Abstract—In this paper, we want to derive achievable secrecy +rate regions for quantum interference channel with classical +inputs under one-shot setting. The main idea to this end is to use +the combination of superposition and rate splitting for encoding +scheme and constructing a decoding scheme based on +simultaneous decoding. +Keywords—Quantum Channel; Mutual Information; Secrecy +Capacity; Multiple Access Channel +I. INTRODUCTION +The physical layer security was introduced by Shannon for +the first time [1]. After that, the wiretap channel was presented +by Wyner, in which a sender transmits its message to a +legitimate receiver in the presence of a passive eavesdropper +[2]. Moreover, Csiszár and Körner introduced the broadcast +channel with confidential messages [3]. +However, the physical layer security problems have been +extended to multi-terminal channels like multiple access +channels (MACs), Interference channels (ICs), relay channels, +etc., due to their importance and their usage in practical systems +[4-10]. +In recent decades, with development in quantum data +processing and its applications, a significant effort has begun to +use the natural features of quantum mechanics to improve +communication. Some of these features are as follows: +entanglement, uncertainty, no-cloning theorem, superposition, +etc. [11]. These natural features help the communication to be +faster and more secure. +Moreover, the security problem plays a critical role in +quantum communication and devotes a considerable part of the +research to itself. In this regard, the quantum wiretap channel +(QWTC) was firstly introduced in [12] and [13]. +Then, secrecy constraints are extended to multi-user +quantum channels such as quantum interference channel (QIC) +[14] and quantum multiple access channel (QMAC) [15-18]. +The interference phenomenon is one of the major problems in +communication systems. +In this paper, we derive some achievable secrecy rate +regions for quantum interference channel with classical inputs. +One of the major open problems in the quantum information +theory is related to the simultaneous decoder for quantum +channels with three or more senders (i.e., jointly typical +decoder). However, this problem has been solved for some +cases, such as the min-entropy case and the case of the quantum +multiple access channels (QMACs), in which the output +systems are commutative [19]. Therefore, in the independent +and identical distributed (i.i.d.) case, we have to use successive +decoding combined with time-sharing. In contrast, for the one- +shot case, we have to use the simultaneous decoder. Sen proved +a joint typicality lemma which is helpful to decode any number +of messages simultaneously in the one-shot case [19]. +In this paper, we want to study secure communication over +a classical-quantum interference wiretap channel (C-QI-WTC) +under the one-shot setting. Up to the best knowledge, it is the +first time that this channel is studied. Even in the classical case, +the security problem of interference channel has been +investigated under a different scenario called interference +channel with confidential messages. Also, another feature of +our problem is that the channel is considered under the one-shot +setting. This choice is due to the fact that there is not a proven +joint typicality lemma in the asymptotic i.i.d. case for general +quantum channels (i.e., quantum channels with any number of +senders). Therefore, all of the obtained results are new, and the +proposed strategies in the paper can be applied to the classical +interference channel. +The paper is organized as follows: In Section II, some +seminal definitions are presented. In Section III, the main +channel and information processing tasks are presented. In +Section IV, the results and main theorems are presented. +Section V is dedicated to discussion and future works. +II. PRELIMINARIES +Let A (Alice) and B (Bob) be two quantum systems. These +quantum systems can be denoted by their corresponding +Hilbert spaces as ℋ�, ℋ�. The states of the above quantum +systems are presented as density operators �� and ��, +respectively, while the shared state between Alice and Bob is +denoted by ���. A density operator is a positive semidefinite +operator with a unit trace. Alice or Bob’s state can be defined +by a partial trace operator over the shared state. The partial trace +is used to model the lack of access to a quantum system. Thus, +Alice’s density operator using partial trace is �� = ���{���}, +and Bob’s density operator is �� = ���{���}. We use |�⟩� to +denote the pure state of system A. The corresponding density +operator is �� = |�⟩⟨�|�. The von Neumann entropy of the +state �� is defined by �(�)� = −��{�� log ��}. For an +arbitrarily state such as ���, the quantum conditional entropy +is defined by �(�|�)� = �(�, �)� − �(�)�. The quantum +mutual information is defined by �(�; �)� = �(�)� + +�(�)� − �(�, �)�, and the conditional quantum mutual +information is defined by: + +�(�; �|�)� = �(�|�)� + �(�|�)� − �(�, �|�)� +Quantum operations can be denoted by completely positive +trace-preserving (CPTP) maps ��→�. The CPTP maps accept +input states in A and output states in B. The distance between +two quantum states, such as A and B is defined by trace +distance. The trace distance between two arbitrarily states such +as � and � is: +‖� − �‖� = ��|� − �| +(1) +where |Ψ| = √Ψ�Ψ. This quantity is zero for two similar and +perfectly distinguishable states. +Fidelity is defined as �(�, �) = ���√��� +�, and purified +distance is a metric on �(ℋ) and is defined as �(�, �) ≔ +�1 − �(�, �)�. Most of the above definitions are given from +[20]. +Definition 1: (Hypothesis testing mutual information) +Hypothesis testing mutual information is denoted by �� +�(�; �) +∶= �� +� (���‖�� ⊗ ��), � ∈ (0,1) and is considered as quantum +hypothesis testing divergence [21] where �� +� (. ‖.) is hypothesis +testing relative entropy [21]. �ℋ�ℋ� is the joint state of input +and output over their Hilbert spaces (ℋ�, ℋ�), and it can be +shown as ���: +��� = � ��(�)|�⟩⟨�|� ⊗ �� +� +� + +where �� is the input distribution. +Definition 2: (Quantum relative entropy [22]): Consider +states ��, �� ∈ �(ℋ�). The Quantum relative entropy is +defined as: +�(��‖��) +≔ ���{���log� �� − log� ���} +����(��) ⊆ ����(��) ++∞ +��ℎ������ + +where ����(��) refers to the set-theoretic support of �. +����(�) is the subspace of ℋ spanned by all eigenvectors of � +with non-zero eigenvalues. +Fact: The following relation exists between the quantum +relative entropy and hypothesis testing relative entropy for � ∈ +(0,1) [21]: +�� +�(��‖��) ≤ +1 +1 − � ��(��‖��) + ℎ�(�)� +where ℎ�(�) ≔ −� log� � − (1 − �) log�(1 − �) is a binary +entropy function. +Definition 3: (Max mutual information [23]) Consider a +bipartite state ��� and a parameter � ∈ (0,1). The max mutual +information can be defined as follows: +����(�; �)� ≔ ����(��� ‖��⨂�� )� +where � refers to the state ��� and ����(∣∣) is the max-relative +entropy [24] for ��, �� ∈ ℋ�: +����(�� ‖��) ≔ inf{� ∈ ℝ: �� ≤ 2���} +Definition 4: (Quantum smooth max relative entropy [24]) +Consider states ��, �� ∈ �(ℋ�) and � ∈ (0,1). The quantum +smooth max relative entropy is defined as: +���� +� +(��‖��) ∶= +inf +�� +� ∈ℬ�(��) ����(�� +� ‖�� ) +where ℬ�(��) ≔ {�� +� ∈ �(ℋ�): �(�� +� , ��) ≤ �} is �-ball for +���. +Definition 5: (Quantum smooth max mutual information +[23]) Consider ��� ∶= ∑ +��(�)|�⟩⟨�|� ⊗ +�∈� +�� +� as a classical- +quantum state and a parameter � ∈ (0,1). The smooth max +mutual information between the systems � and � can be defined +as follows: +���� +� +(�; �) ∶= +inf +��� +� +∈ℬ�(���) ����(��� +� ‖��⨂�� ) += +inf +��� +� +∈ℬ�(���)����(�; �)�� , +where ℬ�(���) ≔ {��� +� +∈ �(ℋ� ⊗ ℋ�): �(��� +� , ���) ≤ �} is +�-ball for ���. +Definition 6: (Conditional smooth hypothesis testing +mutual +information +[25]) +Consider +���� +∶= ∑ +��(�)|�⟩⟨�|� +�∈� +⊗ ��� +� be a tripartite classical-quantum +state and � ∈ (0,1). We define, +�� +�(�; �|�)� ≔ max +�� +min +�∈������� +� � �� +�(�; �)��� +� , +where maximization is over all �� +� = ∑ +��(�)|�⟩⟨�|� +�∈� + +satisfying �(�� +� , ��) ≤ �. +Definition 7: (Conditional smooth max mutual information +[25]) +Consider +���� ∶= ∑ +��(�)|�⟩⟨�|� +�∈� +⊗ ��� +� +be +a +tripartite classical-quantum state and � ∈ (0,1). We define, +���� +� +(�; �|�)� ≔ max +�� +min +�∈������� +� � ���� +� +(�; �)��� +� , +where maximization is over all �� +� = ∑ +��(�)|�⟩⟨�|� +�∈� + +satisfying �(�� +� , ��) ≤ �. +Definition 8: (Quantum Rényi relative entropy of order � +[21]) For a state � ∈ �(ℋ) and a positive semidefinite operator +�, the quantum Rényi relative entropy of order �, where � ∈ +�0,1) ∪ (1, +∞) is defined as: +��(�‖�) ≡ +1 +� − 1 log� ��{������} +Also, Rényi entropy of order � can be defined as follows: +��(�)� ≡ +1 +1 − � log� ��{�� +�} +Definition 9: (One-shot inner bound of a classical-quantum +multiple access channel) [19] A two user C-QMAC under the +one-shot setting is a triple (�� × ��, �����→�(��, ��) ≡ +�� +����, ℋ�), where �� and �� are the alphabet sets of two +classical inputs, and � is the output system. ����� +� + is a quantum +state, and the channel has a completely positive trace- +preserving map (CPTP) �����→�. + + +Figure 1. The C-QI-WTC model +Considering the joint typicality lemma introduced in +[Corollary 4, 19], the one-shot inner bound of a C-QMAC is as +follows: +�� ≤ �� +�(��: ���)� − 2 + log � +�� ≤ �� +�(��: ���)� − 2 + log � +�� + �� ≤ �� +�(����: �)� − 2 + log � +where �� +�(. ) is the hypothesis testing mutual information +defined in Definition 1 with respect to the controlling state: +������� ∶= � �(�)�(��|�)�(��|�)|�����⟩⟨�����|����� +����� +⊗ �� +���� +and � is a time-sharing variable. +Note that �� +�(: ) is the difference between a Rényi entropy +of order two and a conditional quantum entropy. +III. CHANNEL MODEL +A +two-user +C-QI-WTC +is +a +triple +(�� × +��, �����→�����(��, ��) ≡ ����� +�����, ℋ�� ⊗ ℋ�� ⊗ ℋ�), +where ��, � ∈ {1,2} denote the input alphabet sets, and ��, ��, +� denote the output systems (��, �� denote the channel outputs +at the two legitimate receivers and � is the channel outputs at +the eavesdropper). ����� +����� is the system output’s quantum state. +Each user wants to transmit its message as secure as possible +over a C-QI-WTC to its intended receiver. +The main channel (i.i.d. case) is illustrated in Figure 1. +Consider the main channel illustrated in Figure 1 under the +one-shot setting. Each user chooses its message ��; � ∈ {1,2} +from its message set ℳ� = �1: |ℳ�| = 2���; � ∈ {1,2}, and +send it over a C-QI-WTC. The users also use two junk +variables ��; � ∈ {1,2} from two amplification sets �� = +�1: |��| = 2����; � ∈ {1,2} for randomizing Eve’s knowledge. +We have two doubly indexed codebooks ��(��, ��) and +��(��, ��) for user-1 and user-2, respectively. The above +channel can be divided into two sub C-QMA-WTCs (one from +both users to (��, �) and another from both users to (��, �)). +IV. MAIN RESULTS +In this section, we present the main results. +Theorem 1: (One-shot achievable rate region for C-QI- +WTC) Consider a two-user C-QI-WTC which accepts �� and +�� as inputs and ��, �� and � as outputs. ����� +����� is the channel +density operator. For any fixed � ∈ (0,1), �� ∈ (0, ��) and �, �� +such that �, �� > 0, the rate pair �� = ���|ℳ�| + �, � ∈ {1,2} +is achievable to satisfy the following inequalities: +�� ≤ min��� +�(��: ����|�)�, �� +�(��: ����|�)�� +− ���� +� +(��: �|�)� + log � − 1 − log 3 +��� ++ 1 +4 log � +�� ≤ min��� +�(��: ����|�)�, �� +�(��: ����|�)�� +− ���� +� +(��: ���|�)� + log � − 1 − log 3 +��� ++ 1 +4 log � +�� + �� ≤ min��� +�(����: ��|�)�, �� +�(����: ��|�)�� +− ���� +� +(��: �|�)� − ���� +� +(��: ���|�)� ++ log � − 1 − 2 log 3 +��� + 1 +2 log � + �(1) +where � = �� − �� and the union is taken over input distribution +��(�)���|�(��|�)���|�(��|�). Q is the time-sharing random +variable, and all of the mutual information quantities are taken +with respect to the following state: + +����������� ≡ +� ���(�)���|�(��|�)���|�(��|�)|�⟩⟨�|� +�,��,�� +⊗ |��⟩⟨��|�� ⊗ |��⟩⟨��|�� +⊗ ����� +����� + + + + + +(4) +Proof: See Appendix A. +Sketch of proof: The channel can be split into two sub-QMA- +WTCs with classical inputs. One from (��, ��) to (��, �) and +another from (��, ��) to (��, �). Using the proposed method by +El-Gamal and H. Kim [26] helps to prove this theorem. +Theorem 1 gives the simplest achievable rate region for C- +QI-WTC under the one-shot setting. Without considering the +secrecy constraints, Han and Kobayashi obtained the best +achievable rate region for interference channel (i.i.d. setting) +using rate splitting that the messages are split into common and +personal messages. This technique is extended to the quantum +case with some limits [14]. Using the Han-Kobayashi’s +technique, the message �� is split into ��� (common part) and +��� (personal part), where � ∈ {1,2}. +The structure of the C-QI-WTC under the Han-Kobayashi’s +setting is illustrated in Figure 2. The following channel can be +divided into two separate sub 3-user C-QMA-WTCs: one from +(���, ���, ���) to (��, �) and another from (���, ���, ���) to +(��, �). +As mentioned before, there is not a proven quantum +simultaneous decoder for decoding three or more messages in +general and it remains a conjecture (except some cases such as +the commutative version of output states and min-entropy cases +[14]). + +wiretappel +→>X2 +→(M1,M2)Y1 Y2Z +x1x2m1 +>Mi +X1 +C-QI-WTC +Y2Y����� +� +(������������: �)� ≤ ������� +� +(���������: �)� ≤ ��, ���� +� +(���������: �)� ≤ ��� +where ��, �� and �� are arbitrary small numbers. +(5) +�� ≤ �� +�(������: �����)� − ���� +�����(���: �)� − ���� +�����(���: �������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 + �(1) +(6) +�� ≤ �� +�(���: ��������)� + �� +�(���: ��������)� − ���� +�����(���: �)� − ���� +�����(���: �������)� − 2 log 3 +��� + 1 +2 log �� ++ 2 log � − 4 + �(1) +(7) +�� ≤ �� +�(������: �����)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 ++ �(1) +(8) +�� ≤ �� +�(���: ��������)� + �� +�(���: ��������)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 2 log 3 +��� ++ 1 +2 log �� + 2 log � − 4 + �(1) +(9) +�� + �� ≤ �� +�(���: ��������)� + �� +�(���������: ��)� − ���� +�����(���: �)� − ���� +�����(���: �������)� +− ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 4 log 3 +��� + log �� + 2 log � − 4 + �(1) +(10) +�� + �� ≤ �� +�(���: ��������)� + �� +�(���������: ��)� − ���� +�����(���: �)� − ���� +�����(���: �������)� +− ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 4 log 3 +��� + log �� + 2 log � − 4 + �(1) +(11) +�� + �� ≤ �� +�(������: �����)� + �� +�(������: �����)� − ���� +�����(���: �)� − ���� +�����(���: �������)� +− ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 4 log 3 +��� + log �� + 2 log � − 4 + �(1) +(12) +2�� + �� ≤ �� +�(���: ��������)� + �� +�(������: �����)� + �� +�(���������: ��)� − 2���� +�����(���: �)� +− 2���� +�����(���: �������)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 6 log 3 +��� ++ 3 +2 log �� + 3 log � − 6 + �(1) +(13) +�� + 2�� ≤ �� +�(������: �����)� + �� +�(���: ��������)� + �� +�(���������: ��)� − ���� +�����(���: �)� +− ���� +�����(���: �������)� − 2���� +�����(���: ����)� − 2���� +�����(���: ����������)� − 6 log 3 +��� ++ 3 +2 log �� + 3 log � − 6 + �(1) +(14) + + +Figure 2. The structure of the C-QI-WTC under the Han-Kobayashi +settings. +Remark 1: Note that, to take the intersection of the private +regions for two 3-sender MACs raised in Theorem 1, we used +the method of [26]. Another approach can be using Fȕrier- +Motzkin elimination [Appendix D, 26] which gives achievable +rate region similar to the Han-Kobayashi expression. +Remark 2: The Han-Kobayashi technique is based on rate +splitting. It should be noted that the split messages are not +independent of each other. Thus, obtaining secrecy against the +eavesdropper by Wyner's randomizing technique becomes +problematic in this setting. In other words, we cannot +randomize over a block independently. For example, �� should +be randomized using the product of two junk variables +(���. ���). +Conjecture: (An inner bound on the one-shot secrecy +capacity region of the C-QI-WTC) Consider the region: +ℛ(�) = �{(��, ��) ∈ ��|����. (6) − (14) ℎ���} +� + +Proof: In Appendix B. +Sketch of proof: We consider two sub C-QMA-WTCs. +Therefore, from the perspective of the first receiver (��), there + +(m20,m22102 +C1X2 +m2 +m10.m11Y2mare three messages (���, ���, ���) → (��, �) , and for the +second receiver, there are three messages (���, ���, ���) → +(��, �). The paper [27] introduces the same setting, but it +considers a randomized order such as ��� → ��� → ���. For +the first C-QMA-WTC, Alice should randomize over a total +block of size (���. ���). For the second C-QMA-WTC, Bob +should randomize over a total block of size (���. ���). Then, +we can analyze both sub-channels. +Remark 3: The above conjecture holds if and only if +condition (5) holds. Because taking the intersection of the +private regions for two 3-sender C-QMACs is not enough to get +a private region for the full C-QI-WTC. +To overcome the above problem, we should change the +encoding process, which results in the following theorem. +Theorem 2: (An inner bound on the one-shot secrecy +capacity region of the C-QI-WTC) Consider the region: +ℛ(�) = �{(��, ��) ∈ ��|����. (15) − (28) ℎ���} +� + +Proof: In Appendix C. +Sketch of proof: The overall sketch of the proof is the same as +that for the above Conjecture with one difference: Suppose that +both receivers want to decode non-interfering messages. Also, +this setting is similar to Theorem 1. It can be helpful for the +receivers to decode their messages, including the intended +messages and interfering messages. In other words, ��� and +��� can be used as side information. Therefore, the first sub- +channel can be modeled as (��������) → (��, �). All steps, +such as encoding and decoding, are the same as for the above +Conjecture. +Secrecy criterion: The secrecy criterion for the channel can +be defined as follows: +�(��, ��: �) ≤ � +For Theorem 1 +�(���, ���, ���, ���: �) ≤ � +For Conjecture and Theorem 2 +This means that the mutual information between the sent +messages and the wiretapper should be bounded above by an +arbitrarily small number. +V. DISCUSSION AND FUTURE WORKS +In this paper, the problem of secure communication over a +quantum interference channel has been studied. The main +approach for decoding sent messages is simultaneous decoding +(one-shot quantum joint typicality lemma) [19]. Also, we used +the method of [27] to randomize Eve’s knowledge and calculate +leaked information. The mentioned Conjecture gives a one-shot +achievable rate region for C-QI-WTC in the form of the Han- +Kobayashi rate region. Still, it is not clear how we can conclude +secrecy requirement for this channel from secrecy criterion of +sub C-QMA-WTCs. However, Theorem 2 solves this problem +using a new encoding. + +APPENDIX +Appendix A: (Proof of the Theorem 1) +The channel in the Figure 1 can be split into two sub-QMA- +WTCs with classical inputs. One from both users to (��, �) and +another from both users to (��, �) . At last, the overall +achievable secrecy rate region can be calculated as: +ℛ�������� ≤ min�ℛ����������, ℛ����������� +Consider the first sub-channel. From Sen’s jointly typical +decoder [19] and [Lemma 3.2, 27], it is clear that: +�� ≤ �� +�(��: ����|�)� − ���� +� +(��: �|�)� + log � − 1 − log 3 +��� ++ 1 +4 log � +�� ≤ �� +�(��: ����|�)� − ���� +� +(��: ���|�)� + log � − 1 +− log 3 +��� + 1 +4 log � +�� + �� ≤ �� +�(����: ��|�)� − ���� +� +(��: �|�)� +− ���� +� +(��: ���|�)� + log � − 1 − 2 log 3 +��� ++ 1 +2 log � + �(1) +There are similar rates for the second sub-channel. Taking +the intersection of the derived regions for the two sub-channels +completes the proof. +Secrecy criterion: The secrecy constraint requires that Eve +just could be able to decode a negligible information: +���� +� +(����: �)� ≤ � +It is obvious that [Lemma 3.2, 27] guarantees the secrecy +criterion. +Appendix B: (Proof of the Conjecture) +To bypass the problem raised in Remark 1 and recover the +non-corner points in the secrecy rate region, we use rate +splitting. We apply the following setting: +We consider two sub C-QMA-WTCs. Therefore, from the +perspective of the first receiver (��), there are three messages +(���, ���, ���) → (��, �), and for the second receiver, there +are three messages (���, ���, ���) → (��, �). The paper [27] +introduces the same setting, but it considers a randomized order +such as ��� → ��� → ��� . This order has not impact on +decoding the messages, but it is helpful to compute leaked +information. Also, it should be considered that in the one-shot +case, we do not use the successive decoder because the time- +sharing strategy gives only finite achievable rate pair. Instead, +we use the one-shot jointly typical decoder [19] for both sub- +channels. +For the first C-QMA-WTC, Alice should randomize over +total block of size (���. ���). It refers to the fact that the split +messages are dependent. There is a detailed discussion in [28]. +For the C-QI-WTC, the controlling state is as follows: + +�� ≤ min��� +�(������: ����)�, �� +�(��: ��������)�� − ���� +�����(���: �)� − ���� +�����(���: �������)� − 2 log 3 +��� ++ 1 +2 log �� + log � − 2 + �(1) +(15) +�� ≤ ��� +�(���: �������)� + �� +�(���: �������)�, �� +�(�����: �����)�, �� +�(�����: �����)�� − ���� +�����(���: �)� +− ���� +�����(���: �������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 + �(1) +(16-17) +�� ≤ min��� +�(������: ����)�, �� +�(��: ��������)�� − �� +�(������: �����)� − ���� +�����(���: ����)� +− ���� +�����(���: ����������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 + �(1) +(18) +�� ≤ ��� +�(���: �������)� + �� +�(���: �������)�, �� +�(�����: �����)�, �� +�(�����: �����)�� − ���� +�����(���: ����)� +− ���� +�����(���: ����������)� − 2 log 3 +��� + 1 +2 log �� + 2 log � − 4 + �(1) +(19-21) +�� + �� ≤ min��� +�(��������: ��)�, �� +�(��������: ��)�� − ���� +�����(���: �)� − ���� +�����(���: ������)� +− ���� +�����(���: ����)� − ���� +�����(���: ������)� − 4 log 3 +��� + log �� + 2 log � − 4 + �(1) +(22) +�� + �� ≤ ��� +�(���: �������)� + �� +�(������: �����)�, �� +�(���: �������)� + �� +�(������: �����)�, �� +�(�����: �����)� ++ �� +�(���: �������)�, �� +�(�����: �����)� + �� +�(���: �������)�� − ���� +�����(���: �)� +− ���� +�����(���: �������)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 4 log 3 +��� + log �� ++ 2 log � − 4 + �(1) +(23-26) +2�� + �� ≤ �� +�(�����: �����)� + �� +�(�����: �����)� − 2���� +�����(���: �)� − 2���� +�����(���: �������)� +− ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 6 log 3 +��� + 3 +2 log �� + 2 log � − 4 + �(1) +(27) +�� + 2�� ≤ �� +�(�����: �����)� + �� +�(�����: �����)� − ���� +�����(���: �)� − ���� +�����(���: ������)� +− 2���� +�����(���: ����)� − 2���� +�����(���: ����������)� − 6 log 3 +��� + 3 +2 log �� + +2 log � − 4 ++ �(1) +(28) +���������������� +≔ +� +����(���)����(���)����(���)����(���) +���,���∈�� +���,���∈�� + +|���⟩⟨���|��� ⊗ |���⟩⟨���|��� ⊗ |���⟩⟨���|��� +⊗ |���⟩⟨���|��� ⊗ ������ +������������ + + + + + +(29) +To simplify the analysis, we first remove the security +constraint of the problem. From Sen’s one-shot jointly typical +decoder [19], we have the following region for the first C- +QMAC: +��� +� +≤ �� +�(���: ��������)� + log � − 2 +��� +� +≤ �� +�(���: ��������)� + log � − 2 +��� +� +≤ �� +�(���: ��������)� + log � − 2 +��� +� + ��� +� +≤ �� +�(������: �����)� + log � − 2 +��� +� + ��� +� +≤ �� +�(������: �����)� + log � − 2 +��� +� + ��� +� +≤ �� +�(������: �����)� + log � − 2 +��� +� + ��� +� + ��� +� +≤ �� +�(���������: ��)� + log � − 2 +Also, for the second C-QMAC there are similar rates. It +should be noted that �� = ��� + ��� and �� = ��� + ��� . +After eliminating redundant rates and using the Fȕrier-Motzkin +elimination, we have: +ℛ����� = +� +�: ����(���)����(���)����(���)����(���) + +�� +� ≤ �� +�(������: �����)� + log � − 2 +�� +� ≤ �� +�(���: ��������)� + �� +�(���: ��������)� + 2 log � − 4 +�� +� ≤ �� +�(������: �����)� + log � − 2 +�� +� ≤ �� +�(���: ��������)� + �� +�(���: ��������)� + 2 log � − 4 +�� +� + �� +� ≤ �� +�(���: ��������)� + �� +�(���������: ��)� ++ 2 log � − 4 +�� +� + �� +� ≤ �� +�(������: �����)� + �� +�(������: �����)� ++ 2 log � − 4 + + +�� ≤ �� +�(������: ����)� − ���� +�����(���: �)� − ���� +�����(���: ������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 + �(1) +(30) +�� ≤ �� +�(���: �������)� + �� +�(���: �������)� − ���� +�����(���: �)� − ���� +�����(���: ������)� − 2 log 3 +��� + 1 +2 log �� ++ 2 log � − 4 + �(1) +(31) +�� ≤ �� +�(��: ��������)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 ++ �(1) +(32) +�� ≤ �� +�(�����: �����)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 ++ �(1) +(33) +�� ≤ �� +�(�����: �����)� − ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 2 log 3 +��� + 1 +2 log �� + log � − 2 ++ �(1) +(34) +�� + �� ≤ �� +�(���: �������)� + �� +�(������: �����)� − ���� +�����(���: �)� − ���� +�����(���: ������)� +− ���� +�����(���: ����)� − ���� +�����(���: ����������)� − 4 log 3 +��� + log �� + 2 log � − 4 + �(1) +(35) +�� + �� ≤ �� +�(�����: �����)� + �� +�(���: �������)� − ���� +�����(���: �)� − ���� +�����(���: ������)� − ���� +�����(���: ����)� +− ���� +�����(���: ����������)� − 4 log 3 +��� + log �� + 2 log � − 4 + �(1) +(36) +�� + �� ≤ �� +�(��������: ��)� − ���� +�����(���: �)� − ���� +�����(���: ������)� − ���� +�����(���: ����)� +− ���� +�����(���: ����������)� − 4 log 3 +��� + log �� + log � − 2 + �(1) +(37) +�� + 2�� ≤ �� +�(�����: �����)� + �� +�(�����: �����)� − ���� +�����(���: �)� − ���� +�����(���: ������)� +− 2���� +�����(���: ����)� − 2���� +�����(���: ����������)� − 6 log 3 +��� + 3 +2 log �� + +2 log � − 4 ++ �(1) +(38) +�� +� + �� +� ≤ �� +�(���: ��������)� + �� +�(���������: ��)� ++ 2 log � − 4 +2�� +� + �� +� ≤ �� +�(���: ��������)� + �� +�(������: �����)� ++ �� +�(���������: ��)� + 3 log � − 6 +�� +� + 2�� +� ≤ �� +�(������: �����)� + �� +�(���: ��������)� ++ �� +�(���������: ��)� + 3 log � − 6 +This region is called the quantum one-shot Han-Kobayashi +rate region for C-QIC, which is calculated in a special case by +Sen [30]. He considers the case of an interference channel with +independent prior entanglement between sender 1 and its +intended receiver and between sender 2 and its intended +receiver. It should be noted that the quantum Han-Kobayashi +rate region for C-QIC in the i.i.d. case is conjectured in [14]. +Note that, all of the above rates correspond to the C-QMACs +without secrecy constraints. Now we want to consider the +secrecy requirements of the problem. +For a C-QMA-WTC, we need a smooth version of the +tripartite convex split lemma [20]. This runs into the smoothing +bottleneck of quantum information theory. In [27], the authors +suggested a novel lemma that gives the size of the randomized +block in terms of smooth max mutual information. +Lemma 1: Given the control state in (29) and decoding +order such as ��� → ��� → ��� → ��� , �� > 0 and 0 < +�� < �� +,let +���� +(�), … , ��� +(���)� +, +���� +(�), … , ��� +(���)� + and +��� +(�), … , �� +(��)� be i.i.d. samples from the distributions ����, +���� and ���. Then, if +log|���| ≥ ���� +�����(���: �)� + log 3 +��� − 1 +4 log �� +log|���| ≥ ���� +�����(���: ����)� + log 3 +��� − 1 +4 log �� + �(1) +log|���| ≥ ���� +�����(���: �������)� + log 3 +��� − 1 +4 log �� ++ �(1) + +log|���| ≥ ���� +�����(���: ����������)� + log 3 +��� − 1 +4 log �� ++ �(1) +the following holds, +����~���� +���~���� +���~���� +���~���� +� +1 +|��||��| � � � � ���� +� ��� +� ��� +� ��� +� +� +− �� +|���| +��� +|���| +��� +|���| +��� +|���| +��� +� +� +≤ 60��� +� +Proof: The proof is similar to the two-user case explained in +[27]. +As mentioned before, let �� = ���. ��� and �� = ���. ���. +Note that, �� = �� +� − log �� , �� = �� +� − log �� . Using the +above lemma completes the proof. +Appendix C: (Proof of the Theorem 2) As mentioned in +Appendix A, the secrecy constraint requires that Eve just could +be able to decode a negligible information: +���� +� +(������������: �)� ≤ � +(39) +Encoding: Suppose that both receivers want to decode non- +interfering messages. This setting is similar to Theorem 1. It can +be helpful for the receivers to decode their messages, including +the intended messages and interfering messages. In other +words, ��� and ��� can be used as side information. Therefore, +the first sub-channel can be modeled as (��������) → (��, �). +Consider the first C-QMA-WTC (��������) → (��, �) . +From [27], we know that an achievable rate region can be +calculated as stated in (30)-(38). +For the second C-QMA-WTC (��������) → (��, �), there +are similar achievable rates. Taking the intersection of the +secrecy regions for both sub-channels can be calculated as +stated in (15)-(28). Against Conjecture, Lemma 1 guarantees +that the secrecy constraint for this problem (39) holds. This +completes the proof. +REFERENCES +[1] C. Shannon, "Communication Theory of Secrecy Systems," Bell Syst. +Tech. J., vol. 28(4), pp. 656–715, 1949. +[2] A. D. Wyner. “The wire-tap channel,” Bell Syst. Tech. J., vol. 54, no. 8, +pp. 2–10, October 1975. +[3] I. Csiszar and J. Korner. “Broadcast channels with confidential +messages,” IEEE Trans. Inf. Theory, vol. 24, no. 3, pp. 339–348, May +1978. +[4] E. Ekrem and S. Ulukus. “On the secrecy of multiple access wiretap +channel,” in proc. 46th Annual Allerton Conference on Communication, +Control, and Computing, Urbana-Champaign, IL, USA, March 2009. +[5] E. Tekin and A. Yener, “The Gaussian multiple access wiretap channel,” +IEEE Trans. Inf. Theory, vol.54, no.12, pp. 5747-5755, December 2008. +[6] E. Tekin and A. Yener, “The general Gaussian multiple access and two- +way wire-tap channels: Achievable rates and cooperative jamming,” IEEE +Trans. Inf. Theory, vol.54, no. 6, pp. 2735–2751, June 2008. +[7] Y. Liang, H.V. Poor, “Multiple-access channels with confidential +messages,” IEEE Trans. Inf. Theory, vol. 54, no.3, pp. 976–1002, 2008. +[8] X. Tang, R. Liu, P. Spasojevi´c, H.V. Poor, “Multiple access channels +with generalized feedback and confidential messages,” in: Proc. Inf. +Theory Workshop, Lake Tahoe, CA, USA (September 2007) 608–613. +[9] B. Dai and Z.Ma, “Some New Results on the Multiple-Access Wiretap +Channel,” Entropy, pp. 4693-4712, 2014. +[10] R. Liu, I. Mari´c, R. Yates, P. Spasojevi´c, “The discrete memoryless +multiple access channel with confidential messages,” in: Proc. Int. Symp. +Inf. Theory, Seattle, USA (July 2006) 957–961. +[11] L. Gyongyosi, S. Imre, H.V Nguyen, “A Survey on Quantum Channel +Capacities,” IEEE Communications Surveys & Tutorials, vol. 20, Issue 2, +pp. 1149-1205, 2018. +[12] N. Cai, A. Winter, and R. W. Yeung. “Quantum Privacy and Quantum +Wiretap Channels,” Problems of Information Transmission, vol. 40, no. +4, pp. 318-336, 2004. +[13] I. Devetak, “The Private Classical Capacity and Quantum Capacity of a +Quantum Channel,” IEEE Trans. Inf. Theory, vol. 51, no. 1, pp. 44-55, +January 2005. +[14] O. Fawzi, P. Hayden, I. Savov, P. Sen, and M. M. Wilde, “Classical +communication over a quantum interference channel,” IEEE Trans. on +Inf. Theory, vol. 58, no. 6, pp. 3670-3691, June 2012. +[15] H. Aghaee, B. Akhbari, “Classical-Quantum Multiple Access Wiretap +Channel,” +in +Proc. +16th +International +ISC +Conference +on Information Security and Cryptology (ISCISC'19), Mashhad, Iran, +August 2019. +[16] H. Aghaee, B. Akhbari, “Private Classical Information over a Quantum +Multiple Access Channel: One-shot Secrecy Rate Region,” in Proc 10th +International Symposium on Telecommunications (IST'2020), Iran, 2020. +[17] H. Aghaee, B. Akhbari, “Classical-Quantum Multiple Access Channel +with Secrecy Constraint: One-shot Rate Region,” International Journal +of Information and Communication Technology Research (IJICTR), vol. +12, no. 2, pp. 1-10, Spring 2020. +[18] H. Aghaee, B. Akhbari, “Classical-Quantum Multiple Access Wiretap +Channel with Common Message: One-shot Rate Region,” in Proc. 11th +Conference on Information and Knowledge Technology (IKT'2020), Iran, +2020. +[19] P. Sen, “Unions, intersections and a one-shot quantum joint typicality +lemma,” Sadhana, 46(1):1–44, 2021. +[20] A. Anshu, R. Jain and N. A. Warsi, "A Generalized Quantum Slepian– +Wolf," in IEEE Trans. on Inf. Theory, vol. 64, no. 3, pp. 1436-1453, +March 2018, doi: 10.1109/TIT.2017.2786348. +[21] H. Qi, Q. Wang, M. M. Wilde,” Applications of position-based coding to +classical communication over quantum channels,” Journal of Physics A +Mathematical and Theoretical 51(44), september 2018. +[22] M. M. Wilde, “Position-based coding and convex splitting for private +communication over quantum channels,” Quantum Information +Processing. 16(10):264, October 2017. arXiv:1703.01733. +[23] L. Wang and R. Renner, “One-shot classical-quantum capacity and +hypothesis testing,” Phys. Rev. Lett., vol. 108, no. 20, p. 200501, 2012. +[24] H. Umegaki, “Conditional expectations in an operator algebra IV (entropy +and information),” Kodai Math. Sem. Rep., vol. 14, pp. 59-85, 1962. +[25] M. Berta, M. Christandl and R. Renner, “The quantum reverse Shannon +theorem based on one-shot information theory,” Commun. Math. Phys., +vol. 306, no. 3, pp. 579-615, 2011. +[26] A. El Gamal, Y.H. Kim,”Network information theory,” Cambridge univ. +press, 2012. +[27] S. Chakraborty, A. Nema, P. Sen, “One-shot inner bounds for sending +private classical information over a quantum MAC,” arXiv:2105.06100. +[28] O. Fawzi, I. Savov. “Rate-splitting in the presence of multiple receivers.” +ArXiv abs/1207.0543 (2012): n. pag. +[29] N. Datta, “Min- max-relative entropies and a new entanglement +monotone”, IEEE. Trans. Inf. Theory, vol. 59, pp. 2816-2816, 2009. +[30] P. Sen, “Inner bounds via simultaneous decoding in quantum network +information theory,” arXiv e-prints, p. arXiv:1806.07276, Jun 2018. + diff --git a/MtE1T4oBgHgl3EQftQWC/content/tmp_files/load_file.txt b/MtE1T4oBgHgl3EQftQWC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..1577883bfa2b3878c02ce3b9b5aa0018074d8517 --- /dev/null +++ b/MtE1T4oBgHgl3EQftQWC/content/tmp_files/load_file.txt @@ -0,0 +1,749 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf,len=748 +page_content='One-Shot Achievable Secrecy Rate Regions for Quantum Interference Wiretap Channel Hadi Aghaee Faculty of Electrical Engineering K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Toosi University of Technology Tehran, Iran Email: Aghaee_Hadi@email.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='ir Bahareh Akhbari Faculty of Electrical Engineering K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Toosi University of Technology Tehran, Iran Email: akhbari@eetd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='kntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='ir Abstract—In this paper, we want to derive achievable secrecy rate regions for quantum interference channel with classical inputs under one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The main idea to this end is to use the combination of superposition and rate splitting for encoding scheme and constructing a decoding scheme based on simultaneous decoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Keywords—Quantum Channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Mutual Information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Secrecy Capacity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Multiple Access Channel I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' INTRODUCTION The physical layer security was introduced by Shannon for the first time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' After that, the wiretap channel was presented by Wyner, in which a sender transmits its message to a legitimate receiver in the presence of a passive eavesdropper [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Moreover, Csiszár and Körner introduced the broadcast channel with confidential messages [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' However, the physical layer security problems have been extended to multi-terminal channels like multiple access channels (MACs), Interference channels (ICs), relay channels, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', due to their importance and their usage in practical systems [4-10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In recent decades, with development in quantum data processing and its applications, a significant effort has begun to use the natural features of quantum mechanics to improve communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Some of these features are as follows: entanglement, uncertainty, no-cloning theorem, superposition, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' These natural features help the communication to be faster and more secure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Moreover, the security problem plays a critical role in quantum communication and devotes a considerable part of the research to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In this regard, the quantum wiretap channel (QWTC) was firstly introduced in [12] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Then, secrecy constraints are extended to multi-user quantum channels such as quantum interference channel (QIC) [14] and quantum multiple access channel (QMAC) [15-18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The interference phenomenon is one of the major problems in communication systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In this paper, we derive some achievable secrecy rate regions for quantum interference channel with classical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' One of the major open problems in the quantum information theory is related to the simultaneous decoder for quantum channels with three or more senders (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', jointly typical decoder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' However, this problem has been solved for some cases, such as the min-entropy case and the case of the quantum multiple access channels (QMACs), in which the output systems are commutative [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Therefore, in the independent and identical distributed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=') case, we have to use successive decoding combined with time-sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In contrast, for the one- shot case, we have to use the simultaneous decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sen proved a joint typicality lemma which is helpful to decode any number of messages simultaneously in the one-shot case [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In this paper, we want to study secure communication over a classical-quantum interference wiretap channel (C-QI-WTC) under the one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Up to the best knowledge, it is the first time that this channel is studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Even in the classical case, the security problem of interference channel has been investigated under a different scenario called interference channel with confidential messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Also, another feature of our problem is that the channel is considered under the one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This choice is due to the fact that there is not a proven joint typicality lemma in the asymptotic i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' case for general quantum channels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', quantum channels with any number of senders).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Therefore, all of the obtained results are new, and the proposed strategies in the paper can be applied to the classical interference channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The paper is organized as follows: In Section II, some seminal definitions are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In Section III, the main channel and information processing tasks are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In Section IV, the results and main theorems are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Section V is dedicated to discussion and future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' PRELIMINARIES Let A (Alice) and B (Bob) be two quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' These quantum systems can be denoted by their corresponding Hilbert spaces as ℋ�, ℋ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The states of the above quantum systems are presented as density operators �� and ��, respectively, while the shared state between Alice and Bob is denoted by ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' A density operator is a positive semidefinite operator with a unit trace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Alice or Bob’s state can be defined by a partial trace operator over the shared state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The partial trace is used to model the lack of access to a quantum system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Thus, Alice’s density operator using partial trace is �� = ���{���}, and Bob’s density operator is �� = ���{���}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' We use |�⟩� to denote the pure state of system A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The corresponding density operator is �� = |�⟩⟨�|�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The von Neumann entropy of the state �� is defined by �(�)� = −��{�� log ��}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For an arbitrarily state such as ���, the quantum conditional entropy is defined by �(�|�)� = �(�, �)� − �(�)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The quantum mutual information is defined by �(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �)� = �(�)� + �(�)� − �(�, �)�, and the conditional quantum mutual information is defined by: �(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �|�)� = �(�|�)� + �(�|�)� − �(�, �|�)� Quantum operations can be denoted by completely positive trace-preserving (CPTP) maps ��→�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The CPTP maps accept input states in A and output states in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The distance between two quantum states, such as A and B is defined by trace distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The trace distance between two arbitrarily states such as � and � is: ‖� − �‖� = ��|� − �| (1) where |Ψ| = √Ψ�Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This quantity is zero for two similar and perfectly distinguishable states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Fidelity is defined as �(�, �) = ���√��� �, and purified distance is a metric on �(ℋ) and is defined as �(�, �) ≔ �1 − �(�, �)�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Most of the above definitions are given from [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 1: (Hypothesis testing mutual information) Hypothesis testing mutual information is denoted by �� �(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �) ∶= �� � (���‖�� ⊗ ��), � ∈ (0,1) and is considered as quantum hypothesis testing divergence [21] where �� � (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ‖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=') is hypothesis testing relative entropy [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �ℋ�ℋ� is the joint state of input and output over their Hilbert spaces (ℋ�, ℋ�), and it can be shown as ���: ��� = � ��(�)|�⟩⟨�|� ⊗ �� � � where �� is the input distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 2: (Quantum relative entropy [22]): Consider states ��, �� ∈ �(ℋ�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The Quantum relative entropy is defined as: �(��‖��) ≔ ���{���log� �� − log� ���} ����(��) ⊆ ����(��) +∞ ��ℎ������ where ����(��) refers to the set-theoretic support of �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ����(�) is the subspace of ℋ spanned by all eigenvectors of � with non-zero eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Fact: The following relation exists between the quantum relative entropy and hypothesis testing relative entropy for � ∈ (0,1) [21]: �� �(��‖��) ≤ 1 1 − � ��(��‖��) + ℎ�(�)� where ℎ�(�) ≔ −� log� � − (1 − �) log�(1 − �) is a binary entropy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 3: (Max mutual information [23]) Consider a bipartite state ��� and a parameter � ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The max mutual information can be defined as follows: ����(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �)� ≔ ����(��� ‖��⨂�� )� where � refers to the state ��� and ����(∣∣) is the max-relative entropy [24] for ��, �� ∈ ℋ�: ����(�� ‖��) ≔ inf{� ∈ ℝ: �� ≤ 2���} Definition 4: (Quantum smooth max relative entropy [24]) Consider states ��, �� ∈ �(ℋ�) and � ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The quantum smooth max relative entropy is defined as: ���� � (��‖��) ∶= inf �� � ∈ℬ�(��) ����(�� � ‖�� ) where ℬ�(��) ≔ {�� � ∈ �(ℋ�): �(�� � , ��) ≤ �} is �-ball for ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 5: (Quantum smooth max mutual information [23]) Consider ��� ∶= ∑ ��(�)|�⟩⟨�|� ⊗ �∈� �� � as a classical- quantum state and a parameter � ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The smooth max mutual information between the systems � and � can be defined as follows: ���� � (�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �) ∶= inf ��� � ∈ℬ�(���) ����(��� � ‖��⨂�� ) = inf ��� � ∈ℬ�(���)����(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �)�� , where ℬ�(���) ≔ {��� � ∈ �(ℋ� ⊗ ℋ�): �(��� � , ���) ≤ �} is �-ball for ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 6: (Conditional smooth hypothesis testing mutual information [25]) Consider ���� ∶= ∑ ��(�)|�⟩⟨�|� �∈� ⊗ ��� � be a tripartite classical-quantum state and � ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' We define, �� �(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �|�)� ≔ max �� min �∈������� � � �� �(�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �)��� � , where maximization is over all �� � = ∑ ��(�)|�⟩⟨�|� �∈� satisfying �(�� � , ��) ≤ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 7: (Conditional smooth max mutual information [25]) Consider ���� ∶= ∑ ��(�)|�⟩⟨�|� �∈� ⊗ ��� � be a tripartite classical-quantum state and � ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' We define, ���� � (�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �|�)� ≔ max �� min �∈������� � � ���� � (�;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �)��� � , where maximization is over all �� � = ∑ ��(�)|�⟩⟨�|� �∈� satisfying �(�� � , ��) ≤ �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Definition 8: (Quantum Rényi relative entropy of order � [21]) For a state � ∈ �(ℋ) and a positive semidefinite operator �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' the quantum Rényi relative entropy of order �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' where � ∈ �0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='1) ∪ (1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' +∞) is defined as: ��(�‖�) ≡ 1 � − 1 log� ��{������} Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Rényi entropy of order � can be defined as follows: ��(�)� ≡ 1 1 − � log� ��{�� �} Definition 9: (One-shot inner bound of a classical-quantum multiple access channel) [19] A two user C-QMAC under the one-shot setting is a triple (�� × ��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �����→�(��,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ��) ≡ �� ����,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ℋ�),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' where �� and �� are the alphabet sets of two classical inputs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' and � is the output system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ����� � is a quantum state, and the channel has a completely positive trace- preserving map (CPTP) �����→�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The C-QI-WTC model Considering the joint typicality lemma introduced in [Corollary 4, 19], the one-shot inner bound of a C-QMAC is as follows: �� ≤ �� �(��: ���)� − 2 + log � �� ≤ �� �(��: ���)� − 2 + log � �� + �� ≤ �� �(����: �)� − 2 + log � where �� �(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ) is the hypothesis testing mutual information defined in Definition 1 with respect to the controlling state: ������� ∶= � �(�)�(��|�)�(��|�)|�����⟩⟨�����|����� ����� ⊗ �� ���� and � is a time-sharing variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Note that �� �(: ) is the difference between a Rényi entropy of order two and a conditional quantum entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' CHANNEL MODEL A two-user C-QI-WTC is a triple (�� × ��, �����→�����(��, ��) ≡ ����� �����, ℋ�� ⊗ ℋ�� ⊗ ℋ�), where ��, � ∈ {1,2} denote the input alphabet sets, and ��, ��, � denote the output systems (��, �� denote the channel outputs at the two legitimate receivers and � is the channel outputs at the eavesdropper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ����� ����� is the system output’s quantum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Each user wants to transmit its message as secure as possible over a C-QI-WTC to its intended receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The main channel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' case) is illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Consider the main channel illustrated in Figure 1 under the one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Each user chooses its message ��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' � ∈ {1,2} from its message set ℳ� = �1: |ℳ�| = 2���;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' � ∈ {1,2}, and send it over a C-QI-WTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The users also use two junk variables ��;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' � ∈ {1,2} from two amplification sets �� = �1: |��| = 2����;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' � ∈ {1,2} for randomizing Eve’s knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' We have two doubly indexed codebooks ��(��, ��) and ��(��, ��) for user-1 and user-2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The above channel can be divided into two sub C-QMA-WTCs (one from both users to (��, �) and another from both users to (��, �)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' MAIN RESULTS In this section, we present the main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theorem 1: (One-shot achievable rate region for C-QI- WTC) Consider a two-user C-QI-WTC which accepts �� and �� as inputs and ��, �� and � as outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ����� ����� is the channel density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For any fixed � ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ��) and �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� such that �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' the rate pair �� = ���|ℳ�| + �,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' � ∈ {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2} is achievable to satisfy the following inequalities: �� ≤ min��� �(��: ����|�)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(��: ����|�)�� − ���� � (��: �|�)� + log � − 1 − log 3 ��� + 1 4 log � �� ≤ min��� �(��: ����|�)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(��: ����|�)�� − ���� � (��: ���|�)� + log � − 1 − log 3 ��� + 1 4 log � �� + �� ≤ min��� �(����: ��|�)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(����: ��|�)�� − ���� � (��: �|�)� − ���� � (��: ���|�)� + log � − 1 − 2 log 3 ��� + 1 2 log � + �(1) where � = �� − �� and the union is taken over input distribution ��(�)���|�(��|�)���|�(��|�).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Q is the time-sharing random variable, and all of the mutual information quantities are taken with respect to the following state: ����������� ≡ � ���(�)���|�(��|�)���|�(��|�)|�⟩⟨�|� �,��,�� ⊗ |��⟩⟨��|�� ⊗ |��⟩⟨��|�� ⊗ ����� ����� (4) Proof: See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sketch of proof: The channel can be split into two sub-QMA- WTCs with classical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' One from (��, ��) to (��, �) and another from (��, ��) to (��, �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Using the proposed method by El-Gamal and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Kim [26] helps to prove this theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theorem 1 gives the simplest achievable rate region for C- QI-WTC under the one-shot setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Without considering the secrecy constraints, Han and Kobayashi obtained the best achievable rate region for interference channel (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' setting) using rate splitting that the messages are split into common and personal messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This technique is extended to the quantum case with some limits [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Using the Han-Kobayashi’s technique, the message �� is split into ��� (common part) and ��� (personal part), where � ∈ {1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The structure of the C-QI-WTC under the Han-Kobayashi’s setting is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The following channel can be divided into two separate sub 3-user C-QMA-WTCs: one from (���, ���, ���) to (��, �) and another from (���, ���, ���) to (��, �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' As mentioned before, there is not a proven quantum simultaneous decoder for decoding three or more messages in general and it remains a conjecture (except some cases such as the commutative version of output states and min-entropy cases [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' wiretappel →>X2 →(M1,M2)Y1 Y2Z x1x2m1 >Mi X1 C-QI-WTC Y2Y����� � (������������: �)� ≤ ������� � (���������: �)� ≤ ��, ���� � (���������: �)� ≤ ��� where ��, �� and �� are arbitrary small numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + log � − 2 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(6) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(7) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + log � − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(9) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(10) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(11) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(12) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 6 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + 3 log � − 6 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + 2�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 6 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + 3 log � − 6 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The structure of the C-QI-WTC under the Han-Kobayashi settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Remark 1: Note that, to take the intersection of the private regions for two 3-sender MACs raised in Theorem 1, we used the method of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Another approach can be using Fȕrier- Motzkin elimination [Appendix D, 26] which gives achievable rate region similar to the Han-Kobayashi expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Remark 2: The Han-Kobayashi technique is based on rate splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' It should be noted that the split messages are not independent of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=" Thus, obtaining secrecy against the eavesdropper by Wyner's randomizing technique becomes problematic in this setting." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In other words, we cannot randomize over a block independently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For example, �� should be randomized using the product of two junk variables (���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Conjecture: (An inner bound on the one-shot secrecy capacity region of the C-QI-WTC) Consider the region: ℛ(�) = �{(��, ��) ∈ ��|����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' (6) − (14) ℎ���} � Proof: In Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sketch of proof: We consider two sub C-QMA-WTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Therefore, from the perspective of the first receiver (��), there (m20,m22102 C1X2 m2 m10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='m11Y2mare three messages (���, ���, ���) → (��, �) , and for the second receiver, there are three messages (���, ���, ���) → (��, �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The paper [27] introduces the same setting, but it considers a randomized order such as ��� → ��� → ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For the first C-QMA-WTC, Alice should randomize over a total block of size (���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For the second C-QMA-WTC, Bob should randomize over a total block of size (���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Then, we can analyze both sub-channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Remark 3: The above conjecture holds if and only if condition (5) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Because taking the intersection of the private regions for two 3-sender C-QMACs is not enough to get a private region for the full C-QI-WTC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' To overcome the above problem, we should change the encoding process, which results in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theorem 2: (An inner bound on the one-shot secrecy capacity region of the C-QI-WTC) Consider the region: ℛ(�) = �{(��, ��) ∈ ��|����.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' (15) − (28) ℎ���} � Proof: In Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sketch of proof: The overall sketch of the proof is the same as that for the above Conjecture with one difference: Suppose that both receivers want to decode non-interfering messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Also, this setting is similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' It can be helpful for the receivers to decode their messages, including the intended messages and interfering messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In other words, ��� and ��� can be used as side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Therefore, the first sub- channel can be modeled as (��������) → (��, �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' All steps, such as encoding and decoding, are the same as for the above Conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Secrecy criterion: The secrecy criterion for the channel can be defined as follows: �(��, ��: �) ≤ � For Theorem 1 �(���, ���, ���, ���: �) ≤ � For Conjecture and Theorem 2 This means that the mutual information between the sent messages and the wiretapper should be bounded above by an arbitrarily small number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' DISCUSSION AND FUTURE WORKS In this paper, the problem of secure communication over a quantum interference channel has been studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The main approach for decoding sent messages is simultaneous decoding (one-shot quantum joint typicality lemma) [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Also, we used the method of [27] to randomize Eve’s knowledge and calculate leaked information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The mentioned Conjecture gives a one-shot achievable rate region for C-QI-WTC in the form of the Han- Kobayashi rate region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Still, it is not clear how we can conclude secrecy requirement for this channel from secrecy criterion of sub C-QMA-WTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' However, Theorem 2 solves this problem using a new encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' APPENDIX Appendix A: (Proof of the Theorem 1) The channel in the Figure 1 can be split into two sub-QMA- WTCs with classical inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' One from both users to (��, �) and another from both users to (��, �) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' At last, the overall achievable secrecy rate region can be calculated as: ℛ�������� ≤ min�ℛ����������, ℛ����������� Consider the first sub-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' From Sen’s jointly typical decoder [19] and [Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2, 27], it is clear that: �� ≤ �� �(��: ����|�)� − ���� � (��: �|�)� + log � − 1 − log 3 ��� + 1 4 log � �� ≤ �� �(��: ����|�)� − ���� � (��: ���|�)� + log � − 1 − log 3 ��� + 1 4 log � �� + �� ≤ �� �(����: ��|�)� − ���� � (��: �|�)� − ���� � (��: ���|�)� + log � − 1 − 2 log 3 ��� + 1 2 log � + �(1) There are similar rates for the second sub-channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Taking the intersection of the derived regions for the two sub-channels completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Secrecy criterion: The secrecy constraint requires that Eve just could be able to decode a negligible information: ���� � (����: �)� ≤ � It is obvious that [Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2, 27] guarantees the secrecy criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Appendix B: (Proof of the Conjecture) To bypass the problem raised in Remark 1 and recover the non-corner points in the secrecy rate region, we use rate splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' We apply the following setting: We consider two sub C-QMA-WTCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Therefore, from the perspective of the first receiver (��), there are three messages (���, ���, ���) → (��, �), and for the second receiver, there are three messages (���, ���, ���) → (��, �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' The paper [27] introduces the same setting, but it considers a randomized order such as ��� → ��� → ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This order has not impact on decoding the messages, but it is helpful to compute leaked information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Also, it should be considered that in the one-shot case, we do not use the successive decoder because the time- sharing strategy gives only finite achievable rate pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Instead, we use the one-shot jointly typical decoder [19] for both sub- channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For the first C-QMA-WTC, Alice should randomize over total block of size (���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ���).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' It refers to the fact that the split messages are dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' There is a detailed discussion in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For the C-QI-WTC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' the controlling state is as follows: �� ≤ min��� �(������: ����)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(��: ��������)�� − ���� �����(���: �)� − ���� �����(���: �������)� − 2 log 3 ��� + 1 2 log �� + log � − 2 + �(1) (15) �� ≤ ��� �(���: �������)� + �� �(���: �������)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(�����: �����)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(�����: �����)�� − ���� �����(���: �)� − ���� �����(���: �������)� − 2 log 3 ��� + 1 2 log �� + log � − 2 + �(1) (16-17) �� ≤ min��� �(������: ����)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(��: ��������)�� − �� �(������: �����)� − ���� �����(���: ����)� − ���� �����(���: ����������)� − 2 log 3 ��� + 1 2 log �� + log � − 2 + �(1) (18) �� ≤ ��� �(���: �������)� + �� �(���: �������)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(�����: �����)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(�����: �����)�� − ���� �����(���: ����)� − ���� �����(���: ����������)� − 2 log 3 ��� + 1 2 log �� + 2 log � − 4 + �(1) (19-21) �� + �� ≤ min��� �(��������: ��)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(��������: ��)�� − ���� �����(���: �)� − ���� �����(���: ������)� − ���� �����(���: ����)� − ���� �����(���: ������)� − 4 log 3 ��� + log �� + 2 log � − 4 + �(1) (22) �� + �� ≤ ��� �(���: �������)� + �� �(������: �����)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(���: �������)� + �� �(������: �����)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� �(�����: �����)� + �� �(���: �������)�,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: �������)�� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(23-26) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 6 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(27) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + 2�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 6 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + +2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(28) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='���������������� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='≔ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='����(���)����(���)����(���)����(���) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='���,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='���∈�� ���,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='���∈�� |���⟩⟨���|��� ⊗ |���⟩⟨���|��� ⊗ |���⟩⟨���|��� ⊗ |���⟩⟨���|��� ⊗ ������ ������������ (29) To simplify the analysis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' we first remove the security constraint of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' From Sen’s one-shot jointly typical decoder [19],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' we have the following region for the first C- QMAC: ��� � ≤ �� �(���: ��������)� + log � − 2 ��� � ≤ �� �(���: ��������)� + log � − 2 ��� � ≤ �� �(���: ��������)� + log � − 2 ��� � + ��� � ≤ �� �(������: �����)� + log � − 2 ��� � + ��� � ≤ �� �(������: �����)� + log � − 2 ��� � + ��� � ≤ �� �(������: �����)� + log � − 2 ��� � + ��� � + ��� � ≤ �� �(���������: ��)� + log � − 2 Also,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' for the second C-QMAC there are similar rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' It should be noted that �� = ��� + ��� and �� = ��� + ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' After eliminating redundant rates and using the Fȕrier-Motzkin elimination,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' we have: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='ℛ����� = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�: ����(���)����(���)����(���)����(���) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + log � − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + 2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + log � − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + 2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + log � − 2 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(30) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: �������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: �������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(31) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(��: ��������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + log � − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + log � − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(33) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 2 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + log � − 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(34) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: �������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(35) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: �������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� + 2 log � − 4 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(36) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + �� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(��������: ��)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 4 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + log �� + log � − 2 + �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(37) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� + 2�� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(�����: �����)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: �)� − ���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='− 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����)� − 2���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�����(���: ����������)� − 6 log 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='��� + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2 log �� + +2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �(1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='(38) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ 2 log � − 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� + 3 log � − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� + 2�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='� ≤ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(������: �����)� + �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���: ��������)� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='+ �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='�(���������: ��)� + 3 log � − 6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='This region is called the quantum one-shot Han-Kobayashi ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='rate region for C-QIC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' which is calculated in a special case by Sen [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' He considers the case of an interference channel with independent prior entanglement between sender 1 and its intended receiver and between sender 2 and its intended receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' It should be noted that the quantum Han-Kobayashi rate region for C-QIC in the i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' case is conjectured in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Note that, all of the above rates correspond to the C-QMACs without secrecy constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Now we want to consider the secrecy requirements of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For a C-QMA-WTC, we need a smooth version of the tripartite convex split lemma [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This runs into the smoothing bottleneck of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In [27], the authors suggested a novel lemma that gives the size of the randomized block in terms of smooth max mutual information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Lemma 1: Given the control state in (29) and decoding order such as ��� → ��� → ��� → ��� , �� > 0 and 0 < �� < �� ,let ���� (�), … , ��� (���)� , ���� (�), … , ��� (���)� and ��� (�), … , �� (��)� be i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' samples from the distributions ����, ���� and ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' if log|���| ≥ ���� �����(���: �)� + log 3 ��� − 1 4 log �� log|���| ≥ ���� �����(���: ����)� + log 3 ��� − 1 4 log �� + �(1) log|���| ≥ ���� �����(���: �������)� + log 3 ��� − 1 4 log �� + �(1) log|���| ≥ ���� �����(���: ����������)� + log 3 ��� − 1 4 log �� + �(1) the following holds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ����~���� ���~���� ���~���� ���~���� � 1 |��||��| � � � � ���� � ��� � ��� � ��� � � − �� |���| ��� |���| ��� |���| ��� |���| ��� � � ≤ 60��� � Proof: The proof is similar to the two-user case explained in [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' As mentioned before, let �� = ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ��� and �� = ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Note that, �� = �� � − log �� , �� = �� � − log �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Using the above lemma completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Appendix C: (Proof of the Theorem 2) As mentioned in Appendix A, the secrecy constraint requires that Eve just could be able to decode a negligible information: ���� � (������������: �)� ≤ � (39) Encoding: Suppose that both receivers want to decode non- interfering messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This setting is similar to Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' It can be helpful for the receivers to decode their messages, including the intended messages and interfering messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' In other words, ��� and ��� can be used as side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Therefore, the first sub-channel can be modeled as (��������) → (��, �).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Consider the first C-QMA-WTC (��������) → (��, �) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' From [27], we know that an achievable rate region can be calculated as stated in (30)-(38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' For the second C-QMA-WTC (��������) → (��, �), there are similar achievable rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Taking the intersection of the secrecy regions for both sub-channels can be calculated as stated in (15)-(28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Against Conjecture, Lemma 1 guarantees that the secrecy constraint for this problem (39) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' REFERENCES [1] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Shannon, "Communication Theory of Secrecy Systems," Bell Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 28(4), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 656–715, 1949.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [2] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Wyner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' “The wire-tap channel,” Bell Syst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 8, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 2–10, October 1975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [3] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Csiszar and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Korner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' “Broadcast channels with confidential messages,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 24, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 339–348, May 1978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [4] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Ekrem and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Ulukus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' “On the secrecy of multiple access wiretap channel,” in proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 46th Annual Allerton Conference on Communication, Control, and Computing, Urbana-Champaign, IL, USA, March 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [5] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Tekin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Yener, “The Gaussian multiple access wiretap channel,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='12, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 5747-5755, December 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [6] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Tekin and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Yener, “The general Gaussian multiple access and two- way wire-tap channels: Achievable rates and cooperative jamming,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 2735–2751, June 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [7] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Liang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Poor, “Multiple-access channels with confidential messages,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 54, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 976–1002, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [8] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Tang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Liu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Spasojevi´c, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Poor, “Multiple access channels with generalized feedback and confidential messages,” in: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory Workshop, Lake Tahoe, CA, USA (September 2007) 608–613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Dai and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='Ma, “Some New Results on the Multiple-Access Wiretap Channel,” Entropy, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 4693-4712, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [10] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Liu, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Mari´c, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Yates, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Spasojevi´c, “The discrete memoryless multiple access channel with confidential messages,” in: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Symp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, Seattle, USA (July 2006) 957–961.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [11] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Gyongyosi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Imre, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='V Nguyen, “A Survey on Quantum Channel Capacities,” IEEE Communications Surveys & Tutorials, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 20, Issue 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 1149-1205, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Cai, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Winter, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Yeung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' “Quantum Privacy and Quantum Wiretap Channels,” Problems of Information Transmission, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 40, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 4, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 318-336, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [13] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Devetak, “The Private Classical Capacity and Quantum Capacity of a Quantum Channel,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 51, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 44-55, January 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [14] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Fawzi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Hayden, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Savov, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Wilde, “Classical communication over a quantum interference channel,” IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' on Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 58, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 6, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 3670-3691, June 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [15] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Aghaee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Akhbari, “Classical-Quantum Multiple Access Wiretap Channel,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=" 16th International ISC Conference on Information Security and Cryptology (ISCISC'19), Mashhad, Iran, August 2019." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [16] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Aghaee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=" Akhbari, “Private Classical Information over a Quantum Multiple Access Channel: One-shot Secrecy Rate Region,” in Proc 10th International Symposium on Telecommunications (IST'2020), Iran, 2020." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [17] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Aghaee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Akhbari, “Classical-Quantum Multiple Access Channel with Secrecy Constraint: One-shot Rate Region,” International Journal of Information and Communication Technology Research (IJICTR), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 2, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 1-10, Spring 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [18] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Aghaee, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Akhbari, “Classical-Quantum Multiple Access Wiretap Channel with Common Message: One-shot Rate Region,” in Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=" 11th Conference on Information and Knowledge Technology (IKT'2020), Iran, 2020." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sen, “Unions, intersections and a one-shot quantum joint typicality lemma,” Sadhana, 46(1):1–44, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [20] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Anshu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Jain and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Warsi, "A Generalized Quantum Slepian– Wolf," in IEEE Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' on Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 64, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 1436-1453, March 2018, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='1109/TIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='2786348.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Qi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Wilde,” Applications of position-based coding to classical communication over quantum channels,” Journal of Physics A Mathematical and Theoretical 51(44), september 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Wilde, “Position-based coding and convex splitting for private communication over quantum channels,” Quantum Information Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 16(10):264, October 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' arXiv:1703.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='01733.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [23] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Wang and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Renner, “One-shot classical-quantum capacity and hypothesis testing,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 108, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 200501, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [24] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Umegaki, “Conditional expectations in an operator algebra IV (entropy and information),” Kodai Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 14, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 59-85, 1962.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [25] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Berta, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Christandl and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Renner, “The quantum reverse Shannon theorem based on one-shot information theory,” Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=', vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 306, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 3, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 579-615, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' El Gamal, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Kim,”Network information theory,” Cambridge univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' press, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [27] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Chakraborty, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Nema, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sen, “One-shot inner bounds for sending private classical information over a quantum MAC,” arXiv:2105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='06100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [28] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Fawzi, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Savov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' “Rate-splitting in the presence of multiple receivers.” ArXiv abs/1207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='0543 (2012): n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' pag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [29] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Datta, “Min- max-relative entropies and a new entanglement monotone”, IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Inf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Theory, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 59, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' 2816-2816, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' Sen, “Inner bounds via simultaneous decoding in quantum network information theory,” arXiv e-prints, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content=' arXiv:1806.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} +page_content='07276, Jun 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtE1T4oBgHgl3EQftQWC/content/2301.03375v1.pdf'} diff --git a/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf b/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..90c6037f6c2b7b50bcc831e8c4d4b5eeb65ac4ba --- /dev/null +++ b/OtFOT4oBgHgl3EQf3zQR/content/2301.12947v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:e19b0402e3571e14b894d8e1980f15e62bd2967fc4dd6eb45cb0daacc1605d4f +size 925282 diff --git a/OtFOT4oBgHgl3EQf3zQR/vector_store/index.faiss b/OtFOT4oBgHgl3EQf3zQR/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..cf760ebd44ec0a35ba88511b6f028e58af3e9857 --- /dev/null +++ b/OtFOT4oBgHgl3EQf3zQR/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d80b67060447a10c66f4246a5c89755d1b2d02cab8e2aaf1dd2347a17a6c5264 +size 3211309 diff --git a/OtFOT4oBgHgl3EQf3zQR/vector_store/index.pkl b/OtFOT4oBgHgl3EQf3zQR/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..998ddd3fedab11d32869481def382bb1383400c6 --- /dev/null +++ b/OtFOT4oBgHgl3EQf3zQR/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bcd61f8260a9443fe8e6a6bda0d889a5dcb698dd6cb5666ca3e38b426ffff001 +size 113196 diff --git a/PNAyT4oBgHgl3EQftfl6/content/tmp_files/2301.00596v1.pdf.txt b/PNAyT4oBgHgl3EQftfl6/content/tmp_files/2301.00596v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0cae164a57dc679283f2373a039a7ba52f36fb63 --- /dev/null +++ b/PNAyT4oBgHgl3EQftfl6/content/tmp_files/2301.00596v1.pdf.txt @@ -0,0 +1,738 @@ +A contrastive learning approach for +individual re-identification in a wild fish population +Ørjan Langøy Olsen1, Tonje Knutsen Sørdalen2, Morten Goodwin1, Ketil Malde3, +Kristian Muri Knausg˚ard∗4, and Kim Tallaksen Halvorsen3 +1Centre for Artificial Intelligence Research, University of Agder, Norway +2Centre for Coastal Research, University of Agder, Norway +3Institute of Marine Research, Ecosystem Acoustics Group, Bergen, Norway +4Top Research Centre Mechatronics, University of Agder, Norway +Abstract +In both terrestrial and marine ecology, physical tag- +ging is a frequently used method to study popula- +tion dynamics and behavior. However, such tag- +ging techniques are increasingly being replaced by +individual re-identification using image analysis. +This paper introduces a contrastive learning- +based model for identifying individuals. The model +uses the first parts of the Inception v3 network, +supported by a projection head, and we use con- +trastive learning to find similar or dissimilar image +pairs from a collection of uniform photographs. We +apply this technique for corkwing wrasse, Sympho- +dus melops, an ecologically and commercially im- +portant fish species. Photos are taken during re- +peated catches of the same individuals from a wild +population, where the intervals between individual +sightings might range from a few days to several +years. +Our model achieves a one-shot accuracy of 0.35, +a 5-shot accuracy of 0.56, and a 100-shot accuracy +of 0.88, on our dataset. +1 +Introduction +Physical tagging, using external or internal mark- +ings for individual identification, is a widely used +method for monitoring terrestrial and aquatic an- +imal populations. +Information from resightings +or recapture of the same individuals can be used +∗Corresponding kristianmk@ieee.org +to estimate population size, survival and move- +ment patterns. +However, most tagging methods +are costly, intrusive, and labor-intensive. +To our +beneift, many animals have natural markings or +morphological features that are unique to individu- +als that could be used for photo-identification and +replace the need for physical tags [22, 19]. How- +ever, for ecologists, working with fish may mean +keeping track of hundreds or potentially thousands +of individuals in a population, which makes manual +photo-identification challenging, if not impossible. +For this reason, fully- or semi-automatic tools for +re-identification of individuals would be immensely +useful for ecologists. +Re-identification (re-ID) is different from normal +classification in that it is a few-shot learning prob- +lem. Few-shot problems are characterised by hav- +ing few samples per class, but there may be a large +or indefinite number of classes. One way to solve +such problems is a technique called metric learn- +ing, where data is transformed into embeddings +of a lower dimension, that clusters points from +the same class together. +Classification can then +be performed on the embeddings. Metric learning +approaches have been proved to work well for re- +identification of animal species [18]. A crucial ad- +vantage with metric learning approaches is that the +network does not need to be retrained to be able to +add new classes. +Contrastive learning is a technique that can be +used to solve few-shot problems. +Constrastive +learning compares data and identifies whether they +are similar or dissimilar. A siamese network [1] is +https://doi.org/10.7557/18.6824 +© The author(s). Licensee Septentrio Academic Publishing, Tromsø, Norway. This is an open access article distributed +under the terms and conditions of the Creative Commons Attribution license +(http://creativecommons.org/licenses/by/4.0/). +1 +arXiv:2301.00596v1 [cs.CV] 2 Jan 2023 + +the most basic form and takes two inputs through +the same network with the same shared weights +and gets an embedding for both. During training, +it tries to predict whether they are of the same class +or not. A major advantage here is that it does not +need to know which class an input belongs to, nor +how many classes there are. Triplet networks [10] +are an improvement to the siamese network with +three inputs. +The goal of this work was to test the applica- +bility of image based re-ID analysis for a commer- +cially and ecologically important fish species, the +corkwing wrasse (Symphodus melops). The image +dataset consists of standardized photos of captures +and recaptures of individuals in a wild population, +where the time between individual sightings spans +from days to several years. The first step is to de- +tect a fish in an image with an object detector, +followed by a re-identification method. With high +enough precision, computer vision re-ID has the +potential to replace physical tagging for individ- +ual identification and may be applied in monitoring +of survival rates, growth, movement, and popula- +tion size, key knowledge for sustainable manage- +ment and conservation [18] [4]. +2 +Related works +Advancements in machine learning have produced +powerful techniques for extracting ecologically im- +portant information from image and video data. +For instance, machine learning have successfully +been utilized to detect fish wounds [5], count and +categorize organisms in digital photos and real-time +video [14], [12], identify species, [6], and discover, +and count creatures from digital images, [4], and +even quantify their behaviour [3]. +Some work on the topic of re-identification of fish +has been conducted, but work on wild teleost fish +are lacking. Bruslund Haurum et al. [2] achieved +an mAP of 99% on Zebrafish using metric learn- +ing with 15 samples per class of 6 classes. +Mei- +dell and Sjøblom [16] reports a true positive rate +of 96% on 225 thousand images of salmon divided +between 715 individuals. +Li et al. [13] achieved +an accuracy of 92% using 3412 images of 10 indi- +viduals using their novel FFRNet network. These +studies have in common that they were carried out +in captivity and are not using temporally indepen- +dent observations. In other words, the individuals +did not change morphology through growth, matu- +ration, senescence, or similar biological processes. +Moskvyak et al. [17] used a metric learning ap- +proach on a dataset of 1730 images of 120 manta +ray individuals and achieved an accuracy@1 of 62% +and an accuracy@10 of 97%. +3 +Method +3.1 +Data collection +The study species, S. melops, is a commercially and +ecologically important species in coastal ecosystems +in the Northeastern Atlantic [7]. This species have +two distinct male morphs, colourful large males +that build nest and care for the eggs, and smaller +sneaker males, with a more brown coloration resem- +bling the female morphology (brown and gray) [21]. +The dataset was collected in Austevoll, western +Norway, 2018-2021, by catching corkwing wrasse +by fyke nets left in the sea overnight and mark- +ing all captured individuals with uniquely coded +passive integrated transponder (PIT) tags (11 mm +tags, RFID Solutions). The tags were implanted in +the abdominal cavity of the fish, see full sampling +description in [8] and [9]. +This method enabled us to collect independent +observations of each individual across time and for +the dataset to encompass changes in the fish’s mor- +phology. At each capture, a few images were taken +of the fish on both sides and the images were tagged +with an id based on the RFID. The images are cap- +tured with the dorsal side of the fish facing up. Af- +ter some filtration, a dataset that could be used for +the task was compiled. The final dataset consists +of 2113 images from 513 unique individuals. +As +an added statistic, the mean between the first and +last capture-date of all the individuals is 230 days. +Samples from the dataset can be seen in Figure 1. +3.2 +Individual re-identification +The re-identification system consists of a pipeline +of different components, as illustrated in Figure 2. +The components fall into two categories, a prepro- +cessing part and a re-identification part. As part +of a preprocessing step in the pipeline, the system +takes an image as input and feeds it to a object de- +2 + +Figure 1: Samples from the unprocessed dataset. +tection network to get an image crop, only contain- +ing the fish in the frame. Then a different network, +the direction component, classifies whether the fish +is facing right or left and passes this as metadata. +For the re-identification part, the preprocessed data +is fed to a contrastive learning network that learns +to group embeddings for the same individual to- +gether and different apart. Classification can then +be performed on the embeddings. By storing the +embeddings of all previously observed individuals, +re-identification can be achieved by nearest neigh- +bor methods. +The object detector uses YOLOv5 [11] with an +image size of 416x416, a batch size of 32 and is +trained for 50 epochs. +During training, the net- +work was provided with manually annoted bound- +ing boxes enclosing the fish. +The direction network is an Inception v3 [20] +model with all its weights frozen. A global aver- +age pooling layer, a ReLU activated layer with 32 +neurons, and a sigmoid activated output have been +appended to the network. The dataset used for the +training is the images cropped to only contain the +head. The dataset is manually annotated with the +direction. +The embedding network consists of a CNN model +with a projection head. Its constituent parts were +Object +detector +Direction +classifier +Embedding +Classification +Preprocessing +Re-identification +Figure 2: The network pipeline takes an image of a +fish as input and outputs the id of the individual. +found experimentally. The CNN model is an Incep- +tion v3 model pre-trained on ImageNet, with the +layers after the fourth concatenation layer (layer +46, or 132 if counting activation layers) removed. +Appended at the end is a 2D global average pooling +layer and a 128-dimensional linear projection that +is normalized to the unit hypersphere. +The net- +work diagram is shown in Figure 3. The input size +of the network is 224, and the images are resized +accordingly before being fed into it. The network +utilizes letter-boxing to maintain aspect-ratio. For +the training of the embedding network, the dataset +is split into a training and test set with a test set +fraction of 0.3. +The training of the embedding network uses +gradual unfreezing. The first 100 epochs have the +layers before layer 29 frozen and a learning rate of +0.001, and the next 100 epochs have the layers be- +fore layer 18 frozen and a learning rate of 0.0001 +for a total of 200 epochs. Layer 29 and layer 18 +were selected because they are concatenation bot- +tlenecks in the network architecture (green nodes in +Figure 3). The loss function is online hard triplet +mining with a margin of 1.0. Hard triplet mining is +a technique where loss is only backpropagated for +triplets where the negative is closer to the anchor +than the positive. +Thus, the use of online min- +ing circumvents the need for three identical net- +works with shared weights. The training samples +are randomly applied with image augmentations. +A number between -20 and 20 is added to the hue +3 + +3x +Convolution +MaxPool +AvgPool +Concatenation +GlobalAvg2D +Dense +L2 Normalization +Figure 3: The embedding network utilizes the first part of Inception v3 [20] with a custom projection +head. The dashed line marks where the Inception part ends and the custom part starts. The grey part +is repeated three times. +and saturation. The image is rotated by a fraction +between 0 and 0.1 in either direction, and a scale +transformation between 0 and 0.1 is applied. The +batch size used is 32. +Classification, and by extension re-identification, +is done using a nearest neighbor approach. And in +this case it is useful to define the training set as the +support set and the test set as the query set. Near- +est neighbor classification is non-parametric and +does not need to be trained through optimization. +The training step is simply to feed the support im- +ages through the embedding network and store the +associated embeddings for the inference step. To +classify an image, a query image is fed through the +embedding network and then simply select the class +of the nearest point of the query embedding to the +support set embeddings. Source code for our im- +plementation is available at GitHub1. +3.3 +Method for experiments +The Symphodus melops +have a distinct high- +contrast pattern in the head region (particularly on +the operculum). For this reason, it would be useful +to explore whether the network performs better on +head crops than on crops of the whole body. The +experiment is performed by training and evaluating +1https://github.com/orilan93/SiameseFish +the embedding network on images that are cropped +to either part. +The system can also treat each side of the fish +as different classes, and thus valuable information +can be gained by doing inference on both, and then +combining the results in an ensemble classifier. For +this experiment, the dataset is split up into a left- +sided section and a right-sided section such that +there is a pairwise correspondance between the im- +ages. Two models are trained, where one is only +for left-sided images and the other is only for right- +sided images. The embeddings in the support set is +sorted by the distance to the query image for each +side. The predicted class is then the class which ap- +pears first when both sorted collections are taken +into account. +An experiment to evaluate how well the system is +able to distinguish between a re-sighted individual +and an individual that has never been seen before +was also conducted. A query embedding is consid- +ered a new individual if its distance is greater than +a certain distance away from any support embed- +ding. The query set was split into a test set and +a validation set. +A grid search was used to find +a good distance threshold by maximizing the F1 +score when evaluating the test set. The validation +dataset for this experiment contains 317 samples. +4 + +4 +Results +The metrics we use are accuracy@1, accuracy@5, +and mAP@5. +Accuracy@1 shows the correctness +of the highest ranked category, i.e., the percentages +of the highest predicted class are equal to the true +class. Accuracy@5 shows the correctness of the five +highest ranked categories, i.e., how many of the five +highest classes contain the true class. mAP@5 sim- +ilarly shows the precision of the five highest ranked +categories, i.e., how many of the true categories are +among the five highest ranked categories. +4.1 +Re-identification +The re-identifcation system was evaluated against +both the head and body crop datasets. +Table 1 +presents results from accuracy@1 and accuracy@5 +and shows that the model performs best on the +head crops. Figure 4 shows how the model performs +as the number of accumulated attempts increase. +This approach is essential in practice because, in- +stead of having an unsorted catalog of images to go +through, a professional biologist can go through a +sorted catalog and expect to find the correct indi- +vidual after inspecting the k most promising images +sorted based on the distance measure. The larger +k the higher accuracy, and as the number of at- +tempts are approaching the number of images in +the support set, the accuracy is approaching 100%. +Table 1: Results for re-identification on head and +body crops. +Type +Accuracy@1 +Accuracy@5 +mAP@5 +Head +0.3534 +0.5647 +0.4227 +Body +0.2043 +0.3892 +0.2690 +Table 2 shows four random image samples from +the dataset, together with the image the trained +model predicts is the same individual and the +ground truth. The classification rank and the dis- +tance in the embedding space are also shown. +To gain insight into what the model focuses on +when making its inferences, we present some test +set samples and the accompanying SHAP plot [15] +in Figure 5. The colored area shows that the model +is indeed picking up on the pattern of the fish. +Figure 4: The number of accumulated attempts (k) +needed to attain a certain accuracy (accuracy@k). +4.2 +Ensemble classifier +This experiment shows the results of training a new +model for each side of the fish and then combin- +ing their respective classifications. Table 3 shows +that this strategy can significantly increase perfor- +mance. Note that the direction component, that is +required for the ensemble classifier, yielded an ac- +curacy@1 of 99.38% on the validation set using the +head cropped dataset. +4.3 +New observations +As our previous experiments have shown, +re- +identification works relatively well. We aim at us- +ing this model for distinguishing new individuals +from earlier observed individuals. To identify new +individuals with the model, an embedding distance +threshold needs to be decided. Note that this re- +lates to the distance metric in Table 2. Using grid +search, we found a threshold of 0.820 to yield the +best performance score on the validation set. The +system predicted 95 individuals as new sightings +and got a 62.78% accuracy@1 at this task. +5 +Discussion and conclusion +Our experiments, summarized in Table 4, indicate +that the system performs better on the head crops +of the fish than on the whole body. This is likely +5 + +Table 2: The retrieval rank and euclidean distance between the embedding of a query image and a +correct image. +Query +Predicted +Ground truth +Rank +Distance +1 +0.65 +217 +1.27 +1 +0.61 +1012 +1.46 +Table 3: Ensemble classifier results. +Type +Accuracy@1 +Accuracy@5 +mAP@5 +Left +0.3568 +0.5463 +0.4243 +Right +0.4097 +0.5595 +0.4623 +As pair +0.5286 +0.7533 +0.6140 +Table 4: Summary of results. +Experiment +Result +Metric +Re-identification +0.3534 +Accuracy@1 +Object detector +0.9951 +mAP@0.5 +Direction classifier +0.9937 +Accuracy@1 +Ensemble classifier +0.5286 +Accuracy@1 +New observations +0.6278 +Accuracy@1 +because the pattern on the head is most distinct +and thus an important feature, and this will ap- +pear at a higher resolution for the algorithm when +resizing for the network input size. However, the +drawback here is that the network is exposed to less +information available in the data. +By utilizing the existing system in a new way by +training separate models for each side of the fish, +one can make an ensemble classifier. This method +was tested and gained a considerable improvement +from 35% to 53% accuracy. This shows how im- +portant it is to use all the information available to +make good predictions. +The accuracy of this system is not high enough +for a fully automated system with humans out-of- +the-loop, which is required to replace the need for +physical tags in ecological studies. +However, we +believe that continued collection of data can pro- +duce a dataset that is more temporally balanced +to enable the model to account for the growth and +ageing of the individuals. +Automatization can produce great benefits and +is increasingly being adopted by many industries, +and the field of ecology should be no different. A +successful Re-ID algorithm with high precision can +provide a new method with improved fish welfare, +while also being cheaper (only a camera needed) +and potentially more accurate (no tag loss). +In +the future, we envision that re-ID can be applied +6 + +Figure 5: SHAP plot showing which areas in the +images that are most influential for the decisions of +the model. +directly on live streams from under-water video +cameras, removing the need for capture and han- +dling fish altogether. This would be a revolution- +ary method that can drastically change how we can +collect key information for sustainable conservation +and management of fish and other animals. +Acknowledgements +We thank Torkel Larsen, Anne Berit Skiftesvik, +Ovin Holm, Ylva Vik, Nicolai Aasen, Ben Ellis, +Vegard Omestad Berntsen, and Steve Shema for +assistance in collecting the photos and capture- +recapture data in the field. +This study re- +ceived funding from Centre for Artificial Intelli- +gence Research (CAIR), Centre for Coastal Re- +search (CCR), and Top Research Centre Mecha- +tronics (TRCM) at University of Agder, the Insti- +tute of Marine Research (project 15638-01), and +the Research Council of Norway (CoastVision, +project number 325862, and CreateView, project +number 309784). +References +[1] J. Bromley, J. W. Bentz, L. Bottou, I. Guyon, +Y. LeCun, +C. Moore, +E. S¨ackinger, +and +R. Shah. +Signature verification using a +“siamese” time delay neural network. Inter- +national Journal of Pattern Recognition and +Artificial Intelligence, 7(04):669–688, 1993. +[2] J. Bruslund Haurum, A. Karpova, M. Ped- +ersen, S. Hein Bengtson, and T. B. Moes- +lund. Re-identification of zebrafish using met- +ric learning. In Proceedings of the IEEE/CVF +Winter Conference on Applications of Com- +puter Vision Workshops, pages 1–11, 2020. +doi: 10.1109/WACVW50321.2020.9096922. +[3] E. M. Ditria, E. L. Jinks, and R. M. Con- +nolly. +Automating the analysis of fish graz- +ing behaviour from videos using image classi- +fication and optical flow. Animal Behaviour, +177:31–37, jul 2021. +ISSN 00033472. +doi: +10.1016/j.anbehav.2021.04.018. +[4] M. Goodwin, K. T. Halvorsen, L. Jiao, K. M. +Knausg˚ard, A. H. Martin, M. Moyano, R. A. +Oomen, J. H. Rasmussen, T. K. Sørdalen, and +S. H. Thorbjørnsen. Unlocking the potential +of deep learning for marine ecology: overview, +applications, and outlook. +ICES Journal of +Marine Science, 01 2022. ISSN 1054-3139. doi: +10.1093/icesjms/fsab255. +[5] A. Gupta, E. Bringsdal, N. Salbuvik, K. M. +Knausg˚ard, +and +M. +Goodwin. +An +ac- +curate +convolutional +neural +networks +ap- +proach to wound detection for farmed salmon. +In +International +Conference +on +Engineer- +ing Applications of Neural Networks, pages +139–149. Springer, +2022. +doi: +10.1007/ +978-3-031-08223-8 12. +[6] A. Gupta, E. S. Kalhagen, Ø. L. Olsen, and +M. Goodwin. Hierarchical object detection ap- +plied to fish species: Hierarchical object de- +tection of fish species. Nordic Machine Intel- +ligence, 2(1), 2022. doi: 10.5617/nmi.9452. +[7] K. T. Halvorsen, T. K. Sørdalen, C. Durif, +H. Knutsen, E. M. Olsen, A. B. Skiftesvik, +T. E. Rustand, R. M. Bjelland, and L. A. +Vøllestad. Male-biased sexual size dimorphism +7 + +in the nest building corkwing wrasse (Sympho- +dus melops): Implications for a size regulated +fishery. ICES Journal of Marine Science, 73 +(10):2586–2594, nov 2016. ISSN 10959289. doi: +10.1093/icesjms/fsw135. +[8] K. T. Halvorsen, +T. K. Sørdalen, +L. A. +Vøllestad, A. B. Skiftesvik, S. H. Espeland, +and E. M. Olsen. +Sex- and size-selective +harvesting of corkwing wrasse (Symphodus +melops)—a cleaner fish used in salmonid aqua- +culture. ICES Journal of Marine Science, 74 +(3):660–669, mar 2017. ISSN 1054-3139. doi: +10.1093/icesjms/fsw221. +[9] K. T. Halvorsen, T. Larsen, H. I. Browman, +C. Durif, N. Aasen, L. A. Vøllestad, A. Cresci, +T. K. Sørdalen, R. M. Bjelland, and A. B. +Skiftesvik. +Movement patterns of temperate +wrasses ( Labridae ) within a small marine +protected area. +Journal of Fish Biology, 99 +(4):1513–1518, jul 2021. ISSN 0022-1112. doi: +10.1111/jfb.14825. +[10] E. Hoffer and N. Ailon. Deep metric learning +using triplet network. In International work- +shop on similarity-based pattern recognition, +pages 84–92. Springer, 2015. +doi: 10.1007/ +978-3-319-24261-3 7. +[11] G. +Jocher, +A. +Stoken, +A. +Chaurasia, +J. Borovec, et al. +ultralytics/yolov5: +v6.0 +- YOLOv5n ’Nano’ models, +Roboflow in- +tegration, +TensorFlow +export, +OpenCV +DNN support, +Oct. 2021. +URL https: +//doi.org/10.5281/zenodo.5563715. +[12] K. +M. +Knausg˚ard, +A. +Wiklund, +T. +K. +Sørdalen, K. T. Halvorsen, A. R. Kleiven, +L. +Jiao, +and +M. +Goodwin. +Temperate +fish +detection +and +classification: +a +deep +learning based approach. +Applied Intelli- +gence, 52(6):6988–7001, 2022. doi: 10.1007/ +s10489-020-02154-9. +[13] D. Li, H. Su, K. Jiang, D. Liu, and X. Duan. +Fish face identification based on rotated object +detection: Dataset and exploration. Fishes, 7 +(5):219, 2022. doi: 10.3390/fishes7050219. +[14] V. +Lopez-Vazquez, +J. +M. +Lopez-Guede, +S. +Marini, +E. +Fanelli, +E. +Johnsen, +and +J. Aguzzi. +Video image enhancement and +machine learning pipeline for underwater ani- +mal detection and classification at cabled ob- +servatories. +Sensors, 20(3):726, 2020. +doi: +10.3390/s20030726. +[15] S. M. Lundberg and S.-I. Lee. A unified ap- +proach to interpreting model predictions. Ad- +vances in neural information processing sys- +tems, 30, 2017. +[16] E. Meidell and E. S. Sjøblom. Fishnet: A uni- +fied embedding for salmon recognition. Mas- +ter’s thesis, NTNU, 2019. +[17] O. Moskvyak, F. Maire, F. Dayoub, A. O. +Armstrong, and M. Baktashmotlagh. +Ro- +bust re-identification of manta rays from nat- +ural markings by learning pose invariant em- +beddings. In 2021 Digital Image Computing: +Techniques and Applications (DICTA), pages +1–8. IEEE, 2019. doi: 10.1109/DICTA52665. +2021.9647359. +[18] S. Schneider, G. W. Taylor, S. Linquist, and +S. C. Kremer. +Past, present and future ap- +proaches using computer vision for animal re- +identification from camera trap data. Methods +in Ecology and Evolution, 10(4):461–470, 2019. +doi: 10.1111/2041-210X.13133. +[19] S. Schneider, G. W. Taylor, and S. C. Kre- +mer. Similarity learning networks for animal +individual re-identification-beyond the capa- +bilities of a human observer. +In Proceedings +of the IEEE/CVF winter conference on appli- +cations of computer vision workshops, pages +44–52, 2020. +[20] C. Szegedy, V. Vanhoucke, S. Ioffe, J. Shlens, +and Z. Wojna. Rethinking the inception ar- +chitecture for computer vision. In Proceedings +of the IEEE Conference on Computer Vision +and Pattern Recognition (CVPR), June 2016. +[21] I. Uglem, G. Rosenqvist, and H. S. Wasslavik. +Phenotypic variation between dimorphic males +in corkwing wrasse. Journal of Fish Biology, +57(1):1–14, jul 2000. ISSN 00221112. doi: 10. +1006/jfbi.2000.1283. +[22] B. G. Weinstein. A computer vision for ani- +mal ecology. Journal of Animal Ecology, 87(3): +533–545, 2018. doi: 10.1111/1365-2656.12780. +8 + diff --git a/PNAyT4oBgHgl3EQftfl6/content/tmp_files/load_file.txt b/PNAyT4oBgHgl3EQftfl6/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..153775b2416497611b9a2428a8a9837a9c8bcc8c --- /dev/null +++ b/PNAyT4oBgHgl3EQftfl6/content/tmp_files/load_file.txt @@ -0,0 +1,512 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf,len=511 +page_content='A contrastive learning approach for individual re-identification in a wild fish population Ørjan Langøy Olsen1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Tonje Knutsen Sørdalen2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Morten Goodwin1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ketil Malde3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Kristian Muri Knausg˚ard∗4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' and Kim Tallaksen Halvorsen3 1Centre for Artificial Intelligence Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' University of Agder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Norway 2Centre for Coastal Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' University of Agder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Norway 3Institute of Marine Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ecosystem Acoustics Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bergen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Norway 4Top Research Centre Mechatronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' University of Agder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Norway Abstract In both terrestrial and marine ecology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' physical tag- ging is a frequently used method to study popula- tion dynamics and behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' However, such tag- ging techniques are increasingly being replaced by individual re-identification using image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This paper introduces a contrastive learning- based model for identifying individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The model uses the first parts of the Inception v3 network, supported by a projection head, and we use con- trastive learning to find similar or dissimilar image pairs from a collection of uniform photographs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' We apply this technique for corkwing wrasse, Sympho- dus melops, an ecologically and commercially im- portant fish species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Photos are taken during re- peated catches of the same individuals from a wild population, where the intervals between individual sightings might range from a few days to several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Our model achieves a one-shot accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='35, a 5-shot accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='56, and a 100-shot accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='88, on our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 1 Introduction Physical tagging, using external or internal mark- ings for individual identification, is a widely used method for monitoring terrestrial and aquatic an- imal populations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Information from resightings or recapture of the same individuals can be used ∗Corresponding kristianmk@ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='org to estimate population size, survival and move- ment patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' However, most tagging methods are costly, intrusive, and labor-intensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' To our beneift, many animals have natural markings or morphological features that are unique to individu- als that could be used for photo-identification and replace the need for physical tags [22, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' How- ever, for ecologists, working with fish may mean keeping track of hundreds or potentially thousands of individuals in a population, which makes manual photo-identification challenging, if not impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' For this reason, fully- or semi-automatic tools for re-identification of individuals would be immensely useful for ecologists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Re-identification (re-ID) is different from normal classification in that it is a few-shot learning prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Few-shot problems are characterised by hav- ing few samples per class, but there may be a large or indefinite number of classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' One way to solve such problems is a technique called metric learn- ing, where data is transformed into embeddings of a lower dimension, that clusters points from the same class together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Classification can then be performed on the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Metric learning approaches have been proved to work well for re- identification of animal species [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A crucial ad- vantage with metric learning approaches is that the network does not need to be retrained to be able to add new classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Contrastive learning is a technique that can be used to solve few-shot problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Constrastive learning compares data and identifies whether they are similar or dissimilar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A siamese network [1] is https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='7557/18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='6824 © The author(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Licensee Septentrio Academic Publishing, Tromsø, Norway.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='org/licenses/by/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='0/).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='00596v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='CV] 2 Jan 2023 the most basic form and takes two inputs through the same network with the same shared weights and gets an embedding for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' During training, it tries to predict whether they are of the same class or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A major advantage here is that it does not need to know which class an input belongs to, nor how many classes there are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Triplet networks [10] are an improvement to the siamese network with three inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The goal of this work was to test the applica- bility of image based re-ID analysis for a commer- cially and ecologically important fish species, the corkwing wrasse (Symphodus melops).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The image dataset consists of standardized photos of captures and recaptures of individuals in a wild population, where the time between individual sightings spans from days to several years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The first step is to de- tect a fish in an image with an object detector, followed by a re-identification method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' With high enough precision, computer vision re-ID has the potential to replace physical tagging for individ- ual identification and may be applied in monitoring of survival rates, growth, movement, and popula- tion size, key knowledge for sustainable manage- ment and conservation [18] [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 2 Related works Advancements in machine learning have produced powerful techniques for extracting ecologically im- portant information from image and video data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' For instance, machine learning have successfully been utilized to detect fish wounds [5], count and categorize organisms in digital photos and real-time video [14], [12], identify species, [6], and discover, and count creatures from digital images, [4], and even quantify their behaviour [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Some work on the topic of re-identification of fish has been conducted, but work on wild teleost fish are lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bruslund Haurum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [2] achieved an mAP of 99% on Zebrafish using metric learn- ing with 15 samples per class of 6 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Mei- dell and Sjøblom [16] reports a true positive rate of 96% on 225 thousand images of salmon divided between 715 individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [13] achieved an accuracy of 92% using 3412 images of 10 indi- viduals using their novel FFRNet network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' These studies have in common that they were carried out in captivity and are not using temporally indepen- dent observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In other words, the individuals did not change morphology through growth, matu- ration, senescence, or similar biological processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Moskvyak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [17] used a metric learning ap- proach on a dataset of 1730 images of 120 manta ray individuals and achieved an accuracy@1 of 62% and an accuracy@10 of 97%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 3 Method 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1 Data collection The study species, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' melops, is a commercially and ecologically important species in coastal ecosystems in the Northeastern Atlantic [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This species have two distinct male morphs, colourful large males that build nest and care for the eggs, and smaller sneaker males, with a more brown coloration resem- bling the female morphology (brown and gray) [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The dataset was collected in Austevoll, western Norway, 2018-2021, by catching corkwing wrasse by fyke nets left in the sea overnight and mark- ing all captured individuals with uniquely coded passive integrated transponder (PIT) tags (11 mm tags, RFID Solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The tags were implanted in the abdominal cavity of the fish, see full sampling description in [8] and [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This method enabled us to collect independent observations of each individual across time and for the dataset to encompass changes in the fish’s mor- phology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' At each capture, a few images were taken of the fish on both sides and the images were tagged with an id based on the RFID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The images are cap- tured with the dorsal side of the fish facing up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Af- ter some filtration, a dataset that could be used for the task was compiled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The final dataset consists of 2113 images from 513 unique individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' As an added statistic, the mean between the first and last capture-date of all the individuals is 230 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Samples from the dataset can be seen in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2 Individual re-identification The re-identification system consists of a pipeline of different components, as illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The components fall into two categories, a prepro- cessing part and a re-identification part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' As part of a preprocessing step in the pipeline, the system takes an image as input and feeds it to a object de- 2 Figure 1: Samples from the unprocessed dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' tection network to get an image crop, only contain- ing the fish in the frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Then a different network, the direction component, classifies whether the fish is facing right or left and passes this as metadata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' For the re-identification part, the preprocessed data is fed to a contrastive learning network that learns to group embeddings for the same individual to- gether and different apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Classification can then be performed on the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' By storing the embeddings of all previously observed individuals, re-identification can be achieved by nearest neigh- bor methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The object detector uses YOLOv5 [11] with an image size of 416x416, a batch size of 32 and is trained for 50 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' During training, the net- work was provided with manually annoted bound- ing boxes enclosing the fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The direction network is an Inception v3 [20] model with all its weights frozen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A global aver- age pooling layer, a ReLU activated layer with 32 neurons, and a sigmoid activated output have been appended to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The dataset used for the training is the images cropped to only contain the head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The dataset is manually annotated with the direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The embedding network consists of a CNN model with a projection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Its constituent parts were Object detector Direction classifier Embedding Classification Preprocessing Re-identification Figure 2: The network pipeline takes an image of a fish as input and outputs the id of the individual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' found experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The CNN model is an Incep- tion v3 model pre-trained on ImageNet, with the layers after the fourth concatenation layer (layer 46, or 132 if counting activation layers) removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Appended at the end is a 2D global average pooling layer and a 128-dimensional linear projection that is normalized to the unit hypersphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The net- work diagram is shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The input size of the network is 224, and the images are resized accordingly before being fed into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The network utilizes letter-boxing to maintain aspect-ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' For the training of the embedding network, the dataset is split into a training and test set with a test set fraction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The training of the embedding network uses gradual unfreezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The first 100 epochs have the layers before layer 29 frozen and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='001, and the next 100 epochs have the layers be- fore layer 18 frozen and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='0001 for a total of 200 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Layer 29 and layer 18 were selected because they are concatenation bot- tlenecks in the network architecture (green nodes in Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The loss function is online hard triplet mining with a margin of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Hard triplet mining is a technique where loss is only backpropagated for triplets where the negative is closer to the anchor than the positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Thus, the use of online min- ing circumvents the need for three identical net- works with shared weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The training samples are randomly applied with image augmentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A number between -20 and 20 is added to the hue 3 3x Convolution MaxPool AvgPool Concatenation GlobalAvg2D Dense L2 Normalization Figure 3: The embedding network utilizes the first part of Inception v3 [20] with a custom projection head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The dashed line marks where the Inception part ends and the custom part starts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The grey part is repeated three times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' and saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The image is rotated by a fraction between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1 in either direction, and a scale transformation between 0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1 is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The batch size used is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Classification, and by extension re-identification, is done using a nearest neighbor approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' And in this case it is useful to define the training set as the support set and the test set as the query set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Near- est neighbor classification is non-parametric and does not need to be trained through optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The training step is simply to feed the support im- ages through the embedding network and store the associated embeddings for the inference step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' To classify an image, a query image is fed through the embedding network and then simply select the class of the nearest point of the query embedding to the support set embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Source code for our im- plementation is available at GitHub1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3 Method for experiments The Symphodus melops have a distinct high- contrast pattern in the head region (particularly on the operculum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' For this reason, it would be useful to explore whether the network performs better on head crops than on crops of the whole body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The experiment is performed by training and evaluating 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='com/orilan93/SiameseFish the embedding network on images that are cropped to either part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The system can also treat each side of the fish as different classes, and thus valuable information can be gained by doing inference on both, and then combining the results in an ensemble classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' For this experiment, the dataset is split up into a left- sided section and a right-sided section such that there is a pairwise correspondance between the im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Two models are trained, where one is only for left-sided images and the other is only for right- sided images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The embeddings in the support set is sorted by the distance to the query image for each side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The predicted class is then the class which ap- pears first when both sorted collections are taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' An experiment to evaluate how well the system is able to distinguish between a re-sighted individual and an individual that has never been seen before was also conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A query embedding is consid- ered a new individual if its distance is greater than a certain distance away from any support embed- ding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The query set was split into a test set and a validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A grid search was used to find a good distance threshold by maximizing the F1 score when evaluating the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The validation dataset for this experiment contains 317 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 4 4 Results The metrics we use are accuracy@1, accuracy@5, and mAP@5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Accuracy@1 shows the correctness of the highest ranked category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=', the percentages of the highest predicted class are equal to the true class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Accuracy@5 shows the correctness of the five highest ranked categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=', how many of the five highest classes contain the true class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' mAP@5 sim- ilarly shows the precision of the five highest ranked categories, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=', how many of the true categories are among the five highest ranked categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1 Re-identification The re-identifcation system was evaluated against both the head and body crop datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Table 1 presents results from accuracy@1 and accuracy@5 and shows that the model performs best on the head crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Figure 4 shows how the model performs as the number of accumulated attempts increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This approach is essential in practice because, in- stead of having an unsorted catalog of images to go through, a professional biologist can go through a sorted catalog and expect to find the correct indi- vidual after inspecting the k most promising images sorted based on the distance measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The larger k the higher accuracy, and as the number of at- tempts are approaching the number of images in the support set, the accuracy is approaching 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Table 1: Results for re-identification on head and body crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Type Accuracy@1 Accuracy@5 mAP@5 Head 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5647 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='4227 Body 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2690 Table 2 shows four random image samples from the dataset, together with the image the trained model predicts is the same individual and the ground truth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The classification rank and the dis- tance in the embedding space are also shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' To gain insight into what the model focuses on when making its inferences, we present some test set samples and the accompanying SHAP plot [15] in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The colored area shows that the model is indeed picking up on the pattern of the fish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Figure 4: The number of accumulated attempts (k) needed to attain a certain accuracy (accuracy@k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2 Ensemble classifier This experiment shows the results of training a new model for each side of the fish and then combin- ing their respective classifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Table 3 shows that this strategy can significantly increase perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Note that the direction component, that is required for the ensemble classifier, yielded an ac- curacy@1 of 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='38% on the validation set using the head cropped dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3 New observations As our previous experiments have shown, re- identification works relatively well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' We aim at us- ing this model for distinguishing new individuals from earlier observed individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' To identify new individuals with the model, an embedding distance threshold needs to be decided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Note that this re- lates to the distance metric in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Using grid search, we found a threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='820 to yield the best performance score on the validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The system predicted 95 individuals as new sightings and got a 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='78% accuracy@1 at this task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 5 Discussion and conclusion Our experiments, summarized in Table 4, indicate that the system performs better on the head crops of the fish than on the whole body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This is likely 5 Table 2: The retrieval rank and euclidean distance between the embedding of a query image and a correct image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Query Predicted Ground truth Rank Distance 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='65 217 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='27 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='61 1012 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='46 Table 3: Ensemble classifier results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Type Accuracy@1 Accuracy@5 mAP@5 Left 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='4243 Right 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='4097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='4623 As pair 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='7533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='6140 Table 4: Summary of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Experiment Result Metric Re-identification 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3534 Accuracy@1 Object detector 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='9951 mAP@0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5 Direction classifier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='9937 Accuracy@1 Ensemble classifier 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5286 Accuracy@1 New observations 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='6278 Accuracy@1 because the pattern on the head is most distinct and thus an important feature, and this will ap- pear at a higher resolution for the algorithm when resizing for the network input size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' However, the drawback here is that the network is exposed to less information available in the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' By utilizing the existing system in a new way by training separate models for each side of the fish, one can make an ensemble classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This method was tested and gained a considerable improvement from 35% to 53% accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This shows how im- portant it is to use all the information available to make good predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' The accuracy of this system is not high enough for a fully automated system with humans out-of- the-loop, which is required to replace the need for physical tags in ecological studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' However, we believe that continued collection of data can pro- duce a dataset that is more temporally balanced to enable the model to account for the growth and ageing of the individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Automatization can produce great benefits and is increasingly being adopted by many industries, and the field of ecology should be no different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A successful Re-ID algorithm with high precision can provide a new method with improved fish welfare, while also being cheaper (only a camera needed) and potentially more accurate (no tag loss).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In the future, we envision that re-ID can be applied 6 Figure 5: SHAP plot showing which areas in the images that are most influential for the decisions of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' directly on live streams from under-water video cameras, removing the need for capture and han- dling fish altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This would be a revolution- ary method that can drastically change how we can collect key information for sustainable conservation and management of fish and other animals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Acknowledgements We thank Torkel Larsen, Anne Berit Skiftesvik, Ovin Holm, Ylva Vik, Nicolai Aasen, Ben Ellis, Vegard Omestad Berntsen, and Steve Shema for assistance in collecting the photos and capture- recapture data in the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' This study re- ceived funding from Centre for Artificial Intelli- gence Research (CAIR), Centre for Coastal Re- search (CCR), and Top Research Centre Mecha- tronics (TRCM) at University of Agder, the Insti- tute of Marine Research (project 15638-01), and the Research Council of Norway (CoastVision, project number 325862, and CreateView, project number 309784).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' References [1] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bromley, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bentz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bottou, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Guyon, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' LeCun, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Moore, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' S¨ackinger, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Signature verification using a “siamese” time delay neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Inter- national Journal of Pattern Recognition and Artificial Intelligence, 7(04):669–688, 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [2] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bruslund Haurum, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Karpova, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ped- ersen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Hein Bengtson, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Moes- lund.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Re-identification of zebrafish using met- ric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Winter Conference on Applications of Com- puter Vision Workshops, pages 1–11, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1109/WACVW50321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='9096922.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [3] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ditria, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Jinks, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Con- nolly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Automating the analysis of fish graz- ing behaviour from videos using image classi- fication and optical flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Animal Behaviour, 177:31–37, jul 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ISSN 00033472.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='anbehav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Goodwin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Halvorsen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Jiao, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Knausg˚ard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Martin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Moyano, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Oomen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Rasmussen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sørdalen, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Thorbjørnsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Unlocking the potential of deep learning for marine ecology: overview, applications, and outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ICES Journal of Marine Science, 01 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ISSN 1054-3139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1093/icesjms/fsab255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [5] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Gupta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bringsdal, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Salbuvik, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Knausg˚ard, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Goodwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' An ac- curate convolutional neural networks ap- proach to wound detection for farmed salmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In International Conference on Engineer- ing Applications of Neural Networks, pages 139–149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Springer, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1007/ 978-3-031-08223-8 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Gupta, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Kalhagen, Ø.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Olsen, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Goodwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Hierarchical object detection ap- plied to fish species: Hierarchical object de- tection of fish species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Nordic Machine Intel- ligence, 2(1), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5617/nmi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='9452.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [7] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Halvorsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sørdalen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Durif, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Knutsen, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Olsen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Skiftesvik, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Rustand, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bjelland, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Vøllestad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Male-biased sexual size dimorphism 7 in the nest building corkwing wrasse (Sympho- dus melops): Implications for a size regulated fishery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ICES Journal of Marine Science, 73 (10):2586–2594, nov 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ISSN 10959289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1093/icesjms/fsw135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [8] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Halvorsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sørdalen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Vøllestad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Skiftesvik, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Espeland, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Olsen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sex- and size-selective harvesting of corkwing wrasse (Symphodus melops)—a cleaner fish used in salmonid aqua- culture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ICES Journal of Marine Science, 74 (3):660–669, mar 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ISSN 1054-3139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1093/icesjms/fsw221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [9] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Halvorsen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Larsen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Browman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Durif, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Aasen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Vøllestad, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Cresci, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sørdalen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Bjelland, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Skiftesvik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Movement patterns of temperate wrasses ( Labridae ) within a small marine protected area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Journal of Fish Biology, 99 (4):1513–1518, jul 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ISSN 0022-1112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1111/jfb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='14825.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Hoffer and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ailon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Deep metric learning using triplet network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In International work- shop on similarity-based pattern recognition, pages 84–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Springer, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1007/ 978-3-319-24261-3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [11] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Jocher, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Stoken, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Chaurasia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Borovec, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ultralytics/yolov5: v6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='0 YOLOv5n ’Nano’ models, Roboflow in- tegration, TensorFlow export, OpenCV DNN support, Oct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' URL https: //doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='5563715.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [12] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Knausg˚ard, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Wiklund, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sørdalen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Halvorsen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Kleiven, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Jiao, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Goodwin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Temperate fish detection and classification: a deep learning based approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Applied Intelli- gence, 52(6):6988–7001, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1007/ s10489-020-02154-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [13] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Li, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Su, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Jiang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Liu, and X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Duan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Fish face identification based on rotated object detection: Dataset and exploration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Fishes, 7 (5):219, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3390/fishes7050219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [14] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Lopez-Vazquez, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Lopez-Guede, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Marini, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Fanelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Johnsen, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Aguzzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Video image enhancement and machine learning pipeline for underwater ani- mal detection and classification at cabled ob- servatories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sensors, 20(3):726, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='3390/s20030726.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Lundberg and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A unified ap- proach to interpreting model predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ad- vances in neural information processing sys- tems, 30, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [16] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Meidell and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Sjøblom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Fishnet: A uni- fied embedding for salmon recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Mas- ter’s thesis, NTNU, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [17] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Moskvyak, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Maire, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Dayoub, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Armstrong, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Baktashmotlagh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ro- bust re-identification of manta rays from nat- ural markings by learning pose invariant em- beddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In 2021 Digital Image Computing: Techniques and Applications (DICTA), pages 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' IEEE, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1109/DICTA52665.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='9647359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [18] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Schneider, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Taylor, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Linquist, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Kremer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Past, present and future ap- proaches using computer vision for animal re- identification from camera trap data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Methods in Ecology and Evolution, 10(4):461–470, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1111/2041-210X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='13133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [19] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Schneider, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Taylor, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Kre- mer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Similarity learning networks for animal individual re-identification-beyond the capa- bilities of a human observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF winter conference on appli- cations of computer vision workshops, pages 44–52, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [20] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Szegedy, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Vanhoucke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Ioffe, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Shlens, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Wojna.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Rethinking the inception ar- chitecture for computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), June 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [21] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Uglem, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Rosenqvist, and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Wasslavik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Phenotypic variation between dimorphic males in corkwing wrasse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Journal of Fish Biology, 57(1):1–14, jul 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' ISSN 00221112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 1006/jfbi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Weinstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' A computer vision for ani- mal ecology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' Journal of Animal Ecology, 87(3): 533–545, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='1111/1365-2656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content='12780.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} +page_content=' 8' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAyT4oBgHgl3EQftfl6/content/2301.00596v1.pdf'} diff --git a/PNAzT4oBgHgl3EQfIfvC/content/tmp_files/2301.01064v1.pdf.txt b/PNAzT4oBgHgl3EQfIfvC/content/tmp_files/2301.01064v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..dac84c78803a23b46bd81e79da0cd482f13dcf38 --- /dev/null +++ b/PNAzT4oBgHgl3EQfIfvC/content/tmp_files/2301.01064v1.pdf.txt @@ -0,0 +1,1174 @@ +PIE-QG: Paraphrased Information Extraction for Unsupervised Question +Generation from Small Corpora +Dinesh Nagumothu, Bahadorreza Ofoghi, Guangyan Huang, Peter W. Eklund +School of Information Technology, Deakin University, 221 Burwood Hwy, Burwood 3125 +Victoria, Australia +{dnagumot,b.ofoghi,guangyan.huang,peter.eklund}@deakin.edu.au +Abstract +Supervised Question Answering systems (QA +systems) rely on domain-specific human- +labeled data for training. +Unsupervised +QA systems generate their own question- +answer training pairs, typically using sec- +ondary knowledge sources to achieve this out- +come. +Our approach (called PIE-QG) uses +Open Information Extraction (OpenIE) to gen- +erate synthetic training questions from para- +phrased passages and uses the question-answer +pairs as training data for a language model for +a state-of-the-art QA system based on BERT. +Triples in the form of are extracted from each passage, and +questions are formed with subjects (or objects) +and predicates while objects (or subjects) are +considered as answers. Experimenting on five +extractive QA datasets demonstrates that our +technique achieves on-par performance with +existing state-of-the-art QA systems with the +benefit of being trained on an order of mag- +nitude fewer documents and without any re- +course to external reference data sources. +1 +Introduction +Question Answering systems (QA systems) pro- +vide answers to input questions posed in natural lan- +guage. Answering questions from unstructured text +can be performed using Machine Reading Com- +prehension (MRC). Given a passage, several sen- +tences or a paragraph, and a question posed, the +QA system produces the best suitable answer. Ex- +tractive Question Answering systems (EQA sys- +tems) are a subset of QA systems and involve an +MRC task where the predicted answer is a span +of words from the passage. With pre-trained lan- +guage models (Radford et al., 2018), EQA systems +achieve excellent results, surpassing even human +performance. Pre-trained language models, such +as BERT (Devlin et al., 2019) and GPT (Radford +et al.), can be fine-tuned to perform downstream +tasks such as QA. However, huge amounts of data +are required to train these models for specific do- +mains, making the task labor-intensive, in terms +of the effort required to assemble suitable domain- +specific training data. +A single training instance for an EQA system +dataset requires a question, a passage, and an an- +swer. Domain-relevant documents can be collected +with advanced information retrieval tools, and pas- +sages are formed by splitting documents into sev- +eral related sentences or a paragraph. However, +generating the question and answer pairs, that pro- +vide the training set for the QA system from a given +passage, is considered the most difficult challenge, +an approach known as unsupervised QA (Cui et al., +2004). +Existing unsupervised QA system techniques +such as (Lewis et al., 2019) and (Lyu et al., 2021) +use an out-of-domain dataset for question gen- +eration, namely, they require additional training +sources beyond what can be provided by the target +corpus and a pre-trained generic model. On the +other hand, rule-based QA system methods, those +constrained to generate question-answer pairs from +only the corpus itself, run the risk of generating +questions with high lexical overlap with the pas- +sage, at risk of forcing the model to learn word +matching patterns. The work of (Fabbri et al., 2020) +and (Li et al., 2020) use information retrieval-based +methods, such as elastic search and citation naviga- +tion, to create questions from passages other than +those presented within the target dataset. However, +these methods may not generate sufficient training +questions, especially when the corpus is small and +has no citation or inter-document linking structure. +In this paper, we focus on addressing the limita- +tions of EQA systems using a novel unsupervised +Paraphrased Information Extraction for Question +Generation (PIE-QG) method that generates syn- +thetic training data through the extraction of triples from a given corpus. +We use the original passage to produce question- +arXiv:2301.01064v1 [cs.CL] 3 Jan 2023 + +Paraphrasing +The European Commission , +which is responsible for +regulation competition in the +European Union , is +concerned that these deals +could violate EU antitrust laws. +Open Information Extraction +European Commission is worried that the deals could violate EU antitrust laws. +Q: What is worried that the deals could violate EU antitrust laws? +Answer: The European Commission +Question Formation + + +Context, Question, Answer +Context +Filtered +Figure 1: Question Generation from a context (left) by paraphrasing followed by information extraction using +OpenIE. Note: The text in green indicates the selected answer. +answer training pairs by generating a paraphrased +version of the original passage to avoid lexical over- +lap between the passage and the question-answers. +We adopt Open Information Extraction (Kolluru +et al., 2020) to extract +triples from every sentence of the paraphrased pas- +sage. These triples are rich in semantics and rep- +resent raw facts; therefore, generating question- +answer pairs from triples results in well-formed +and effective training data. Furthermore, many sen- +tences in the passage contribute to generating mean- +ingful extractions, thus helping to pose questions +in different ways from a single passage. An exam- +ple of the question generation process we propose +(called PIE-QG for Paraphrasing, Information Ex- +traction Question Generation) is shown in Figure 1. +The contributions of this paper are as follows: +1. We describe the PIE-QG method in which +paraphrased passages from the original cor- +pus are used to generate question-answer +pairs without reliance on external reference +data sources, such as retrieval-based or inter- +document link navigation methods. Paraphras- +ing passages reduces the effect of lexical over- +lap between the passage and the question. +2. We generate multiple questions from a sin- +gle paraphrased passage by adopting Open +Information Extraction to extract facts, thus in- +creasing the number of question-answer pairs +extracted from the corpus. +We have conducted experiments on four Extrac- +tive QA datasets and demonstrate that the proposed +PIE-QG method achieves comparable performance +in terms of Exact Match (EM) and F1 score while +requiring significantly fewer passages. +The remainder of this paper is organized as fol- +lows. We present related work in Section 2. In Sec- +tion 3, we describe the proposed PIE-QG method. +Section 4 discusses the experimental setup. In Sec- +tion 5, we evaluate the performance of our method. +Section 6 presents the limitations of the proposed +method and Section 7 offers some concluding re- +marks. +2 +Related Work +Pre-trained language models, such as BERT (De- +vlin et al., 2019), can be fine-tuned for downstream +tasks like Extractive QA systems (EQA systems). +A comprehensive natural language (NL) passage, +which might be several sentences or a paragraph of +NL-text, is considered as the context where the +model finds the answer span. +The input ques- +tion and the context are represented as a single +sequence, passed to a pre-trained model and the +answer is predicted by calculating the probabili- +ties of the first and last tokens of the answer span. +Pre-trained language models such as BERT (De- +vlin et al., 2019), T5 transformer (Raffel et al., +2020) and XLNet (Yang et al., 2019), achieve ex- +ceptional performance in EQA systems, however +at the cost of reliance on large human-annotated +supervised datasets. The Stanford Question An- +swering Dataset (SQuAD) (Rajpurkar et al., 2016) +is a widely used dataset for EQA systems. +Lewis et al. (2019), Fabbri et al. (2020), Li et al. +(2020), and Lyu et al. (2021) used randomly sam- +pled passages from Wikipedia, where named en- +tities, or noun chunks, are identified as answers +as these tend to be useful for question answering. +The questions are then formed in natural language +according to the passage and a selected answer +phrase. + +w1, w2, w3,....wn +w1, w2, w3,....wm +w1, w2, w3,..wi,..wm + +........... + + +........... + +Context, Question, Answer +, > +........... + +q1: Wh + v1+o1 + v2 + o2? | a1: s1 +........... +qk: Wh + sk+vk? | ak: ok +Passage +Paraphrased Passage +Coreference Resolution +Open Information Extraction +Named Entity Filtering +Generate Question and Answers Pairs +Merging Triples +Question Formation +Answer +NER +Question +Context +Figure 2: The general pipeline of PIE-QG for question generation using paraphrasing and OpenIE. Note: Blue +indicates named entities, red merged triples with a common subject and green the selected answers. +Unsupervised EQA is achieved using the cloze- +translation method (Lewis et al., 2019) by forming +passage, question-answer triples from a given tar- +get corpus. The answers present in the passages are +masked to form “fill in the blanks” styled questions, +so-called cloze questions. The authors translate +natural language questions using a neural machine +translation (NMT) model trained with different cor- +pora that contain cloze questions and natural ques- +tion pairs. +Questions generated directly from the passage +can only answer simple cloze questions by match- +ing text within the passage, an approach that can +not give correct answers for differently phrased +questions. In an effort to broaden the questions +used to train an EQA system, Fabbri et al. (2020) +generated questions using a similar sentence taken +from a different passage. The actual passage is con- +sidered a query and sentences are retrieved using +elastic search. The most similar sentence, which +contains the answer but excludes the original query +passage, and with less than 95% similarity, to avoid +plagiarised sentences, is used to form the question- +answer pairs. The answer from these sentences is +masked and a question in the form of a “Wh+B+A?” +rule, where “wh” (one of what, when, or who) is +selected based on the answer-entity type (“B” is a +fragment of the sentence that comes after the an- +swer mask, and “A” is the fragment that is present +before the answer mask). +Li et al. (2020) uses citations to form a summary +of the passage. The cited passage is considered +the context, and the sentence where the citation +appeared is used for question generation, to avoid +lexical overlap. The question generation process +involves masking the answer with a cloze mask, +where the mask mentions only the type of the an- +swer entity. The dependency tree for the sentence is +altered in such a way that the cloze mask is brought +to the beginning. The question is then created by +replacing the cloze mask with the suitable “wh” +word, again determined by the type of the answer +entities. +Lyu et al. (2021) perform unsupervised QA by +creating a question generation model from text +summaries. +The model uses dependency trees +and semantic role labels extracted from the sum- +mary to generate a question. A neural encoder- +decoder model is then trained to translate articles to +summary-informed questions. The trained model +is applied to the actual passages to create ques- +tions. However, we consider this method as a trans- +fer learning task rather than unsupervised ques- +tion generation due to its dependency on a text- +summary dataset. Our method compares to Fabbri +et al. (2020) and Li et al. (2020), avoids the sen- +tence and citation-based retrieval, and minimizes +the requirement of having a large corpus to gener- +ate question-answer pairs. +3 +Paraphrased Information Extraction +for Question Generation +To overcome the reliance on external reference data +sources with a large number of passages, we made + +use of OpenIE and paraphrased passages for unsu- +pervised synthetic question generation. The actual +passages are first altered to a paraphrased form +and triples are then +extracted from the paraphrased passages. These +triples, combined with certain heuristics, form +question-answer pairs which are then used along- +side the original passage as context to fine-tune the +QA model. +The pipeline of our proposed EQA question gen- +eration process is illustrated in Figure 2. The steps +in this pipeline are detailed as follows. +(i) Paraphrasing: Question-answer pairs gen- +erated directly from the passage result in inferior +QA system performance, as they produce models +that have little ability to generalize (Fabbri et al., +2020). Paraphrasing is therefore adopted to al- +ter the passage without changing its actual mean- +ing. The intuition behind this is to create ques- +tions from passages that are semantically similar +but lexicographically different from the original +passage. Paraphrasing question-answer pairs them- +selves has been shown to cause semantic drift (Pan +et al., 2021). By contrast, in our approach, the pas- +sage is paraphrased, rather than question-answer +pair. This improves the model’s performance. The +effect of paraphrasing is discussed in Section 5. +(ii) Co-reference resolution: As we aim to +make use of every sentence in the passage to gener- +ate questions, some sentences are ineffective due +to the presence of pronouns (Ma et al., 2021). This +problem is solved by implementing co-reference +resolution, replacing pronouns in the paraphrased +passages with the proper name of the referring +noun. +(iii) Information Extraction: OpenIE is applied +on paraphrased passages to generate extractions in +the form of arguments and relations from natural +language text (Mausam, 2016). Given a sentence +wi in the passage, {w1, w2, w3, ..wN}, OpenIE +generates extractions {T1, T2, T3, ...TM}, where +each extraction is in the form , namely triples. OpenIE is proven to be +an efficient solution for downstream tasks such as +complex question answering (Khot et al., 2017). +(v) Question formation: OpenIE extractions +produced from a passage are used to form ques- +tions as a synthetic training set for QA system fine- +tuning. +(vi) Named entity filtering: Since triples ex- +tracted from a passage have different types of ex- +Algorithm 1: PIE-QG: Question genera- +tion from passages. +Input +:Given a passage P from the corpus +Output :A list of Question-Answer Pairs +P ′ = Paraphrase(P) +CP = Coreference_Resolution(P ′) +T = Open_IE(CP) +named_entities = NER(CP) +Tne = NE_filter(T, named_entities ) +TIF = IdenticalTriple_filter(Tne) +TM = Merge_Triples(TIF ) +TIF = Remove_Merged_Triples(TIF , TM) +QA_Pairs ←− newlist +for tn in TM do +A = Select_Answer(tn) +Q = Wh +for in tn do +Q = Q + relation + object/subject +QA_Pairs ←− append(⟨Q, A, P⟩) +end +end +for in TIF do +Q = Wh + relation + object? +A = subject +QA_Pairs ←− append(⟨Q, A, P⟩) +Q = Wh + subject + relation? +A = object +QA_Pairs ←− append(⟨Q, A, P⟩) +end +return QA_Pairs +tractions, we select the triples that contain named +entities in the answer. In other words, the subject +(or object) is selected as an answer only if it is a +named entity. +(vii) Eliminating duplicate triples: One down- +side of open information extraction is the presence +of duplicate or semantically redundant triples. Gen- +erating separate questions from similar or duplicate +triples causes inferior performance in the EQA sys- +tem model, hence redundant triples are sorted and +the longest triple from the sort is selected as the +single source for final question generation. +(viii) Merging triples: +Questions generated +from the triples using the above methods result +in simple and easy-to-answer questions. For robust +model training, we generate more complex ques- +tions from multiple triples by grouping triples with +the same subject or object. For instance, if there are +two triples of the form {⟨s1, r1, o1⟩, ⟨s2, r2, o2⟩} + +and s1 = s2, we form a question-answer pair with +“Wh + r1 + o1, r2 + o2?” as the question and s1 (or +s2) as the answer. +Each triple extracted from a paraphrased passage +can form two questions with either subject or ob- +ject as an answer. When a subject is selected as +an answer, the question is formulated as “Wh + +relation + object?”. Conversely, when an object +is selected as the answer, the question generated +is of the form “Wh + subject + relation?”. “Wh” +is the question word in these formulations and the +appropriate form is selected from a list, based on +the answer entity type as earlier described. +4 +Experimental Platform +Datasets +The performance of our question gener- +ation method is evaluated in terms of Exact-Match +(EM) and F-1 score using existing EQA datasets, +namely SQuAD v1.1 (Rajpurkar et al., 2016) devel- +opment set, and NewsQA (Trischler et al., 2016), +BioASQ (Tsatsaronis et al., 2015) and DuoRC +(Saha et al., 2018) test sets. SQuAD version 1.1 is +acquired from the official version1 while the Fisch +et al. (2019) published versions of test sets are +considered for NewsQA, BioASQ, and DuoRC. A +more recent SQuAD v2.0 (Rajpurkar et al., 2018) +is considered unsuitable for our experiments as the +synthetic training set does not contain unanswer- +able questions. +Question Generation +We take a relatively small +subset of 30,000 passages from the (Li et al., 2020) +sampled Wikipedia dataset for question generation +and for training the model. The pseudo-code for +the proposed question generation technique is pre- +sented in Algorithm 1. +Some of the questions resulting from this pro- +cess can be grammatically incorrect. We rely on +questions posed to the model during inference to +be in natural language with correct grammar, we +experiment by introducing a grammar correction +module in the pipeline to synthesize syntactically +accurate questions but later removed this due to its +effect discussed in Section 5. +Sourced +Wikipedia +passages +are +trans- +formed +into +paraphrased +passages +with +a +pre-trained model2 based on the PEGASUS +transformer (Zhang et al., 2020). +Pronouns in +the paraphrased passage are replaced with the +1https://rajpurkar.github.io/SQuAD-explorer/ +2https://huggingface.co/tuner007/pegasus_ +paraphrase +nouns they refer to. We used neuralcoref3 for this +purpose, the spaCy implementation of pre-trained +co-referent resolution based on reinforcement +learning (Clark and Manning, 2016). OpenIE6 is +used to extract triples +from the pronoun-replaced paraphrased passages. +OpenIE6 uses Iterative Grid Labeling and is +based on BERT. A spaCy-based named-entity +recognition (NER) module (Honnibal et al., 2020) +is used to generate a list of named-entities from +the passage. +Named-entity recognition (NER) +is particularly helpful for filtering triples and +determining the answer-entity type for appropriate +“wh” word selection. The simplest version of “Wh” +word is selected for a particular named entity based +on Fabbri et al. (2020). Questions generated from +this process are grammatically corrected using a +RoBERTa-based (Liu et al., 2019) grammar cor- +rection module named “GECToR” (Omelianchuk +et al., 2020). All models are applied from the +above-mentioned sources out-of-the-box, namely +with no domain specific fine-tuning. +QA fine-tuning +We use pre-trained BERT mod- +els from Devlin et al. (2019) as the baseline and +fine-tune the models for downstream QA system +tasks with the generated training data. The gen- +erated question, and its context (the actual NL- +passage that contains both the question and its an- +swer), are represented as a single sequence, sepa- +rated by different segment masks and the “[SEP]” +token. The final linear layer of the model is trained, +to identify the start and end spans of the answer, by +computing log-likelihood for each token. All ex- +periments are performed on the uncased version of +the BERT-base model with a learning rate of 3e-5, +a maximum sequence length of 384, a batch size +of 12, a document stride of 128 for 2 epochs, and +a check-point at every 500 steps. The best check- +point was selected by validating each against 5000 +QA pairs randomly sampled from the synthetic +training data. We use the Huggingface4 implemen- +tation for input tokenization, model initialization, +and training. For comparison with the state-of- +the-art EQA models, we also experimented on the +BERT-large whole-word masking version with the +same training data. All models are trained and +validated on a single NVIDIA Tesla A100 GPU. +3https://spacy.io/universe/project/neuralcoref +4https://huggingface.co + +SQuAD1.1 +NewsQA +BioASQ +DuoRC +PIE-QG Heurisitcs +EM +F-1 +EM +F-1 +EM +F-1 +EM +F-1 +Open IE +22.8 +36.5 +13.0 +23.9 +16.6 +24.3 +22.0 +28.5 ++ Paraphrasing +37.7 +53.6 +19.9 +32.3 +20.3 +31.6 +32.6 +40.7 ++ Co-reference Resolution +44.2 +53.4 +21.1 +31.8 +26.5 +34.7 +34.8 +40.4 ++ Named-Entity filter +46.6 +56.5 +21.7 +32.5 +30.3 +36.9 +36.9 +42.4 ++ Filtering Identical Triples +47.5 +57.8 +21.8 +32.2 +30.1 +37.1 +35.1 +41.1 ++ Merging Triples +48.6 +58.7 +21.8 +32.8 +29.6 +37.5 +34.3 +40.1 ++ Grammar Correction +47.1 +56.8 +21.9 +32.3 +29.1 +36.5 +35.2 +40.7 +Table 1: Ablation study of the different techniques used in PIE-QG and their subsequent impacts on the EM and +F-1 after fine-tuning the BERT-base model. Note: Each step represents an incremental upgrade to the previous +step in question generation. +5 +Results and Discussion +The effectiveness of the question-answer data gen- +erated using the PIE-QG method is measured by +training the BERT-base model and evaluating it +against existing EQA development and test sets. +The Exact Match (EM) and F-1 scores are selected +as the metrics to evaluate the effectiveness of each +component in the QA models. The initial set of +questions is created using OpenIE, where the pas- +sage is directly used to form triples and generate +questions as described in Section 3. The intuition +behind using OpenIE is to generate multiple ques- +tions from a single passage. However, as previously +described, such a simple-minded approach suffers +from having pronouns as answers, ungrammatical +questions, and high degrees lexical similarity be- +tween passage and question, making most extracted +triples suitable for word matching only. +Effect of Paraphrasing +Using paraphrased pas- +sages for question generation avoids lexical over- +laps with the passage and improves model perfor- +mance. Ten different paraphrases are generated +for each sentence in the passage using the PEGA- +SUS (Zhang et al., 2020) paraphrasing generation +model. Jensen-Shannon Divergence (JSD) is cal- +culated for each paraphrase against the original +sentence. JSD calculates a divergence score based +on the word distributions between two sentences, a +higher value for JSD accounts for a more different +sentence, while a lower value JSD score represents +higher lexical overlap. In our PIE-QG pipeline, sen- +tences with the highest JSD values are selected for +question generation to make the question syntacti- +cally different. Paraphrasing has a strong positive +effect on the model, improving the EM & F-1 score +by at least 4% and 7% respectively on all evaluation +sets. +Effect of Co-reference Resolution +The pres- +ence of pronouns in passages results in meaning- +less question-answer pairs. For instance, “Vaso +Sepashvili (; born 17 December 1969) is a retired +Georgian professional footballer. He made his pro- +fessional debut in the Soviet Second League B in +1990 for FC Aktyubinets Aktyubinsk” is the pas- +sage. This produces a triple “”. While the +relation and object form a question “Who made his +professional debut in the Soviet Second League B +in 1990 for FC Aktyubinets Aktyubinsk?” with +the subject “He” selected as the answer. The best +answer for this question is found co-referenced +in the previous sentence where the pronoun “He” +refers to “Vaso Sepashvili”. To address this we +alter the passage with co-reference resolution to +replace all pronouns with the referring proper noun. +The above sentence is changed in such a way that +the extracted triple becomes “” +and the ideal answer is selected. Pronouns were re- +placed with their referring nouns using this method +to generate meaningful questions while the original +passage is retained for training the QA model. In +this way, co-referent resolution has a positive im- +pact on the model performance increasing the EM +by 2%-6% across all the sets. +Named-Entity Filtering +As triples are the di- +rect source of training questions, the quality of +triples leads to better training questions for the PIE- +QG model. In general, OpenIE6 returns all possi- +ble triples from a sentence, but selecting suitable +triples, to generate better question-answer pairs, be- +comes important. To assist in identifying the best +set of triples, we filter triples that do not contain +named entities. We use Named Entity Recogni- + +SQuAD1.1 +NewsQA +BioASQ +DuoRC +Fine-tuning Models +EM +F-1 +EM +F-1 +EM +F-1 +EM +F-1 +#Training +Contexts +BERT-base +Sentence Retrieval (Fabbri et al., 2020) +46.1† +56.8† +20.1 +31.1 +29.4 +38.1 +28.8 +35.0 +45K +PIE-QG (Ours) +48.6 +58.7 +21.8 +32.5 +29.6 +37.5 +34.3 +40.1 +20-28K +BERT-large +Cloze Translation (Lewis et al., 2019) † +45.4 +55.6 +19.6 +28.5 +18.9 +27.0 +26.0 +32.6 +782K +RefQA (Li et al., 2020) +57.1 † +66.8 † +27.6 +41.0 +42.0 +54.9 +41.6 +49.7 +178K ++ Iterative Data Refinement +62.5 † +72.6 † +32.1 +45.1 +44.1 +57.4 +45.7 +54.2 +240K +PIE-QG (Ours) +61.2 +72.6 +29.7 +44.1 +43.6 +55.1 +44.6 +52.9 +20-28K +Table 2: Comparison of PIE-QG with state-of-the-art unsupervised QA models. Note: Iterative refinement achieves +the best performance through structural analysis of the corpus via citation and intra-document links, a model that +requires ×8 as many contexts as the PIE-QG model we propose.‘†’ indicates results taken from the existing litera- +ture, and all other figures are evaluated with published synthetic training data (or) pre-trained models. “#Training +Contexts” are measured based on respective published synthetic datasets. Each model uses the same synthetic +training data sourced from Wikipedia for fine-tuning and is evaluated against the standard EQA datasets. +tion (NER) to extract all named entities from the +passage. To become a candidate to be selected for +the question generation process, either the subject +or object from the triple must contain at least one +named entity. This NER filtering method is bene- +ficial to the model, it eliminates many impractical +question-answer pairs from the training set and im- +proves the overall Exact Match (EM) and F-1 score +by 2% except for NewsQA. +Effect of Filtering Identical Triples +Semanti- +cally similar triples are formed using OpenIE6 with +a high degree of lexical overlap. Constructing ques- +tions from these triples causes question duplication +and has the potential to deteriorate model perfor- +mance and even result in over-fitting. To filter sim- +ilar or duplicate triples, each triple is verified with +other triples extracted from the passage to discover +lexical overlaps between them. If a triple formed +as a sentence is a sub-string of another, the shorter +is removed from the training set to avoid the pro- +duction of redundant questions. From Figure 1, +triples such as and convey the same meaning with a high degree +of lexical overlap, hence the former is removed. +Filtering identical triples in this way has a small +but favorable impact on the model as shown in the +ablation summary in Table 1. +Effect of Merging Triples +A subject (or object) +in a passage can exhibit relations to multiple ob- +jects (or subjects). Triples with common subjects +are merged to form complex questions such that +QA model can understand complex relationships. +Merging triples has a small but positive effect on +the model performance improving EM by 1.2% and +F-1 by 0.9% as shown in Table 1. +Effect of Grammar Correction +Questions gen- +erated from the above process often contain gram- +matical errors which can negatively impact model +performances. We experimented with “GECToR” +5, a grammar correction module that tags and cor- +rects input questions with grammar errors. For +instance, the question “What is is worried that the +deals could violate EU antitrust laws?” is formu- +lated. The repeat occurrence of the verb “is” is +an obvious error. The grammar correction mod- +ule alters the question where the final question is +formulated correctly as “What is worried that the +deals could violate EU antitrust laws?”. Based on +heuristics presented in Table 1, all incremental up- +grades until “Merging Triples” improve the model +performance, but Grammar correction does not and +is hence removed from the pipeline. +Effect of Training Data Size +Experiments were +conducted to measure the EM and F-score at dif- +ferent synthetic data sizes to identify the optimal +number of training questions. Figure 3 presents +the results of these experiments and reveals that +PIE-QG achieves peak performance between 30K- +50K training questions using BERT-large model +and begin to over-fit beyond that number. The +same effect is also observed in (Fabbri et al., 2020). +The method to determine the optimal number of +training questions is to split the generated question- +answer pairs into blocks each of 10K. These are +then split into training and validation sets. At fixed +points of 500 training steps, the validation set is +5https://github.com/grammarly/gector + +10 +20 +40 +50 +70 +80 +# Training Questions (in K) +69 +70 +71 +72 +F-score +SQuAD +10 +20 +40 +50 +70 +80 +# Training Questions (in K) +40 +42 +44 +F-score +NewsQA +10 +20 +40 +50 +70 +80 +# Training Questions (in K) +50 +52 +54 +F-score +BioASQ +10 +20 +40 +50 +70 +80 +# Training Questions (in K) +48 +50 +52 +F-score +DuoRC +Figure 3: Evaluation of the PIE-QG model F-score for +different datasets against the number of questions in the +training set using the BERT-large model, the optimal +number for each dataset is in the range 30-50K. +measured against the QA model. This incremen- +tally informs the process of when the model opti- +mizes against the number of question-answer pairs +used to train it. It is observed, shown in Figure 3, +that this occurs for each of the datasets in the range +30-50K. Increasing the number of template-styled +training questions negatively affects the evaluation +performance after a certain point because of mem- +orisation of synthetic data patterns. +Comparison with the State-of-the-Art +Fabbri +et al. (2020) use a BERT-base model as the back- +bone for their experiments while Lewis et al. (2019) +and Li et al. (2020) employed the BERT-large +whole word masking pre-trained model. Questions +generated from the PIE-QG model performed bet- +ter than the information retrieval-based method pre- +sented by Fabbri et al. (2020) and produced an +absolute improvement of 2.5% on EM and 1.9% +on F-1 on the SQuAD 1.1 development set. Com- +paring BERT-large models, the PIE-QG model out- +performs citation retrieval-based RefQA, a method +that involves dependency tree reconstruction. How- +ever, RefQA, which includes a refinement tech- +nique, achieves the best performance, achieving +1-2.5% higher F-1 score than that of PIE-QG, but +at the cost of using 8× more passages and 10× +more training questions. Also, refinements in Re- +fQA are performed on the training data through +iterative cross-validations on the SQuAD 1.1 devel- +opment set, whereas the PIE-QG model does not +involve such a process. The number of passages +and questions used by each method are presented +in detail in Table 3. PIE-QG outperforms retrieval- +System +#Contexts +#Questions +Fabbri et al. (2020) +45K +50K +RefQA Li et al. (2020) +178K +300K ++ IDR +240K +480K +PIE-QG +20-28K +30-50K +Table 3: Comparison of statistics of the synthetic train- +ing data generated by existing unsupervised question +generation methods with PIE-QG. +based question generation on every dataset and pro- +duces comparable performance with RefQA with +8× fewer passages. +To summarise, the experimental results demon- +strate the advantages of the PIE-QG method; +1. Paraphrasing the original passage eliminates +the need of using external knowledge sources +to avoid lexical overlap; +2. Multiple questions generated using OpenIE +with our proposed method minimizes the re- +quirement of a large corpus without having to +sacrifice the performance. +6 +Limitations +The downside of the PIE-QG unsupervised ques- +tion generation pipeline is the use of external mod- +ules like paraphrasing, OpenIE, and NER, which +may not exist in languages other than English. The +quality of question-answer pairs generated to train +the QA model is therefore dependent on the perfor- +mance of these modules on the selected corpus. It +is however anticipated that PIE-QG will perform +similarly well on any English language corpus. It +is future work to apply these modules within the +PIE-QG pipeline to other languages where compa- +rable language-specific models can be sourced and +performance outcomes analyzed. +7 +Conclusion +With no reliance on any external reference cor- +pora, the PIE-QG model uses paraphrasing and +Open Information Extraction (OpenIE) to gener- +ate synthetic training questions for fine-tuning the +language model in a QA system based on BERT. +Triples in the form of are extracted from paraphrased passages, and +questions are formed with subjects (or objects) as +answers. Pronoun co-referents are resolved and +where possible, triples are merged, and duplicate +and highly similar triples are removed. Further- +more, triples that do not contain named entities + +P +Georgia Tech undergraduate programs continue to excel, and I’m pleased that we’ve been able +to maintain this measure of excellence for so long, ” said Interim President and Provost Gary +Schuster. +Q +Who was said that he was pleased that Georgia Tech undergraduate programs continued to excel? +A +Gary Schuster +P +“We’re very upset, very angry,” said Raphael Felli, 35, a U.S.- based attorney and son of executed +Colonel Roger Felli, who was foreign minister in the Acheampong administration. +Q +Who was is an attorney based in the U.S., is the son of executed Colonel Roger Felli? +A +Raphael Felli +P +Liberty and Tyranny Sells a Million. Politics Radio host Mark R. Levin’s bestselling Liberty +and Tyranny : A Conservative Manifesto has sold one million copies, according to publisher +Threshold Editions.....Published on March 24, 2009, Liberty debuted at # 1 on the New York +Times bestseller list. +Q +What sell a million, made it to the New York Times bestsellers list? +A +Liberty +Table 4: Example synthetic question-answer pairs generated using PIE-QG. Note: P represents the passage ex- +tracted from a document. Q and A are the generated question and the selected answer from the passage, respec- +tively. +are eliminated. The PIE-QG pipeline results in a +high-quality question-answer training set that in- +forms the QA model. Using the PIE-QG pipeline +results in a QA model that achieves performance +comparable to the state-of-the-art performance us- +ing significantly fewer passages. It is only narrowly +outperformed by RefQA, an approach that uses it- +erative data refinement, and therefore relies on the +citation structure of corpora and ×10 more training +questions. +References +Kevin Clark and Christopher D. Manning. 2016. Deep +reinforcement learning for mention-ranking corefer- +ence models. +In Proceedings of the 2016 Confer- +ence on Empirical Methods in Natural Language +Processing, pages 2256–2262, Austin, Texas. Asso- +ciation for Computational Linguistics. +Hang Cui, Min-Yen Kan, and Tat-Seng Chua. 2004. +Unsupervised learning of soft patterns for defi- +nitional question answering. +In Proceedings of +the Thirteenth World Wide Web conference (WWW +2004), pages 90–99. +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and +Kristina Toutanova. 2019. +BERT: Pre-training of +deep bidirectional transformers for language under- +standing. +In Proceedings of the 2019 Conference +of the North American Chapter of the Association +for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), +pages 4171–4186, Minneapolis, Minnesota. Associ- +ation for Computational Linguistics. +Alexander Fabbri, Patrick Ng, Zhiguo Wang, Ramesh +Nallapati, and Bing Xiang. 2020. Template-based +question generation from retrieved sentences for im- +proved unsupervised question answering. +In Pro- +ceedings of the 58th Annual Meeting of the Asso- +ciation for Computational Linguistics, pages 4508– +4513, Online. Association for Computational Lin- +guistics. +Adam Fisch, Alon Talmor, Robin Jia, Minjoon Seo, Eu- +nsol Choi, and Danqi Chen. 2019. +MRQA 2019 +shared task: Evaluating generalization in reading +comprehension. +In Proceedings of 2nd Machine +Reading for Reading Comprehension (MRQA) Work- +shop at EMNLP. +Matthew Honnibal, Ines Montani, Sofie Van Lan- +deghem, +and Adriane Boyd. 2020. +spaCy: +Industrial-strength Natural Language Processing in +Python. +Tushar Khot, Ashish Sabharwal, and Peter Clark. 2017. +Answering complex questions using open informa- +tion extraction. In Proceedings of the 55th Annual +Meeting of the Association for Computational Lin- +guistics (Volume 2: Short Papers), pages 311–316, +Vancouver, Canada. Association for Computational +Linguistics. +Keshav Kolluru, Vaibhav Adlakha, Samarth Aggarwal, +Mausam, and Soumen Chakrabarti. 2020. OpenIE6: +Iterative Grid Labeling and Coordination Analysis +for Open Information Extraction. In Proceedings of +the 2020 Conference on Empirical Methods in Nat- +ural Language Processing (EMNLP), pages 3748– +3761, Online. Association for Computational Lin- +guistics. +Patrick Lewis, Ludovic Denoyer, and Sebastian Riedel. +2019. +Unsupervised question answering by cloze + +translation. In Proceedings of the 57th Annual Meet- +ing of the Association for Computational Linguistics, +pages 4896–4910, Florence, Italy. Association for +Computational Linguistics. +Zhongli Li, Wenhui Wang, Li Dong, Furu Wei, and +Ke Xu. 2020. +Harvesting and refining question- +answer pairs for unsupervised QA. In Proceedings +of the 58th Annual Meeting of the Association for +Computational Linguistics, pages 6719–6728, On- +line. Association for Computational Linguistics. +Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- +dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, +Luke Zettlemoyer, and Veselin Stoyanov. 2019. +Roberta: A robustly optimized bert pretraining ap- +proach. arXiv preprint arXiv:1907.11692. +Chenyang Lyu, Lifeng Shang, Yvette Graham, Jennifer +Foster, Xin Jiang, and Qun Liu. 2021. Improving un- +supervised question answering via summarization- +informed question generation. In Proceedings of the +2021 Conference on Empirical Methods in Natural +Language Processing, pages 4134–4148, Online and +Punta Cana, Dominican Republic. Association for +Computational Linguistics. +Kaixin Ma, Filip Ilievski, Jonathan Francis, Yonatan +Bisk, Eric Nyberg, and Alessandro Oltramari. 2021. +Knowledge-driven data construction for zero-shot +evaluation in commonsense question answering. In +Proceedings of the AAAI Conference on Artificial In- +telligence, volume 35, pages 13507–13515. +Mausam Mausam. 2016. Open information extraction +systems and downstream applications. In Proceed- +ings of the twenty-fifth international joint conference +on artificial intelligence, pages 4074–4077. +Kostiantyn Omelianchuk, Vitaliy Atrasevych, Artem +Chernodub, and Oleksandr Skurzhanskyi. 2020. +GECToR – grammatical error correction: Tag, not +rewrite. In Proceedings of the Fifteenth Workshop +on Innovative Use of NLP for Building Educational +Applications, pages 163–170, Seattle, WA, USA → +Online. Association for Computational Linguistics. +Liangming Pan, Wenhu Chen, Wenhan Xiong, Min- +Yen Kan, and William Yang Wang. 2021. Unsuper- +vised multi-hop question answering by question gen- +eration. In Proceedings of the 2021 Conference of +the North American Chapter of the Association for +Computational Linguistics: Human Language Tech- +nologies, pages 5866–5880, Online. Association for +Computational Linguistics. +Alec Radford, Karthik Narasimhan, Tim Salimans, and +Ilya Sutskever. 2018. +Improving language under- +standing by generative pre-training. +Alec Radford, Jeffrey Wu, Rewon Child, David Luan, +Dario Amodei, and Ilya Sutskever. Language mod- +els are unsupervised multitask learners. +Colin Raffel, Noam Shazeer, Adam Roberts, Katherine +Lee, Sharan Narang, Michael Matena, Yanqi Zhou, +Wei Li, Peter J Liu, et al. 2020. Exploring the limits +of transfer learning with a unified text-to-text trans- +former. J. Mach. Learn. Res., 21(140):1–67. +Pranav Rajpurkar, Robin Jia, and Percy Liang. 2018. +Know what you don’t know: Unanswerable ques- +tions for SQuAD. In Proceedings of the 56th An- +nual Meeting of the Association for Computational +Linguistics (Volume 2: Short Papers), pages 784– +789, Melbourne, Australia. Association for Compu- +tational Linguistics. +Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and +Percy Liang. 2016. SQuAD: 100,000+ questions for +machine comprehension of text. In Proceedings of +the 2016 Conference on Empirical Methods in Natu- +ral Language Processing, pages 2383–2392, Austin, +Texas. Association for Computational Linguistics. +Amrita Saha, Rahul Aralikatte, Mitesh M. Khapra, and +Karthik Sankaranarayanan. 2018. DuoRC: Towards +complex language understanding with paraphrased +reading comprehension. In Proceedings of the 56th +Annual Meeting of the Association for Computa- +tional Linguistics (Volume 1: Long Papers), pages +1683–1693, Melbourne, Australia. Association for +Computational Linguistics. +Adam Trischler, Tong Wang, Xingdi (Eric) Yuan, +Justin D. Harris, Alessandro Sordoni, Philip Bach- +man, and Kaheer Suleman. 2016. Newsqa: A ma- +chine comprehension dataset. +George Tsatsaronis, Georgios Balikas, Prodromos +Malakasiotis, Ioannis Partalas, Matthias Zschunke, +Michael R Alvers, Dirk Weissenborn, Anastasia +Krithara, Sergios Petridis, Dimitris Polychronopou- +los, et al. 2015. An overview of the bioasq large- +scale biomedical semantic indexing and question an- +swering competition. BMC bioinformatics, 16(1):1– +28. +Zhilin Yang, Zihang Dai, Yiming Yang, Jaime Car- +bonell, Russ R Salakhutdinov, and Quoc V Le. 2019. +Xlnet: Generalized autoregressive pretraining for +language understanding. Advances in neural infor- +mation processing systems, 32. +Jingqing Zhang, Yao Zhao, Mohammad Saleh, and Pe- +ter Liu. 2020. Pegasus: Pre-training with extracted +gap-sentences for abstractive summarization. In In- +ternational Conference on Machine Learning, pages +11328–11339. PMLR. + diff --git a/PNAzT4oBgHgl3EQfIfvC/content/tmp_files/load_file.txt b/PNAzT4oBgHgl3EQfIfvC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..08006ae6c741742ccc91a31500d369e1b91e0729 --- /dev/null +++ b/PNAzT4oBgHgl3EQfIfvC/content/tmp_files/load_file.txt @@ -0,0 +1,637 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf,len=636 +page_content='PIE-QG: Paraphrased Information Extraction for Unsupervised Question Generation from Small Corpora Dinesh Nagumothu, Bahadorreza Ofoghi, Guangyan Huang, Peter W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Eklund School of Information Technology, Deakin University, 221 Burwood Hwy, Burwood 3125 Victoria, Australia {dnagumot,b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='ofoghi,guangyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='huang,peter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='eklund}@deakin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='au Abstract Supervised Question Answering systems (QA systems) rely on domain-specific human- labeled data for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Unsupervised QA systems generate their own question- answer training pairs, typically using sec- ondary knowledge sources to achieve this out- come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Our approach (called PIE-QG) uses Open Information Extraction (OpenIE) to gen- erate synthetic training questions from para- phrased passages and uses the question-answer pairs as training data for a language model for a state-of-the-art QA system based on BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Triples in the form of are extracted from each passage, and questions are formed with subjects (or objects) and predicates while objects (or subjects) are considered as answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Experimenting on five extractive QA datasets demonstrates that our technique achieves on-par performance with existing state-of-the-art QA systems with the benefit of being trained on an order of mag- nitude fewer documents and without any re- course to external reference data sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' 1 Introduction Question Answering systems (QA systems) pro- vide answers to input questions posed in natural lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Answering questions from unstructured text can be performed using Machine Reading Com- prehension (MRC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Given a passage, several sen- tences or a paragraph, and a question posed, the QA system produces the best suitable answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Ex- tractive Question Answering systems (EQA sys- tems) are a subset of QA systems and involve an MRC task where the predicted answer is a span of words from the passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' With pre-trained lan- guage models (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2018), EQA systems achieve excellent results, surpassing even human performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Pre-trained language models, such as BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2019) and GPT (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' ), can be fine-tuned to perform downstream tasks such as QA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' However, huge amounts of data are required to train these models for specific do- mains, making the task labor-intensive, in terms of the effort required to assemble suitable domain- specific training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' A single training instance for an EQA system dataset requires a question, a passage, and an an- swer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Domain-relevant documents can be collected with advanced information retrieval tools, and pas- sages are formed by splitting documents into sev- eral related sentences or a paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' However, generating the question and answer pairs, that pro- vide the training set for the QA system from a given passage, is considered the most difficult challenge, an approach known as unsupervised QA (Cui et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Existing unsupervised QA system techniques such as (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2019) and (Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2021) use an out-of-domain dataset for question gen- eration, namely, they require additional training sources beyond what can be provided by the target corpus and a pre-trained generic model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' On the other hand, rule-based QA system methods, those constrained to generate question-answer pairs from only the corpus itself, run the risk of generating questions with high lexical overlap with the pas- sage, at risk of forcing the model to learn word matching patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The work of (Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2020) and (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2020) use information retrieval-based methods, such as elastic search and citation naviga- tion, to create questions from passages other than those presented within the target dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' However, these methods may not generate sufficient training questions, especially when the corpus is small and has no citation or inter-document linking structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' In this paper, we focus on addressing the limita- tions of EQA systems using a novel unsupervised Paraphrased Information Extraction for Question Generation (PIE-QG) method that generates syn- thetic training data through the extraction of triples from a given corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' We use the original passage to produce question- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='01064v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='CL] 3 Jan 2023 Paraphrasing The European Commission , which is responsible for regulation competition in the European Union , is concerned that these deals could violate EU antitrust laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Open Information Extraction European Commission is worried that the deals could violate EU antitrust laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Q: What is worried that the deals could violate EU antitrust laws?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Answer: The European Commission Question Formation Context, Question, Answer Context Filtered Figure 1: Question Generation from a context (left) by paraphrasing followed by information extraction using OpenIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Note: The text in green indicates the selected answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' answer training pairs by generating a paraphrased version of the original passage to avoid lexical over- lap between the passage and the question-answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' We adopt Open Information Extraction (Kolluru et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2020) to extract triples from every sentence of the paraphrased pas- sage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' These triples are rich in semantics and rep- resent raw facts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' therefore, generating question- answer pairs from triples results in well-formed and effective training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Furthermore, many sen- tences in the passage contribute to generating mean- ingful extractions, thus helping to pose questions in different ways from a single passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' An exam- ple of the question generation process we propose (called PIE-QG for Paraphrasing, Information Ex- traction Question Generation) is shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The contributions of this paper are as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' We describe the PIE-QG method in which paraphrased passages from the original cor- pus are used to generate question-answer pairs without reliance on external reference data sources, such as retrieval-based or inter- document link navigation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Paraphras- ing passages reduces the effect of lexical over- lap between the passage and the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' We generate multiple questions from a sin- gle paraphrased passage by adopting Open Information Extraction to extract facts, thus in- creasing the number of question-answer pairs extracted from the corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' We have conducted experiments on four Extrac- tive QA datasets and demonstrate that the proposed PIE-QG method achieves comparable performance in terms of Exact Match (EM) and F1 score while requiring significantly fewer passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The remainder of this paper is organized as fol- lows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' We present related work in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' In Sec- tion 3, we describe the proposed PIE-QG method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Section 4 discusses the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' In Sec- tion 5, we evaluate the performance of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Section 6 presents the limitations of the proposed method and Section 7 offers some concluding re- marks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' 2 Related Work Pre-trained language models, such as BERT (De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2019), can be fine-tuned for downstream tasks like Extractive QA systems (EQA systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' A comprehensive natural language (NL) passage, which might be several sentences or a paragraph of NL-text, is considered as the context where the model finds the answer span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The input ques- tion and the context are represented as a single sequence, passed to a pre-trained model and the answer is predicted by calculating the probabili- ties of the first and last tokens of the answer span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Pre-trained language models such as BERT (De- vlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2019), T5 transformer (Raffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2020) and XLNet (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2019), achieve ex- ceptional performance in EQA systems, however at the cost of reliance on large human-annotated supervised datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The Stanford Question An- swering Dataset (SQuAD) (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2016) is a widely used dataset for EQA systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2019), Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2020), Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2020), and Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2021) used randomly sam- pled passages from Wikipedia, where named en- tities, or noun chunks, are identified as answers as these tend to be useful for question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The questions are then formed in natural language according to the passage and a selected answer phrase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' w1, w2, w3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='.wn w1, w2, w3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='.wm w1, w2, w3,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='.wi,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='.wm .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Context, Question, Answer , > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' q1: Wh + v1+o1 + v2 + o2?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' | a1: s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' qk: Wh + sk+vk?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' | ak: ok Passage Paraphrased Passage Coreference Resolution Open Information Extraction Named Entity Filtering Generate Question and Answers Pairs Merging Triples Question Formation Answer NER Question Context Figure 2: The general pipeline of PIE-QG for question generation using paraphrasing and OpenIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Note: Blue indicates named entities, red merged triples with a common subject and green the selected answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Unsupervised EQA is achieved using the cloze- translation method (Lewis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2019) by forming passage, question-answer triples from a given tar- get corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The answers present in the passages are masked to form “fill in the blanks” styled questions, so-called cloze questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The authors translate natural language questions using a neural machine translation (NMT) model trained with different cor- pora that contain cloze questions and natural ques- tion pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Questions generated directly from the passage can only answer simple cloze questions by match- ing text within the passage, an approach that can not give correct answers for differently phrased questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' In an effort to broaden the questions used to train an EQA system, Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2020) generated questions using a similar sentence taken from a different passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The actual passage is con- sidered a query and sentences are retrieved using elastic search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The most similar sentence, which contains the answer but excludes the original query passage, and with less than 95% similarity, to avoid plagiarised sentences, is used to form the question- answer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The answer from these sentences is masked and a question in the form of a “Wh+B+A?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' rule, where “wh” (one of what, when, or who) is selected based on the answer-entity type (“B” is a fragment of the sentence that comes after the an- swer mask, and “A” is the fragment that is present before the answer mask).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2020) uses citations to form a summary of the passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The cited passage is considered the context, and the sentence where the citation appeared is used for question generation, to avoid lexical overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The question generation process involves masking the answer with a cloze mask, where the mask mentions only the type of the an- swer entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The dependency tree for the sentence is altered in such a way that the cloze mask is brought to the beginning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The question is then created by replacing the cloze mask with the suitable “wh” word, again determined by the type of the answer entities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Lyu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2021) perform unsupervised QA by creating a question generation model from text summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The model uses dependency trees and semantic role labels extracted from the sum- mary to generate a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' A neural encoder- decoder model is then trained to translate articles to summary-informed questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The trained model is applied to the actual passages to create ques- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' However, we consider this method as a trans- fer learning task rather than unsupervised ques- tion generation due to its dependency on a text- summary dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Our method compares to Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2020) and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (2020), avoids the sen- tence and citation-based retrieval, and minimizes the requirement of having a large corpus to gener- ate question-answer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' 3 Paraphrased Information Extraction for Question Generation To overcome the reliance on external reference data sources with a large number of passages, we made use of OpenIE and paraphrased passages for unsu- pervised synthetic question generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The actual passages are first altered to a paraphrased form and triples are then extracted from the paraphrased passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' These triples, combined with certain heuristics, form question-answer pairs which are then used along- side the original passage as context to fine-tune the QA model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The pipeline of our proposed EQA question gen- eration process is illustrated in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The steps in this pipeline are detailed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (i) Paraphrasing: Question-answer pairs gen- erated directly from the passage result in inferior QA system performance, as they produce models that have little ability to generalize (Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Paraphrasing is therefore adopted to al- ter the passage without changing its actual mean- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The intuition behind this is to create ques- tions from passages that are semantically similar but lexicographically different from the original passage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Paraphrasing question-answer pairs them- selves has been shown to cause semantic drift (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' By contrast, in our approach, the pas- sage is paraphrased, rather than question-answer pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' This improves the model’s performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' The effect of paraphrasing is discussed in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (ii) Co-reference resolution: As we aim to make use of every sentence in the passage to gener- ate questions, some sentences are ineffective due to the presence of pronouns (Ma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' This problem is solved by implementing co-reference resolution, replacing pronouns in the paraphrased passages with the proper name of the referring noun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (iii) Information Extraction: OpenIE is applied on paraphrased passages to generate extractions in the form of arguments and relations from natural language text (Mausam, 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Given a sentence wi in the passage, {w1, w2, w3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='.wN}, OpenIE generates extractions {T1, T2, T3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content='TM}, where each extraction is in the form , namely triples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' OpenIE is proven to be an efficient solution for downstream tasks such as complex question answering (Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (v) Question formation: OpenIE extractions produced from a passage are used to form ques- tions as a synthetic training set for QA system fine- tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' (vi) Named entity filtering: Since triples ex- tracted from a passage have different types of ex- Algorithm 1: PIE-QG: Question genera- tion from passages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' Input :Given a passage P from the corpus Output :A list of Question-Answer Pairs P ′ = Paraphrase(P) CP = Coreference_Resolution(P ′) T = Open_IE(CP) named_entities = NER(CP) Tne = NE_filter(T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' named_entities ) TIF = IdenticalTriple_filter(Tne) TM = Merge_Triples(TIF ) TIF = Remove_Merged_Triples(TIF ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNAzT4oBgHgl3EQfIfvC/content/2301.01064v1.pdf'} +page_content=' TM) QA_Pairs ←− newlist for tn in TM do A = Select_Answer(tn) Q = Wh for rg, and +T = +� +1 − r/rge +r +2rg cosh +� ct +2rg +� +, +(4) +X = +� +1 − r/rge +r +2rg sinh +� ct +2rg +� +, +(5) +for 0 < r < rg. +In these coordinates, the BH center +(r = 0) is a space-like line T = +√ +1 + X2. +This line +sets a boundary of the Schwarzschild spacetime in the +Kruskal-Szekeres coordinates (see Fig. 1). The boundary +appears because coordinate transformation (2)-(5) maps +the region −∞ < t < ∞, r ≥ 0 into T ≤ +√ +1 + X2, +T ≥ −X. In the Kruskal-Szekeres coordinates, in 1+1 +dimension, the Schwarzschild metric +ds2 = 4r3 +g +r e−r/rg � +dT 2 − dX2� +(6) +is conformally invariant to the Minkowski metric and, +thus, a massless scalar field φ obeys the same wave equa- +tion as in the Minkowski spacetime +� ∂2 +∂T 2 − ∂2 +∂X2 +� +φ = 0. +(7) +For the present problem only the right-moving field in +Fig. 1 is important. It is convenient to describe such a +field using Rindler modes [20] +φ1Ω(T, X) = (X − T )iΩθ(X − T ), +(8) +φ2Ω(T, X) = (T − X)−iΩθ(T − X), +(9) +where Ω > 0 is a parameter, and θ is the Heaviside step +function. Rindler modes φ1Ω and φ2Ω are solutions of +the wave equation (7), and for Ω > 0 have positive norm +(defined as the Klein–Gordon inner product). The mode +functions (8) and (9) are non-zero outside and inside the +BH event horizon (line T = X) respectively (see Fig. +1). These two regions are causally disconnected for the +right-moving field. Annihilation operators of the Rindler +photons we denote as ˆb1Ω and ˆb2Ω. +It is believed that, to a good approximation, Unruh +vacuum |0U⟩ describes state of the field produced by a +gravitational collapse of a star into a BH. In this state, +there are no left-moving Rindler photons and no right- +moving Minkowski photons [3]. +That is, Unruh vac- +uum is Rindler vacuum for the left-moving photons and +Minkowski vacuum for the right-moving photons. +In +terms of the right-moving Rindler photons, which are +relevant for the present discussion, the Unruh vacuum is +a squeezed state [21] +|0U⟩ = +� +Ω>0 +� +1 − γ2eγˆb† +1Ωˆb† +2Ω |0R⟩ , +(10) +where +γ = e−πΩ, +(11) +|0R⟩ refers to the Rindler vacuum, ˆb† +1Ω and ˆb† +2Ω are cre- +ation operators of the right-moving Rindler photons. + +=√1+X2 +Schwarzschild spacetime +Image +T +W +1 +2 +Boundary3 +That is, Unruh vacuum is filled with the right-moving +Rindler photons, but it looks empty if the right-moving +field is described by means of the Minkowski photons. +If a hypothetical observer is located at the BH center +(r = 0), then Schwarzschild coordinate t is the proper +time for such observer. Recall that proper time of an +object is the coordinate which changes in the object’s +frame. If the object is held fixed at r = const then t is +the proper time. In the region 0 < r < rg, it is phys- +ically impossible to hold particles fixed at r = const, +that’s why we use the word “hypothetical”. Eqs. (4), (5) +and (9) yield that at the BH center the non-zero Rindler +mode φ2Ω oscillates as a function of t as φ2Ω ∝ eiΩct/2rg. +That is, from the observer’s perspective the Rindler pho- +tons behave as if they have negative frequency −Ωc/2rg +[19]. Hence, in the Unruh vacuum, there is a flux of the +negative frequency (energy) Rindler photons into the BH +center. Absorption of such photons near the BH center +decreases energy (mass) of the BH, leading to BH evap- +oration. +If an observer is held fixed outside the event horizon +at a constant Schwarzschild coordinate r, then at the ob- +server’s location the non-zero Rindler modes φ1Ω oscil- +late as φ1Ω ∝ e−iΩct/2rg, where t is the observer’s proper +time. That is, from the external observer perspective the +Rindler photons behave as if they have positive frequency +ν = Ωc +2rg +(12) +and, thus, they can excite a detector. Photons φ1Ω prop- +agate away from the BH. +For simplicity, we will assume that the field has only +modes with one “frequency” Ω. We denote such Rindler +modes as φ1 and φ2. Then Unruh vacuum can be written +as +|0U⟩ = +� +1 − γ2eγˆb† +1ˆb† +2 |0R⟩ = +� +1 − γ2 +∞ +� +n=0 +γn |nn⟩ , +(13) +where |nn⟩ is a state with n Rindler photons in the modes +φ1 and φ2. +If modes φ1 and φ2 are considered separately, then +tracing over or absorbing one of the modes leaves the +remaining mode in a thermal state. Namely, if we trace +over the Rindler modes under the event horizon φ2, which +are not accessible to the external observer, the reduced +density operator for the field φ1 is thermal +ˆρ1 = Tr2 (|0U⟩ ⟨0U|) = +� +1 − γ2� ∞ +� +n=0 +γ2n |n⟩ ⟨n| +(14) +with the average number of photons +¯n1 = +γ2 +1 − γ2 . +(15) +Thus, an observer held fixed outside the BH horizon +feels thermal radiation coming out from the BH, which is +known as Hawking radiation. Using Eqs. (11), (12) and +(15), one can write ¯n1 as a Planck factor +¯n1 = +1 +e +4πrgν +c +− 1 += +1 +e +ℏν +kB TH − 1 +(16) +with the Hawking temperature TH = ℏc/4πkBrg. +FIG. 2. +Light rays of Rindler photons φ1 and φ2 in +Schwarzschild coordinates. +Figure 2 shows light rays of Rindler photons (8) and +(9) in the Schwarzschild coordinates. It looks like the +negative (φ2) and positive (φ1) frequency Rindler pho- +tons are generated at the event horizon. This is consis- +tent with the interpretation of the Hawking radiation as +a continuous creation of particle-antiparticle pairs near +the event horizon, with one carrying positive energy to +infinity and the other carrying negative energy into the +BH [22]. Calculations of the energy-momentum tensor +for the field near an evaporating BH directly show that +there is a negative-energy flux into the BH center and a +positive-energy flux far away from the BH [23]. +III. +MODEL OF AN EVAPORATING BLACK +HOLE TAKING INTO ACCOUNT +NON-UNITARY PHOTON ABSORPTION AT +THE CENTER +According to Eq. (7), in the Kruskal-Szekeres coordi- +nates the field evolves following the same wave equation +as in Minkowski spacetime. For the latter, absorption or +emission of photons in the region T > X cannot affect the +state of the field in the region T < X. However, the BH +spacetime has a space-like boundary at T = +√ +1 + X2. +We will show below that absorption of photons at the +boundary changes the state of the field outside the event + +3 +5 +2 +Event horizon +2 +5 +0 +2 +5 +0.0 +Singularity +tc/4 +horizon and radiation of the evaporating BH is not ther- +mal. We will assume that Unruh vacuum is the state of +the field only at the onset of evaporation and calculate +how the field evolves. +According to general relativity, spacetime disappears +at the BH center (spacetime boundary). It is assumed +that matter disappears together with the spacetime, but +state of matter (mass, angular momentum, etc.) +is +recorded in the gravitational field near the BH center. +This process transfers characteristics of the accreting +matter into the BH internal gravitational field. +The worldlines of the Rindler photons ˆb2 terminate +at the spacetime boundary (see Fig. 1). But we can’t +just say that photons disappear. +One should describe +this process quantum mechanically using a Hamiltonian. +Space-like boundary breaks the symmetry between emis- +sion and absorption of Rindler photons ˆb2. Namely, if +backward in time propagation is not allowed, Rindler +photons ˆb2 cannot be emitted at the boundary because +such a process means emission of particles outside the +spacetime. +Next we consider a simple toy model of BH evaporation +modeling the boundary as a set of harmonic oscillators +that totally absorb the ingoing field. The oscillators fol- +low the worldline of the boundary which is not geodesic. +We do not associate the oscillators with ordinary par- +ticles. Rather, the oscillators provide a physical model +of the gravitational field near the BH center that car- +ries information about the state of the BH interior. In +our model, the oscillator’s energy is the origin of the BH +mass. +As we showed above, from the oscillator’s per- +spective, Rindler photons have negative energy. Thus, +absorption of Rindler photons reduces the energy of the +oscillators (BH mass decreases). +Since oscillators are under the BH horizon, they can +interact only with photons ˆb2. +In the toy model, the +interaction Hamiltonian describing BH evaporation reads +ˆV2(t) = gˆσe−iωtφ2(t)ˆb2, +(17) +where ˆσ is the lowering operator for the oscillator of fre- +quency ω, g is the coupling constant and the field mode +function φ2(T, X) is taken at the location of the oscilla- +tor φ2(t) = φ2(T (t, 0), X(t, 0)) = ei cΩt +2rg . In Eq. (17), t +is the proper time of the oscillator which coincides with +the Schwarzschild coordinate t because oscillators are lo- +cated at fix r = 0. Since oscillators cannot emit Rindler +photons, the Hamiltonian (17) is not Hermitian. +We will consider evolution of the system as a function +of the oscillator proper time t. Schr¨odinger equation for +the system’s state vector +iℏ ∂ +∂t |ψ(t)⟩ = ˆV2(t) |ψ(t)⟩ +yields +|ψ(t)⟩ = eβ(t)ˆσˆb2 |0U⟩ |A⟩ , +(18) +where |0U⟩ and |A⟩ are the initial state vectors of the +field and the oscillator, and +β(t) = −ig +ℏ +� t +0 +dt′e−iωt′φ2(t′) = −ig +ℏ +� t +0 +dt′ei +� +cΩ +2rg −ω +� +t′ +. +Plug |0U⟩ in Eq. (18) gives +|ψ(t)⟩ = +� +1 − γ2eβ(t)ˆσˆb2eγˆb† +1ˆb† +2 |0R⟩ |A⟩ . +(19) +Using the Baker–Hausdorff formula e ˆ +Ae ˆ +B = e[ ˆ +A, ˆ +B]e ˆ +Be ˆ +A, +we obtain +|ψ(t)⟩ = +� +1 − γ2eγβ(t)ˆσˆb† +1eγˆb† +1ˆb† +2eβ(t)ˆσˆb2 |0R⟩ |A⟩ , +or +|ψ(t)⟩ = eγβ(t)ˆσˆb† +1 |0U⟩ |A⟩ . +(20) +Equation (20) shows that non-unitary field absorption +at the spacetime boundary yields generation of photons +outside the BH event horizon (into the Rindler mode +1). +Taking time derivative of Eq. +(20) leads to the +Schr¨odinger equation with the interaction Hamiltonian +ˆV1(t) = γgˆσe−iωtφ∗ +1(t)ˆb† +1, +where we used φ2(t) = φ∗ +1(t) = φ∗ +1(−T (t, 0), X(t, 0)). +That is, the process looks like as if there is a mir- +ror “image” of the oscillator located along the line T = +− +√ +1 + X2 (see Fig. 1) which is coupled with the exter- +nal mode φ1 with a reduced coupling constant γg. The +oscillator’s image produces field outside the event hori- +zon which propagates away from the BH. Such field is +not thermal. E.g., if the oscillator is in a coherent state, +the generated field is coherent. The information stored +in the oscillators is recorded in the outgoing field. +BH radiation is not thermal because evolution of the +field under the horizon is described by the non-Hermitian +Hamiltonian (17). Indeed, if the Hamiltonian would be +Hermitian and depends only on ˆb2 and ˆb† +2, the Heisenberg +equation of motion for the operator ˆb1(t) +dˆb1(t) +dt += i +ℏ +� +ˆH†ˆb1(t) − ˆb1(t) ˆH +� += i +ℏ +� +ˆH†(t) − ˆH(t) +� +ˆb1 +(21) +would yield ˆb1(t) = const. That is field outside the BH +event horizon would not change. However, if ˆH† ̸= ˆH, +the right-hand-side of Eq. (21) is no longer zero and the +external field can be altered. +In the present model of BH evaporation the von Neu- +mann entropy is preserved. Namely, since evolution of +the system is described by a Hamiltonian, the system re- +mains in a pure state and, thus, the net entropy remains +equal to zero. +This is true even if the Hamiltonian is +not Hermitian. For the latter, the system’s state vector +should be normalized such that ⟨ψ |ψ⟩ = 1. +The toy model Hamiltonian (17) explains why non- +unitary absorption of photons at the BH center alters + +5 +radiation outside the BH. However, it does not describe +the system’s dynamics correctly. The point is that, non- +Hermitian Hamiltonians don’t preserve the expectation +value of an operator ˆQ with which they commute. This +is the reason why the norm of the state vector is not +conserved (in this case ˆQ = 1). To incorporate a con- +servation law +� +ˆQ +� += const into the model, we must +replace the non-Hermitian Hamiltonian ˆH with a con- +strained Hamiltonian [24, 25] +ˆH − λ(t) ˆQ, +(22) +where λ(t) is a Lagrange multiplier whose value is to be +chosen so as to honor the constraint condition +� +ˆQ +� += +const. +We will impose a constraint that during BH evapora- +tion the average energy is conserved. Operators describ- +ing conserved quantities must commute with the Hamil- +tonian. +Such “energy” operators commuting with the +Hamiltonian (17) are +ˆσ†ˆσ − ˆb† +2ˆb2, and ˆb† +1ˆb1, +and the constraints read +� +ˆσ†ˆσ − ˆb† +2ˆb2 +� += const and +� +ˆb† +1ˆb1 +� += const. +(23) +The constrained interaction Hamiltonian is +ˆV (t) = gˆσe−iωtφ2(t)ˆb2 + iℏ ˆC(t), +(24) +where +ˆC(t) = ˙µ1(t)ˆb† +1ˆb1 + ˙µ2(t) +� +ˆσ†ˆσ − ˆb† +2ˆb2 +� ++ ˙µ3(t) +and the dot denotes derivative over t. The latter is in- +troduced for convenience. We assume that the resonance +condition ω = cΩ/2rg is satisfied, which yields +ˆV (t) = gˆσˆb2 + iℏ ˆC(t). +(25) +The Lagrange multiplier ˙µ3(t) takes into account the nor- +malization condition ⟨ψ |ψ⟩ = 1. For the present prob- +lem, Lagrange multipliers ˙µ1,2,3(t) are real functions. +We assume that initially the oscillator is in a coher- +ent state |A⟩ and the field is in the Unruh vacuum |0U⟩. +Schr¨odinger equation with the constrained Hamiltonian +(25) yields (see Appendix A and B) +|ψ(t)⟩ = N(t)e− i +ℏ γgAeµ1(t)tˆb† +1eeµ1(t)−µ2(t)γˆb† +1ˆb† +2 |0R⟩ +���eµ2(t)A +� +, +(26) +where N(t) is a normalization factor and the Lagrange +multipliers are obtained from the constraint equations +e2µ2A2 − +˜γ2 +1 − ˜γ2 − e2µ1˜γ2 (γΛt)2 +(1 − ˜γ2)2 += A2 − +γ2 +1 − γ2 , (27) +˜γ2 +1 − ˜γ2 + e2µ1 (γΛt)2 +(1 − ˜γ2)2 += +γ2 +1 − γ2 , +(28) +where ˜γ = eµ1−µ2γ and Λ = gA/ℏ is the Rabi frequency. +For t → ∞, Eqs. (26)-(28) give +|ψ(∞)⟩ = Ne +− +iγ +√ +1−γ2 ˆb† +1 |0R⟩ |A∞⟩ , +(29) +where +A2 +∞ = A2 − +γ2 +1 − γ2 +(30) +is the mean number of oscillator excitations in the final +state. The present model of the spacetime boundary is +self-consistent if the oscillators absorb all ingoing pho- +tons, which implies A2 +∞ > 0. +Otherwise, photon flux +through the boundary would be nonzero. +Eq. (29) shows that the final state of the field is the +Rindler vacuum for photons ˆb2 and a coherent state for +photons ˆb1 with the average photon number γ2/(1 − γ2). +The oscillator remains in the coherent state, but the oscil- +lator’s mean excitation number is reduced by an amount +γ2/(1 − γ2) due to absorption of all ˆb2 photons. +For in our model +� +ˆb† +1ˆb1 +� += const, the radiation power +of an evaporating BH is given by the Hawking’s formula, +but photon statistics is not thermal and the outgoing +radiation carries information about the BH interior. In +particular, coherent oscillations of the BH interior lead +to a coherent outgoing radiation. +In the limit γ ≪ 1, Eqs. (27) and (28) can be solved +analytically yielding the following expression for the sys- +tem’s state vector as a function of t +|ψ(t)⟩ = N(t)e +− +iγΛtˆb† +1 +√ +1+Λ2t2 e +γˆb† +1ˆb† +2 +√ +1+Λ2t2 |0R⟩ |A(t)⟩ , +(31) +where +A2(t) = A2 − +Λ2t2 +1 + Λ2t2 γ2. +According to Eq. (31), initial thermal Hawking radiation +evolves into the coherent state e−iγˆb† +1 |0R⟩ on a time scale +1/Λ, while the oscillator’s energy (BH mass) decreases as +ℏωA2(t). +IV. +INSIGHTS FROM QUANTUM GRAVITY +MODELS +Here we show that present mechanism of nonthermal +emission of evaporating BHs holds for an effective metric +obtained in quantum gravity models. Most of such mod- +els suggest that the classical singularity at r = 0 should +be replaced by a regular timelike boundary. To be spe- +cific, we consider an effective BH metric obtained from +scale-dependent effective average action which takes into +account the effect of all loops [26–28]. As a function of + +6 +this scale, the effective average action satisfies a renor- +malization group equation yielding the effective metric +[29] +ds2 = f(r)c2dt2 − +1 +f(r)dr2 − r2 � +dθ2 + sin2 θdϕ2� +, (32) +where +f(r) = 1 − rg +r +1 +1 + +¯ωr2 +g +r2 +, +(33) +and ¯ω > 0 is a constant that involves the quantum grav- +ity correction to the BH geometry coming from the renor- +malization group approach. +The metric (32) is regular at r = 0 and has two hori- +zons which can be found by setting f(r) = 0 in Eq. (33). +The position of the outer and inner horizons is +r± = rg +2 +� +1 ± +√ +1 − 4¯ω +� +. +In terms of r±, one can write +1 +f(r) = 1 + +rgr +(r − r−)(r − r+). +A massless scalar field φ obeys the covariant wave +equation +1 +√−g +∂ +∂xµ +�√−ggµν ∂φ +∂xν +� += 0, +(34) +where gµν is the spacetime metric given by the interval +(32), namely +gtt = +1 +f(r), +grr = −f(r). +For the truncated 1+1 dimensional spacetime √−g = 1, +and the wave equation (34) reduces to +1 +c2 +∂2φ +∂t2 − f(r) ∂ +∂r +� +f(r)∂φ +∂r +� += 0. +(35) +Solutions of Eq. (35) read +φν(t, r) = e−iν[t± r +c ∓χ(r)], +(36) +where +χ(r) = r− ln |r − r−| − r+ ln |r − r+| +c√1 − 4¯ω +. +Using Eq. +(36), one can construct mode functions +analogous to the Rindler modes (8) and (9) in the +Schwarzschild coordinates, namely, +φ1ν(t, r) = e−iν[t− r +c +χ(r)]θ(r − r+), +(37) +φ2ν(t, r) = eiν[t− r +c +χ(r)]θ(r+ − r)θ(r − r−), +(38) +where ν > 0. For r− = 0, the mode functions (37) and +(38) reduce to Eqs. (8) and (9) with Ω = 2rgν/c. +Eqs. (37) and (38) show that if an observer is held fixed +outside the outer event horizon at a constant r > r+, then +at the observer’s location the non-zero Rindler modes +φ1ν oscillate as φ1ν ∝ e−iνt, where t is the observer’s +proper time. That is, from the observer’s perspective, the +Rindler photons φ1ν behave as if they have positive fre- +quency ν. However, if a hypothetical observer is located +at fixed r− < r < r+, the non-zero Rindler mode φ2ν +oscillates as a function of the proper time t as φ2ν ∝ eiνt. +That is, from the observer’s perspective, the Rindler pho- +tons φ2ν behave as if they have negative frequency −ν. +Absorption of photons φ2ν decreases energy (mass) of the +BH, leading to BH evaporation. +Photons falling into the BH from BH exterior are de- +scribed by the mode functions +φ3ν(t, r) = e−iν[t+ r +c −χ(r)] − eiϕ0e−iν[t− r +c +χ(r)]θ(r− − r), +(39) +where the last term describes a wave reflected from the +timelike spacetime boundary r = 0, and ϕ0 is a phase +shift introduced to satisfy the reflective boundary con- +dition, e.g., ∂φ3ν/∂r|r=0 = 0. From the perspective of +an observer held fixed at r = const the mode functions +φ3ν have positive frequency. Thus, absorption of such +photons increases the BH mass. +FIG. 3. Light rays of photons φ1ν, φ2ν (solid line) and φ3ν +(dash line) in the metric (32) for r− = 0.2rg and r+ = 0.8rg. +In Fig. +3 we plot light rays of photons (37), (38) +and (39) in the Schwarzschild coordinates. The figure +shows that the negative (φ2ν) and positive (φ1ν) fre- +quency Rindler photons are generated at the outer hori- +zon. These photons are produced in pairs and are en- +tangled. Photons φ1ν carry energy away from BH, while +the negative energy photons φ2ν propagate toward the +BH center and are absorbed at the inner horizon. The +positive energy photons φ3ν carry energy into the BH + +8 +1.61 +0.6 0.8 1.0 1.2 1.4 1 +Outer horizon +horizon +0.4( +Inner +0.2( +0.0 +3 +2 +2 +3 +4 +tc/17 +from the BH exterior. They cross both outer and inner +horizons, and after reflection from the BH center are ab- +sorbed at the inner horizon. In the region r− < r < r+, +the coordinate r plays the role of time for particles which +move unidirectionally along the r coordinate in this re- +gion. For r < r− and r > r+ the particles move unidi- +rectionally along the t coordinate. +Spacetime described by the metric (32) is non-singular +and matter does not disappear. Figure 3 shows that mat- +ter and energy (infalling photons φ3ν) are concentrated +in the vicinity of the inner horizon. Since Rindler pho- +tons φ2ν can only be annihilated and not created at the +inner horizon, the non-unitary absorption of the Rindler +photons φ2ν at the inner horizon, combined with the en- +tanglement of photon pairs φ1ν and φ2ν generated at +the outer horizon, leads to nonthermal outgoing radia- +tion that carries information about the BH interior. One +can model this process by the same Hamiltonian (24) of +the previous section, but now the oscillators absorbing +the ingoing photons φ2ν follow the worldline of the in- +ner horizon and can be a model of matter rather than +gravitational field. +The picture becomes more intuitive if we describe BH +evaporation in terms of particles and antiparticles that +can annihilate with each other. In this picture, particle +(φ1ν) and antiparticle (φ2ν) are generated as entangled +pairs at the outer horizon. The particles φ1ν carry en- +ergy away from BH. The antiparticles move towards BH +center and at the inner horizon annihilate with particles +φ3ν which have been accumulated at the inner horizon +during BH formation. Due to entanglement between φ1ν +and φ2ν, the information about state of particles φ3ν is +recorded into the outgoing flux of particles φ1ν. +V. +SUMMARY AND DISCUSSION +Evaporation of a classical Schwarzschild BH is caused +by +creation +of +entangled +particle-antiparticle +pairs +(Rindler photons in the present discussion) near the event +horizon, with one carrying positive energy to infinity and +the other carrying negative energy into the BH. This is +the mechanism of Hawking radiation. Absorption of the +negative energy photons at the center of the classical BH +reduces the BH mass. +Here we argue that previous models of Hawking radi- +ation are lacking an important ingredient. Namely, the +process of photon absorption at the BH center must be +properly described quantum mechanically by construct- +ing a Hamiltonian. Since under the BH event horizon, +light can propagate only towards the BH center, the +symmetry between absorption and emission is broken. +Namely, BH center can only absorb photons, but do not +emit. As a result, the Hamiltonian describing BH evap- +oration is not Hermitian. +To describe absorption of photons at the BH center, +we assume that the latter consists of harmonic oscilla- +tors which absorb the ingoing radiation, but do not emit. +In our model, the oscillators follow the worldline of the +BH center, rather than geodesics, and carry information +about the BH interior. +We show that due to entanglement between photons +moving inside and outside the BH event horizon, the +non-unitary absorption of the radiation under the hori- +zon alters the state of the field outside the BH. As a +consequence, radiation produced by the evaporating BH +is not thermal and carries information about the BH in- +terior. After the BH has evaporated, the information is +recorded in the remaining non-thermal field. Since evolu- +tion is governed by a Hamiltonian, the state of the system +remains pure and during BH evaporation the von Neu- +mann entropy is preserved. In our model we impose a +constraint that energy is conserved during BH evapora- +tion. As a consequence, our model yields that luminosity +of an evaporating BH coincides with that for Hawking +radiation. +Erasing information at the BH center produced by pho- +ton absorption is a non-unitary process which leads to a +change of the field outside the horizon. +This is some- +what analogous to the quantum eraser experiments in +which the interference pattern can be destroyed or re- +stored by manipulating entangled photon partners [30– +32]. In these experiments, after two entangled photons +are created, each is directed into different section of the +apparatus and an interference pattern for one of them is +examined. A measurement done on the entangled partner +to learn about the photon path influences the interference +pattern. +Similarly to BH evaporation, non-unitarity of the mea- +surement process alters the state of the entangled part- +ner. However, the state vector collapse brought about +by a measurement is a probabilistic and discontinuous +change, while BH evaporation is a deterministic, contin- +uous time evolution of an isolated system that obeys the +Schr¨odinger equation. +Our findings show that quantum mechanical evolution, +governed by the Schr¨odinger equation, allows informa- +tion to leak out from the BH. This is the case because +BH center breaks the emission-absorption symmetry and +photons external to the horizon are entangled with those +inside it. Such entanglement is an inherent property of +the field for evaporating BHs. +We also show that present mechanism of nonthermal +emission of evaporating BHs holds for spacetimes ob- +tained in quantum gravity models in which the classi- +cal singularity at r = 0 is replaced by a regular time- +like boundary. +For such spacetimes the metric has an +inner and outer horizons, and matter does not disap- +pear. +Instead, particles are accumulated in the vicin- +ity of the inner horizon. For this spacetime, the entan- +gled particle-antiparticle pairs are generated at the outer +horizon. The generated particles carry energy away from +BH, while antiparticles move towards the BH center and +annihilate at the inner horizon with particles that form +the BH interior. Due to entanglement of the particle- +antiparticle pairs produced at the outer horizon, the in- + +8 +formation about the BH interior is recorded in the out- +going particle flux. +One should mention that if our findings are correct, +and radiation of evaporating BHs is nonthermal, the +Bekenstein-Hawking formula [33, 34] does not describe +the BH entropy. Recall that the latter formula assumes +thermal BH emission with the Hawking temperature. +Our results demonstrate that quantum mechanics +works in an exotic spacetime geometry of a BH. However, +BHs might have only a mathematical significance. The +point is that there is an evidence that general relativity +is ruled out by gravitational waves detection experiments +in favor of the vector theory of gravity [35]. The latter +theory [36, 37] agrees with all available tests of gravity, +including detection of gravitational waves and observa- +tions of supermassive objects at galactic centers [35, 38]. +In addition, vector gravity predicts no BHs and yields the +measured value of the cosmological constant [39] with no +free parameters [36, 37]. +ACKNOWLEDGMENTS +This work was supported by the Air Force Office of +Scientific Research (Grant No. FA9550-20-1-0366 DEF), +the Robert A. Welch Foundation (Grant No. A-1261), +and the National Science Foundation (Grant No. PHY- +2013771). +Appendix A: Operator identities and expectation +values +Operators of Rindler photons ˆb1 and ˆb2 obey bosonic +commutation relations +[ˆb1,ˆb† +1] = 1, +[ˆb2,ˆb† +2] = 1, +and all other commutators are equal to zero. First we +prove an operator identity +ˆb2eγˆb† +1ˆb† +2 = eγˆb† +1ˆb† +2ˆb2 + γˆb† +1eγˆb† +1ˆb† +2, +(A1) +where γ is a complex number. Introducing operator +ˆF(γ) = ˆb2eγˆb† +1ˆb† +2 − eγˆb† +1ˆb† +2ˆb2, +we have +d ˆF(γ) +dγ += ˆb2ˆb† +1ˆb† +2eγˆb† +1ˆb† +2−ˆb† +1ˆb† +2eγˆb† +1ˆb† +2ˆb2 = ˆb† +1ˆb† +2 ˆF(γ)+ˆb† +1eγˆb† +1ˆb† +2. +Solution of this differential equation, subject to the con- +dition ˆF(0) = 0, is +ˆF(γ) = γˆb† +1eγˆb† +1ˆb† +2, +which proves the identity (A1). +Next we prove an identity +eλˆb† +2ˆb2ˆb† +2 = eλˆb† +2eλˆb† +2ˆb2, +(A2) +where λ is a complex number. Introducing operator +ˆF(λ) = eλˆb† +2ˆb2ˆb† +2 − ˆb† +2eλˆb† +2ˆb2, +we have +d ˆF(λ) +dλ += ˆb† +2ˆb2eλˆb† +2ˆb2ˆb† +2−ˆb† +2ˆb† +2ˆb2eλˆb† +2ˆb2 = ˆb† +2ˆb2 ˆF(λ)+ˆb† +2eλˆb† +2ˆb2. +Solution of this differential equation, subject to the con- +dition ˆF(0) = 0, is +ˆF(λ) = +� +eλ − 1 +�ˆb† +2eλˆb† +2ˆb2, +which proves the identity (A2). +Next we prove an identity +eλˆb† +2ˆb2eγˆb† +1ˆb† +2 = eeλγˆb† +1ˆb† +2eλˆb† +2ˆb2. +(A3) +Introducing operator +ˆF(λ) = eλˆb† +2ˆb2eγˆb† +1ˆb† +2e−λˆb† +2ˆb2, +and taking derivative over λ, we have +d ˆF(λ) +dλ += eλˆb† +2ˆb2ˆb† +2ˆb2eγˆb† +1ˆb† +2e−λˆb† +2ˆb2−eλˆb† +2ˆb2eγˆb† +1ˆb† +2ˆb† +2ˆb2e−λˆb† +2ˆb2. +Taking into account identities (A1) and (A2), we obtain +d ˆF(λ) +dλ += γˆb† +1eλˆb† +2ˆb2ˆb† +2eγˆb† +1ˆb† +2e−λˆb† +2ˆb2 = γeλˆb† +1ˆb† +2 ˆF(λ). +Solution of this differential equation, subject to the con- +dition ˆF(0) = eγˆb† +1ˆb† +2, is +ˆF(λ) = eeλγˆb† +1ˆb† +2, +which proves the identity (A3). +Next we calculate a matrix element ⟨ψ|ψ⟩, where state +vector |ψ⟩ is +|ψ⟩ = +� +1 − γ2eβˆb† +1eγˆb† +1ˆb† +2 |0R⟩ , +(A4) +|0R⟩ stands for the Rindler vacuum, β is a complex num- +ber and γ is a real number. The state vector (A4) can +be written as +|ψ⟩ = eβˆb† +1 |0M⟩ , +where +|0M⟩ = +� +1 − γ2eγˆb† +1ˆb† +2 |0R⟩ +(A5) +is the Minkowski vacuum. +Using a relation between +operators of the Rindler photons ˆb1,2 and the Unruh- +Minkowski photons ˆa1,2 [40] +ˆb† +1 = ˆa† +1 + γˆa2 +� +1 − γ2 , + +9 +and the property ˆa1,2 |0M⟩ = 0, we obtain +|ψ⟩ = e +βˆa† +1 +√ +1−γ2 |0M⟩ . +Taking into account that +eαˆa† +1 |0M⟩ = e +|α|2 +2 +|α0⟩ , +where |α0⟩ stands for a coherent state |α⟩ for the Unruh- +Minkowski photons ˆa1 and the vacuum state for the +Unruh-Minkowski photons ˆa2, we find +|ψ⟩ = e +|β|2 +2(1−γ2) |α0⟩ , +(A6) +where α = β/ +� +1 − γ2. Therefore +⟨ψ|ψ⟩ = e +|β|2 +1−γ2 . +(A7) +Next we calculate the average number of Rindler +photons +ˆb1 +in +the +state +|ψ⟩, +that +is +� +ˆb† +1ˆb1 +� +≡ +⟨ψ|ˆb† +1ˆb1 |ψ⟩ / ⟨ψ|ψ⟩. Taking derivative of Eq. (A7) with +respect to β and β∗, and using Eq. (A4), we have +⟨ψ|ˆb1ˆb† +1 |ψ⟩ = +∂2 +∂β∂β∗ e +|β|2 +1−γ2 = 1 − γ2 + |β|2 +(1 − γ2)2 +e +|β|2 +1−γ2 . +Therefore +� +ˆb† +1ˆb1 +� += ⟨ψ|ˆb1ˆb† +1 |ψ⟩ +⟨ψ|ψ⟩ +− 1 = +γ2 +1 − γ2 + +|β|2 +(1 − γ2)2 . (A8) +To find +� +ˆb† +2ˆb2 +� +we use the relations between operators +of the Rindler photons ˆb1,2 and the Unruh-Minkowski +photons ˆa1,2 [40] +ˆb2 = ˆa2 + γˆa† +1 +� +1 − γ2 , +ˆb† +2 = ˆa† +2 + γˆa1 +� +1 − γ2 , +which yield +ˆb† +2ˆb2 = +1 +1 − γ2 +� +ˆa† +2ˆa2 + γ2ˆa1ˆa† +1 + γˆa1ˆa2 + γˆa† +2ˆa† +1 +� +. +Using Eq. (A6), we obtain +⟨ψ|ˆb† +2ˆb2 |ψ⟩ = γ2 � +1 + |α|2� +1 − γ2 +e +|β|2 +1−γ2 , +where we took into account that ⟨α0| ˆa1ˆa† +1 |α0⟩ = 1+|α|2. +As a result, +� +ˆb† +2ˆb2 +� += ⟨ψ|ˆb† +2ˆb2 |ψ⟩ +⟨ψ|ψ⟩ += +γ2 +1 − γ2 + +γ2|β|2 +(1 − γ2)2 . +(A9) +Appendix B: State vector evolution during black +hole evaporation +For our model of black hole evaporation, the con- +strained interaction Hamiltonian is +ˆV (t) = gˆσˆb2+iℏ ˙µ1(t)ˆb† +1ˆb1+iℏ ˙µ2(t) +� +ˆσ†ˆσ − ˆb† +2ˆb2 +� ++iℏ ˙µ3(t), +where functions µ1,2,3(t) are real, and the oscillator’s +lowering and raising operators ˆσ and ˆσ† obey the +same bosonic commutation relations as the operators of +Rindler photons. Schrodinger equation for the evolution +of the field state vector +iℏ ∂ +∂t |ψ(t)⟩ = ˆV (t) |ψ(t)⟩ +yields +|ψ(t)⟩ = eµ3eµ1ˆb† +1ˆb1+µ2(ˆσ†ˆσ−ˆb† +2ˆb2)− i +ℏ gtˆσˆb2 |0M⟩ |A⟩ , (B1) +where |0M⟩ and |A⟩ are the initial state vectors of the +field and the oscillator respectively. We assume that the +latter is a coherent state |A⟩, where A is real, and the +former is the Minkowski vacuum |0M⟩. Recall that Unruh +vacuum coincides with the Minkowski vacuum for the +right-moving photons. +Taking into account that ˆσˆb2 commutes with ˆb† +1ˆb1 and +ˆσ†ˆσ−ˆb† +2ˆb2, and plugging |0M⟩ from Eq. (A5) in Eq. (B1), +we obtain +|ψ(t)⟩ = +� +1 − γ2eµ3e− i +ℏ gtˆσˆb2eµ1ˆb† +1ˆb1+µ2(ˆσ†ˆσ−ˆb† +2ˆb2)eγˆb† +1ˆb† +2 |0R⟩ |A⟩ . +Using identity (A3), we have +|ψ(t)⟩ = +� +1 − γ2eµ3e− i +ℏ gtˆσˆb2eeµ1−µ2 γˆb† +1ˆb† +2eµ2ˆσ†ˆσ |0R⟩ |A⟩ . +Taking into account that +eµ2ˆσ† ˆσ |A⟩ = e +|A|2 +2 (e2µ2 −1) |eµ2A⟩ , +we find +|ψ(t)⟩ = +� +1 − γ2eµ3+ |A|2 +2 (e2µ2 −1)e− i +ℏ gtˆσˆb2eeµ1−µ2 γˆb† +1ˆb† +2 |0R⟩ |eµ2A⟩ . +Since the initial state of the oscillator is the coherent +state |A⟩, and ˆσ |A⟩ = A |A⟩, one can write +|ψ(t)⟩ = +� +1 − γ2eµ3+ |A|2 +2 (e2µ2 −1)e− i +ℏ geµ2 Atˆb2eeµ1−µ2 γˆb† +1ˆb† +2 |0R⟩ |eµ2A⟩ . + +10 +Using the Baker–Hausdorff formula e ˆ +Ae ˆ +B = e[ ˆ +A, ˆ +B]e ˆ +Be ˆ +A, +we finally obtain +|ψ(t)⟩ = +� +1 − γ2eµ3+ |A|2 +2 (e2µ2 −1)e− i +ℏ γgAeµ1 tˆb† +1eeµ1−µ2 γˆb† +1ˆb† +2 |0R⟩ |eµ2A⟩ . +(B2) +Using Eqs. (A8) and (A9), we find that the average +number of Rindler photons in the state (B2) is +� +ˆb† +1ˆb1 +� += +˜γ2 +1 − ˜γ2 + (γgAt)2 +ℏ2 +e2µ1 +(1 − ˜γ2)2 , +� +ˆb† +2ˆb2 +� += +˜γ2 +1 − ˜γ2 + (γgAt)2 +ℏ2 +e2µ1˜γ2 +(1 − ˜γ2)2 , +where +˜γ = eµ1−µ2γ. +The average number of oscillator excitations in the state +(B2) is +� +ˆσ†ˆσ +� += e2µ2A2. +Constraints +� +ˆσ†ˆσ − ˆb† +2ˆb2 +� += const and +� +ˆb† +1ˆb1 +� += const +give equations +e2µ2A2 − +˜γ2 +1 − ˜γ2 − (γgAt)2 +ℏ2 +e2µ1˜γ2 +(1 − ˜γ2)2 = A2 − +γ2 +1 − γ2 , +˜γ2 +1 − ˜γ2 + (γgAt)2 +ℏ2 +e2µ1 +(1 − ˜γ2)2 = +γ2 +1 − γ2 , +which for t → ∞ yield +˜γ → 0, +1 +ℏγgAeµ1 ≈ +γ +� +1 − γ2t +, +e2µ2A2 → A2− +γ2 +1 − γ2 . +Therefore, for t → ∞ +� +ˆb† +1ˆb1 +� += +γ2 +1 − γ2 , +� +ˆb† +2ˆb2 +� += 0, +� +ˆσ†ˆσ +� += A2 − +γ2 +1 − γ2 , +and the normalized state vector of the system is +|ψ(∞)⟩ = e +− +γ2 +2(1−γ2) e +−i +γ +√ +1−γ2 ˆb† +1 |0R⟩ +����� +� +A2 − +γ2 +1 − γ2 +� +. +The final state is the Rindler vacuum for photons ˆb2, a +coherent state for photons ˆb1 with the average photon +number γ2/(1 − γ2), and the oscillator remains in the +coherent state with a reduced average excitation number +A2 − γ2/(1 − γ2). +[1] S.W. Hawking, BH explosions?, Nature (London) 248, +30 (1974). +[2] S.W. Hawking, Particle creation by black holes, Commu- +nications in Mathematical Physics 43, 199 (1975). +[3] V. Mukhanov, and S. Winizki, Introduction to Quantum +Effects in Gravity (Cambridge University Press, Cam- +bridge, England, 2007). +[4] V.P. Frolov, +and I.D. Novikov, +Black hole physics +(Kluwer Academic Publishers, The Netherlands, 1998). +[5] S.W. Hawking, Breakdown of predictability in gravita- +tional collapse, Phys. Rev. D 14, 2460 (1976). +[6] D. Marolf, The black hole information problem: +past, +present, and future, Rep. Prog. Phys. 80, 092001 (2017). +[7] X. Calmet, and S.D.H. Hsu, A brief history of Hawking’s +information paradox, EPL 139, 49001 (2022). +[8] G. ’t Hooft, Dimensional reduction in quantum gravity, +Conf. Proc. C 930308, 284 (1993). +[9] L. Susskind, The World as a hologram, J. Math. Phys. +36, 6377 (1995). +[10] A. Laddha, S. G. Prabhu, S. Raju and P. Shrivastava, +The holographic nature of null infinity, SciPost Phys. 10, +041 (2021). +[11] C. Chowdhury, O. Papadoulaki and S. Raju, A physical +protocol for observers near the boundary to obtain bulk +information in quantum gravity, SciPost Phys. 10, 106 +(2021). +[12] X. Calmet and S. D. H. Hsu, Quantum hair and black +hole information, Phys. Lett. B 827, 136995 (2022). +[13] X. Calmet, R. Casadio, S.D.H. Hsu, and F. Kuipers, +Quantum hair from gravity, Phys. Rev. Lett. 128, 111301 +(2022). +[14] A. Strominger, On BMS invariance of gravitational scat- +tering, J. High Energy Phys. 2014, 152 (2014). +[15] S.W. Hawking, M.J. Perry, and A. Strominger, Soft hair +on black holes, Phys. Rev. Lett. 116, 231301 (2016). +[16] S.W. Hawking, M.J. Perry, and A. Strominger, Superro- + +11 +tation charge and supertranslation hair on black holes, J. +High Energ. Phys. 2017, 161 (2017). +[17] J.M. Maldacena, The Large N Limit of Superconformal +field theories and supergravity, Adv. Theor. Math. Phys. +2, 231 (1998). +[18] A. Almheiri, T. Hartman, J. Maldacena, E. Shaghoulian, +and A. Tajdini, The entropy of Hawking radiation, Rev. +Mod. Phys. 93, 035002 (2021). +[19] A.A. Svidzinsky, A. Azizi, J.S. Ben-Benjamin, M.O. +Scully, and W. Unruh, Unruh and Cherenkov radiation +from a negative frequency perspective, Phys. Rev. Lett. +126, 063603 (2021). +[20] W. Rindler, Kruskal space and the uniformly accelerated +frame, Am. J. Phys. 34, 1174 (1966). +[21] W.G. Unruh and R.M. Wald, What happens when an ac- +celerating observer detects a Rindler particle, Phys. Rev. +D 29, 1047 (1984). +[22] S.W. Hawking, The quantum mechanics of black holes, +Scientific American 236, 34 (1977). +[23] P.C.W. Davies, S.A. Fulling, and W.G. Unruh, Energy- +momentum tensor near an evaporating black hole, Phys. +Rev. D 13, 2720 (1976). +[24] E.P. Gross, Collective rotations of nuclei, Nucl. Phys. 14, +389 (1959). +[25] P. Amiot, and J.J. Griffin, Constraints in Quantum Me- +chanics via Lagrangians, Hamiltonians, and Canonical +Equations of Motion, Annals of Phys. 95, 295 (1975). +[26] C. Wetterich, Exact evolution equation for the effective +potential, Phys. Lett. B 301, 90 (1993). +[27] M. Reuter and C. Wetterich, Effective average action for +gauge theories and exact evolution equations, Nucl. Phys. +B417, 181 (1994). +[28] M. Reuter, Nonperturbative evolution equation for quan- +tum gravity, Phys. Rev. D 57, 971 (1998). +[29] A. Bonanno and M. Reuter, Renormalization group im- +proved black hole spacetimes, Phys. Rev. D 62, 043008 +(2000). +[30] M.O. Scully and K. Dr¨uhl, Quantum eraser: +A pro- +posed photon correlation experiment concerning observa- +tion and ”delayed choice” in quantum mechanics, Phys. +Rev. A 25, 2208 (1982). +[31] Y.-H. Kim, R. Yu, S.P. Kulik, Y.H. Shih, M.O. Scully, A +Delayed Choice Quantum Eraser, Phys. Rev. Lett. 84, 1 +(2000). +[32] S.P. Walborn, M.O. Terra Cunha, S. Padua, and C.H. +Monken, Double-slit quantum eraser, Phys. Rev. A 65, +033818 (2002). +[33] J.D. Bekenstein, Black holes and the second law, Lett. +Nuovo Cimento 4, 737 (1972). +[34] S.W. Hawking, Black holes and thermodynamics, Phys. +Rev. D 13, 191 (1976). +[35] A.A. Svidzinsky, and R.C. Hilborn, GW170817 event +rules out general relativity in favor of vector gravity, Eur. +Phys. J. Spec. Top. 230, 1149 (2021). +[36] A.A. Svidzinsky, Vector theory of gravity: universe with- +out black holes and solution of dark energy problem, Phys. +Scr. 92, 125001 (2017). +[37] A.A. Svidzinsky, Simplified equations for gravitational +field in the vector theory of gravity and new insights into +dark energy, Phys. Dark Universe 25, 100321 (2019). +[38] A.A. Svidzinsky, Oscillating axion bubbles as an alterna- +tive to supermassive black holes at galactic centers, JCAP +10, 018 (2007). +[39] Planck Collaboration, P.A.R. Ade, N. Aghanim, C. +Armitage-Caplan et al., Planck 2013 results. XVI. Cos- +mological parameters, A&A 571, A16 (2014). +[40] A.A. Svidzinsky, A. Azizi, J.S. Ben-Benjamin, M.O. +Scully, and W. Unruh, Causality in quantum optics and +entanglement of Minkowski vacuum, Phys. Rev. Research +3, 013202 (2021). + diff --git a/PdFPT4oBgHgl3EQfnjXp/content/tmp_files/load_file.txt b/PdFPT4oBgHgl3EQfnjXp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..106c22cc2d748d0125ef0cfa19a1f0491f5b6aec --- /dev/null +++ b/PdFPT4oBgHgl3EQfnjXp/content/tmp_files/load_file.txt @@ -0,0 +1,573 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf,len=572 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='13131v1 [gr-qc] 30 Jan 2023 Nonthermal radiation of evaporating black holes Anatoly A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky Texas A&M University, College Station, TX 77843 (Dated: January 31, 2023) Black hole (BH) evaporation is caused by creation of entangled particle-antiparticle pairs near the event horizon, with one carrying positive energy to infinity and the other carrying negative energy into the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since under the event horizon, particles always move toward the BH center, they can only be absorbed but not emitted at the center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This breaks absorption-emission symmetry and, as a result, annihilation of the particle at the BH center is described by a non-Hermitian Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We show that due to entanglement between photons moving inside and outside the event horizon, non-unitary absorption of the negative energy photons near the BH center, alters the outgoing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a result, radiation of the evaporating BH is not thermal, it carries information about BH interior and entropy is preserved during evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' INTRODUCTION According to principles of quantum mechanics, state of an isolated system remains pure during evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is the case for both types of quantum mechanical evolution - a unitary evolution governed by the Schr¨odinger equation and a non-unitary state vector collapse brought about by a measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' If the system remains in a pure state the von Neumann entropy is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Computations of Hawking radiation, which is believed to be produced by an evaporating black hole (BH), indi- cated that it is completely thermal [1–4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Therefore, an evaporating BH would eventually leave behind a cloud of thermal radiation, independently of the initial state from which it was formed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, one could imagine forming a BH from a pure state that seems to evolve to a mixed thermal state which amounts to a loss of informa- tion and thus is incompatible with quantum mechanical evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is known as the BH information paradox [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For proposals to resolve the BH information problem see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=', [6, 7] and references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' According to the holographic principle, the bulk in- formation in models of gravity in d-dimensions might be available on the d − 1 dimensional boundary of space- time [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Holography of information implies that the internal quantum state of a BH must be encoded in the asymptotic quantum state of its graviton field, since oth- erwise the information would not be recoverable at the boundary [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In [12, 13] it has been shown explic- itly that information about the BH internal state is avail- able in the quantum state of its gravity field (quantum hair).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Moreover, it has been argued that long wavelength gravitons can give rise to an infinite number of conserved charges which preserve an infinite amount of information outside BHs [14] which could give a new perspective on the information problem [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Holography was given an explicit realization in the AdS/CFT correspondence of Maldacena [17], which sug- gests that BH evaporation can be unitary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Recently there has been considerable progress in directly computing the entanglement entropy of evaporation using AdS methods, and these results suggest that the process is unitary [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' While both holography of information and AdS/CFT du- ality suggest that the BH information paradox is some- how resolved in favor of unitarity, neither yield a specific description of the physical process by which BH informa- tion is encoded in Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Here we show that entanglement of particle pairs generated during BH evaporation, combined with non- unitary absorption of particles near the BH center, leads to nonthermal outgoing radiation that carries informa- tion about the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' BHs possess an event horizon - the boundary under which no particles, at least if they are treated classically and moving forward in time, can escape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This leads to a believe that an observer outside the BH has no access to the interior part of the total quantum system, and information about the internal degrees of freedom is lost during BH evaporation leaving the system in a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, BH spacetime has another inherent feature, namely, under the event horizon, particles always move toward the BH center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The latter probably has a Planck scale and is described by yet not well-understood physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' What matters for the present discussion is that, since particles can move only towards the center, photons with a wavelength much greater than the Plank length can only be absorbed, but not emitted at the BH center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Emitted particles move away from the source and, since particles cannot move away from the BH center, they cannot be emitted in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This breaks the symmetry between absorption and emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a result, annihilation of the particle at the BH center is described by a non-Hermitian Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Here we show that if the emission-absorption symme- try is broken, quantum mechanics predicts that radiation of an evaporating BH is nonthermal and it carries infor- mation about the state of matter in the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Next we briefly discuss the physics of Hawking radiation from a negative frequency perspective [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Schwarzschild spacetime in the Kruskal-Szekeres co- ordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' A space-like line T = √ 1 + X2 sets the space- time boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Unruh vacuum is filled with entangled right- moving Rindler photons φ1 and φ2 which are localized outside and inside the BH event horizon (line T = X) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Absorption of Rindler photons φ2 at the boundary reduces BH mass and leads to BH evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We model the bound- ary as a set of harmonic oscillators that absorb all ingoing photons, but do not emit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Due to vacuum entanglement, the process looks like as if there is a mirror “image” of the oscil- lators located along the line T = − √ 1 + X2 which emit (but do not absorb) light outside the BH event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' HAWKING RADIATION FROM A NEGATIVE FREQUENCY PERSPECTIVE According to general relativity, a static BH of mass M in 3+1 dimension in Schwarzschild coordinates is de- scribed by a metric ds2 = � 1 − rg r � c2dt2− 1 1 − rg r dr2−r2 � dθ2 + sin2 θdϕ2� , (1) where rg = 2GM/c2 is the gravitational radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For sim- plicity, we truncate the spacetime to 1+1 dimension (t and r, where r ≥ 0) and use Kruskal-Szekeres coordinates T and X that are defined in terms of the Schwarzschild coordinates t and r as T = � r/rg − 1e r 2rg sinh � ct 2rg � , (2) X = � r/rg − 1e r 2rg cosh � ct 2rg � , (3) for r > rg, and T = � 1 − r/rge r 2rg cosh � ct 2rg � , (4) X = � 1 − r/rge r 2rg sinh � ct 2rg � , (5) for 0 < r < rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In these coordinates, the BH center (r = 0) is a space-like line T = √ 1 + X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This line sets a boundary of the Schwarzschild spacetime in the Kruskal-Szekeres coordinates (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The boundary appears because coordinate transformation (2)-(5) maps the region −∞ < t < ∞, r ≥ 0 into T ≤ √ 1 + X2, T ≥ −X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In the Kruskal-Szekeres coordinates, in 1+1 dimension, the Schwarzschild metric ds2 = 4r3 g r e−r/rg � dT 2 − dX2� (6) is conformally invariant to the Minkowski metric and, thus, a massless scalar field φ obeys the same wave equa- tion as in the Minkowski spacetime � ∂2 ∂T 2 − ∂2 ∂X2 � φ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (7) For the present problem only the right-moving field in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 1 is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' It is convenient to describe such a field using Rindler modes [20] φ1Ω(T, X) = (X − T )iΩθ(X − T ), (8) φ2Ω(T, X) = (T − X)−iΩθ(T − X), (9) where Ω > 0 is a parameter, and θ is the Heaviside step function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rindler modes φ1Ω and φ2Ω are solutions of the wave equation (7), and for Ω > 0 have positive norm (defined as the Klein–Gordon inner product).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The mode functions (8) and (9) are non-zero outside and inside the BH event horizon (line T = X) respectively (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' These two regions are causally disconnected for the right-moving field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Annihilation operators of the Rindler photons we denote as ˆb1Ω and ˆb2Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' It is believed that, to a good approximation, Unruh vacuum |0U⟩ describes state of the field produced by a gravitational collapse of a star into a BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In this state, there are no left-moving Rindler photons and no right- moving Minkowski photons [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is, Unruh vac- uum is Rindler vacuum for the left-moving photons and Minkowski vacuum for the right-moving photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In terms of the right-moving Rindler photons, which are relevant for the present discussion, the Unruh vacuum is a squeezed state [21] |0U⟩ = � Ω>0 � 1 − γ2eγˆb† 1Ωˆb† 2Ω |0R⟩ , (10) where γ = e−πΩ, (11) |0R⟩ refers to the Rindler vacuum, ˆb† 1Ω and ˆb† 2Ω are cre- ation operators of the right-moving Rindler photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' =√1+X2 Schwarzschild spacetime Image T W 1 2 Boundary3 That is, Unruh vacuum is filled with the right-moving Rindler photons, but it looks empty if the right-moving field is described by means of the Minkowski photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' If a hypothetical observer is located at the BH center (r = 0), then Schwarzschild coordinate t is the proper time for such observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Recall that proper time of an object is the coordinate which changes in the object’s frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' If the object is held fixed at r = const then t is the proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In the region 0 < r < rg, it is phys- ically impossible to hold particles fixed at r = const, that’s why we use the word “hypothetical”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (4), (5) and (9) yield that at the BH center the non-zero Rindler mode φ2Ω oscillates as a function of t as φ2Ω ∝ eiΩct/2rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is, from the observer’s perspective the Rindler pho- tons behave as if they have negative frequency −Ωc/2rg [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hence, in the Unruh vacuum, there is a flux of the negative frequency (energy) Rindler photons into the BH center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Absorption of such photons near the BH center decreases energy (mass) of the BH, leading to BH evap- oration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' If an observer is held fixed outside the event horizon at a constant Schwarzschild coordinate r, then at the ob- server’s location the non-zero Rindler modes φ1Ω oscil- late as φ1Ω ∝ e−iΩct/2rg, where t is the observer’s proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is, from the external observer perspective the Rindler photons behave as if they have positive frequency ν = Ωc 2rg (12) and, thus, they can excite a detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Photons φ1Ω prop- agate away from the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For simplicity, we will assume that the field has only modes with one “frequency” Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We denote such Rindler modes as φ1 and φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Then Unruh vacuum can be written as |0U⟩ = � 1 − γ2eγˆb† 1ˆb† 2 |0R⟩ = � 1 − γ2 ∞ � n=0 γn |nn⟩ , (13) where |nn⟩ is a state with n Rindler photons in the modes φ1 and φ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' If modes φ1 and φ2 are considered separately, then tracing over or absorbing one of the modes leaves the remaining mode in a thermal state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Namely, if we trace over the Rindler modes under the event horizon φ2, which are not accessible to the external observer, the reduced density operator for the field φ1 is thermal ˆρ1 = Tr2 (|0U⟩ ⟨0U|) = � 1 − γ2� ∞ � n=0 γ2n |n⟩ ⟨n| (14) with the average number of photons ¯n1 = γ2 1 − γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (15) Thus, an observer held fixed outside the BH horizon feels thermal radiation coming out from the BH, which is known as Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (11), (12) and (15), one can write ¯n1 as a Planck factor ¯n1 = 1 e 4πrgν c − 1 = 1 e ℏν kB TH − 1 (16) with the Hawking temperature TH = ℏc/4πkBrg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Light rays of Rindler photons φ1 and φ2 in Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Figure 2 shows light rays of Rindler photons (8) and (9) in the Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' It looks like the negative (φ2) and positive (φ1) frequency Rindler pho- tons are generated at the event horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is consis- tent with the interpretation of the Hawking radiation as a continuous creation of particle-antiparticle pairs near the event horizon, with one carrying positive energy to infinity and the other carrying negative energy into the BH [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Calculations of the energy-momentum tensor for the field near an evaporating BH directly show that there is a negative-energy flux into the BH center and a positive-energy flux far away from the BH [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' MODEL OF AN EVAPORATING BLACK HOLE TAKING INTO ACCOUNT NON-UNITARY PHOTON ABSORPTION AT THE CENTER According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (7), in the Kruskal-Szekeres coordi- nates the field evolves following the same wave equation as in Minkowski spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For the latter, absorption or emission of photons in the region T > X cannot affect the state of the field in the region T < X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, the BH spacetime has a space-like boundary at T = √ 1 + X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We will show below that absorption of photons at the boundary changes the state of the field outside the event 3 5 2 Event horizon 2 5 0 2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='0 Singularity tc/4 horizon and radiation of the evaporating BH is not ther- mal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We will assume that Unruh vacuum is the state of the field only at the onset of evaporation and calculate how the field evolves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' According to general relativity, spacetime disappears at the BH center (spacetime boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' It is assumed that matter disappears together with the spacetime, but state of matter (mass, angular momentum, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=') is recorded in the gravitational field near the BH center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This process transfers characteristics of the accreting matter into the BH internal gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The worldlines of the Rindler photons ˆb2 terminate at the spacetime boundary (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' But we can’t just say that photons disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' One should describe this process quantum mechanically using a Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Space-like boundary breaks the symmetry between emis- sion and absorption of Rindler photons ˆb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Namely, if backward in time propagation is not allowed, Rindler photons ˆb2 cannot be emitted at the boundary because such a process means emission of particles outside the spacetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Next we consider a simple toy model of BH evaporation modeling the boundary as a set of harmonic oscillators that totally absorb the ingoing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The oscillators fol- low the worldline of the boundary which is not geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We do not associate the oscillators with ordinary par- ticles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rather, the oscillators provide a physical model of the gravitational field near the BH center that car- ries information about the state of the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In our model, the oscillator’s energy is the origin of the BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As we showed above, from the oscillator’s per- spective, Rindler photons have negative energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Thus, absorption of Rindler photons reduces the energy of the oscillators (BH mass decreases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since oscillators are under the BH horizon, they can interact only with photons ˆb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In the toy model, the interaction Hamiltonian describing BH evaporation reads ˆV2(t) = gˆσe−iωtφ2(t)ˆb2, (17) where ˆσ is the lowering operator for the oscillator of fre- quency ω, g is the coupling constant and the field mode function φ2(T, X) is taken at the location of the oscilla- tor φ2(t) = φ2(T (t, 0), X(t, 0)) = ei cΩt 2rg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (17), t is the proper time of the oscillator which coincides with the Schwarzschild coordinate t because oscillators are lo- cated at fix r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since oscillators cannot emit Rindler photons, the Hamiltonian (17) is not Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We will consider evolution of the system as a function of the oscillator proper time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Schr¨odinger equation for the system’s state vector iℏ ∂ ∂t |ψ(t)⟩ = ˆV2(t) |ψ(t)⟩ yields |ψ(t)⟩ = eβ(t)ˆσˆb2 |0U⟩ |A⟩ , (18) where |0U⟩ and |A⟩ are the initial state vectors of the field and the oscillator, and β(t) = −ig ℏ � t 0 dt′e−iωt′φ2(t′) = −ig ℏ � t 0 dt′ei � cΩ 2rg −ω � t′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Plug |0U⟩ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (18) gives |ψ(t)⟩ = � 1 − γ2eβ(t)ˆσˆb2eγˆb† 1ˆb† 2 |0R⟩ |A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (19) Using the Baker–Hausdorff formula e ˆ Ae ˆ B = e[ ˆ A, ˆ B]e ˆ Be ˆ A, we obtain |ψ(t)⟩ = � 1 − γ2eγβ(t)ˆσˆb† 1eγˆb† 1ˆb† 2eβ(t)ˆσˆb2 |0R⟩ |A⟩ , or |ψ(t)⟩ = eγβ(t)ˆσˆb† 1 |0U⟩ |A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (20) Equation (20) shows that non-unitary field absorption at the spacetime boundary yields generation of photons outside the BH event horizon (into the Rindler mode 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Taking time derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (20) leads to the Schr¨odinger equation with the interaction Hamiltonian ˆV1(t) = γgˆσe−iωtφ∗ 1(t)ˆb† 1, where we used φ2(t) = φ∗ 1(t) = φ∗ 1(−T (t, 0), X(t, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is, the process looks like as if there is a mir- ror “image” of the oscillator located along the line T = − √ 1 + X2 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 1) which is coupled with the exter- nal mode φ1 with a reduced coupling constant γg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The oscillator’s image produces field outside the event hori- zon which propagates away from the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Such field is not thermal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=', if the oscillator is in a coherent state, the generated field is coherent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The information stored in the oscillators is recorded in the outgoing field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' BH radiation is not thermal because evolution of the field under the horizon is described by the non-Hermitian Hamiltonian (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Indeed, if the Hamiltonian would be Hermitian and depends only on ˆb2 and ˆb† 2, the Heisenberg equation of motion for the operator ˆb1(t) dˆb1(t) dt = i ℏ � ˆH†ˆb1(t) − ˆb1(t) ˆH � = i ℏ � ˆH†(t) − ˆH(t) � ˆb1 (21) would yield ˆb1(t) = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is field outside the BH event horizon would not change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, if ˆH† ̸= ˆH, the right-hand-side of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (21) is no longer zero and the external field can be altered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In the present model of BH evaporation the von Neu- mann entropy is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Namely, since evolution of the system is described by a Hamiltonian, the system re- mains in a pure state and, thus, the net entropy remains equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is true even if the Hamiltonian is not Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For the latter, the system’s state vector should be normalized such that ⟨ψ |ψ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The toy model Hamiltonian (17) explains why non- unitary absorption of photons at the BH center alters 5 radiation outside the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, it does not describe the system’s dynamics correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The point is that, non- Hermitian Hamiltonians don’t preserve the expectation value of an operator ˆQ with which they commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is the reason why the norm of the state vector is not conserved (in this case ˆQ = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' To incorporate a con- servation law � ˆQ � = const into the model, we must replace the non-Hermitian Hamiltonian ˆH with a con- strained Hamiltonian [24, 25] ˆH − λ(t) ˆQ, (22) where λ(t) is a Lagrange multiplier whose value is to be chosen so as to honor the constraint condition � ˆQ � = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We will impose a constraint that during BH evapora- tion the average energy is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Operators describ- ing conserved quantities must commute with the Hamil- tonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Such “energy” operators commuting with the Hamiltonian (17) are ˆσ†ˆσ − ˆb† 2ˆb2, and ˆb† 1ˆb1, and the constraints read � ˆσ†ˆσ − ˆb† 2ˆb2 � = const and � ˆb† 1ˆb1 � = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (23) The constrained interaction Hamiltonian is ˆV (t) = gˆσe−iωtφ2(t)ˆb2 + iℏ ˆC(t), (24) where ˆC(t) = ˙µ1(t)ˆb† 1ˆb1 + ˙µ2(t) � ˆσ†ˆσ − ˆb† 2ˆb2 � + ˙µ3(t) and the dot denotes derivative over t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The latter is in- troduced for convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We assume that the resonance condition ω = cΩ/2rg is satisfied, which yields ˆV (t) = gˆσˆb2 + iℏ ˆC(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (25) The Lagrange multiplier ˙µ3(t) takes into account the nor- malization condition ⟨ψ |ψ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For the present prob- lem, Lagrange multipliers ˙µ1,2,3(t) are real functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We assume that initially the oscillator is in a coher- ent state |A⟩ and the field is in the Unruh vacuum |0U⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Schr¨odinger equation with the constrained Hamiltonian (25) yields (see Appendix A and B) |ψ(t)⟩ = N(t)e− i ℏ γgAeµ1(t)tˆb† 1eeµ1(t)−µ2(t)γˆb† 1ˆb† 2 |0R⟩ ���eµ2(t)A � , (26) where N(t) is a normalization factor and the Lagrange multipliers are obtained from the constraint equations e2µ2A2 − ˜γ2 1 − ˜γ2 − e2µ1˜γ2 (γΛt)2 (1 − ˜γ2)2 = A2 − γ2 1 − γ2 , (27) ˜γ2 1 − ˜γ2 + e2µ1 (γΛt)2 (1 − ˜γ2)2 = γ2 1 − γ2 , (28) where ˜γ = eµ1−µ2γ and Λ = gA/ℏ is the Rabi frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For t → ∞, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (26)-(28) give |ψ(∞)⟩ = Ne − iγ √ 1−γ2 ˆb† 1 |0R⟩ |A∞⟩ , (29) where A2 ∞ = A2 − γ2 1 − γ2 (30) is the mean number of oscillator excitations in the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The present model of the spacetime boundary is self-consistent if the oscillators absorb all ingoing pho- tons, which implies A2 ∞ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Otherwise, photon flux through the boundary would be nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (29) shows that the final state of the field is the Rindler vacuum for photons ˆb2 and a coherent state for photons ˆb1 with the average photon number γ2/(1 − γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The oscillator remains in the coherent state, but the oscil- lator’s mean excitation number is reduced by an amount γ2/(1 − γ2) due to absorption of all ˆb2 photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For in our model � ˆb† 1ˆb1 � = const, the radiation power of an evaporating BH is given by the Hawking’s formula, but photon statistics is not thermal and the outgoing radiation carries information about the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In particular, coherent oscillations of the BH interior lead to a coherent outgoing radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In the limit γ ≪ 1, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (27) and (28) can be solved analytically yielding the following expression for the sys- tem’s state vector as a function of t |ψ(t)⟩ = N(t)e − iγΛtˆb† 1 √ 1+Λ2t2 e γˆb† 1ˆb† 2 √ 1+Λ2t2 |0R⟩ |A(t)⟩ , (31) where A2(t) = A2 − Λ2t2 1 + Λ2t2 γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (31), initial thermal Hawking radiation evolves into the coherent state e−iγˆb† 1 |0R⟩ on a time scale 1/Λ, while the oscillator’s energy (BH mass) decreases as ℏωA2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' INSIGHTS FROM QUANTUM GRAVITY MODELS Here we show that present mechanism of nonthermal emission of evaporating BHs holds for an effective metric obtained in quantum gravity models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Most of such mod- els suggest that the classical singularity at r = 0 should be replaced by a regular timelike boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' To be spe- cific, we consider an effective BH metric obtained from scale-dependent effective average action which takes into account the effect of all loops [26–28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a function of 6 this scale, the effective average action satisfies a renor- malization group equation yielding the effective metric [29] ds2 = f(r)c2dt2 − 1 f(r)dr2 − r2 � dθ2 + sin2 θdϕ2� , (32) where f(r) = 1 − rg r 1 1 + ¯ωr2 g r2 , (33) and ¯ω > 0 is a constant that involves the quantum grav- ity correction to the BH geometry coming from the renor- malization group approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The metric (32) is regular at r = 0 and has two hori- zons which can be found by setting f(r) = 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The position of the outer and inner horizons is r± = rg 2 � 1 ± √ 1 − 4¯ω � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In terms of r±, one can write 1 f(r) = 1 + rgr (r − r−)(r − r+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' A massless scalar field φ obeys the covariant wave equation 1 √−g ∂ ∂xµ �√−ggµν ∂φ ∂xν � = 0, (34) where gµν is the spacetime metric given by the interval (32), namely gtt = 1 f(r), grr = −f(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For the truncated 1+1 dimensional spacetime √−g = 1, and the wave equation (34) reduces to 1 c2 ∂2φ ∂t2 − f(r) ∂ ∂r � f(r)∂φ ∂r � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (35) Solutions of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (35) read φν(t, r) = e−iν[t± r c ∓χ(r)], (36) where χ(r) = r− ln |r − r−| − r+ ln |r − r+| c√1 − 4¯ω .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (36), one can construct mode functions analogous to the Rindler modes (8) and (9) in the Schwarzschild coordinates, namely, φ1ν(t, r) = e−iν[t− r c +χ(r)]θ(r − r+), (37) φ2ν(t, r) = eiν[t− r c +χ(r)]θ(r+ − r)θ(r − r−), (38) where ν > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For r− = 0, the mode functions (37) and (38) reduce to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (8) and (9) with Ω = 2rgν/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (37) and (38) show that if an observer is held fixed outside the outer event horizon at a constant r > r+, then at the observer’s location the non-zero Rindler modes φ1ν oscillate as φ1ν ∝ e−iνt, where t is the observer’s proper time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is, from the observer’s perspective, the Rindler photons φ1ν behave as if they have positive fre- quency ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, if a hypothetical observer is located at fixed r− < r < r+, the non-zero Rindler mode φ2ν oscillates as a function of the proper time t as φ2ν ∝ eiνt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' That is, from the observer’s perspective, the Rindler pho- tons φ2ν behave as if they have negative frequency −ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Absorption of photons φ2ν decreases energy (mass) of the BH, leading to BH evaporation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Photons falling into the BH from BH exterior are de- scribed by the mode functions φ3ν(t, r) = e−iν[t+ r c −χ(r)] − eiϕ0e−iν[t− r c +χ(r)]θ(r− − r), (39) where the last term describes a wave reflected from the timelike spacetime boundary r = 0, and ϕ0 is a phase shift introduced to satisfy the reflective boundary con- dition, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=', ∂φ3ν/∂r|r=0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' From the perspective of an observer held fixed at r = const the mode functions φ3ν have positive frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Thus, absorption of such photons increases the BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Light rays of photons φ1ν, φ2ν (solid line) and φ3ν (dash line) in the metric (32) for r− = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='2rg and r+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='8rg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 3 we plot light rays of photons (37), (38) and (39) in the Schwarzschild coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The figure shows that the negative (φ2ν) and positive (φ1ν) fre- quency Rindler photons are generated at the outer hori- zon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' These photons are produced in pairs and are en- tangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Photons φ1ν carry energy away from BH, while the negative energy photons φ2ν propagate toward the BH center and are absorbed at the inner horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The positive energy photons φ3ν carry energy into the BH 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='4 1 Outer horizon horizon 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='4( Inner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='2( 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='0 3 2 2 3 4 tc/17 from the BH exterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' They cross both outer and inner horizons, and after reflection from the BH center are ab- sorbed at the inner horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In the region r− < r < r+, the coordinate r plays the role of time for particles which move unidirectionally along the r coordinate in this re- gion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For r < r− and r > r+ the particles move unidi- rectionally along the t coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Spacetime described by the metric (32) is non-singular and matter does not disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Figure 3 shows that mat- ter and energy (infalling photons φ3ν) are concentrated in the vicinity of the inner horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since Rindler pho- tons φ2ν can only be annihilated and not created at the inner horizon, the non-unitary absorption of the Rindler photons φ2ν at the inner horizon, combined with the en- tanglement of photon pairs φ1ν and φ2ν generated at the outer horizon, leads to nonthermal outgoing radia- tion that carries information about the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' One can model this process by the same Hamiltonian (24) of the previous section, but now the oscillators absorbing the ingoing photons φ2ν follow the worldline of the in- ner horizon and can be a model of matter rather than gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The picture becomes more intuitive if we describe BH evaporation in terms of particles and antiparticles that can annihilate with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In this picture, particle (φ1ν) and antiparticle (φ2ν) are generated as entangled pairs at the outer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The particles φ1ν carry en- ergy away from BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The antiparticles move towards BH center and at the inner horizon annihilate with particles φ3ν which have been accumulated at the inner horizon during BH formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Due to entanglement between φ1ν and φ2ν, the information about state of particles φ3ν is recorded into the outgoing flux of particles φ1ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' SUMMARY AND DISCUSSION Evaporation of a classical Schwarzschild BH is caused by creation of entangled particle-antiparticle pairs (Rindler photons in the present discussion) near the event horizon, with one carrying positive energy to infinity and the other carrying negative energy into the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is the mechanism of Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Absorption of the negative energy photons at the center of the classical BH reduces the BH mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Here we argue that previous models of Hawking radi- ation are lacking an important ingredient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Namely, the process of photon absorption at the BH center must be properly described quantum mechanically by construct- ing a Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since under the BH event horizon, light can propagate only towards the BH center, the symmetry between absorption and emission is broken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Namely, BH center can only absorb photons, but do not emit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a result, the Hamiltonian describing BH evap- oration is not Hermitian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' To describe absorption of photons at the BH center, we assume that the latter consists of harmonic oscilla- tors which absorb the ingoing radiation, but do not emit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In our model, the oscillators follow the worldline of the BH center, rather than geodesics, and carry information about the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We show that due to entanglement between photons moving inside and outside the BH event horizon, the non-unitary absorption of the radiation under the hori- zon alters the state of the field outside the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a consequence, radiation produced by the evaporating BH is not thermal and carries information about the BH in- terior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' After the BH has evaporated, the information is recorded in the remaining non-thermal field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since evolu- tion is governed by a Hamiltonian, the state of the system remains pure and during BH evaporation the von Neu- mann entropy is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In our model we impose a constraint that energy is conserved during BH evapora- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a consequence, our model yields that luminosity of an evaporating BH coincides with that for Hawking radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Erasing information at the BH center produced by pho- ton absorption is a non-unitary process which leads to a change of the field outside the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is some- what analogous to the quantum eraser experiments in which the interference pattern can be destroyed or re- stored by manipulating entangled photon partners [30– 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In these experiments, after two entangled photons are created, each is directed into different section of the apparatus and an interference pattern for one of them is examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' A measurement done on the entangled partner to learn about the photon path influences the interference pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Similarly to BH evaporation, non-unitarity of the mea- surement process alters the state of the entangled part- ner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, the state vector collapse brought about by a measurement is a probabilistic and discontinuous change, while BH evaporation is a deterministic, contin- uous time evolution of an isolated system that obeys the Schr¨odinger equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Our findings show that quantum mechanical evolution, governed by the Schr¨odinger equation, allows informa- tion to leak out from the BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' This is the case because BH center breaks the emission-absorption symmetry and photons external to the horizon are entangled with those inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Such entanglement is an inherent property of the field for evaporating BHs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We also show that present mechanism of nonthermal emission of evaporating BHs holds for spacetimes ob- tained in quantum gravity models in which the classi- cal singularity at r = 0 is replaced by a regular time- like boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For such spacetimes the metric has an inner and outer horizons, and matter does not disap- pear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Instead, particles are accumulated in the vicin- ity of the inner horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' For this spacetime, the entan- gled particle-antiparticle pairs are generated at the outer horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The generated particles carry energy away from BH, while antiparticles move towards the BH center and annihilate at the inner horizon with particles that form the BH interior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Due to entanglement of the particle- antiparticle pairs produced at the outer horizon, the in- 8 formation about the BH interior is recorded in the out- going particle flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' One should mention that if our findings are correct, and radiation of evaporating BHs is nonthermal, the Bekenstein-Hawking formula [33, 34] does not describe the BH entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Recall that the latter formula assumes thermal BH emission with the Hawking temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Our results demonstrate that quantum mechanics works in an exotic spacetime geometry of a BH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' However, BHs might have only a mathematical significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The point is that there is an evidence that general relativity is ruled out by gravitational waves detection experiments in favor of the vector theory of gravity [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The latter theory [36, 37] agrees with all available tests of gravity, including detection of gravitational waves and observa- tions of supermassive objects at galactic centers [35, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' In addition, vector gravity predicts no BHs and yields the measured value of the cosmological constant [39] with no free parameters [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by the Air Force Office of Scientific Research (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' FA9550-20-1-0366 DEF), the Robert A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Welch Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' A-1261), and the National Science Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' PHY- 2013771).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Appendix A: Operator identities and expectation values Operators of Rindler photons ˆb1 and ˆb2 obey bosonic commutation relations [ˆb1,ˆb† 1] = 1, [ˆb2,ˆb† 2] = 1, and all other commutators are equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' First we prove an operator identity ˆb2eγˆb† 1ˆb† 2 = eγˆb† 1ˆb† 2ˆb2 + γˆb† 1eγˆb† 1ˆb† 2, (A1) where γ is a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Introducing operator ˆF(γ) = ˆb2eγˆb† 1ˆb† 2 − eγˆb† 1ˆb† 2ˆb2, we have d ˆF(γ) dγ = ˆb2ˆb† 1ˆb† 2eγˆb† 1ˆb† 2−ˆb† 1ˆb† 2eγˆb† 1ˆb† 2ˆb2 = ˆb† 1ˆb† 2 ˆF(γ)+ˆb† 1eγˆb† 1ˆb† 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Solution of this differential equation, subject to the con- dition ˆF(0) = 0, is ˆF(γ) = γˆb† 1eγˆb† 1ˆb† 2, which proves the identity (A1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Next we prove an identity eλˆb† 2ˆb2ˆb† 2 = eλˆb† 2eλˆb† 2ˆb2, (A2) where λ is a complex number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Introducing operator ˆF(λ) = eλˆb† 2ˆb2ˆb† 2 − ˆb† 2eλˆb† 2ˆb2, we have d ˆF(λ) dλ = ˆb† 2ˆb2eλˆb† 2ˆb2ˆb† 2−ˆb† 2ˆb† 2ˆb2eλˆb† 2ˆb2 = ˆb† 2ˆb2 ˆF(λ)+ˆb† 2eλˆb† 2ˆb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Solution of this differential equation, subject to the con- dition ˆF(0) = 0, is ˆF(λ) = � eλ − 1 �ˆb† 2eλˆb† 2ˆb2, which proves the identity (A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Next we prove an identity eλˆb† 2ˆb2eγˆb† 1ˆb† 2 = eeλγˆb† 1ˆb† 2eλˆb† 2ˆb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A3) Introducing operator ˆF(λ) = eλˆb† 2ˆb2eγˆb† 1ˆb† 2e−λˆb† 2ˆb2, and taking derivative over λ, we have d ˆF(λ) dλ = eλˆb† 2ˆb2ˆb† 2ˆb2eγˆb† 1ˆb† 2e−λˆb† 2ˆb2−eλˆb† 2ˆb2eγˆb† 1ˆb† 2ˆb† 2ˆb2e−λˆb† 2ˆb2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Taking into account identities (A1) and (A2), we obtain d ˆF(λ) dλ = γˆb† 1eλˆb† 2ˆb2ˆb† 2eγˆb† 1ˆb† 2e−λˆb† 2ˆb2 = γeλˆb† 1ˆb† 2 ˆF(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Solution of this differential equation, subject to the con- dition ˆF(0) = eγˆb† 1ˆb† 2, is ˆF(λ) = eeλγˆb† 1ˆb† 2, which proves the identity (A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Next we calculate a matrix element ⟨ψ|ψ⟩, where state vector |ψ⟩ is |ψ⟩ = � 1 − γ2eβˆb† 1eγˆb† 1ˆb† 2 |0R⟩ , (A4) |0R⟩ stands for the Rindler vacuum, β is a complex num- ber and γ is a real number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The state vector (A4) can be written as |ψ⟩ = eβˆb† 1 |0M⟩ , where |0M⟩ = � 1 − γ2eγˆb† 1ˆb† 2 |0R⟩ (A5) is the Minkowski vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Using a relation between operators of the Rindler photons ˆb1,2 and the Unruh- Minkowski photons ˆa1,2 [40] ˆb† 1 = ˆa† 1 + γˆa2 � 1 − γ2 , 9 and the property ˆa1,2 |0M⟩ = 0, we obtain |ψ⟩ = e βˆa† 1 √ 1−γ2 |0M⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Taking into account that eαˆa† 1 |0M⟩ = e |α|2 2 |α0⟩ , where |α0⟩ stands for a coherent state |α⟩ for the Unruh- Minkowski photons ˆa1 and the vacuum state for the Unruh-Minkowski photons ˆa2, we find |ψ⟩ = e |β|2 2(1−γ2) |α0⟩ , (A6) where α = β/ � 1 − γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Therefore ⟨ψ|ψ⟩ = e |β|2 1−γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A7) Next we calculate the average number of Rindler photons ˆb1 in the state |ψ⟩, that is � ˆb† 1ˆb1 � ≡ ⟨ψ|ˆb† 1ˆb1 |ψ⟩ / ⟨ψ|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Taking derivative of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A7) with respect to β and β∗, and using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A4), we have ⟨ψ|ˆb1ˆb† 1 |ψ⟩ = ∂2 ∂β∂β∗ e |β|2 1−γ2 = 1 − γ2 + |β|2 (1 − γ2)2 e |β|2 1−γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Therefore � ˆb† 1ˆb1 � = ⟨ψ|ˆb1ˆb† 1 |ψ⟩ ⟨ψ|ψ⟩ − 1 = γ2 1 − γ2 + |β|2 (1 − γ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A8) To find � ˆb† 2ˆb2 � we use the relations between operators of the Rindler photons ˆb1,2 and the Unruh-Minkowski photons ˆa1,2 [40] ˆb2 = ˆa2 + γˆa† 1 � 1 − γ2 , ˆb† 2 = ˆa† 2 + γˆa1 � 1 − γ2 , which yield ˆb† 2ˆb2 = 1 1 − γ2 � ˆa† 2ˆa2 + γ2ˆa1ˆa† 1 + γˆa1ˆa2 + γˆa† 2ˆa† 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A6), we obtain ⟨ψ|ˆb† 2ˆb2 |ψ⟩ = γ2 � 1 + |α|2� 1 − γ2 e |β|2 1−γ2 , where we took into account that ⟨α0| ˆa1ˆa† 1 |α0⟩ = 1+|α|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' As a result, � ˆb† 2ˆb2 � = ⟨ψ|ˆb† 2ˆb2 |ψ⟩ ⟨ψ|ψ⟩ = γ2 1 − γ2 + γ2|β|2 (1 − γ2)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A9) Appendix B: State vector evolution during black hole evaporation For our model of black hole evaporation, the con- strained interaction Hamiltonian is ˆV (t) = gˆσˆb2+iℏ ˙µ1(t)ˆb† 1ˆb1+iℏ ˙µ2(t) � ˆσ†ˆσ − ˆb† 2ˆb2 � +iℏ ˙µ3(t), where functions µ1,2,3(t) are real, and the oscillator’s lowering and raising operators ˆσ and ˆσ† obey the same bosonic commutation relations as the operators of Rindler photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Schrodinger equation for the evolution of the field state vector iℏ ∂ ∂t |ψ(t)⟩ = ˆV (t) |ψ(t)⟩ yields |ψ(t)⟩ = eµ3eµ1ˆb† 1ˆb1+µ2(ˆσ†ˆσ−ˆb† 2ˆb2)− i ℏ gtˆσˆb2 |0M⟩ |A⟩ , (B1) where |0M⟩ and |A⟩ are the initial state vectors of the field and the oscillator respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' We assume that the latter is a coherent state |A⟩, where A is real, and the former is the Minkowski vacuum |0M⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Recall that Unruh vacuum coincides with the Minkowski vacuum for the right-moving photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Taking into account that ˆσˆb2 commutes with ˆb† 1ˆb1 and ˆσ†ˆσ−ˆb† 2ˆb2, and plugging |0M⟩ from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A5) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (B1), we obtain |ψ(t)⟩ = � 1 − γ2eµ3e− i ℏ gtˆσˆb2eµ1ˆb† 1ˆb1+µ2(ˆσ†ˆσ−ˆb† 2ˆb2)eγˆb† 1ˆb† 2 |0R⟩ |A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Using identity (A3), we have |ψ(t)⟩ = � 1 − γ2eµ3e− i ℏ gtˆσˆb2eeµ1−µ2 γˆb† 1ˆb† 2eµ2ˆσ†ˆσ |0R⟩ |A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Taking into account that eµ2ˆσ† ˆσ |A⟩ = e |A|2 2 (e2µ2 −1) |eµ2A⟩ , we find |ψ(t)⟩ = � 1 − γ2eµ3+ |A|2 2 (e2µ2 −1)e− i ℏ gtˆσˆb2eeµ1−µ2 γˆb† 1ˆb† 2 |0R⟩ |eµ2A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Since the initial state of the oscillator is the coherent state |A⟩, and ˆσ |A⟩ = A |A⟩, one can write |ψ(t)⟩ = � 1 − γ2eµ3+ |A|2 2 (e2µ2 −1)e− i ℏ geµ2 Atˆb2eeµ1−µ2 γˆb† 1ˆb† 2 |0R⟩ |eµ2A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 10 Using the Baker–Hausdorff formula e ˆ Ae ˆ B = e[ ˆ A, ˆ B]e ˆ Be ˆ A, we finally obtain |ψ(t)⟩ = � 1 − γ2eµ3+ |A|2 2 (e2µ2 −1)e− i ℏ γgAeµ1 tˆb† 1eeµ1−µ2 γˆb† 1ˆb† 2 |0R⟩ |eµ2A⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (B2) Using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' (A8) and (A9), we find that the average number of Rindler photons in the state (B2) is � ˆb† 1ˆb1 � = ˜γ2 1 − ˜γ2 + (γgAt)2 ℏ2 e2µ1 (1 − ˜γ2)2 , � ˆb† 2ˆb2 � = ˜γ2 1 − ˜γ2 + (γgAt)2 ℏ2 e2µ1˜γ2 (1 − ˜γ2)2 , where ˜γ = eµ1−µ2γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The average number of oscillator excitations in the state (B2) is � ˆσ†ˆσ � = e2µ2A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Constraints � ˆσ†ˆσ − ˆb† 2ˆb2 � = const and � ˆb† 1ˆb1 � = const give equations e2µ2A2 − ˜γ2 1 − ˜γ2 − (γgAt)2 ℏ2 e2µ1˜γ2 (1 − ˜γ2)2 = A2 − γ2 1 − γ2 , ˜γ2 1 − ˜γ2 + (γgAt)2 ℏ2 e2µ1 (1 − ˜γ2)2 = γ2 1 − γ2 , which for t → ∞ yield ˜γ → 0, 1 ℏγgAeµ1 ≈ γ � 1 − γ2t , e2µ2A2 → A2− γ2 1 − γ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Therefore, for t → ∞ � ˆb† 1ˆb1 � = γ2 1 − γ2 , � ˆb† 2ˆb2 � = 0, � ˆσ†ˆσ � = A2 − γ2 1 − γ2 , and the normalized state vector of the system is |ψ(∞)⟩ = e − γ2 2(1−γ2) e −i γ √ 1−γ2 ˆb† 1 |0R⟩ ����� � A2 − γ2 1 − γ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' The final state is the Rindler vacuum for photons ˆb2, a coherent state for photons ˆb1 with the average photon number γ2/(1 − γ2), and the oscillator remains in the coherent state with a reduced average excitation number A2 − γ2/(1 − γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, BH explosions?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=', Nature (London) 248, 30 (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [2] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, Particle creation by black holes, Commu- nications in Mathematical Physics 43, 199 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [3] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Mukhanov, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Winizki, Introduction to Quantum Effects in Gravity (Cambridge University Press, Cam- bridge, England, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [4] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Frolov, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Novikov, Black hole physics (Kluwer Academic Publishers, The Netherlands, 1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [5] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, Breakdown of predictability in gravita- tional collapse, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D 14, 2460 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [6] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Marolf, The black hole information problem: past, present, and future, Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 80, 092001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [7] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Calmet, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hsu, A brief history of Hawking’s information paradox, EPL 139, 49001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [8] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' ’t Hooft, Dimensional reduction in quantum gravity, Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' C 930308, 284 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [9] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Susskind, The World as a hologram, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 36, 6377 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [10] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Laddha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Prabhu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Raju and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Shrivastava, The holographic nature of null infinity, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 10, 041 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Chowdhury, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Papadoulaki and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Raju, A physical protocol for observers near the boundary to obtain bulk information in quantum gravity, SciPost Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 10, 106 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [12] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Calmet and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hsu, Quantum hair and black hole information, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' B 827, 136995 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [13] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Calmet, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Casadio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hsu, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Kuipers, Quantum hair from gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 128, 111301 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [14] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Strominger, On BMS invariance of gravitational scat- tering, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' High Energy Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 2014, 152 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [15] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Perry, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Strominger, Soft hair on black holes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 116, 231301 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [16] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Perry, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Strominger, Superro- 11 tation charge and supertranslation hair on black holes, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 2017, 161 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [17] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Maldacena, The Large N Limit of Superconformal field theories and supergravity, Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Theor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 2, 231 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [18] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Almheiri, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hartman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Maldacena, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Shaghoulian, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Tajdini, The entropy of Hawking radiation, Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 93, 035002 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [19] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Azizi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Ben-Benjamin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Scully, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Unruh, Unruh and Cherenkov radiation from a negative frequency perspective, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 126, 063603 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [20] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rindler, Kruskal space and the uniformly accelerated frame, Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 34, 1174 (1966).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [21] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Unruh and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Wald, What happens when an ac- celerating observer detects a Rindler particle, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D 29, 1047 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [22] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, The quantum mechanics of black holes, Scientific American 236, 34 (1977).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [23] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Davies, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Fulling, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Unruh, Energy- momentum tensor near an evaporating black hole, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D 13, 2720 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [24] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Gross, Collective rotations of nuclei, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 14, 389 (1959).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [25] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Amiot, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Griffin, Constraints in Quantum Me- chanics via Lagrangians, Hamiltonians, and Canonical Equations of Motion, Annals of Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 95, 295 (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [26] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Wetterich, Exact evolution equation for the effective potential, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' B 301, 90 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [27] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Reuter and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Wetterich, Effective average action for gauge theories and exact evolution equations, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' B417, 181 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [28] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Reuter, Nonperturbative evolution equation for quan- tum gravity, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D 57, 971 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [29] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Bonanno and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Reuter, Renormalization group im- proved black hole spacetimes, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D 62, 043008 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [30] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Scully and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Dr¨uhl, Quantum eraser: A pro- posed photon correlation experiment concerning observa- tion and ”delayed choice” in quantum mechanics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' A 25, 2208 (1982).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [31] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Kim, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Yu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Kulik, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Shih, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Scully, A Delayed Choice Quantum Eraser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 84, 1 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [32] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Walborn, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Terra Cunha, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Padua, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Monken, Double-slit quantum eraser, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' A 65, 033818 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [33] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Bekenstein, Black holes and the second law, Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Nuovo Cimento 4, 737 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [34] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hawking, Black holes and thermodynamics, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' D 13, 191 (1976).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [35] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Hilborn, GW170817 event rules out general relativity in favor of vector gravity, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Spec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 230, 1149 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [36] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky, Vector theory of gravity: universe with- out black holes and solution of dark energy problem, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' 92, 125001 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [37] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky, Simplified equations for gravitational field in the vector theory of gravity and new insights into dark energy, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Dark Universe 25, 100321 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [38] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky, Oscillating axion bubbles as an alterna- tive to supermassive black holes at galactic centers, JCAP 10, 018 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [39] Planck Collaboration, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Ade, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Aghanim, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Armitage-Caplan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=', Planck 2013 results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' XVI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Cos- mological parameters, A&A 571, A16 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' [40] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Svidzinsky, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Azizi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Ben-Benjamin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Scully, and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Unruh, Causality in quantum optics and entanglement of Minkowski vacuum, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} +page_content=' Research 3, 013202 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdFPT4oBgHgl3EQfnjXp/content/2301.13131v1.pdf'} diff --git a/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf b/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..796c919c5345110eb76eae38f39d19e75539345e --- /dev/null +++ b/QdA0T4oBgHgl3EQfDf80/content/2301.02003v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bcfe13c8fd78e7a3142276f4c803112b91366db0f031aae99dcad363724fcdfa +size 622311 diff --git a/QdE5T4oBgHgl3EQfZQ86/content/tmp_files/2301.05579v1.pdf.txt b/QdE5T4oBgHgl3EQfZQ86/content/tmp_files/2301.05579v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1684719810c424c4a41ba4e9bbec58333bb9a2aa --- /dev/null +++ b/QdE5T4oBgHgl3EQfZQ86/content/tmp_files/2301.05579v1.pdf.txt @@ -0,0 +1,1597 @@ +A survey and taxonomy of loss functions in machine +learning +LORENZO CIAMPICONI, ADAM ELWOOD, MARCO LEONARDI, ASHRAF MOHAMED, +and ALESSANDRO ROZZA, lastminute.com group, Switzerland +Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions. Defining +appropriate loss functions is therefore critical to successfully solving problems in this field. We present a +survey of the most commonly used loss functions for a wide range of different applications, divided into +classification, regression, ranking, sample generation and energy based modelling. Overall, we introduce +33 different loss functions and we organise them into an intuitive taxonomy. Each loss function is given a +theoretical backing and we describe where it is best used. This survey aims to provide a reference of the most +essential loss functions for both beginner and advanced machine learning practitioners. +Additional Key Words and Phrases: loss functions, machine learning, neural networks, survey +1 +INTRODUCTION +In the last few decades there has been an explosion in interest in machine learning [52, 76]. This +field focuses on the definition and application of algorithms that can be trained on data to model +underlying patterns [11, 73, 77, 88]. Machine learning approaches can be applied to many different +research fields, including biomedical science [59, 84, 95, 126], natural language understanding +[22, 83], [97] anomaly detection [17], image classification [71], database knowledge discovery [32], +robot learning [3], online advertising [86], time series forecasting [13], brain computer interfacing +[78] and many more [98]. To train these algorithms, it is necessary to define an objective function, +which gives a scalar measure of the algorithm’s performance [77, 116]. They can then be trained +by optimising the value of the objective function. +Within the machine learning literature, such objective functions are usually defined in the form +of loss functions, which are optimal when they are minimised. The exact form of the loss function +depends on the nature of the problem to be solved, the data available and the type of machine +learning algorithm being optimised. Finding appropriate loss functions is therefore one of the most +important research endeavours in machine learning. +As the field of machine learning has developed, lots of different loss functions have been proposed. +It is therefore very useful to summarise and understand them. However, there are few works that +attempt to do this for the whole field [119]. The existing reviews of loss functions in the literature +either lack a good taxonomy to structure and contextualise the different losses, or are specifically +focused on a particular subset of machine learning applications [49, 117]. There is also no single +source that puts the most commonly used loss functions in the same formal setting, listing the +advantages and drawbacks of each one. +For this reason, we have worked to build a proper taxonomy of loss functions, where we show +the advantages and disadvantages for each technique. We hope this will be useful for new users +who want to familiarise themselves with the most common loss functions used in the machine +learning literature and find one that is suitable for a problem that they are trying to solve. We also +hope this summary will be useful as a comprehensive reference for advanced users, allowing them +to quickly find the best loss function without having to broadly search the literature. Additionally, +this can be helpful for researchers to find possible avenues for further research, or to understand +where to place any new techniques that they have proposed. They could, for example, use this +Authors’ address: Lorenzo Ciampiconi, lorenzo.ciampiconi@lastminute.com; Adam Elwood, adam.elwood@lastminute.com; +Marco Leonardi, marco.leonardi@lastminute.com; Ashraf Mohamed, ashraf.mohamed@lastminute.com; Alessandro Rozza, +alessandro.rozza@lastminute.com, lastminute.com group, Vicolo de’ Calvi, 2, Chiasso, Switzerland. +, Vol. 1, No. 1, Article . Publication date: January 2023. +arXiv:2301.05579v1 [cs.LG] 13 Jan 2023 + +survey to understand if their new proposals fit somewhere inside the taxonomy we present, or if +they are in a completely new category, maybe combining disparate ideas in novel ways. +Overall, we have included 33 of the most widely used loss functions. In each section of this work, +we break down the losses based on the broad classification of tasks that they can be used for. Each +loss function will be defined mathematically, and its most common applications listed highlighting +advantages and drawbacks. +The main contribution of this work can be found in the proposed taxonomy depicted in Fig. 1. +Each loss function is first divided according the specific task on which they are exploited: regression, +classification, ranking, sample generation and energy-based modelling. Furthermore, we divide +them by the type of learning paradigm on which they can be applied to, from supervised to +unsupervised. Finally, we classify them according to the underling strategy on which they are based, +such as if they rely on a probabilistic formalization, or are based on errors or a margin between the +prediction and the actual values. +This work is organized as follows: In Section 2, we provide a formal definition of a loss function +and introduce our taxonomy. In Section 3, we describe the most common regularization methods +used to reduce model complexity. In Section 4, we describe the regression task and the key loss +functions used to train regression models. In Section 5, we introduce the classification problem and +the associated loss functions. In Section 6, we present generative models and their losses. Ranking +problems and their loss functions are introduced in Section 7, and energy based models and their +losses are described in Section 8. Finally, we draw conclusions in Section 9. +2 +DEFINITION OF OUR LOSS FUNCTION TAXONOMY +In a general machine learning problem, the aim is to learn a function 𝑓 that transforms an input, +defined by the input space Φ into a desirable output, defined by the output space Y: +𝑓 : Φ → Y +Where 𝑓 is a function that can be approximated by a model, 𝑓Θ, parameterised by the parameters +Θ. +Given a set of inputs {x0, ..., x𝑁 } ∈ Φ, they are used to train the model with reference to target +variables in the output space, {y0, ..., y𝑁 } ∈ Y. Notice that, in some cases (such as autoencoders) +Y = Φ. +A loss function, 𝐿, is defined as a mapping of 𝑓 (x𝑖) with it’s corresponding y𝑖 to a real number +𝑙 ∈ R, which captures the similarity between 𝑓 (x𝑖) and y𝑖. Aggregating over all the points of the +dataset we find the overall loss, L: +L(𝑓 |{x0, ..., x𝑁 }, {y0, ..., y𝑁 }) = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝐿(𝑓 (x𝑖), y𝑖) +(1) +The optimisation function to be solved is defined as: +min +𝑓 +L(𝑓 |{x0, ..., x𝑁 }, {y0, ..., y𝑁 }) +(2) +Notice that, it is often convenient to explicitly introduce a regularisation term (𝑅) which maps 𝑓 +to a real number 𝑟 ∈ R. This term is usually used for penalising the complexity of the model in the +optimisation [77]: +2 + +min +𝑓 +1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝐿(𝑓 (x𝑖), y𝑖) + 𝑅(𝑓 ) +(3) +In practice, the family of functions chosen for the optimisation can be parameterised by a +parameter vector Θ, which allows the minimisation to be defined as an exploration in the parameter +space: +min +Θ +1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝐿(𝑓Θ(x𝑖), y𝑖) + 𝑅(Θ) +(4) +2.1 +Optimisation techniques for loss functions +2.1.1 +Loss functions and optimisation methods. In this section, we list out the most common +mathematical properties that a loss may or may not satisfy and then we briefly discuss the main +optimisation methods employed to minimise them. For the sake of simplicity, visualisation and +understanding we define such properties in a two dimensional space, but they can be easily +generalised to a d-dimensional one. +• Continuity (CONT): A real function, that is a function from real numbers to real numbers, +can be represented by a graph in the Cartesian plane; such a function is continuous if the +graph is a single unbroken curve belonging to the real domain. A more mathematically +rigorous definition can be given by defining continuity in terms of limits. A function 𝑓 with +variable 𝑥 is continuous at the real number 𝑐, if lim𝑥→𝑐 𝑓 (𝑥) = 𝑓 (𝑐). +• Differentiability (DIFF): A differentiable function 𝑓 on a real variable is a function derivable +in each point of its domain. A differentiable function is smooth (the function is locally well +approximated as a linear function at each interior point) and does not contain any break, +angle, or cusp. A continuous function is not necessarily differentiable, but a differentiable +function is necessarily continuous. +• Lipschitz Continuity (L-CONT): A Lipschitz continuous function is limited in how fast it +can change. More formally, there exists a real number such that, for every pair of points on +the graph of this function, the absolute value of the slope of the line connecting them is not +greater than this real number; this value is called the Lipschitz constant of the function. +To understand the robustness of a model, such as a neural network, some research papers +[39, 115] have tried to train the underlying model by defining an input-output map with a +small Lipschitz constant. The intuition is that if a model is robust, it should not be too affected +by perturbations in the input, 𝑓 (𝑥 + 𝛿𝑥) ≈ 𝑓 (𝑥), and this would be ensured by having 𝑓 be +ℓ-Lipschitz where ℓ is small [85]. +• Convexity (CONV): a real-valued function 𝑓 is convex if each segment between any two +points on the graph of the function lies above the graph between the two points. Convexity is +a key feature, since the local minima of convex function is also the global minima. Whenever +the second derivative of a function exists, then the convexity is easy to check, since the +Hessian of the function must be positive semi-definite. +• Strict Convexity (S-CONV): a real-valued function is stricly convex if the segment between +any two points on the graph of the function lies above the graph between the two points, +except for the intersection points between the straight line and the curve. Strictly convex +functions have a positive definitive Hessian. Positive-definite matrices are invertible and the +optimisation problem can be so solved in a closed form. +3 + +Algorithm 2.1 Gradient Descent +Input: initial parameters Θ(0), number of iterations 𝑇, learning rate 𝛼 +Output: final learning Θ(𝑇) +1. +for 𝑡 = 0 to 𝑇 − 1 +2. +estimate ∇L(Θ(𝑡)) +3. +compute ΔΘ(𝑡) = −∇L(Θ(𝑡)) +4. +Θ(𝑡+1) := Θ(𝑡) + 𝛼ΔΘ(𝑡) +5. +return Θ(𝑇) +2.1.2 +Relevant optimisation methods. An optimisation method is a technique that, given a for- +malised optimisation problem with an objective function, returns the solution to obtain the optimal +value of that optimisation problem. Most of the optimisation methods presented in this work +rely on algorithms that may not guarantee the optimality of the solution, but imply a degree of +approximation. +• Closed form solutions are systems of equations that can be solved analytically by finding +the values of Θ that lead to a zero value for the derivative of the loss function. An optimization +problem is closed-form solvable if its objective function is differentiable with respect to Θ +and the differentiation can be solved for Θ. In general differentiability and strict convexity +are required to have a closed form solution. Closed-form solutions should always be used +instead of iterative algorithms if they’re available and computationally feasible. +• Gradient Descent is a first-order1 iterative optimization algorithm for finding a local mini- +mum of a differentiable function. The procedure takes repeated steps in the opposite direction +of the gradient of the function at the current point, with a step-size defined by a parameter 𝛼, +often called the learning rate. +The loss function employed must be differentiable, so that the gradient can be computed. In +order to overcome this limitation and employ also non-differentiable loss function, approxi- +mation of gradient and other techniques can be used [56, 99]. The procedure for gradient +descent is formalized in Algorithm 1. +• Stochastic Gradient Descent (SGD [77]) is a stochastic approximation of gradient descent +optimization. It replaces the actual gradient, calculated from the entire dataset, by an estimate, +which is calculated from a randomly selected subset of the data. The stochastic gradient is an +unbiased estimate of the real gradient. +In high-dimensional optimization problems, such as in artificial neural networks, this reduces +the time cost. The stochasticity of this method reduces the probability of the optimisation to +get stuck in a local minimum. SGD shares the same constraints (i.e. differentiability, convexity +for optimal solution) of traditional Gradient Descent. +• Derivative Free Optimisation In some cases the derivative of the objective function may +not exist, or may not be easy to calculate. This is where derivative-free optimisation comes +into the picture. Classical simulated annealing arithmetic, genetic algorithms and particle +swarm optimisation are a few such examples. Conventional derivative free optimisation +methods are usually difficult to scale to large-size problems. To learn more about derivative +free optimisation you can refer to [24, 91]. +• Zeroth Order optimisation Zeroth-Order (ZOO) optimisation is a subset of gradient-free +optimisation that emerges in various signal processing as well as machine learning appli- +cations [70]. ZOO optimisation methods are the gradient-free counterparts of first-order +1In numerical analysis, methods that have at most linear local error are called first order methods. They are frequently +based on finite differences, a local linear approximation. +4 + +optimisation techniques. ZOO approximates the full gradients or stochastic gradients through +function value-based gradient estimates. Some recent important applications include gen- +eration of prediction-evasive, black-box adversarial attacks on deep neural networks [19], +generation of model-agnostic explanation from machine learning systems [26], and design of +gradient or curvature regularised robust ML systems in a computationally-efficient manner +[70]. Zeroth optimisation can be a convenient option, compared to the conventional derivative +free optimisation approach, as it’s easy to implement inside commonly used gradient based +algorithm (e.g SGD), it approximates derivatives efficiently and has comparable convergence +rates to first-order algorithms. +5 + +Probabilistic +Error based +Margin based +GENERATIVE +CLASSIFICATION +RANKING +ENERGY BASED +SUPERVISED +SEMI SUPERVISED +UNSUPERVISED +Regularization Methods +|𝟂| - weight-normbased +𝞖 - entropy based +Lasso +Ridge +|𝟂| +Cosine +similarity +Quadratically +Smoothed +Modified +Huber +Cross +Entropy +Kullback-Leibler +Divergence +Ramp loss +Energy loss +Generalized +Perceptron +REGRESSION +Zero-One +Smoothed +Hinge +Negative +Log-Likelihood +MinMax +Wesserstein +Pairwise +Ranking +Triplet +Ranking +Diffusion +Generalized +Margin +Log loss +Minimum +classification error +Square square +Square exponential +Hinge +Mean Squared +Error +Mean Bias +Error +Huber +Log cosh +Root Mean +Squared Error +Smooth L1 +Mean Absolute +Error +Root Mean Squared +Logarithmic Error +Fig. 1. The proposed taxonomy. Five major tasks are identified on which loss function are applied to, namely +regression, classification, ranking, generating samples (generative) and energy based. With different colors +we specify the type of learning paradigm, from supervised to unsupervised of each loss function. Finally the +underlying strategy to optimize them, namely margin based, probabilistic and error based is illustrated under +each group of losses. +6 + +2.2 +Our taxonomy +Our taxonomy is summarized in Fig 1 . To define it, we started by categorizing the losses depending +on which machine learning problem they are best suited to solve. We have identified the following +categories: +• Regression (Sec. 4) +• Classification (Sec. 5) +• Generative modelling (Sec. 6) +• Ranking (Sec. 7) +• Energy based modelling (Sec. 8) +We also made a distinction based on the mathematical concepts used to define the loss obtaining +the following sub-categories: +• Error based +• Probabilistic +• Margin based +Exploiting this approach we find a compact and intuitive taxonomy, with little redundancy or +overlap between the different sections. We have employed well known terminology to define the +taxonomy, which will make it easier for any user to intuitively understand it. +3 +REGULARISATION METHODS +Regularisation methods can be applied to almost all loss functions. They are employed to reduce +model complexity, simplifying the trained model and reducing it’s propensity to overfit the training +data [5, 30, 61]. Model complexity, is usually measured by the number of parameters and their +magnitude [5, 77, 79]. There are many techniques which fall under the umbrella of regularisation +method and a significant number of them are based on the augmentation of the loss function [30, 77]. +An intuitive justification for regularization is that it imposes Occam’s razor on the complexity of +the final model. More theoretically, many loss-based regularization techniques are equivalent to +imposing certain prior distributions on the model parameters. +3.1 +Regularisation by Loss Augmentation +One can design the loss function to penalise the magnitude of model parameters, thus learning the +best trade-off between bias and variance of the model and reducing the generalization error without +affecting the training error too much. This prevents overfitting, while avoiding underfitting, and +can be done by augmenting the loss function with a term that explicitly controls the magnitude of +the parameters, or implicitly reduces the number of them. The general way of augmenting a loss +function in order to regularise the result is formalized in the following equation: +�𝐿(𝑓 (x𝑖), y𝑖) = 𝐿(𝑓 (x𝑖), y𝑖) + 𝜆𝜌(Θ) +(5) +where 𝜌(Θ) is called regularization function and 𝜆 defines the amount of regularisation (the trade-off +between fit and generalisation). +This general definition makes it clear that we can employ regularization on any of the losses +proposed in this paper. +We are now going to describe the most common regularisation methods based on loss augmen- +tation. +3.1.1 +L2-norm regularisation. In 𝐿2 regularization the loss is augmented to include the weighted +𝐿2 norm of the weights [12, 77], so the regularisation function is 𝜌(Θ) = ∥Θ∥2 +2: +�𝐿(𝑓 (x𝑖), y𝑖) = 𝐿(𝑓 (x𝑖), y𝑖) + 𝜆 ∥Θ∥2 +2 +(6) +7 + +when this is employed to regression problems it is also known as Ridge regression [46, 77]. +3.1.2 +𝐿1-norm regularisation. In 𝐿1 regularization the loss is augmented to to include the weighted +𝐿1 norm of the weights [12, 77], so the regularisation function is 𝜌(Θ) = ∥Θ∥ +�𝐿(𝑓 (x𝑖), y𝑖) = 𝐿(𝑓 (x𝑖), y𝑖) + 𝜆 ∥Θ∥1 +(7) +when this is employed to regression problems it is also known as Lasso regression [77, 109]. +3.2 +Comparison between 𝐿2 and 𝐿1 norm regularisations +𝐿1 and 𝐿2 regularisations are both based on the same concept of penalising the magnitude of the +weights composing the models. Despite that, the two methods have important differences in their +employability and their effects on the result. +One of the most crucial differences is that 𝐿1, when optimised, is able to shrink weights to 0, while +𝐿2 results in non-zeros (smoothed) values [7, 12, 73, 77, 81]. This allows 𝐿1 to reduce the dimension +of a model’s parameter space and perform an implicit feature selection. Indeed, it has been shown +by [81] that by employing 𝐿1 regularization on logistic regression, the sample complexity (i.e., the +number of training examples required to learn “well”) grows logarithmically in the number of +irrelevant features. On the contrary, the authors show that any rotationally invariant algorithm +(including logistic regression) with 𝐿2 regularization has a worst case sample complexity that grows +at least linearly in the number of irrelevant features. Moreover, 𝐿2 is more sensitive to the outliers +than 𝐿1-norm since it squares the error. +𝐿2 is continuous, while 𝐿1 is a piece-wise function. The main advantage of 𝐿2 is that it is +differentiable, while 𝐿1 is non-differentiable at 0, which has some strong implications. Precisely, +the 𝐿2 norm can be easily trained with gradient descent, while 𝐿1 sometimes cannot be efficiently +applied. The first problem is the inefficiency of applying the 𝐿1 penalty to the weights of all the +features, especially when the dimension of the feature space tends to be very large [110], producing +a significant slow down of the weights updating process. Finally the naive application of 𝐿1 penalty +in SGD does not always lead to compact models, because the approximate gradient used at each +update could be very noisy, so the weights of the features can be easily moved away from zero by +those fluctuations and 𝐿1 looses its main advantages with respect to 𝐿2 [110]. +4 +REGRESSION LOSSES +The aim of a regression model is to predict the outcome of a continuous variable 𝑦 (the dependent +variable) based on the value of one or multiple predictor variables x (the independent variables). +More precisely, let 𝑓Θ be a generic model parameterized by Θ, which maps the independent +variables x ∈ {x0, ..., x𝑁 }, x𝑖 ∈ R𝐷 into the dependent variable 𝑦 ∈ R. The final goal is to estimate +the parameters of the model Θ that most closely fits the data by minimizing a loss function 𝐿. +8 + +Mean Squared +Error +Mean Bias +Error +Huber +Log cosh +Root Mean +Squared Error +Smooth L1 +Mean Absolute +Error +Root Mean Squared +Logarithmic Error +Fig. 2. Schematic overview of the regression losses showing the connection +All the losses considered for the regression task are based on functions of the residuals, i.e. the +difference between the observed value 𝑦 and the predicted value 𝑓 (x). In the following, let 𝑓 (x𝑖) +be the outcome of the prediction over x𝑖, and 𝑦 be the ground truth of the 𝑖𝑡ℎ variable of interest. +As highlighted by Fig. 2 the Mean Bias Error (𝑀𝐵𝐸) loss can be considered a base pillar for +regression losses, characterized by many variations. Among them the most relevant are: Mean +Absolute Error (𝑀𝐴𝐸), Mean Squared Error (𝑀𝑆𝐸), and Root Mean Squared Error (𝑅𝑀𝑆𝐸) losses. +In this section we are also going to introduce the Huber loss and the smooth L1, which are a blend +between the 𝑀𝐴𝐸 and the 𝑀𝑆𝐸. Finally, the Log-cosh and the Root Mean Squared Logarithmic +Error losses are presented. +4.0.1 +Mean Bias Error Loss (CONT, DIFF). The most straightforward loss function is the Mean +Bias Error loss, illustrated in Equation 8. It captures the average bias in the prediction, but is rarely +adopted as loss function to train regression models, because positive errors may cancel out the +negative ones, leading to a potential erroneous estimation of the parameters. Nevertheless, it is +the starting point of the loss functions defined in the next subsections and it is commonly used to +evaluate the performances of the models [60, 111, 112]. +L𝑀𝐵𝐸 = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝑦𝑖 − 𝑓 (x𝑖) +(8) +Directly connected to 𝑀𝐵𝐸 there are respectively the Mean Absolute Error, the Mean Squared Error +and the Log-cosh losses, which basically differs from 𝑀𝐵𝐸 in how they exploit the bias. +4.0.2 +Mean Absolute Error Loss (L-CONT,CONV). The Mean Absolute Error loss or L1 loss is one +of the most basic loss functions for regression, it measures the average of the absolute bias in the +prediction. The absolute value overcomes the problem of the 𝑀𝐵𝐸 ensuring that positive errors do +not cancel the negative ones. Therefore each error contributes to 𝑀𝐴𝐸 in proportion to the absolute +value of the error. Notice that, the contribution of the errors follows a linear behavior, meaning +that many small errors are important as a big one. This implies that the gradient magnitude is not +dependent on the error size, thus may leading into convergence problems when the error is small. +A model trained to minimize the MAE is more effective when the target data conditioned on the +input is symmetric. It is important to highlight that the derivative of the absolute value at zero is +not defined. +As for MBE, MAE is also used to evaluate the performances of the models [68, 121]. +L𝑀𝐴𝐸 = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +|𝑦𝑖 − 𝑓 (x𝑖)| +(9) +9 + +4.0.3 +Mean Squared Error Loss (CONT, DIFF, CONV). The Mean Squared Error loss, or L2 loss, is +the average of squared distances between the observed value 𝑦 and the predicted value ˆ𝑦. As for +𝑀𝐴𝐸, it is a well-known and straightforward loss function for regression. The squared term makes +all the biases positive and magnifies the contribution made by outliers, making it more suitable for +problems where noise in the observations follows a normal distribution. The main drawback is the +sensitivity to the outliers. +L𝑀𝑆𝐸 = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +(𝑦𝑖 − 𝑓 (x𝑖))2 +(10) +4.0.4 +Root Mean Squared Error Loss(CONT,DIFF,CONV). Directly connected to MSE, we have the +Root Mean Squared Error loss, which is similar to MSE except for the square root term. The main +advantage is to make sure that the loss has the same units and scale of the variable of interest. Since +the only difference between the MSE and the RMSE consists in the application of the root term, the +minimization process converge to the same optimal value. However, depending on the optimisation +technique used, the RMSE may take different gradient steps. As the previously presented loss +functions, it is also used as a metric to compare the performances of the model [68, 112], and it +shares the same limitations. +L𝑅𝑀𝑆𝐸 = +� +� +� +1 +𝑁 +𝑁 +∑︁ +𝑖=1 +(𝑦𝑖 − 𝑓 (x𝑖))2 +(11) +4.0.5 +Huber loss (L-CONT,DIFF,S-CONV). The Huber loss [48] is a variant of the MAE that becomes +MSE when the residuals are small. It is parameterized by 𝛿, which defines the transition point +from MAE to MSE. When |yi − 𝑓 (xi)| ≤ 𝛿 the Huber loss follows the MSE, otherwise it follows the +MAE. This allows it to combine the advantages of both the MAE and the MSE, when the difference +between the prediction and the output of the model is huge errors are linear, make the Huber +loss less sensitive to the outliers. Conversely, when the error is small, it follows the MSE making +the convergence much faster and differentiable at 0. The choice of 𝛿 is fundamental and it can be +constantly adjusted during the training procedure based on what is considered an outlier. The main +limitation of the Huber loss resides in the additional extra hyperparameter 𝛿. +𝐿𝐻𝑢𝑏𝑒𝑟𝑙𝑜𝑠𝑠 = +� +1 +2 (𝑦𝑖 − 𝑓 (x𝑖))2 +𝑓 𝑜𝑟 |𝑦𝑖 − 𝑓 (x𝑖)| ≤ 𝛿, +𝛿 �|𝑦𝑖 − 𝑓 (x𝑖)| − 1 +2𝛿� +𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 +(12) +Notice that, when 𝛿 = 1, we obtain the smooth L1 loss. +4.0.6 +Log-cosh loss(CONT, DIFF). The log-cosh loss is the logarithm of the hyperbolic cosine of +the residuals between the observed value 𝑦 and the predicted value ˆ𝑦. It has all the advantages of +the Huber loss, without the requirement of setting a hyperparameter, at the cost of being more +computationally expensive. Furthermore, another benefit of the log-cosh loss is related to the fact +that is differentiable twice everywhere, making it suitable for methods that requires solving the +second derivative. As 𝑙𝑜𝑔 (𝑐𝑜𝑠ℎ (x)) is approximately equal to x2 +2 for small values of x it behave +similarly to the MSE. For larger value of x instead, is nearly equivalent to |x| − 𝑙𝑜𝑔 (2) making it +similar to MAE. +L𝑙𝑜𝑔𝑐𝑜𝑠ℎ = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝑙𝑜𝑔 (𝑐𝑜𝑠ℎ (𝑓 (x𝑖) − 𝑦𝑖)) +(13) +Another drawback of L𝑙𝑜𝑔𝑐𝑜𝑠ℎ is related to the fact that, compared to the Huber loss, it is less +customizable. +10 + +4.0.7 +Root Mean Squared Logarithmic Error Loss(CONT,DIFF,CONV). The Root Mean Squared +Logarithmic Error (RMSLE) loss (formalized in Eq. 14) is the RMSE of the log-transformed observed +value 𝑦 and log-transformed predicted value ˆ𝑦. The only difference with respect to RMSE is that +the logarithm is applied to both the predicted and the observed values. The plus one term inside +the logarithm allows values of 𝑓 (x𝑖) to be zero. +Due to the properties of the logarithm, the error between the predicted and the actual values is +relative, making the RMSLE more robust to outliers. Precisely, the magnitude of the RMLSE does +not scale accordingly to the magnitude of the error. Indeed, data points with big residuals are less +penalized when the predicted and the actual values have high values too. This make the RMSLE +suitable for problems where targets have an exponential relationship, or it is preferable to penalize +more under estimates than over estimates. However, this loss is not appropriate for problems that +allows negative values. +L𝑅𝑀𝑆𝐿𝐸 = +� +� +� +1 +𝑁 +𝑁 +∑︁ +𝑖=1 +(log(𝑦𝑖 + 1) − log(𝑓 (x𝑖) + 1))2 +(14) +5 +CLASSIFICATION LOSSES +5.1 +Problem Formulation and Notation +Classification is a subset of problems belonging to supervised learning. The goal is to assign an input +x to one of 𝐾 discrete classes. This goal can be pursued by training a model 𝑓Θ and its parameters Θ +by minimizing a loss function 𝐿. Let the target space of 𝑓 discrete and consider a model returning +the output label, 𝑓 can be defined as: +𝑓 : Φ → Λ𝐾 +Λ = {0, 1} +The above definition is working also for multi-label classification, since more than one label could +be associated to a sample, e.g. 𝑓 (x) = [0, 1, 0, 1, 0, 0]. In order to define single label classification we +need to add the constraint that the output sum up to 1, � +𝑘 𝜆𝑘 = 1. +We can also consider models with continuous outputs, in case they return a probability 𝑝𝑘 (x) ∈ [0, 1] +to a sample x for each possible assignable label 𝑘 ∈ 1, ..., 𝐾: +𝑓 : Φ → 𝑃𝐾 +𝑃 = [0, 1] +As before, to switch between multi-label and single-label classification, we need to constraint the +probabilities output to sum up to one, � +𝑘 𝑝𝑘 = 1, if we want to force a single label assignment. +A more narrow notation for classification can be introduced in order to describe binary classi- +fication problems. This notation is useful in this work because margin based losses are designed +to solve binary classification problems and cannot be generalised to multi-class or multi label +classification. For the subset of binary classification problems the target space of 𝑓 is discrete and it +is defined as follows: +𝑓 : Φ → 𝐵 +𝐵 = {−1, 1} +We define two different macro categories of classification losses accordingly to the underlying +strategy employed to optimize them, namely the margin based and the probabilistic ones as +illustrated in Fig. 3. In the next section, we introduce the margin based loss function starting +by the most basic and intuitive one, the Zero-One loss. Subsequently, we present the Hinge loss +11 + +and its variants (the Smoothed and Quadratically Smoothed Hinge losses). Then, the Modified +Huber loss, the Ramp loss, and the Cosine Similarity loss are described. Moreover, we introduce +the probabilistic loss by introducing the Cross Entropy loss and Negative Log-Likelihood loss, +which, from a mathematical point of view, coincides. Finally, the Kullback-Leibler Divergence loss +is presented. +Probabilistic +Margin based +Cosine +similarity +Hinge +Quadratically +Smoothed +Modified +Huber +Cross +Entropy +Kullback-Leibler +Divergence +Ramp loss +Zero-One +Smoothed +Hinge +Negative +Log-Likelihood +Fig. 3. Overview of the classification losses divided in two major groups: margin based losses and probabilistic +ones. +5.2 +Margin Based Loss Functions +In this section, we introduce the most known margin based loss functions. +5.2.1 +Zero-One loss. The basic and more intuitive margin based classification loss is the Zero-One +loss. It assigns 1 to a misclassified observation and 0 to a correctly classified one. +𝐿ZeroOne(𝑓 (x),𝑦) = +� +1 +if 𝑓 (x) · 𝑦 < 0 +0 +otherwise +(15) +ZeroOne loss is not directly usable since it lacks convexity and differentiability. However, it is +possible to derive employable surrogate losses that are classification calibrated, which means +that they are a relaxation of 𝐿𝑍𝑒𝑟𝑜𝑂𝑛𝑒, or an upper bound, or an approximation of such loss. A +significant achievement of the recent literature on binary classification has been the identification +of necessary and sufficient conditions under which such relaxations yield Fisher consistency +[6, 50, 72, 74, 105, 124]. All the following losses satisfy such conditions. +5.2.2 +Hinge loss and Perceptron loss (L-CONT,CONV). The most famous surrogated loss is the +Hinge loss [35], which linearly penalizes every prediction where the resulting agreement is <= 1. +𝐿Hinge(𝑓 (x),𝑦) = max(0, 1 − (𝑓 (x) · 𝑦)) +(16) +The Hinge loss is not strictly convex, but it is Lipschitz continuous and convex, so many of the +usual convex optimizers used in machine learning can work with it. The Hinge loss is commonly +employed to optimise the Support Vector Machine (SVM [14, 75]). +To train the Perceptron [93] a variation of this loss, the Perceptron loss, is employed. This +loss slightly differs from the Hinge loss, because it does not penalise samples inside the margin, +surrounding the separating hyperplane, but just the ones that are mislabeled by this hyperplane +with the same linear penalisation. +𝐿Perceptron(𝑓 (x),𝑦) = max(0, −(𝑓 (x) · 𝑦)) +(17) +12 + +There are two main drawbacks using the hinge loss. Firstly, its adoption use to make the model +sensible to outliers in the training data. Secondly, due to the discontinuity of the derivative at +(𝑓 (x) · 𝑦) = 1, i.e. the fact that is not continuously differentiable, Hinge loss results difficult to +optimise. +5.2.3 +Smoothed Hinge loss (L-CONT,CONV). A smoothed version of the Hinge loss was defined in +[90] with the goal of obtaining a function easier to optimise as shown by the following equation: +𝐿SmoothedHinge(𝑓 (x),𝑦) = + + +1 +2 − (𝑓 (x) · 𝑡) +(𝑓 (x) · 𝑦) <= 0 +1 +2 (1 − (𝑓 (x) · 𝑡))2 +0 < (𝑓 (x) · 𝑦) < 1 +0 +(𝑓 (x) · 𝑦) >= 1 +(18) +This smoothed version of the Hinge loss is differentiable. Clearly, this is not the only possible +smooth version of the Hinge loss. However, it is a canonical one that has the important property of +being zero for 𝑧 >= 1 and it has constant (negative) slope for 𝑧 <= 0. Moreover, for 0 < 𝑧 < 1, the +loss smoothly transitions from zero slope to a constant negative one. This loss inherit sensibility to +outliers from the original Hinge loss. +5.2.4 +Quadratically Smoothed Hinge loss (L-CONT,CONV,DIFF). With the same goal of the Smoothed +Hinge loss a quadratically smoothed version has been defined in [125], to make it easier to be +optimised: +𝐿QSmoothedHinge(𝑓 (x),𝑦) = +� 1 +2𝛾 max(0, −(𝑓 (x) · 𝑦))2 +(𝑓 (x) · 𝑦) >= 1 − 𝛾 +1 − 𝛾 +2 − (𝑓 (x) · 𝑦) +otherwise +(19) +The hyperparameter 𝛾 determines the degree of smoothing, for 𝛾 → 0 the loss becomes the original +hinge. In contrast with the Smoothed Hinge loss, this version is not differentiable in the whole +domain. +5.2.5 +Modified Huber loss (L-CONT, DIFF, S-CONV). The Modified Huber loss is a slight variation +of the Huber loss for regression and a special case of the Quadratic Smoothed Hinge loss with 𝛾 = 2 +(For more details refer to section 4.0.5): +𝐿ModHuber(𝑓 (x),𝑦) = +� +1 +4 max(0, −(𝑓 (x) · 𝑦))2 +(𝑓 (x) · 𝑦) >= −1 +−(𝑓 (x) · 𝑦) +otherwise +(20) +5.2.6 +Ramp loss (CONT,CONV). The Ramp loss, or Truncated Hinge, is a piece-wise linear, contin- +uous and convex loss that has been presented in [122]. Under multi-class setting, this loss is more +robust to outliers. When employed in SVM, it produces more accurate classifiers using a smaller, +and more stable, set of support vectors than the multi-class SVM that employes 𝐿Hinge [66], also +preserving fisher consistency. +𝐿Ramp(𝑓 (x),𝑦) = +� +𝐿Hinge(𝑓 (x),𝑦)) +(𝑓 (x) · 𝑦) <= 1 +1 +otherwise +(21) +5.2.7 +Cosine Similarity loss (L-CONT,DIFF). Cosine similarity is generally used as a metric to +measure distance when the magnitude of vectors is not important [12]. A typical example is related +to text data representation by means of word counts [12, 77]. When the label and output can be +13 + +interpreted as vectors it is possible to derive a distance metric between them, which can be adapted +into a loss function as follows: +𝐿𝑐𝑜𝑠−𝑠𝑖𝑚(𝑓 (x), y) = 1 − +y · f(x) +∥y∥ ∥f(x)∥ +(22) +It is important to underline that, when using Cosine Similarity loss, the range of possible values +is restricted to the interval [-1, 1], which may not be suitable for all types of data or applications, +particularly when interpretability is a key requirement. +5.3 +Probabilistic loss Functions +Let 𝑞 be the probability distribution underlying the dataset and 𝑓Θ the function generating the +output, probabilistic loss functions provide some distance function between𝑞 and 𝑓Θ. By minimizing +that distance, the model output distribution converges to the ground truth one. Usually, models +trained with probabilistic loss functions can provide a measure of how likely a sample is labeled +with one class instead of another [12, 43, 77] providing richer information w.r.t. margin based . +5.3.1 +Cross Entropy loss and Negative Log-Likelihood loss (CONT,DIFF,CONV). Maximum likelihood +estimation (MLE) is a method to estimate the parameters of a probability distribution by maximizing +the likelihood [12, 77, 80]. From the point of view of Bayesian inference, MLE can be considered a +special case of maximum a-posteriori estimation (MAP) that assumes a uniform prior distribution +of the parameters. Formally, it means that, given a dataset of samples D, we are maximizing the +following quantity: +𝑃(D|Θ) = +𝑁 +� +𝑛=1 +𝑓Θ(x𝑖)𝑦𝑖 · (1 − 𝑓Θ(x𝑖))1−𝑦𝑖 +(23) +The aim is to find the maximum likelihood estimate by minimizing a loss function. To maximize +Eq. 23, we can turn it into a minimisation problem by employing the negative log likelihood. To +achieve this goal we need to define the following quantity: +𝑙𝑜𝑔(𝑃(D|Θ)) = +𝑁 +∑︁ +𝑖=1 +(𝑦𝑖 log(𝑓Θ(x𝑖)) + (1 − 𝑦𝑖) log(1 − 𝑓Θ(x𝑖)))) +(24) +and we can obtain the loss function by taking the negative of the log: +L𝑁𝐿𝐿 = − +𝑁 +∑︁ +𝑖=1 +(𝑦𝑖 log(𝑓Θ(x𝑖)) + (1 − 𝑦𝑖) log(1 − 𝑓Θ(x𝑖))) +(25) +Often, the above loss is also called the cross-entropy loss, because it can be derived by minimising +the cross entropy between 𝑓Θ and 𝑞. +𝐻 (𝑞, 𝑓Θ) = − +∫ +𝑞(x) log(𝑓Θ(x))𝑑x +(26) +For the discrete case (which is the one we are interested in) the definition of the cross entropy is: +𝐻 (𝑞, 𝑓Θ) = − +𝑁 +∑︁ +𝑖=1 +𝑞(x𝑖) log(𝑓Θ(x𝑖)) +(27) +14 + +Maximizing the likelihood with respect to the parameters Θ is the same as minimizing the cross- +entropy as shown by the following equations: +𝑁 +∑︁ +𝑖=1 +(𝑦𝑖 log(𝑓Θ(x𝑖)) + (1 − 𝑦𝑖) log(1 − 𝑓Θ(x𝑖))) = 1 +𝑁 +𝑁 +� +𝑛=1 +𝑓Θ(x𝑖)𝑁 𝑦𝑖 +(28) += +𝑁 +∑︁ +𝑖=1 +𝑞(x𝑖) log(𝑓Θ(x𝑖) +(29) += − 𝐻 (𝑞, 𝑓Θ) +(30) +The classical approach to extend this loss to the multi-class scenario is to add as a final activation of +the model a softmax function, defined accordingly to the number of (𝐾) classes considered. Given a +score for each class 𝑓𝑘 (x) = 𝑠, it’s output can be squashed to sum up to 1 by mean of a softmax +function 𝑓𝑆 obtaining: +�𝑓𝑘 (x𝑖) = 𝑓𝑆 (𝑓𝑘 (x)) +(31) +where, the softmax is defined as follows: +𝑓𝑆 (𝑠𝑖) = +𝑒𝑠𝑖 +�𝐾 +𝑗=1 𝑒𝑠𝑗 +(32) +The final loss (usually named as categorical cross entropy) is: +𝐿𝐶𝐶𝐸 = − 1 +𝐾 +𝐾 +∑︁ +𝑗=1 +log( �𝑓𝑘 (x)) +(33) +5.3.2 +Kullback-Leibler divergence (CONT, CONV, DIFF). The Kullback-Leibler (KL) divergence is +an information-based measure of disparity among probability distributions. Precisely, it is a non- +symmetrical measurement of how one probability distribution differs from another one [12, 53, 77]. +Technically speaking, KL divergence is not a distance metric because it doesn’t obey to the triangle +inequality (𝐾𝐿(𝑞||𝑓Θ) is not equal to 𝐾𝐿(𝑓Θ||𝑞)). It is important to notice that, in the classification +use case, minimizing the KL divergence is the same as minimising the cross entropy. Precisely, the +KL between two continuous distributions is defined as: +KL(𝑞||𝑓Θ) = +∫ +𝑞(x) log( 𝑞(x) +𝑓Θ(x) )𝑑x = − +∫ +𝑞(x) log(𝑓Θ(x))𝑑x + +∫ +𝑞(x) log(𝑞(x))𝑑x +(34) +If we want to minimise KL on the parameter Θ, since the second integral is non-dependant on Θ, +we obtain: +min +Θ KL(𝑞||𝑓Θ) = min +Θ − +∫ +𝑞(x) log(𝑓Θ(x))𝑑x = min +Θ 𝐻 (𝑞, 𝑓Θ) +(35) +In general, it is preferable using cross entropy, instead of KL divergence, because it is typically +easier to compute and optimize. The cross entropy only involves a single sum over the data, whereas +the KL divergence involves a double sum. This can make it more computationally efficient, especially +when working with large datasets. +6 +GENERATIVE LOSSES +In recent years, generative models have become particularly useful for both understanding the +complexity of data distributions and being able to regenerate them [36, 37]. In this section, as +shown in the Fig. 4, we describe the losses relevant to Generative Adversarial Networks (GANs) and +Diffusion Models. However, generative models are not limited to these cases, but extend to include +15 + +more models. For example, Variational Auto-Encoders (VAEs) [2, 55, 87] in which the KL-divergence, +described in section 5.3.2, is the loss function employed. The objective of the VAE loss is to reduce +the discrepancy between the original distribution and its predicted distribution. Other models such +as the pixel recurrent neural networks [113] and real-valued Non-Volume Preserving (realNVP) +models [27] are not considered in this survey. +MinMax +Wesserstein +Diffusion +Fig. 4. Overview of the generative losses. +6.1 +Generative Adversarial Networks +Generative Adversarial Networks (GANs) are used to create new data instances that are sampled +from the training data. GANs have two main components: +• The generator, referred as 𝐺({z0, · · · , z𝑁 }), which generates data starting from random +noise and tries to replicate real data distributions +• The discriminator, referred as 𝐷({x0, · · · , x𝑁 }), which learns to distinguish the generator’s +fake data from real one. It applies penalties in the generator loss for producing distinguishable +fake data compared with real data. +The GAN architecture is relatively straightforward, although one aspect remains challenging: GAN +loss functions. Precisely, the discriminator is trained to provide the loss function for the generator. +If generator training goes well, the discriminator gets worse at telling the difference between real +and fake samples. It starts to classify fake data as real, and its accuracy decreases. +Both the generator and the discriminator components are typically neural networks, where the +generator output is connected directly to the discriminator input. The discriminator’s classification +provides a signal that the generator uses to update its weights through back-propagation. +As GANs try to replicate a probability distribution, they should use loss functions that reflect +the distance between the distribution of the data generated by the GAN and the distribution of the +real data. +Two common GAN loss functions are typically used: minimax loss [38] and Wasserstein +loss [4]. The generator and discriminator losses derive from a single distance measure between +the two aforementioned probability distributions. The generator can only affect one term in the +distance measure: the term that reflects the distribution of the fake data. During generator training, +we drop the other term, which reflects the real data distribution. The generator and discriminator +losses look different, even though they derive from a single formula. +In the following, both minimax and Wasserstein losses are written in a general form. The +properties of the loss function (CONT, DIFF, etc.) are identified based on the function chosen for +the generator or discriminator. +6.1.1 +Minimax loss. The generative model 𝐺 learns the data distributions and is trained simultane- +ously with the discriminative model 𝐷. The latter estimates the probability that a given sample is +identical to the training data rather than𝐺.𝐺 is trained to maximize the likelihood of tricking 𝐷 [38]. +In other words, the generator tries to minimize the following function while the discriminator tries +16 + +to maximize it: +L𝑚𝑖𝑛𝑖𝑚𝑎𝑥 (𝐷,𝐺) = 𝐸{x0,···,x𝑁 }[log(𝐷({x0, · · · , x𝑁 }))] + 𝐸{z0,···,z𝑁 }[log(1 − 𝐷(𝐺({z0, · · · , z𝑁 })))], +(36) +where: +• 𝐷({x0, · · · , x𝑁 }) is the discriminator’s estimate of the probability that real data instance +{x0, · · · , x𝑁 } is real, +• 𝐸{x0,···,x𝑁 } is the expected value over all real data instances, +• 𝐺({z0, · · · , z𝑁 }) is the generator’s output when given noise {z0, · · · , z𝑁 }, +• 𝐷(𝐺(𝑧)) is the discriminator’s estimate of the probability that a fake instance is real, +• 𝐸{z0,···,z𝑁 } is the expected value over all random inputs to the generator (in effect, the expected +value over all generated fake instances 𝐺({z0, · · · , z𝑁 })). +The loss function above directly represents the cross-entropy between real and generated data +distributions. The generator can’t directly affect the log(𝐷({x0, · · · , x𝑁 })) term in the function, +and it only minimizes the term log(1 − 𝐷(𝐺({z0, · · · , z𝑁 }))). A disadvantage of this formulation of +the loss function is that the above minimax loss function can cause the GAN to get stuck in the +early stages of the training when the discriminator received trivial tasks. Therefore, a suggested +modification to the loss [38] is to allow the generator to maximize log(𝐷(𝐺({z0, · · · , z𝑁 }))). +6.1.2 +Wasserstein loss. The Wasserstein distance gives an alternative method of training the +generator to better approximate the distribution of the training dataset. In this setup, the training +of the generator itself is responsible for minimizing the distance between the distribution of the +training and generated datasets. The possible solutions are to use distribution distance measures, +like Kullback-Leibler (KL) divergence, Jensen-Shannon (JS) divergence, and the Earth-Mover (EM) +distance (also called Wasserstein distance). The main advantage of using Wasserstein distance is +due to its differentiability and having a continuous linear gradient [4]. +A GAN that uses a Wasserstein loss, known as a WGAN, does not discriminate between real and +generated distributions in the same way as other GANs. Instead, the WGAN discriminator is called +a "critic," and it scores each instance with a real-valued score rather than predicting the probability +that it is fake. This score is calculated so that the distance between scores for real and fake data is +maximised. +The advantage of the WGAN is that the training procedure is more stable and less sensitive to +model architecture and selection of hyperparameters. +The two loss functions can be written as: +L𝑐𝑟𝑖𝑡𝑖𝑐 = 𝐷({x0, · · · , x𝑁 }) − 𝐷(𝐺({z0, · · · , z𝑁 })), L𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 = 𝐷(𝐺({z0, · · · , z𝑁 })) +(37) +The discriminator tries to maximize L𝑐𝑟𝑖𝑡𝑖𝑐. In other words, it tries to maximize the difference +between its output on real instances and its output on fake instances. The generator tries to +maximize L𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟. In other words, It tries to maximize the discriminator’s output for its fake +instances. +The benefit of Wasserstein loss is that it provides a useful gradient almost everywhere, allowing +for the continued training of the models. It also means that a lower Wasserstein loss correlates with +better generator image quality, meaning that it explicitly seeks a minimization of generator loss. +Finally, it is less vulnerable to getting stuck in a local minimum than minimax-based GANs [4]. +However, accurately estimating the Wasserstein distance using batches requires unaffordable batch +size, which significantly increases the amount of data needed [104]. +17 + +6.2 +Diffusion models +Diffusion Models are generative models that rely on probabilistic likelihood estimation to generate +data samples. They were originally inspired by the physical phenomena called diffusion. Diffusion +Modelling is a learning procedure in which a model can learn the systematic decay of information +due to adding a slight noise factor iteratively [100, 101]. The learning procedure enables the +possibility of recovering the decayed information from the noise again. They model a series of noise +distributions in a Markov Chain and they decode the original data by systematically removing the +noises hierarchically [20, 44]. +The diffusion process consists of two steps, the forward process and the reconstruction (reverse) +process. Gaussian noise is gradually added during the forward diffusion process until the data +sample loses its distinguishable features. The reverse process removes the noise and restores the +original data by utilizing a neural network model to learn the conditional probability densities. +6.2.1 +Forward diffusion process. Given a data point x0 sampled from a real data distribution 𝑞(x). +The forward process aims to add a small noise iteratively 𝑇 times. For every data point x0 the +process produces a sequence of noisy samples x1, · · · , xT. The noise distributions is usually chosen +to be Gaussian. Because the forecast of probability density at time 𝑡 is only dependent on the +immediate predecessor at time 𝑡 − 1, the conditional probability density can be calculated as: +𝑞(x𝑡 |x𝑡−1) = N (x𝑡; +√︁ +1 − 𝛽𝑡x𝑡−1, 𝛽𝑡I) +𝑞(x1:𝑇 |x0) = +𝑇 +� +𝑡=1 +𝑞(x𝑡 |x𝑡−1) +(38) +With a 𝛽𝑡 ∈ (0, 1) is a hyperparameter that can be taken as constant or variable and 𝑡 ∈ [1,𝑇]. +As 𝑇 → ∞, xT becomes a pure Gaussian distribution. We can only sample the new states of the +system using the gradient of the density function in Markov Chain updates, according to stochastic +gradient Langevin dynamics [120]. The following formula can then be employed to calculate the +sampling of a new data point at time 𝑡 with a step size 𝛿 dependent on the prior point at time 𝑡 − 1: +x𝑡 = x𝑡−1 + 𝛿 +2∇x log𝑞(x𝑡−1) + +√ +𝛿𝝐𝑡, +where 𝝐𝑡 ∼ N (0, I) +(39) +when 𝑇 → ∞, 𝝐 → 0 equals to the true probability density 𝑝(x). The forward step does not require +training a neural network; it only requires an iterative process for adding random Gaussian noises +to the original data distribution. +6.2.2 +Reverse diffusion process. Given the system’s current state, the reverse process requires +the calculation of probability density at a previous time step. Calculating the 𝑞(x𝑡−1|x𝑡) at the +time 𝑡 = 𝑇 is equal to producing data from an isotropic Gaussian noise. However, contrary to the +forward process, the estimation of the past state from the present state requires the knowledge of +every previous gradient, which is not feasible without the aid of a learning model that can forecast +such estimates. This problem is resolved by training a neural network model that, using the learnt +weights Θ and the current state at time 𝑡, estimates the value of 𝑝Θ(x𝑡−1|x𝑡) as formalized by the +following equation: +𝑝Θ(x0:𝑇 ) = 𝑝(x𝑇 ) +𝑇 +� +𝑡=1 +𝑝Θ(x𝑡−1|x𝑡) +𝑝Θ(x𝑡−1|x𝑡) = N (x𝑡−1; 𝝁Θ(x𝑡,𝑡), 𝚺Θ(x𝑡,𝑡)) +(40) +where 𝝁Θ(x𝑡,𝑡) is the mean function proposed in [44]. The sample at time 𝑡 − 1 can then be +computed as: +x𝑡−1 = N (x𝑡−1; +1 +√𝛼𝑡 +� +x𝑡 − 1 − 𝛼𝑡 +√1 − ¯𝛼𝑡 +𝝐Θ(x𝑡,𝑡) +� +, 𝚺Θ(x𝑡,𝑡)) +(41) +18 + +6.2.3 +Diffusion model loss function (CONT, DIFF). The main task of the network model is to +minimize the following loss: +𝐿𝑑𝑖𝑓 𝑓 𝑢𝑠𝑖𝑜𝑛 = E𝑡 +� +log𝑝(x𝑇 ) − +∑︁ +𝑡 ≥1 +log 𝑝Θ(x𝑡−1|x𝑡) +𝑞(x𝑡 |x𝑡−1) +� +(42) +In [44, 100], it is shown that a simplified version of 𝐿𝑑𝑖𝑓 𝑓 𝑢𝑠𝑖𝑜𝑛 is able to achieve better results: +𝐿simple +𝑑𝑖𝑓 𝑓 𝑢𝑠𝑖𝑜𝑛 = E𝑡∼[1,𝑇 ],x0,𝝐𝑡 +� +∥𝝐𝑡 − 𝝐Θ(√ ¯𝛼𝑡x0 + √1 − ¯𝛼𝑡𝝐𝑡,𝑡)∥2� +(43) +It is important to notice that diffusion Models perform significantly better in some applications +despite being computationally more expensive than alternative deep network structures [25, 45, 89, +92, 94, 102]. +7 +RANKING LOSSES +Machine learning can be employed to solve ranking problems, which have important industrial +applications, especially in information retrieval systems. These problems can be typically solved by +employing supervised, semi-supervised, or reinforcement learning [15, 47]. +The goal of ranking losses, in contrast to other loss functions like the cross-entropy loss or MSE +loss, is to anticipate the relative distances between inputs rather than learning to predict a label, a +value, or a set of values given an input. This is also sometimes called metric learning. Nevertheless, +the cross-entropy loss can be used in the top-one probability ranking. In this scenario, given the +scores of all the objects, the top-one probability of an object in this model indicates the likelihood +that it will be ranked first [16]. +Ranking loss functions for training data can be highly customizable because they require only +a method to measure the similarity between two data points, i.e., similarity score. For example, +consider a face verification dataset, pairs of photographs which belong to the same person will +have a high similarity score, whereas those that don’t will have a low score [118].2 +In general, ranking loss functions require a feature extraction for two (or three) data instances, +which returns an embedded representation for each of them. A metric function can then be defined +to measure the similarity between those representations, such as the euclidean distance. Finally, +the feature extractors are trained to produce similar representations for both inputs in case the +inputs are similar or distant representations in case of dissimilarity. +Similar to Section 6, both pairwise and triplet ranking losses are presented in a general form, as +shown in the Fig. 5. The properties of the loss function (CONT, DIFF, etc.) are identified based on +the metric function chosen. +Pairwise Ranking +Triplet Ranking +Fig. 5. Overview of the ranking losses. +2Different tasks, applications, and neural network configurations use ranking losses (like Siamese Nets or Triplet Nets). +Because of this, there are various losses can be used, including Contrastive loss, Margin loss, Hinge loss, and Triplet loss. +19 + +7.1 +Pairwise Ranking loss +In the context of Pairwise Ranking loss, positive and negative pairs of training data points are +used [15, 21, 42, 69]. Positive pairs are composed of an anchor sample x𝑎 and a positive sample x𝑝, +which is similar to x𝑎 in the metric. Negative pairs are composed of an anchor sample x𝑎 and a +negative sample x𝑛, which is dissimilar to x𝑎 in that metric. The objective is to learn representations +with a small distance 𝑑 between them for positive pairs and a greater distance than some margin +value 𝑚 for negative pairs. Pairwise Ranking loss forces representations to have a 0 distance for +positive pairs and a distance greater than a margin for negative pairs. +Given r𝑎, r𝑝, and r𝑛 the embedded representations (the output of a feature extractor) of the input +samples x𝑎, x𝑝, x𝑛 respectively and 𝑑 as a distance function the loss function can be written as: +𝐿𝑝𝑎𝑖𝑟𝑤𝑖𝑠𝑒 = +� +𝑑(r𝑎, r𝑏) +if positive pair +max(0,𝑚 − 𝑑(r𝑎, r𝑛)) +if negative pair +(44) +For positive pairs, the loss will vanish if the distance between the embedding representations +of the two elements in the pair is 0; instead, the loss will increase as the distance between the +two representations increases. For negative pairs, the loss will vanish if the distance between the +embedding representations of the two elements is greater than the margin 𝑚. However, if the +distance is less than 𝑚, the loss will be positive, and the model parameters will be updated to +provide representations for the two items that are farther apart. When the distance between r𝑎 and +r𝑛 is 0, the loss value will be at most 𝑚. The purpose of the margin is to create representations for +negative pairs that are far enough, thus implicitly stopping the training on these pairs and allowing +the model to focus on more challenging ones. If r0 and r1 are the pair elements representations, 𝑦 +is a binary flag equal to 0 for a negative pair and to 1 for a positive pair, and the distance 𝑑 is the +euclidean distance: +𝐿𝑝𝑎𝑖𝑟𝑤𝑖𝑠𝑒 (r0, r1,𝑦) = 𝑦||r0 − r1|| + (1 − 𝑦) max(0,𝑚 − ||r0 − r1||) +(45) +Unlike typical classification learning, this loss requires more training data and time because it +requires access to all the data of all potential pairs during training. Additionally, because training +involves pairwise learning, it will output the binary distance from each class, which is more +computationally expensive if there is incorrect classification [58]. +7.2 +Triplet Ranking loss +Employing triplets of training data instances instead of pairs can produce better performance [18, +47, 118]. The resultant loss is called Triplet Ranking loss. A triplet consists of an anchor sample +x𝑎, a positive sample x𝑝, and a negative sample x𝑛. The objective is that the distance between +the anchor sample and the negative sample representations 𝑑(r𝑎, r𝑛) is greater (and bigger than a +margin 𝑚) than the distance between the anchor and positive representations 𝑑(r𝑎, r𝑝). The same +notation applies: +𝐿𝑡𝑟𝑖𝑝𝑙𝑒𝑡 (r𝑎, r𝑝, r𝑛) = max(0,𝑚 + 𝑑(r𝑎, r𝑝) − 𝑑(r𝑎, r𝑛)) +(46) +This loss is characterized by three different scenarios based on the values of r𝑎, r𝑝, r𝑛, and 𝑚: +• Easy Triplets: 𝑑(r𝑎, r𝑛) > 𝑑(r𝑎, r𝑝) + 𝑚. In the embedding space, the distance between the +negative sample and the anchor sample is already large enough. The model parameters are +not changed, and the loss is 0. +20 + +• Hard Triplets: 𝑑(r𝑎, r𝑛) < 𝑑(r𝑎, r𝑝).Compared to the positive sample, the negative sample is +closer to the anchor. The loss is positive (and > 𝑚). The model’s parameters are subject to +change. +• Semi-Hard Triplets: 𝑑(r𝑎, r𝑝) < 𝑑(r𝑎, r𝑛) < 𝑑(r𝑎, r𝑝) + 𝑚. The loss is still positive (and < 𝑚) +nevertheless, the negative sample is further away from the anchor than the positive sample. +The model’s parameters are subject to change, and the loss is not 0. +This loss is sensitive to small changes in the input samples, so it cannot be generalized. This +means that once the model has been trained on a specific dataset, it cannot be applied to other +datasets [58]. +8 +ENERGY-BASED LOSSES +An Energy-Based Model (EBM) is a probabilistic model that uses a scalar energy function to describe +the dependencies of the model variables [28, 31, 33, 34, 40, 41, 64]. An EBM can be formalised as +𝐹 : X × Y → R, where 𝐹 (x, y) stands for the relationship between the (x, y) pairings. +Given an energy function and the input x, the best fitting value of y is computed with the +following inference procedure: +˜y = 𝑎𝑟𝑔𝑚𝑖𝑛y{𝐹 (x, {y0, · · · , y𝑁 })} +(47) +Energy-based models provide fully generative models that can be used as an alternative to +probabilistic estimation for prediction, classification, or decision-making tasks [29, 40, 41, 64, 82]. +The energy function 𝐸(x, y) ≡ 𝐹 (x, y) can be explicitly defined for all the values of y ∈ Y = +{y0, · · · , y𝑁 } if and only if the size of the set Y is small enough. In contrast, when the space of Y +is sufficiently large, a specific strategy, known as the inference procedure, must be employed to +find the y that minimizes 𝐸(x, {y0, · · · , y𝑁 }). +In many real situations, the inference procedure can produce an approximate result, which may +or may not be the global minimum of 𝐸(x, {y0, · · · , y𝑁 }) for a given x. Moreover, it is possible that +𝐸(x, {y0, · · · , y𝑁 }) has several equivalent minima. The best inference procedure to use often de- +pends on the internal structure of the model. For example, if Y is continuous and 𝐸(x, {y0, · · · , y𝑁 }) +is smooth and differentiable everywhere concerning y, a gradient-based optimization algorithm +can be employed [8]. +In general, any probability density function 𝑝(x) for x ∈ R𝐷 can be rewritten as an EBM: +𝑝𝜽 (x) = +exp(−𝐸𝜽 (x)) +∫ +x′ exp(−𝐸𝜽 (x′))𝑑x′, +(48) +where the energy function (𝐸𝜽) can be any function parameterised by 𝜽 ∈ Θ (such as a neural +network). In these models, a prediction (e.g. finding 𝑝(x0|x1, x2, ...)) is done by fixing the values +of the conditional variables, and estimating the remaining variables, (e.g. x0), by minimizing the +energy function [106, 108, 123]. +An EBM is trained by finding an energy function that associates low energies to values of x +drawn from the underlying data distribution, 𝑝𝜃 (x) ∼ 𝑝𝐷 (x), and high energies for values of x not +close to the underlying distribution. +8.1 +Training +Given the aforementioned conceptual framework, the training can be thought as finding the model +parameters that define the good match between the output (y ∈ Y) and the input (x ∈ X) for +every step. This is done by estimating the best energy function from the set of energy functions +(E) by scanning all the model parameters Θ [62, 103], where E = {𝐹 (Θ, X, Y) : Θ ∈ W = Θ}. A +loss function should behave similarly to the energy function described in Equation 47, i.e., lower +21 + +energy for a correct answer must be modeled by low losses, instead, higher energy, to all incorrect +answers, by a higher loss. +Considering a set of training samples S = {(x𝑖, y𝑖) : 𝑖 = 1, · · · , 𝑁 }, during the training procedure, +the loss function should have the effect of pushing-down 𝐸(Θ, y𝑖, x𝑖) and pulling-up 𝐸(Θ, ˜y𝑖, x𝑖), +i.e., finding the parameters Θ that minimize the loss: +Θ∗ = 𝑚𝑖𝑛Θ∈W𝐿𝑒𝑏𝑚(Θ, S) +(49) +The general form of the loss function L𝑒𝑏𝑚 is defined as: +L𝑒𝑏𝑚(Θ, S) = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +𝐿𝑒𝑏𝑚(y𝑖, 𝐸(Θ, Y, x𝑖)) + regularizer term +(50) +Where: +• 𝐿𝑒𝑏𝑚(y𝑖, 𝐸(Θ, Y, x𝑖)) is the per-sample loss +• y𝑖 is the desired output +• 𝐸(Θ, Y, x𝑖)) is energy surface for a given x𝑖 as y ∈ Y varies +This is an average of a per-sample loss functional, denoted 𝐿𝑒𝑏𝑚(y𝑖, 𝐸(Θ, Y, x𝑖)), over the training +set. This function depends on the desired output y𝑖 and on the energies derived by holding the +input sample x𝑖 constant and changing the output scanning over the sample Y. With this definition, +the loss remains unchanged when the training samples are mixed up and when the training set +is repeated numerous times [64]. As the size of the training set grows, the model is less likely to +overfit [114]. +8.2 +Loss Functions for EBMs +Following Fig. 6, in this section, we introduce the energy loss first since it is the most straightforward. +Then we present common losses in machine learning that can be adapted to the energy based +models, such as the Negative Log-Likelihood loss, the Hinge loss, and the Log loss. Subsequently, +we introduce more sophisticated losses like the Generalized Perceptron loss and the Generalized +Margin loss. Finally, the Minimum classification error loss, the Square-square loss, and its variation +Square-exponential loss are presented. +Energy loss +Generalized +Perceptron +Generalized +Margin +Log loss +Minimum +classification error +Square square +Square exponential +Negative +Log-Likelihood +Hinge +Fig. 6. Schematic overview of the energy based losses with their connection. +8.2.1 +Energy loss. The so-called energy loss is the most straightforward loss due to its simplicity. +It can be simply defined by using the energy function as the per-sample loss: +𝐿𝑒𝑛𝑒𝑟𝑔𝑦(y𝑖, 𝐸(Θ, Y, x𝑖)) = 𝐸(Θ, x𝑖, y𝑖) +(51) +This loss is often used in regression tasks. Accordingly to its definition, it pulls the energy +function down for values that are close to the correct data distribution. However, the energy +22 + +function is not pulled up for incorrect values. The assumption is that, by lowering the energy +function in the correct location, the energy for incorrect values is left higher as a result. Due to this +assumption, the training is sensitive to the model design and may result in energy collapse, leading +to a largely flat energy function. +8.2.2 +Generalized Perceptron loss (L-CONT, CONV). The Generalized Perceptron loss is defined as: +𝐿𝑝𝑒𝑟𝑐𝑒𝑝𝑡𝑟𝑜𝑛(y𝑖, 𝐸(Θ, Y, x𝑖)) = 𝐸(Θ, y𝑖, x𝑖) − [min +y∈Y]{𝐸(Θ, {y0, · · · , y𝑁 }, x𝑖)} +(52) +This loss is positive definite as the second term is the lower bound of the first one, i.e., 𝐸(Θ, y𝑖, x𝑖) − +[𝑚𝑖𝑛y∈Y]{𝐸(Θ, {y0, · · · , y𝑁 }, x𝑖)} ≥ 0. By minimizing this loss, the first term is pushed down +and the energy of the model prediction is raised. Although it is widely used [23, 63], this loss is +suboptimal as it does not detect the gap between the correct output and the incorrect ones, and it +doesn’t restrict the function from assigning the same value to each wrong output y𝑖 and it may +produce flat energy distributions [64]. +8.2.3 +Negative Log-Likelihood loss (CONT,DIFF,CONV). In analogy with the description in Sec- +tion 5.3.1, the Negative Log-Likelihood loss (NLL) in the energy-based context is defined as: +L𝑁𝐿𝐿(Θ, S) = 1 +𝑁 +𝑁 +∑︁ +𝑖=1 +� +𝐸(Θ, y𝑖, x𝑖) + 1 +𝛽 log +∫ +y∈Y +𝑒𝛽𝐸(Θ,y,x𝑖) +� +(53) +where S is the training set. +This loss reduces to the perceptron loss when 𝛽 → ∞ and to the log loss in case Y has only two +labels (i.e., binary classification). Since the integral above is intractable, considerable efforts have +been devoted to finding approximation methods, including Monte-Carlo sampling methods [96], +and variational methods [51]. While these methods have been devised as approximate ways of +minimizing the NLL loss, they can be viewed in the energy-based framework as different strategies +for choosing the y’s whose energies will be pulled up [64, 65]. The NLL is also known as the cross- +entropy [67] loss and is widely used in many applications, including energy-based models [9, 10]. +This loss function formulation is subject to the same limitations listed in section 5.3.1. +8.2.4 +Generalized Margin loss. The generalized margin loss is a more reliable version of the +generalized perceptron loss. The general form of the generalized margin loss in the context of +energy-based training is defined as: +𝐿𝑚𝑎𝑟𝑔𝑖𝑛(Θ, x𝑖, y𝑖) = 𝑄𝑚(𝐸(Θ, x𝑖, y𝑖), 𝐸(Θ, x𝑖, ¯y𝑖)) +(54) +Where ¯y𝑖 is the so-called "most-offending incorrect output" which is the output that has the lowest +energy among all possible outputs that are incorrect [64], 𝑚 is a positive margin parameter, and 𝑄𝑚 +is a convex function which ensures that the loss receives low values for 𝐸(Θ, x𝑖, y𝑖) and high values +for 𝐸(Θ, x𝑖, ¯y𝑖). In other words, the loss function can ensure that the energy of the most offending +incorrect output is greater by some arbitrary margin than the energy of the correct output. +This loss function is written in the general form and a wide variety of losses that use specific +margin function 𝑄𝑚 to produce a gap between the correct output and the wrong output are +formalised in the following part of the section. +Hinge loss (L-CONT,CONV). Already explained in section 5.2.2, the hinge loss can be rewritten +as: +𝐿ℎ𝑖𝑛𝑔𝑒 (Θ, x𝑖, y𝑖) = 𝑚𝑎𝑥(0,𝑚 + 𝐸(Θ, x𝑖, y𝑖) − 𝐸(Θ, x𝑖, ¯y𝑖)) +(55) +This loss enforces that the difference between the correct answer and the most offending incorrect +answer be at least 𝑚 [1, 107]. Individual energies are not required to take a specific value because +23 + +the hinge loss depends on energy differences. This loss function shares limitations with the original +Hinge loss defined in eq. 16. +Log loss (DIFF,CONT,CONV). This loss is similar to the hinge loss, but it sets a softer margin +between the correct output and the most offending outputs. The log loss is defined as: +𝐿𝑙𝑜𝑔(Θ, x𝑖, y𝑖) = log(1 + 𝑒𝐸(Θ,x𝑖,y𝑖)−𝐸(Θ,x𝑖,¯y𝑖)) +(56) +This loss is also called soft hinge and it may produce overfitting on high dimensional datasets [57]. +Minimum classification error loss (CONT, DIFF, CONV). A straightforward function that roughly +counts the total number of classification errors while being smooth and differentiable is known as +the Minimum Classification Error (MCE) loss [54]. The MCE is written as a sigmoid function: +𝐿𝑚𝑐𝑒 (Θ, x𝑖, y𝑖) = 𝜎(𝐸(Θ, x𝑖, y𝑖) − 𝐸(Θ, x𝑖, ¯y𝑖)) +(57) +Where 𝜎 is defined as 𝜎(𝑥) = (1+𝑒−𝑥)−1. While this function lacks an explicit margin, it nevertheless +produces an energy difference between the most offending incorrect output and the correct output. +Square-square loss (CONT,CONV). Square-square loss deals differently with the energy of the +correct output 𝐸(Θ, x𝑖, y𝑖) and the energy of the most offensive output 𝐸(Θ, x𝑖, ¯y𝑖) as: +𝐿𝑠𝑞−𝑠𝑞(Θ, x𝑖, y𝑖) = 𝐸(Θ, x𝑖, y𝑖)2 + (max(0,𝑚 − 𝐸(Θ, x𝑖, ¯y𝑖)))2 +(58) +The combination aims to minimize the energy of the correct output while enforcing a margin of at +least 𝑚 on the most offending incorrect outputs. This loss is a modified version of the margin loss. +This loss can be only used when there is a lower bound on the energy function [42, 65]. +Square-exponential loss (CONT, DIFF, CONV). This loss is similar to the square-square loss +function, and it only differs in the second term: +𝐿𝑠𝑞−𝑒𝑥𝑝 (Θ, x𝑖, y𝑖) = 𝐸(Θ, x𝑖, y𝑖)2 + 𝛾𝑒−𝐸(Θ,x𝑖,¯y𝑖) +(59) +While 𝛾 is a positive constant, the combination aims to minimize the energy of the correct output +while pushing the energy of the most offending incorrect output to an infinite margin [21, 65, 82]. +This loss is considered a regularized version of the aforementioned square-square loss. This loss, +as for the Square-square loss, can be only used when there is a lower bound on the energy +function [42, 65]. +9 +CONCLUSION +The definition of an appropriate loss function is a critical part of solving many machine learning +problems. In this survey we have described 33 of the most commonly used loss functions from +across the machine learning literature. These functions are appropriate for solving a wide range of +problems, including classification, regression, sample generation and ranking. Overall, we made an +effort to provide a useful resource for newcomers to the machine learning literature and advanced +practitioners alike. +Each of the loss functions we describe have also been put into context and compared in a novel +taxonomy, introduced in Sec. 2. This gives practitioners a quick view of the different techniques +and how they fit together, allowing them to decide and choose the most appropriate approach for +their technique quickly and efficiently. We also hope this framing helps researchers to contextualise +newly developed loss functions. +24 + +REFERENCES +[1] Yasemin Altun, Ioannis Tsochantaridis, and Thomas Hofmann. 2003. Hidden markov support vector machines. In +Proceedings of the 20th international conference on machine learning (ICML-03). 3–10. +[2] Jinwon An and Sungzoon Cho. 2015. Variational autoencoder based anomaly detection using reconstruction probability. +Special Lecture on IE 2, 1 (2015), 1–18. +[3] Brenna D Argall, Sonia Chernova, Manuela Veloso, and Brett Browning. 2009. A survey of robot learning from +demonstration. Robotics and autonomous systems 57, 5 (2009), 469–483. +[4] Martin Arjovsky, Soumith Chintala, and Léon Bottou. 2017. Wasserstein GAN. https://doi.org/10.48550/ARXIV.1701. +07875 +[5] Peter Bartlett, Stéphane Boucheron, and Gábor Lugosi. 2002. Model Selection and Error Estimation. Machine Learning +48 (01 2002), 85–113. https://doi.org/10.1023/A:1013999503812 +[6] Peter L. Bartlett, Michael I. Jordan, and Jon D. McAuliffe. 2006. Convexity, Classification, and Risk Bounds. J. Amer. +Statist. Assoc. 101 (2006), 138 – 156. +[7] Sebahattin Bektaş and Yasemin Şişman. 2010. The comparison of L1 and L2-norm minimization methods. International +Journal of the Physical Sciences 5, 11 (2010), 1721–1727. +[8] Yoshua Bengio. 2000. Gradient-based optimization of hyperparameters. Neural computation 12, 8 (2000), 1889–1900. +[9] Yoshua Bengio, Renato De Mori, Giovanni Flammia, and Ralf Kompe. 1992. Global optimization of a neural network- +hidden Markov model hybrid. IEEE transactions on Neural Networks 3, 2 (1992), 252–259. +[10] Yoshua Bengio, Réjean Ducharme, and Pascal Vincent. 2000. A neural probabilistic language model. Advances in +neural information processing systems 13 (2000). +[11] Christopher M Bishop et al. 1995. Neural networks for pattern recognition. Oxford university press. +[12] Christopher M Bishop and Nasser M Nasrabadi. 2006. Pattern recognition and machine learning. Vol. 4. Springer. +[13] Gianluca Bontempi, Souhaib Ben Taieb, and Yann-Aël Le Borgne. 2012. Machine learning strategies for time series +forecasting. In European business intelligence summer school. Springer, 62–77. +[14] Bernhard E Boser, Isabelle M Guyon, and Vladimir N Vapnik. [n. d.]. A training algorithm for optimal margin +classifiers. In Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory. 144–152. +[15] Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Säckinger, and Roopak Shah. 1993. Signature verification using a" +siamese" time delay neural network. Advances in neural information processing systems 6 (1993). +[16] Zhe Cao, Tao Qin, Tie-Yan Liu, Ming-Feng Tsai, and Hang Li. 2007. Learning to rank: from pairwise approach to +listwise approach. In Proceedings of the 24th international conference on Machine learning. 129–136. +[17] Varun Chandola, Arindam Banerjee, and Vipin Kumar. 2009. Anomaly detection: A survey. ACM computing surveys +(CSUR) 41, 3 (2009), 1–58. +[18] Gal Chechik, Varun Sharma, Uri Shalit, and Samy Bengio. 2010. Large Scale Online Learning of Image Similarity +Through Ranking. Journal of Machine Learning Research 11, 3 (2010). +[19] Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh. 2017. Zoo: Zeroth order optimization +based black-box attacks to deep neural networks without training substitute models. In Proceedings of the 10th ACM +workshop on artificial intelligence and security. 15–26. +[20] Wai-Ki Ching and Michael K Ng. 2006. Markov chains. Models, algorithms and applications (2006). +[21] Sumit Chopra, Raia Hadsell, and Yann LeCun. 2005. Learning a similarity metric discriminatively, with application to +face verification. In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), +Vol. 1. IEEE, 539–546. +[22] KR1442 Chowdhary. 2020. Natural language processing. Fundamentals of artificial intelligence (2020), 603–649. +[23] Michael Collins. 2002. Discriminative training methods for hidden markov models: Theory and experiments with +perceptron algorithms. In Proceedings of the 2002 conference on empirical methods in natural language processing +(EMNLP 2002). 1–8. +[24] Andrew R Conn, Katya Scheinberg, and Luis N Vicente. 2009. Introduction to derivative-free optimization. SIAM. +[25] Prafulla Dhariwal and Alexander Nichol. 2021. Diffusion models beat gans on image synthesis. Advances in Neural +Information Processing Systems 34 (2021), 8780–8794. +[26] Amit Dhurandhar, Tejaswini Pedapati, Avinash Balakrishnan, Pin-Yu Chen, Karthikeyan Shanmugam, and Ruchir +Puri. 2019. Model agnostic contrastive explanations for structured data. arXiv preprint arXiv:1906.00117 (2019). +[27] Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio. 2016. Density estimation using real nvp. arXiv preprint +arXiv:1605.08803 (2016). +[28] Yilun Du, Toru Lin, and Igor Mordatch. 2019. Model Based Planning with Energy Based Models. ArXiv abs/1909.06878 +(2019). +[29] Yilun Du and Igor Mordatch. 2019. Implicit generation and modeling with energy based models. Advances in Neural +Information Processing Systems 32 (2019). +25 + +[30] Bradley Efron and Trevor Hastie. 2016. Computer Age Statistical Inference: Algorithms, Evidence, and Data Science. +Cambridge University Press. https://doi.org/10.1017/CBO9781316576533 +[31] Chelsea Finn, Paul Christiano, Pieter Abbeel, and Sergey Levine. 2016. A connection between generative adversarial +networks, inverse reinforcement learning, and energy-based models. arXiv preprint arXiv:1611.03852 (2016). +[32] William J Frawley, Gregory Piatetsky-Shapiro, and Christopher J Matheus. 1992. Knowledge discovery in databases: +An overview. AI magazine 13, 3 (1992), 57–57. +[33] Karl Friston. 2009. The free-energy principle: a rough guide to the brain? Trends in Cognitive Sciences 13, 7 (2022/08/04 +2009), 293–301. https://doi.org/10.1016/j.tics.2009.04.005 +[34] Karl Friston, James Kilner, and Lee Harrison. 2006. A free energy principle for the brain. Journal of Physiology-Paris +100, 1 (2006), 70–87. https://doi.org/10.1016/j.jphysparis.2006.10.001 Theoretical and Computational Neuroscience: +Understanding Brain Functions. +[35] Claudio Gentile and Manfred K. K Warmuth. 1998. Linear Hinge Loss and Average Margin. In Advances in Neural +Information Processing Systems, M. Kearns, S. Solla, and D. Cohn (Eds.), Vol. 11. MIT Press. +https://proceedings. +neurips.cc/paper/1998/file/a14ac55a4f27472c5d894ec1c3c743d2-Paper.pdf +[36] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and +Yoshua Bengio. 2020. Generative adversarial networks. Commun. ACM 63, 11 (2020), 139–144. +[37] Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, +and Yoshua Bengio. 2014. Generative adversarial networks. arXiv preprint arXiv:1406.2661 (2014). +[38] Ian J. Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, +and Yoshua Bengio. 2014. Generative Adversarial Networks. https://doi.org/10.48550/ARXIV.1406.2661 +[39] Henry Gouk, Eibe Frank, Bernhard Pfahringer, and Michael J Cree. 2021. Regularisation of neural networks by +enforcing lipschitz continuity. Machine Learning 110, 2 (2021), 393–416. +[40] Will Grathwohl, Kuan-Chieh Wang, Jörn-Henrik Jacobsen, David Duvenaud, Mohammad Norouzi, and Kevin Swersky. +2019. Your classifier is secretly an energy based model and you should treat it like one. arXiv preprint arXiv:1912.03263 +(2019). +[41] Tuomas Haarnoja, Haoran Tang, Pieter Abbeel, and Sergey Levine. 2017. Reinforcement learning with deep energy- +based policies. In International conference on machine learning. PMLR, 1352–1361. +[42] Raia Hadsell, Sumit Chopra, and Yann LeCun. 2006. Dimensionality reduction by learning an invariant mapping. In +2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), Vol. 2. IEEE, 1735–1742. +[43] GM Harshvardhan, Mahendra Kumar Gourisaria, Manjusha Pandey, and Siddharth Swarup Rautaray. 2020. A +comprehensive survey and analysis of generative models in machine learning. Computer Science Review 38 (2020), +100285. +[44] Jonathan Ho, Ajay Jain, and Pieter Abbeel. 2020. Denoising diffusion probabilistic models. Advances in Neural +Information Processing Systems 33 (2020), 6840–6851. +[45] Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans. 2022. Cascaded +Diffusion Models for High Fidelity Image Generation. J. Mach. Learn. Res. 23 (2022), 47–1. +[46] Arthur E. Hoerl and Robert W. Kennard. 2000. Ridge Regression: Biased Estimation for Nonorthogonal Problems. +Technometrics 42, 1 (2000), 80–86. http://www.jstor.org/stable/1271436 +[47] Elad Hoffer and Nir Ailon. 2015. Deep metric learning using triplet network. In International workshop on similarity- +based pattern recognition. Springer, 84–92. +[48] Peter J Huber. 1965. A robust version of the probability ratio test. The Annals of Mathematical Statistics (1965), +1753–1758. +[49] Shruti Jadon. 2020. A survey of loss functions for semantic segmentation. In 2020 IEEE Conference on Computational +Intelligence in Bioinformatics and Computational Biology (CIBCB). IEEE, 1–7. +[50] Wenxin Jiang. 2004. Process consistency for AdaBoost. Annals of Statistics 32, 1 (Feb. 2004), 13–29. https://doi.org/10. +1214/aos/1079120128 +[51] Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul. 1999. An introduction to variational +methods for graphical models. Machine learning 37, 2 (1999), 183–233. +[52] Michael I Jordan and Tom M Mitchell. 2015. Machine learning: Trends, perspectives, and prospects. Science 349, 6245 +(2015), 255–260. +[53] James M Joyce. 2011. Kullback-leibler divergence. In International encyclopedia of statistical science. Springer, 720–722. +[54] Biing-Hwang Juang, Wu Hou, and Chin-Hui Lee. 1997. Minimum classification error rate methods for speech +recognition. IEEE Transactions on Speech and Audio processing 5, 3 (1997), 257–265. +[55] Diederik P Kingma and Max Welling. 2013. Auto-encoding variational bayes. arXiv preprint arXiv:1312.6114 (2013). +[56] Krzysztof C Kiwiel. 2006. Methods of descent for nondifferentiable optimization. Vol. 1133. Springer. +[57] David G Kleinbaum, K Dietz, M Gail, Mitchel Klein, and Mitchell Klein. 2002. Logistic regression. Springer. +26 + +[58] Gregory Koch, Richard Zemel, Ruslan Salakhutdinov, et al. 2015. Siamese neural networks for one-shot image +recognition. In ICML deep learning workshop, Vol. 2. Lille, 0. +[59] Konstantina Kourou, Themis P Exarchos, Konstantinos P Exarchos, Michalis V Karamouzis, and Dimitrios I Fotiadis. +2015. Machine learning applications in cancer prognosis and prediction. Computational and structural biotechnology +journal 13 (2015), 8–17. +[60] T Krishnaiah, S Srinivasa Rao, K Madhumurthy, and KS Reddy. 2007. Neural network approach for modelling global +solar radiation. Journal of Applied Sciences Research 3, 10 (2007), 1105–1111. +[61] Jan Kukačka, Vladimir Golkov, and Daniel Cremers. [n. d.]. https://doi.org/10.48550/ARXIV.1710.10686 +[62] Rithesh Kumar, Sherjil Ozair, Anirudh Goyal, Aaron Courville, and Yoshua Bengio. 2019. Maximum entropy generators +for energy-based models. arXiv preprint arXiv:1901.08508 (2019). +[63] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. 1998. Gradient-based learning applied to document +recognition. Proc. IEEE 86, 11 (1998), 2278–2324. +[64] Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, and F Huang. 2006. A tutorial on energy-based learning. +Predicting structured data 1, 0 (2006). +[65] Yann LeCun and Fu Jie Huang. 2005. Loss functions for discriminative training of energy-based models. In International +workshop on artificial intelligence and statistics. PMLR, 206–213. +[66] Y. Lee, Yoonkyung Lee, Yoonkyung Lee, Yi Lin, Yi Lin, Grace Wahba, and Grace Wahba. 2002. Multicategory Support +Vector Machines, Theory, and Application to the Classification of Microarray Data and Satellite Radiance Data. +[67] Esther Levin and Michael Fleisher. 1988. Accelerated learning in layered neural networks. Complex systems 2, 625-640 +(1988), 3. +[68] Gong Li and Jing Shi. 2010. On comparing three artificial neural networks for wind speed forecasting. Applied Energy +87, 7 (2010), 2313–2320. +[69] Yuncheng Li, Yale Song, and Jiebo Luo. 2017. Improving pairwise ranking for multi-label image classification. In +Proceedings of the IEEE conference on computer vision and pattern recognition. 3617–3625. +[70] Sijia Liu, Pin-Yu Chen, Bhavya Kailkhura, Gaoyuan Zhang, Alfred O Hero III, and Pramod K Varshney. 2020. A primer +on zeroth-order optimization in signal processing and machine learning: Principals, recent advances, and applications. +IEEE Signal Processing Magazine 37, 5 (2020), 43–54. +[71] Dengsheng Lu and Qihao Weng. 2007. A survey of image classification methods and techniques for improving +classification performance. International journal of Remote sensing 28, 5 (2007), 823–870. +[72] Gábor Lugosi and Nicolas Vayatis. 2003. On the Bayes-risk consistency of regularized boosting methods. Annals of +Statistics 32 (2003), 30–55. +[73] Batta Mahesh. 2020. +Machine learning algorithms-a review. +International Journal of Science and Research +(IJSR).[Internet] 9 (2020), 381–386. +[74] Shie Mannor, Ron Meir, and Tong Zhang. 2003. Greedy Algorithms for Classification – Consistency, Convergence +Rates, and Adaptivity. J. Mach. Learn. Res. 4 (2003), 713–741. +[75] Ajay Mathur and Giles M Foody. 2008. Multiclass and binary SVM classification: Implications for training and +classification users. IEEE Geoscience and remote sensing letters 5, 2 (2008), 241–245. +[76] T Mitchell, B Buchanan, G DeJong, T Dietterich, P Rosenbloom, and A Waibel. 1990. Machine Learning. Annual +Review of Computer Science 4, 1 (1990), 417–433. https://doi.org/10.1146/annurev.cs.04.060190.002221 +[77] Tom M Mitchell and Tom M Mitchell. 1997. Machine learning. Vol. 1. McGraw-hill New York. +[78] Klaus-Robert Müller, Matthias Krauledat, Guido Dornhege, Gabriel Curio, and Benjamin Blankertz. 2004. Machine +learning techniques for brain-computer interfaces. Biomed. Tech 49, 1 (2004), 11–22. +[79] In Jae Myung. 2000. The Importance of Complexity in Model Selection. Journal of Mathematical Psychology 44, 1 +(2000), 190–204. https://doi.org/10.1006/jmps.1999.1283 +[80] In Jae Myung. 2003. Tutorial on maximum likelihood estimation. Journal of mathematical Psychology 47, 1 (2003), +90–100. +[81] Andrew Y. Ng. 2004. Feature Selection, L1 vs. L2 Regularization, and Rotational Invariance. In Proceedings of the +Twenty-First International Conference on Machine Learning (Banff, Alberta, Canada) (ICML ’04). Association for +Computing Machinery, New York, NY, USA, 78. https://doi.org/10.1145/1015330.1015435 +[82] Margarita Osadchy, Matthew Miller, and Yann Cun. 2004. Synergistic face detection and pose estimation with +energy-based models. Advances in neural information processing systems 17 (2004). +[83] Daniel W Otter, Julian R Medina, and Jugal K Kalita. 2020. A survey of the usages of deep learning for natural +language processing. IEEE transactions on neural networks and learning systems 32, 2 (2020), 604–624. +[84] Shashikant Patil, Kalpesh R Patil, Chandragouda R Patil, and Smit S Patil. 2020. Performance overview of an artificial +intelligence in biomedics: a systematic approach. International Journal of Information Technology 12, 3 (2020), 963–973. +[85] Patricia Pauli, Anne Koch, Julian Berberich, Paul Kohler, and Frank Allgöwer. 2021. Training robust neural networks +using Lipschitz bounds. IEEE Control Systems Letters 6 (2021), 121–126. +27 + +[86] Claudia Perlich, Brian Dalessandro, Troy Raeder, Ori Stitelman, and Foster Provost. 2014. Machine learning for +targeted display advertising: Transfer learning in action. Machine learning 95, 1 (2014), 103–127. +[87] Yunchen Pu, Zhe Gan, Ricardo Henao, Xin Yuan, Chunyuan Li, Andrew Stevens, and Lawrence Carin. 2016. Variational +autoencoder for deep learning of images, labels and captions. Advances in neural information processing systems 29 +(2016). +[88] J. Ross Quinlan. 1986. Induction of decision trees. Machine learning 1, 1 (1986), 81–106. +[89] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen. 2022. Hierarchical text-conditional +image generation with clip latents. arXiv preprint arXiv:2204.06125 (2022). +[90] Jason D. M. Rennie. 2013. Smooth Hinge Classication. +[91] Luis Miguel Rios and Nikolaos V Sahinidis. 2013. Derivative-free optimization: a review of algorithms and comparison +of software implementations. Journal of Global Optimization 56, 3 (2013), 1247–1293. +[92] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer. 2022. High-resolution image +synthesis with latent diffusion models. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern +Recognition. 10684–10695. +[93] Frank Rosenblatt. 1958. The perceptron: a probabilistic model for information storage and organization in the brain. +Psychological review 65, 6 (1958), 386. +[94] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed +Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al. [n. d.]. Photorealistic Text-to- +Image diffusion models with deep language understanding (2022). URL https://arxiv. org/abs/2205.11487 ([n. d.]). +[95] HR Shally and Rejimol Robinson RR. [n. d.]. Survey for Mining Biomedic. ([n. d.]). +[96] Alexander Shapiro. 2003. Monte Carlo sampling methods. Handbooks in operations research and management science +10 (2003), 353–425. +[97] Kamran Shaukat, Suhuai Luo, Vijay Varadharajan, Ibrahim A Hameed, and Min Xu. 2020. A survey on machine +learning techniques for cyber security in the last decade. IEEE Access 8 (2020), 222310–222354. +[98] Pramila P Shinde and Seema Shah. 2018. A review of machine learning and deep learning applications. In 2018 Fourth +international conference on computing communication control and automation (ICCUBEA). IEEE, 1–6. +[99] Naum Zuselevich Shor. 2012. Minimization methods for non-differentiable functions. Vol. 3. Springer Science & Business +Media. +[100] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli. 2015. Deep unsupervised learning +using nonequilibrium thermodynamics. In International Conference on Machine Learning. PMLR, 2256–2265. +[101] Yang Song and Stefano Ermon. 2019. Generative modeling by estimating gradients of the data distribution. Advances +in Neural Information Processing Systems 32 (2019). +[102] Yang Song and Stefano Ermon. 2020. Improved techniques for training score-based generative models. Advances in +neural information processing systems 33 (2020), 12438–12448. +[103] Yang Song and Diederik P Kingma. 2021. How to train your energy-based models. arXiv preprint arXiv:2101.03288 +(2021). +[104] Jan Stanczuk, Christian Etmann, Lisa Maria Kreusser, and Carola-Bibiane Schönlieb. 2021. Wasserstein GANs work +because they fail (to approximate the Wasserstein distance). arXiv preprint arXiv:2103.01678 (2021). +[105] I. Steinwart. 2005. Consistency of support vector machines and other regularized kernel classifiers. IEEE Transactions +on Information Theory 51, 1 (2005), 128–142. https://doi.org/10.1109/TIT.2004.839514 +[106] Kevin Swersky, Marc’Aurelio Ranzato, David Buchman, Nando D Freitas, and Benjamin M Marlin. 2011. On autoen- +coders and score matching for energy based models. In Proceedings of the 28th international conference on machine +learning (ICML-11). 1201–1208. +[107] Ben Taskar, Carlos Guestrin, and Daphne Koller. 2003. Max-margin Markov networks. Advances in neural information +processing systems 16 (2003). +[108] Yee Whye Teh, Max Welling, Simon Osindero, and Geoffrey E Hinton. 2003. Energy-based models for sparse +overcomplete representations. Journal of Machine Learning Research 4, Dec (2003), 1235–1260. +[109] Robert Tibshirani. 1996. Regression Shrinkage and Selection via the Lasso. Journal of the Royal Statistical Society. +Series B (Methodological) 58, 1 (1996), 267–288. http://www.jstor.org/stable/2346178 +[110] Yoshimasa Tsuruoka, Jun’ichi Tsujii, and Sophia Ananiadou. 2009. Stochastic Gradient Descent Training for L1- +regularized Log-linear Models with Cumulative Penalty. 477–485. https://doi.org/10.3115/1687878.1687946 +[111] Fath U Min Ullah, Amin Ullah, Ijaz Ul Haq, Seungmin Rho, and Sung Wook Baik. 2019. Short-term prediction of +residential power energy consumption via CNN and multi-layer bi-directional LSTM networks. IEEE Access 8 (2019), +123369–123380. +[112] Mohammad Valipour, Mohammad Ebrahim Banihabib, and Seyyed Mahmood Reza Behbahani. 2013. Comparison of +the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez +dam reservoir. Journal of hydrology 476 (2013), 433–441. +28 + +[113] Aaron Van Oord, Nal Kalchbrenner, and Koray Kavukcuoglu. 2016. Pixel recurrent neural networks. In International +conference on machine learning. PMLR, 1747–1756. +[114] Vladimir Vapnik. 1999. The nature of statistical learning theory. Springer science & business media. +[115] Aladin Virmaux and Kevin Scaman. 2018. Lipschitz regularity of deep neural networks: analysis and efficient +estimation. Advances in Neural Information Processing Systems 31 (2018). +[116] John Von Neumann and Oskar Morgenstern. 2007. Theory of games and economic behavior. Princeton university +press. +[117] Jun Wang, Suncheng Feng, Yong Cheng, and Najla Al-Nabhan. 2021. Survey on the Loss Function of Deep Learning +in Face Recognition. Journal of Information Hiding and Privacy Protection 3, 1 (2021), 29. +[118] Jiang Wang, Yang Song, Thomas Leung, Chuck Rosenberg, Jingbin Wang, James Philbin, Bo Chen, and Ying Wu. 2014. +Learning fine-grained image similarity with deep ranking. In Proceedings of the IEEE conference on computer vision +and pattern recognition. 1386–1393. +[119] Qi Wang, Yue Ma, Kun Zhao, and Yingjie Tian. 2020. A comprehensive survey of loss functions in machine learning. +Annals of Data Science (2020), 1–26. +[120] Max Welling and Yee W Teh. 2011. Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of +the 28th international conference on machine learning (ICML-11). Citeseer, 681–688. +[121] Cort J Willmott and Kenji Matsuura. 2005. Advantages of the mean absolute error (MAE) over the root mean square +error (RMSE) in assessing average model performance. Climate research 30, 1 (2005), 79–82. +[122] Yichao Wu and Yufeng Liu. 2007. Robust Truncated Hinge Loss Support Vector Machines. J. Amer. Statist. Assoc. 102 +(2007), 974 – 983. +[123] Shuangfei Zhai, Yu Cheng, Weining Lu, and Zhongfei Zhang. 2016. Deep structured energy based models for anomaly +detection. In International conference on machine learning. PMLR, 1100–1109. +[124] Tong Zhang. 2001. Statistical Behavior and Consistency of Classification Methods based on Convex Risk Minimization. +[125] Tong Zhang. 2004. Solving Large Scale Linear Prediction Problems Using Stochastic Gradient Descent Algorithms. +In ICML 2004: PROCEEDINGS OF THE TWENTY-FIRST INTERNATIONAL CONFERENCE ON MACHINE LEARNING. +OMNIPRESS. 919–926. +[126] Xiang Zhang, Lina Yao, Xianzhi Wang, Jessica Monaghan, David Mcalpine, and Yu Zhang. 2019. A survey on deep +learning based brain computer interface: Recent advances and new frontiers. arXiv preprint arXiv:1905.04149 66 +(2019). +29 + diff --git a/QdE5T4oBgHgl3EQfZQ86/content/tmp_files/load_file.txt b/QdE5T4oBgHgl3EQfZQ86/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..eba1bf4b155daa08ff87a8e40904aaaf5ce86635 --- /dev/null +++ b/QdE5T4oBgHgl3EQfZQ86/content/tmp_files/load_file.txt @@ -0,0 +1,1411 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf,len=1410 +page_content='A survey and taxonomy of loss functions in machine learning LORENZO CIAMPICONI, ADAM ELWOOD, MARCO LEONARDI, ASHRAF MOHAMED, and ALESSANDRO ROZZA, lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com group, Switzerland Most state-of-the-art machine learning techniques revolve around the optimisation of loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Defining appropriate loss functions is therefore critical to successfully solving problems in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We present a survey of the most commonly used loss functions for a wide range of different applications, divided into classification, regression, ranking, sample generation and energy based modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Overall, we introduce 33 different loss functions and we organise them into an intuitive taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Each loss function is given a theoretical backing and we describe where it is best used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This survey aims to provide a reference of the most essential loss functions for both beginner and advanced machine learning practitioners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Additional Key Words and Phrases: loss functions, machine learning, neural networks, survey 1 INTRODUCTION In the last few decades there has been an explosion in interest in machine learning [52, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This field focuses on the definition and application of algorithms that can be trained on data to model underlying patterns [11, 73, 77, 88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning approaches can be applied to many different research fields, including biomedical science [59, 84, 95, 126], natural language understanding [22, 83], [97] anomaly detection [17], image classification [71], database knowledge discovery [32], robot learning [3], online advertising [86], time series forecasting [13], brain computer interfacing [78] and many more [98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To train these algorithms, it is necessary to define an objective function, which gives a scalar measure of the algorithm’s performance [77, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' They can then be trained by optimising the value of the objective function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Within the machine learning literature, such objective functions are usually defined in the form of loss functions, which are optimal when they are minimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The exact form of the loss function depends on the nature of the problem to be solved, the data available and the type of machine learning algorithm being optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finding appropriate loss functions is therefore one of the most important research endeavours in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As the field of machine learning has developed, lots of different loss functions have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It is therefore very useful to summarise and understand them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, there are few works that attempt to do this for the whole field [119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The existing reviews of loss functions in the literature either lack a good taxonomy to structure and contextualise the different losses, or are specifically focused on a particular subset of machine learning applications [49, 117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' There is also no single source that puts the most commonly used loss functions in the same formal setting, listing the advantages and drawbacks of each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For this reason, we have worked to build a proper taxonomy of loss functions, where we show the advantages and disadvantages for each technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We hope this will be useful for new users who want to familiarise themselves with the most common loss functions used in the machine learning literature and find one that is suitable for a problem that they are trying to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We also hope this summary will be useful as a comprehensive reference for advanced users, allowing them to quickly find the best loss function without having to broadly search the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Additionally, this can be helpful for researchers to find possible avenues for further research, or to understand where to place any new techniques that they have proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' They could, for example, use this Authors’ address: Lorenzo Ciampiconi, lorenzo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='ciampiconi@lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Adam Elwood, adam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='elwood@lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Marco Leonardi, marco.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='leonardi@lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Ashraf Mohamed, ashraf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='mohamed@lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Alessandro Rozza, alessandro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='rozza@lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com, lastminute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='com group, Vicolo de’ Calvi, 2, Chiasso, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' , Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1, Article .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Publication date: January 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='05579v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='LG] 13 Jan 2023 survey to understand if their new proposals fit somewhere inside the taxonomy we present, or if they are in a completely new category, maybe combining disparate ideas in novel ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Overall, we have included 33 of the most widely used loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In each section of this work, we break down the losses based on the broad classification of tasks that they can be used for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Each loss function will be defined mathematically, and its most common applications listed highlighting advantages and drawbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main contribution of this work can be found in the proposed taxonomy depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Each loss function is first divided according the specific task on which they are exploited: regression, classification, ranking, sample generation and energy-based modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Furthermore, we divide them by the type of learning paradigm on which they can be applied to, from supervised to unsupervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, we classify them according to the underling strategy on which they are based, such as if they rely on a probabilistic formalization, or are based on errors or a margin between the prediction and the actual values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This work is organized as follows: In Section 2, we provide a formal definition of a loss function and introduce our taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Section 3, we describe the most common regularization methods used to reduce model complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Section 4, we describe the regression task and the key loss functions used to train regression models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Section 5, we introduce the classification problem and the associated loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Section 6, we present generative models and their losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Ranking problems and their loss functions are introduced in Section 7, and energy based models and their losses are described in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, we draw conclusions in Section 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2 DEFINITION OF OUR LOSS FUNCTION TAXONOMY In a general machine learning problem, the aim is to learn a function 𝑓 that transforms an input, defined by the input space Φ into a desirable output, defined by the output space Y: 𝑓 : Φ → Y Where 𝑓 is a function that can be approximated by a model, 𝑓Θ, parameterised by the parameters Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Given a set of inputs {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', x𝑁 } ∈ Φ, they are used to train the model with reference to target variables in the output space, {y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', y𝑁 } ∈ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Notice that, in some cases (such as autoencoders) Y = Φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A loss function, 𝐿, is defined as a mapping of 𝑓 (x𝑖) with it’s corresponding y𝑖 to a real number 𝑙 ∈ R, which captures the similarity between 𝑓 (x𝑖) and y𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Aggregating over all the points of the dataset we find the overall loss, L: L(𝑓 |{x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', x𝑁 }, {y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', y𝑁 }) = 1 𝑁 𝑁 ∑︁ 𝑖=1 𝐿(𝑓 (x𝑖), y𝑖) (1) The optimisation function to be solved is defined as: min 𝑓 L(𝑓 |{x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', x𝑁 }, {y0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', y𝑁 }) (2) Notice that, it is often convenient to explicitly introduce a regularisation term (𝑅) which maps 𝑓 to a real number 𝑟 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This term is usually used for penalising the complexity of the model in the optimisation [77]: 2 min 𝑓 1 𝑁 𝑁 ∑︁ 𝑖=1 𝐿(𝑓 (x𝑖), y𝑖) + 𝑅(𝑓 ) (3) In practice, the family of functions chosen for the optimisation can be parameterised by a parameter vector Θ, which allows the minimisation to be defined as an exploration in the parameter space: min Θ 1 𝑁 𝑁 ∑︁ 𝑖=1 𝐿(𝑓Θ(x𝑖), y𝑖) + 𝑅(Θ) (4) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Optimisation techniques for loss functions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Loss functions and optimisation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In this section, we list out the most common mathematical properties that a loss may or may not satisfy and then we briefly discuss the main optimisation methods employed to minimise them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For the sake of simplicity, visualisation and understanding we define such properties in a two dimensional space, but they can be easily generalised to a d-dimensional one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Continuity (CONT): A real function, that is a function from real numbers to real numbers, can be represented by a graph in the Cartesian plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' such a function is continuous if the graph is a single unbroken curve belonging to the real domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A more mathematically rigorous definition can be given by defining continuity in terms of limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A function 𝑓 with variable 𝑥 is continuous at the real number 𝑐, if lim𝑥→𝑐 𝑓 (𝑥) = 𝑓 (𝑐).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Differentiability (DIFF): A differentiable function 𝑓 on a real variable is a function derivable in each point of its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A continuous function is not necessarily differentiable, but a differentiable function is necessarily continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Lipschitz Continuity (L-CONT): A Lipschitz continuous function is limited in how fast it can change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' More formally, there exists a real number such that, for every pair of points on the graph of this function, the absolute value of the slope of the line connecting them is not greater than this real number;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' this value is called the Lipschitz constant of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To understand the robustness of a model, such as a neural network, some research papers [39, 115] have tried to train the underlying model by defining an input-output map with a small Lipschitz constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The intuition is that if a model is robust, it should not be too affected by perturbations in the input, 𝑓 (𝑥 + 𝛿𝑥) ≈ 𝑓 (𝑥), and this would be ensured by having 𝑓 be ℓ-Lipschitz where ℓ is small [85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Convexity (CONV): a real-valued function 𝑓 is convex if each segment between any two points on the graph of the function lies above the graph between the two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Convexity is a key feature, since the local minima of convex function is also the global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Whenever the second derivative of a function exists, then the convexity is easy to check, since the Hessian of the function must be positive semi-definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Strict Convexity (S-CONV): a real-valued function is stricly convex if the segment between any two points on the graph of the function lies above the graph between the two points, except for the intersection points between the straight line and the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Strictly convex functions have a positive definitive Hessian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Positive-definite matrices are invertible and the optimisation problem can be so solved in a closed form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3 Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Gradient Descent Input: initial parameters Θ(0), number of iterations 𝑇, learning rate 𝛼 Output: final learning Θ(𝑇) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' for 𝑡 = 0 to 𝑇 − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' estimate ∇L(Θ(𝑡)) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' compute ΔΘ(𝑡) = −∇L(Θ(𝑡)) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Θ(𝑡+1) := Θ(𝑡) + 𝛼ΔΘ(𝑡) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' return Θ(𝑇) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Relevant optimisation methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' An optimisation method is a technique that, given a for- malised optimisation problem with an objective function, returns the solution to obtain the optimal value of that optimisation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Most of the optimisation methods presented in this work rely on algorithms that may not guarantee the optimality of the solution, but imply a degree of approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Closed form solutions are systems of equations that can be solved analytically by finding the values of Θ that lead to a zero value for the derivative of the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' An optimization problem is closed-form solvable if its objective function is differentiable with respect to Θ and the differentiation can be solved for Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In general differentiability and strict convexity are required to have a closed form solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Closed-form solutions should always be used instead of iterative algorithms if they’re available and computationally feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Gradient Descent is a first-order1 iterative optimization algorithm for finding a local mini- mum of a differentiable function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The procedure takes repeated steps in the opposite direction of the gradient of the function at the current point, with a step-size defined by a parameter 𝛼, often called the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The loss function employed must be differentiable, so that the gradient can be computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In order to overcome this limitation and employ also non-differentiable loss function, approxi- mation of gradient and other techniques can be used [56, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The procedure for gradient descent is formalized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Stochastic Gradient Descent (SGD [77]) is a stochastic approximation of gradient descent optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It replaces the actual gradient, calculated from the entire dataset, by an estimate, which is calculated from a randomly selected subset of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The stochastic gradient is an unbiased estimate of the real gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In high-dimensional optimization problems, such as in artificial neural networks, this reduces the time cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The stochasticity of this method reduces the probability of the optimisation to get stuck in a local minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' SGD shares the same constraints (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' differentiability, convexity for optimal solution) of traditional Gradient Descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Derivative Free Optimisation In some cases the derivative of the objective function may not exist, or may not be easy to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This is where derivative-free optimisation comes into the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Classical simulated annealing arithmetic, genetic algorithms and particle swarm optimisation are a few such examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Conventional derivative free optimisation methods are usually difficult to scale to large-size problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To learn more about derivative free optimisation you can refer to [24, 91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Zeroth Order optimisation Zeroth-Order (ZOO) optimisation is a subset of gradient-free optimisation that emerges in various signal processing as well as machine learning appli- cations [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ZOO optimisation methods are the gradient-free counterparts of first-order 1In numerical analysis, methods that have at most linear local error are called first order methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' They are frequently based on finite differences, a local linear approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4 optimisation techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ZOO approximates the full gradients or stochastic gradients through function value-based gradient estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Some recent important applications include gen- eration of prediction-evasive, black-box adversarial attacks on deep neural networks [19], generation of model-agnostic explanation from machine learning systems [26], and design of gradient or curvature regularised robust ML systems in a computationally-efficient manner [70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Zeroth optimisation can be a convenient option, compared to the conventional derivative free optimisation approach, as it’s easy to implement inside commonly used gradient based algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='g SGD), it approximates derivatives efficiently and has comparable convergence rates to first-order algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Probabilistic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Error based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Margin based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='GENERATIVE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='CLASSIFICATION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='RANKING ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='ENERGY BASED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='SUPERVISED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='SEMI SUPERVISED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='UNSUPERVISED ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Regularization Methods ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='|𝟂| - weight-normbased ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='𝞖 - entropy based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Lasso ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Ridge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='|𝟂| ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Cosine ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='similarity ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Quadratically ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Smoothed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Modified ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Huber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Cross ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Entropy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Kullback-Leibler ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Ramp loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Energy loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Generalized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Perceptron ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='REGRESSION ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Zero-One ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Smoothed ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Hinge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Negative ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Log-Likelihood ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='MinMax ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Wesserstein ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Pairwise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Ranking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Triplet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Ranking ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Diffusion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Generalized ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Margin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Log loss ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Minimum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='classification error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Square square ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Square exponential ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Hinge ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Mean Squared ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Mean Bias ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Huber ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Log cosh ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Root Mean ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Squared Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Smooth L1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Mean Absolute ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Root Mean Squared ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Logarithmic Error ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The proposed taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Five major tasks are identified on which loss function are applied to, namely regression, classification, ranking, generating samples (generative) and energy based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' With different colors we specify the type of learning paradigm, from supervised to unsupervised of each loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally the underlying strategy to optimize them, namely margin based, probabilistic and error based is illustrated under each group of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Our taxonomy Our taxonomy is summarized in Fig 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To define it, we started by categorizing the losses depending on which machine learning problem they are best suited to solve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We have identified the following categories: Regression (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4) Classification (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5) Generative modelling (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6) Ranking (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 7) Energy based modelling (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8) We also made a distinction based on the mathematical concepts used to define the loss obtaining the following sub-categories: Error based Probabilistic Margin based Exploiting this approach we find a compact and intuitive taxonomy, with little redundancy or overlap between the different sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We have employed well known terminology to define the taxonomy, which will make it easier for any user to intuitively understand it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3 REGULARISATION METHODS Regularisation methods can be applied to almost all loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' They are employed to reduce model complexity, simplifying the trained model and reducing it’s propensity to overfit the training data [5, 30, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Model complexity, is usually measured by the number of parameters and their magnitude [5, 77, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' There are many techniques which fall under the umbrella of regularisation method and a significant number of them are based on the augmentation of the loss function [30, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' An intuitive justification for regularization is that it imposes Occam’s razor on the complexity of the final model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' More theoretically, many loss-based regularization techniques are equivalent to imposing certain prior distributions on the model parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Regularisation by Loss Augmentation One can design the loss function to penalise the magnitude of model parameters, thus learning the best trade-off between bias and variance of the model and reducing the generalization error without affecting the training error too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This prevents overfitting, while avoiding underfitting, and can be done by augmenting the loss function with a term that explicitly controls the magnitude of the parameters, or implicitly reduces the number of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The general way of augmenting a loss function in order to regularise the result is formalized in the following equation: �𝐿(𝑓 (x𝑖), y𝑖) = 𝐿(𝑓 (x𝑖), y𝑖) + 𝜆𝜌(Θ) (5) where 𝜌(Θ) is called regularization function and 𝜆 defines the amount of regularisation (the trade-off between fit and generalisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This general definition makes it clear that we can employ regularization on any of the losses proposed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We are now going to describe the most common regularisation methods based on loss augmen- tation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 L2-norm regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In 𝐿2 regularization the loss is augmented to include the weighted 𝐿2 norm of the weights [12, 77], so the regularisation function is 𝜌(Θ) = ∥Θ∥2 2: �𝐿(𝑓 (x𝑖), y𝑖) = 𝐿(𝑓 (x𝑖), y𝑖) + 𝜆 ∥Θ∥2 2 (6) 7 when this is employed to regression problems it is also known as Ridge regression [46, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 𝐿1-norm regularisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In 𝐿1 regularization the loss is augmented to to include the weighted 𝐿1 norm of the weights [12, 77], so the regularisation function is 𝜌(Θ) = ∥Θ∥ �𝐿(𝑓 (x𝑖), y𝑖) = 𝐿(𝑓 (x𝑖), y𝑖) + 𝜆 ∥Θ∥1 (7) when this is employed to regression problems it is also known as Lasso regression [77, 109].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Comparison between 𝐿2 and 𝐿1 norm regularisations 𝐿1 and 𝐿2 regularisations are both based on the same concept of penalising the magnitude of the weights composing the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Despite that, the two methods have important differences in their employability and their effects on the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' One of the most crucial differences is that 𝐿1, when optimised, is able to shrink weights to 0, while 𝐿2 results in non-zeros (smoothed) values [7, 12, 73, 77, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This allows 𝐿1 to reduce the dimension of a model’s parameter space and perform an implicit feature selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Indeed, it has been shown by [81] that by employing 𝐿1 regularization on logistic regression, the sample complexity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', the number of training examples required to learn “well”) grows logarithmically in the number of irrelevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' On the contrary, the authors show that any rotationally invariant algorithm (including logistic regression) with 𝐿2 regularization has a worst case sample complexity that grows at least linearly in the number of irrelevant features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Moreover, 𝐿2 is more sensitive to the outliers than 𝐿1-norm since it squares the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐿2 is continuous, while 𝐿1 is a piece-wise function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main advantage of 𝐿2 is that it is differentiable, while 𝐿1 is non-differentiable at 0, which has some strong implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Precisely, the 𝐿2 norm can be easily trained with gradient descent, while 𝐿1 sometimes cannot be efficiently applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The first problem is the inefficiency of applying the 𝐿1 penalty to the weights of all the features, especially when the dimension of the feature space tends to be very large [110], producing a significant slow down of the weights updating process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally the naive application of 𝐿1 penalty in SGD does not always lead to compact models, because the approximate gradient used at each update could be very noisy, so the weights of the features can be easily moved away from zero by those fluctuations and 𝐿1 looses its main advantages with respect to 𝐿2 [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4 REGRESSION LOSSES The aim of a regression model is to predict the outcome of a continuous variable 𝑦 (the dependent variable) based on the value of one or multiple predictor variables x (the independent variables).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' More precisely, let 𝑓Θ be a generic model parameterized by Θ, which maps the independent variables x ∈ {x0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', x𝑁 }, x𝑖 ∈ R𝐷 into the dependent variable 𝑦 ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The final goal is to estimate the parameters of the model Θ that most closely fits the data by minimizing a loss function 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8 Mean Squared Error Mean Bias Error Huber Log cosh Root Mean Squared Error Smooth L1 Mean Absolute Error Root Mean Squared Logarithmic Error Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Schematic overview of the regression losses showing the connection All the losses considered for the regression task are based on functions of the residuals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' the difference between the observed value 𝑦 and the predicted value 𝑓 (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In the following, let 𝑓 (x𝑖) be the outcome of the prediction over x𝑖, and 𝑦 be the ground truth of the 𝑖𝑡ℎ variable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As highlighted by Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2 the Mean Bias Error (𝑀𝐵𝐸) loss can be considered a base pillar for regression losses, characterized by many variations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Among them the most relevant are: Mean Absolute Error (𝑀𝐴𝐸), Mean Squared Error (𝑀𝑆𝐸), and Root Mean Squared Error (𝑅𝑀𝑆𝐸) losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In this section we are also going to introduce the Huber loss and the smooth L1, which are a blend between the 𝑀𝐴𝐸 and the 𝑀𝑆𝐸.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, the Log-cosh and the Root Mean Squared Logarithmic Error losses are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Mean Bias Error Loss (CONT, DIFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The most straightforward loss function is the Mean Bias Error loss, illustrated in Equation 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It captures the average bias in the prediction, but is rarely adopted as loss function to train regression models, because positive errors may cancel out the negative ones, leading to a potential erroneous estimation of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Nevertheless, it is the starting point of the loss functions defined in the next subsections and it is commonly used to evaluate the performances of the models [60, 111, 112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L𝑀𝐵𝐸 = 1 𝑁 𝑁 ∑︁ 𝑖=1 𝑦𝑖 − 𝑓 (x𝑖) (8) Directly connected to 𝑀𝐵𝐸 there are respectively the Mean Absolute Error, the Mean Squared Error and the Log-cosh losses, which basically differs from 𝑀𝐵𝐸 in how they exploit the bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Mean Absolute Error Loss (L-CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Mean Absolute Error loss or L1 loss is one of the most basic loss functions for regression, it measures the average of the absolute bias in the prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The absolute value overcomes the problem of the 𝑀𝐵𝐸 ensuring that positive errors do not cancel the negative ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Therefore each error contributes to 𝑀𝐴𝐸 in proportion to the absolute value of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Notice that, the contribution of the errors follows a linear behavior, meaning that many small errors are important as a big one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This implies that the gradient magnitude is not dependent on the error size, thus may leading into convergence problems when the error is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A model trained to minimize the MAE is more effective when the target data conditioned on the input is symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It is important to highlight that the derivative of the absolute value at zero is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As for MBE, MAE is also used to evaluate the performances of the models [68, 121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L𝑀𝐴𝐸 = 1 𝑁 𝑁 ∑︁ 𝑖=1 |𝑦𝑖 − 𝑓 (x𝑖)| (9) 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3 Mean Squared Error Loss (CONT, DIFF, CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Mean Squared Error loss, or L2 loss, is the average of squared distances between the observed value 𝑦 and the predicted value ˆ𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As for 𝑀𝐴𝐸, it is a well-known and straightforward loss function for regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The squared term makes all the biases positive and magnifies the contribution made by outliers, making it more suitable for problems where noise in the observations follows a normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main drawback is the sensitivity to the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L𝑀𝑆𝐸 = 1 𝑁 𝑁 ∑︁ 𝑖=1 (𝑦𝑖 − 𝑓 (x𝑖))2 (10) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='4 Root Mean Squared Error Loss(CONT,DIFF,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Directly connected to MSE, we have the Root Mean Squared Error loss, which is similar to MSE except for the square root term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main advantage is to make sure that the loss has the same units and scale of the variable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Since the only difference between the MSE and the RMSE consists in the application of the root term, the minimization process converge to the same optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, depending on the optimisation technique used, the RMSE may take different gradient steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As the previously presented loss functions, it is also used as a metric to compare the performances of the model [68, 112], and it shares the same limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L𝑅𝑀𝑆𝐸 = � � � 1 𝑁 𝑁 ∑︁ 𝑖=1 (𝑦𝑖 − 𝑓 (x𝑖))2 (11) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='5 Huber loss (L-CONT,DIFF,S-CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Huber loss [48] is a variant of the MAE that becomes MSE when the residuals are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It is parameterized by 𝛿, which defines the transition point from MAE to MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' When |yi − 𝑓 (xi)| ≤ 𝛿 the Huber loss follows the MSE, otherwise it follows the MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This allows it to combine the advantages of both the MAE and the MSE, when the difference between the prediction and the output of the model is huge errors are linear, make the Huber loss less sensitive to the outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Conversely, when the error is small, it follows the MSE making the convergence much faster and differentiable at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The choice of 𝛿 is fundamental and it can be constantly adjusted during the training procedure based on what is considered an outlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main limitation of the Huber loss resides in the additional extra hyperparameter 𝛿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐿𝐻𝑢𝑏𝑒𝑟𝑙𝑜𝑠𝑠 = � 1 2 (𝑦𝑖 − 𝑓 (x𝑖))2 𝑓 𝑜𝑟 |𝑦𝑖 − 𝑓 (x𝑖)| ≤ 𝛿, 𝛿 �|𝑦𝑖 − 𝑓 (x𝑖)| − 1 2𝛿� 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 (12) Notice that, when 𝛿 = 1, we obtain the smooth L1 loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='6 Log-cosh loss(CONT, DIFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The log-cosh loss is the logarithm of the hyperbolic cosine of the residuals between the observed value 𝑦 and the predicted value ˆ𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It has all the advantages of the Huber loss, without the requirement of setting a hyperparameter, at the cost of being more computationally expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Furthermore, another benefit of the log-cosh loss is related to the fact that is differentiable twice everywhere, making it suitable for methods that requires solving the second derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As 𝑙𝑜𝑔 (𝑐𝑜𝑠ℎ (x)) is approximately equal to x2 2 for small values of x it behave similarly to the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For larger value of x instead, is nearly equivalent to |x| − 𝑙𝑜𝑔 (2) making it similar to MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L𝑙𝑜𝑔𝑐𝑜𝑠ℎ = 1 𝑁 𝑁 ∑︁ 𝑖=1 𝑙𝑜𝑔 (𝑐𝑜𝑠ℎ (𝑓 (x𝑖) − 𝑦𝑖)) (13) Another drawback of L𝑙𝑜𝑔𝑐𝑜𝑠ℎ is related to the fact that, compared to the Huber loss, it is less customizable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='7 Root Mean Squared Logarithmic Error Loss(CONT,DIFF,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Root Mean Squared Logarithmic Error (RMSLE) loss (formalized in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 14) is the RMSE of the log-transformed observed value 𝑦 and log-transformed predicted value ˆ𝑦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The only difference with respect to RMSE is that the logarithm is applied to both the predicted and the observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The plus one term inside the logarithm allows values of 𝑓 (x𝑖) to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Due to the properties of the logarithm, the error between the predicted and the actual values is relative, making the RMSLE more robust to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Precisely, the magnitude of the RMLSE does not scale accordingly to the magnitude of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Indeed, data points with big residuals are less penalized when the predicted and the actual values have high values too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This make the RMSLE suitable for problems where targets have an exponential relationship, or it is preferable to penalize more under estimates than over estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, this loss is not appropriate for problems that allows negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L𝑅𝑀𝑆𝐿𝐸 = � � � 1 𝑁 𝑁 ∑︁ 𝑖=1 (log(𝑦𝑖 + 1) − log(𝑓 (x𝑖) + 1))2 (14) 5 CLASSIFICATION LOSSES 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Problem Formulation and Notation Classification is a subset of problems belonging to supervised learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The goal is to assign an input x to one of 𝐾 discrete classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This goal can be pursued by training a model 𝑓Θ and its parameters Θ by minimizing a loss function 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Let the target space of 𝑓 discrete and consider a model returning the output label, 𝑓 can be defined as: 𝑓 : Φ → Λ𝐾 Λ = {0, 1} The above definition is working also for multi-label classification, since more than one label could be associated to a sample, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝑓 (x) = [0, 1, 0, 1, 0, 0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In order to define single label classification we need to add the constraint that the output sum up to 1, � 𝑘 𝜆𝑘 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We can also consider models with continuous outputs, in case they return a probability 𝑝𝑘 (x) ∈ [0, 1] to a sample x for each possible assignable label 𝑘 ∈ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', 𝐾: 𝑓 : Φ → 𝑃𝐾 𝑃 = [0, 1] As before, to switch between multi-label and single-label classification, we need to constraint the probabilities output to sum up to one, � 𝑘 𝑝𝑘 = 1, if we want to force a single label assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A more narrow notation for classification can be introduced in order to describe binary classi- fication problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This notation is useful in this work because margin based losses are designed to solve binary classification problems and cannot be generalised to multi-class or multi label classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For the subset of binary classification problems the target space of 𝑓 is discrete and it is defined as follows: 𝑓 : Φ → 𝐵 𝐵 = {−1, 1} We define two different macro categories of classification losses accordingly to the underlying strategy employed to optimize them, namely the margin based and the probabilistic ones as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In the next section, we introduce the margin based loss function starting by the most basic and intuitive one, the Zero-One loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Subsequently, we present the Hinge loss 11 and its variants (the Smoothed and Quadratically Smoothed Hinge losses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Then, the Modified Huber loss, the Ramp loss, and the Cosine Similarity loss are described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Moreover, we introduce the probabilistic loss by introducing the Cross Entropy loss and Negative Log-Likelihood loss, which, from a mathematical point of view, coincides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, the Kullback-Leibler Divergence loss is presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Probabilistic Margin based Cosine similarity Hinge Quadratically Smoothed Modified Huber Cross Entropy Kullback-Leibler Divergence Ramp loss Zero-One Smoothed Hinge Negative Log-Likelihood Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Overview of the classification losses divided in two major groups: margin based losses and probabilistic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Margin Based Loss Functions In this section, we introduce the most known margin based loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Zero-One loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The basic and more intuitive margin based classification loss is the Zero-One loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It assigns 1 to a misclassified observation and 0 to a correctly classified one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐿ZeroOne(𝑓 (x),𝑦) = � 1 if 𝑓 (x) · 𝑦 < 0 0 otherwise (15) ZeroOne loss is not directly usable since it lacks convexity and differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, it is possible to derive employable surrogate losses that are classification calibrated, which means that they are a relaxation of 𝐿𝑍𝑒𝑟𝑜𝑂𝑛𝑒, or an upper bound, or an approximation of such loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A significant achievement of the recent literature on binary classification has been the identification of necessary and sufficient conditions under which such relaxations yield Fisher consistency [6, 50, 72, 74, 105, 124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' All the following losses satisfy such conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Hinge loss and Perceptron loss (L-CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The most famous surrogated loss is the Hinge loss [35], which linearly penalizes every prediction where the resulting agreement is <= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐿Hinge(𝑓 (x),𝑦) = max(0, 1 − (𝑓 (x) · 𝑦)) (16) The Hinge loss is not strictly convex, but it is Lipschitz continuous and convex, so many of the usual convex optimizers used in machine learning can work with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Hinge loss is commonly employed to optimise the Support Vector Machine (SVM [14, 75]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To train the Perceptron [93] a variation of this loss, the Perceptron loss, is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss slightly differs from the Hinge loss, because it does not penalise samples inside the margin, surrounding the separating hyperplane, but just the ones that are mislabeled by this hyperplane with the same linear penalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐿Perceptron(𝑓 (x),𝑦) = max(0, −(𝑓 (x) · 𝑦)) (17) 12 There are two main drawbacks using the hinge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Firstly, its adoption use to make the model sensible to outliers in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Secondly, due to the discontinuity of the derivative at (𝑓 (x) · 𝑦) = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' the fact that is not continuously differentiable, Hinge loss results difficult to optimise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3 Smoothed Hinge loss (L-CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A smoothed version of the Hinge loss was defined in [90] with the goal of obtaining a function easier to optimise as shown by the following equation: 𝐿SmoothedHinge(𝑓 (x),𝑦) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 1 2 − (𝑓 (x) · 𝑡) (𝑓 (x) · 𝑦) <= 0 1 2 (1 − (𝑓 (x) · 𝑡))2 0 < (𝑓 (x) · 𝑦) < 1 0 (𝑓 (x) · 𝑦) >= 1 (18) This smoothed version of the Hinge loss is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Clearly, this is not the only possible smooth version of the Hinge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, it is a canonical one that has the important property of being zero for 𝑧 >= 1 and it has constant (negative) slope for 𝑧 <= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Moreover, for 0 < 𝑧 < 1, the loss smoothly transitions from zero slope to a constant negative one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss inherit sensibility to outliers from the original Hinge loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='4 Quadratically Smoothed Hinge loss (L-CONT,CONV,DIFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' With the same goal of the Smoothed Hinge loss a quadratically smoothed version has been defined in [125], to make it easier to be optimised: 𝐿QSmoothedHinge(𝑓 (x),𝑦) = � 1 2𝛾 max(0, −(𝑓 (x) · 𝑦))2 (𝑓 (x) · 𝑦) >= 1 − 𝛾 1 − 𝛾 2 − (𝑓 (x) · 𝑦) otherwise (19) The hyperparameter 𝛾 determines the degree of smoothing, for 𝛾 → 0 the loss becomes the original hinge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In contrast with the Smoothed Hinge loss, this version is not differentiable in the whole domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='5 Modified Huber loss (L-CONT, DIFF, S-CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Modified Huber loss is a slight variation of the Huber loss for regression and a special case of the Quadratic Smoothed Hinge loss with 𝛾 = 2 (For more details refer to section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='5): 𝐿ModHuber(𝑓 (x),𝑦) = � 1 4 max(0, −(𝑓 (x) · 𝑦))2 (𝑓 (x) · 𝑦) >= −1 −(𝑓 (x) · 𝑦) otherwise (20) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='6 Ramp loss (CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Ramp loss, or Truncated Hinge, is a piece-wise linear, contin- uous and convex loss that has been presented in [122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Under multi-class setting, this loss is more robust to outliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' When employed in SVM, it produces more accurate classifiers using a smaller, and more stable, set of support vectors than the multi-class SVM that employes 𝐿Hinge [66], also preserving fisher consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐿Ramp(𝑓 (x),𝑦) = � 𝐿Hinge(𝑓 (x),𝑦)) (𝑓 (x) · 𝑦) <= 1 1 otherwise (21) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='7 Cosine Similarity loss (L-CONT,DIFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Cosine similarity is generally used as a metric to measure distance when the magnitude of vectors is not important [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A typical example is related to text data representation by means of word counts [12, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' When the label and output can be 13 interpreted as vectors it is possible to derive a distance metric between them, which can be adapted into a loss function as follows: 𝐿𝑐𝑜𝑠−𝑠𝑖𝑚(𝑓 (x), y) = 1 − y · f(x) ∥y∥ ∥f(x)∥ (22) It is important to underline that, when using Cosine Similarity loss, the range of possible values is restricted to the interval [-1, 1], which may not be suitable for all types of data or applications, particularly when interpretability is a key requirement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3 Probabilistic loss Functions Let 𝑞 be the probability distribution underlying the dataset and 𝑓Θ the function generating the output, probabilistic loss functions provide some distance function between𝑞 and 𝑓Θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' By minimizing that distance, the model output distribution converges to the ground truth one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Usually, models trained with probabilistic loss functions can provide a measure of how likely a sample is labeled with one class instead of another [12, 43, 77] providing richer information w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' margin based .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Cross Entropy loss and Negative Log-Likelihood loss (CONT,DIFF,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Maximum likelihood estimation (MLE) is a method to estimate the parameters of a probability distribution by maximizing the likelihood [12, 77, 80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' From the point of view of Bayesian inference, MLE can be considered a special case of maximum a-posteriori estimation (MAP) that assumes a uniform prior distribution of the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Formally, it means that, given a dataset of samples D, we are maximizing the following quantity: 𝑃(D|Θ) = 𝑁 � 𝑛=1 𝑓Θ(x𝑖)𝑦𝑖 · (1 − 𝑓Θ(x𝑖))1−𝑦𝑖 (23) The aim is to find the maximum likelihood estimate by minimizing a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To maximize Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 23, we can turn it into a minimisation problem by employing the negative log likelihood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' To achieve this goal we need to define the following quantity: 𝑙𝑜𝑔(𝑃(D|Θ)) = 𝑁 ∑︁ 𝑖=1 (𝑦𝑖 log(𝑓Θ(x𝑖)) + (1 − 𝑦𝑖) log(1 − 𝑓Θ(x𝑖)))) (24) and we can obtain the loss function by taking the negative of the log: L𝑁𝐿𝐿 = − 𝑁 ∑︁ 𝑖=1 (𝑦𝑖 log(𝑓Θ(x𝑖)) + (1 − 𝑦𝑖) log(1 − 𝑓Θ(x𝑖))) (25) Often, the above loss is also called the cross-entropy loss, because it can be derived by minimising the cross entropy between 𝑓Θ and 𝑞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐻 (𝑞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝑓Θ) = − ∫ 𝑞(x) log(𝑓Θ(x))𝑑x (26) For the discrete case (which is the one we are interested in) the definition of the cross entropy is: 𝐻 (𝑞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝑓Θ) = − 𝑁 ∑︁ 𝑖=1 𝑞(x𝑖) log(𝑓Θ(x𝑖)) (27) 14 Maximizing the likelihood with respect to the parameters Θ is the same as minimizing the cross- entropy as shown by the following equations: 𝑁 ∑︁ 𝑖=1 (𝑦𝑖 log(𝑓Θ(x𝑖)) + (1 − 𝑦𝑖) log(1 − 𝑓Θ(x𝑖))) = 1 𝑁 𝑁 � 𝑛=1 𝑓Θ(x𝑖)𝑁 𝑦𝑖 (28) = 𝑁 ∑︁ 𝑖=1 𝑞(x𝑖) log(𝑓Θ(x𝑖) (29) = − 𝐻 (𝑞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝑓Θ) (30) The classical approach to extend this loss to the multi-class scenario is to add as a final activation of the model a softmax function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' defined accordingly to the number of (𝐾) classes considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Given a score for each class 𝑓𝑘 (x) = 𝑠, it’s output can be squashed to sum up to 1 by mean of a softmax function 𝑓𝑆 obtaining: �𝑓𝑘 (x𝑖) = 𝑓𝑆 (𝑓𝑘 (x)) (31) where, the softmax is defined as follows: 𝑓𝑆 (𝑠𝑖) = 𝑒𝑠𝑖 �𝐾 𝑗=1 𝑒𝑠𝑗 (32) The final loss (usually named as categorical cross entropy) is: 𝐿𝐶𝐶𝐸 = − 1 𝐾 𝐾 ∑︁ 𝑗=1 log( �𝑓𝑘 (x)) (33) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Kullback-Leibler divergence (CONT, CONV, DIFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Kullback-Leibler (KL) divergence is an information-based measure of disparity among probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Precisely, it is a non- symmetrical measurement of how one probability distribution differs from another one [12, 53, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Technically speaking, KL divergence is not a distance metric because it doesn’t obey to the triangle inequality (𝐾𝐿(𝑞||𝑓Θ) is not equal to 𝐾𝐿(𝑓Θ||𝑞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It is important to notice that, in the classification use case, minimizing the KL divergence is the same as minimising the cross entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Precisely, the KL between two continuous distributions is defined as: KL(𝑞||𝑓Θ) = ∫ 𝑞(x) log( 𝑞(x) 𝑓Θ(x) )𝑑x = − ∫ 𝑞(x) log(𝑓Θ(x))𝑑x + ∫ 𝑞(x) log(𝑞(x))𝑑x (34) If we want to minimise KL on the parameter Θ, since the second integral is non-dependant on Θ, we obtain: min Θ KL(𝑞||𝑓Θ) = min Θ − ∫ 𝑞(x) log(𝑓Θ(x))𝑑x = min Θ 𝐻 (𝑞, 𝑓Θ) (35) In general, it is preferable using cross entropy, instead of KL divergence, because it is typically easier to compute and optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The cross entropy only involves a single sum over the data, whereas the KL divergence involves a double sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This can make it more computationally efficient, especially when working with large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6 GENERATIVE LOSSES In recent years, generative models have become particularly useful for both understanding the complexity of data distributions and being able to regenerate them [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In this section, as shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4, we describe the losses relevant to Generative Adversarial Networks (GANs) and Diffusion Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, generative models are not limited to these cases, but extend to include 15 more models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For example, Variational Auto-Encoders (VAEs) [2, 55, 87] in which the KL-divergence, described in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2, is the loss function employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The objective of the VAE loss is to reduce the discrepancy between the original distribution and its predicted distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Other models such as the pixel recurrent neural networks [113] and real-valued Non-Volume Preserving (realNVP) models [27] are not considered in this survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' MinMax Wesserstein Diffusion Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Overview of the generative losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Generative Adversarial Networks Generative Adversarial Networks (GANs) are used to create new data instances that are sampled from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' GANs have two main components: The generator, referred as 𝐺({z0, · · · , z𝑁 }), which generates data starting from random noise and tries to replicate real data distributions The discriminator, referred as 𝐷({x0, · · · , x𝑁 }), which learns to distinguish the generator’s fake data from real one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It applies penalties in the generator loss for producing distinguishable fake data compared with real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The GAN architecture is relatively straightforward, although one aspect remains challenging: GAN loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Precisely, the discriminator is trained to provide the loss function for the generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' If generator training goes well, the discriminator gets worse at telling the difference between real and fake samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It starts to classify fake data as real, and its accuracy decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Both the generator and the discriminator components are typically neural networks, where the generator output is connected directly to the discriminator input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The discriminator’s classification provides a signal that the generator uses to update its weights through back-propagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As GANs try to replicate a probability distribution, they should use loss functions that reflect the distance between the distribution of the data generated by the GAN and the distribution of the real data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Two common GAN loss functions are typically used: minimax loss [38] and Wasserstein loss [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generator and discriminator losses derive from a single distance measure between the two aforementioned probability distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generator can only affect one term in the distance measure: the term that reflects the distribution of the fake data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' During generator training, we drop the other term, which reflects the real data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generator and discriminator losses look different, even though they derive from a single formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In the following, both minimax and Wasserstein losses are written in a general form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The properties of the loss function (CONT, DIFF, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=') are identified based on the function chosen for the generator or discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Minimax loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generative model 𝐺 learns the data distributions and is trained simultane- ously with the discriminative model 𝐷.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The latter estimates the probability that a given sample is identical to the training data rather than𝐺.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='𝐺 is trained to maximize the likelihood of tricking 𝐷 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In other words,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' the generator tries to minimize the following function while the discriminator tries 16 to maximize it: L𝑚𝑖𝑛𝑖𝑚𝑎𝑥 (𝐷,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='𝐺) = 𝐸{x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='x𝑁 }[log(𝐷({x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' x𝑁 }))] + 𝐸{z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='z𝑁 }[log(1 − 𝐷(𝐺({z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' z𝑁 })))],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' (36) where: 𝐷({x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' x𝑁 }) is the discriminator’s estimate of the probability that real data instance {x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' x𝑁 } is real,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐸{x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='x𝑁 } is the expected value over all real data instances,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐺({z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' z𝑁 }) is the generator’s output when given noise {z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' z𝑁 },' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐷(𝐺(𝑧)) is the discriminator’s estimate of the probability that a fake instance is real,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝐸{z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='···,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='z𝑁 } is the expected value over all random inputs to the generator (in effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' the expected value over all generated fake instances 𝐺({z0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' z𝑁 })).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The loss function above directly represents the cross-entropy between real and generated data distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generator can’t directly affect the log(𝐷({x0, · · · , x𝑁 })) term in the function, and it only minimizes the term log(1 − 𝐷(𝐺({z0, · · · , z𝑁 }))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A disadvantage of this formulation of the loss function is that the above minimax loss function can cause the GAN to get stuck in the early stages of the training when the discriminator received trivial tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Therefore, a suggested modification to the loss [38] is to allow the generator to maximize log(𝐷(𝐺({z0, · · · , z𝑁 }))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Wasserstein loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Wasserstein distance gives an alternative method of training the generator to better approximate the distribution of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In this setup, the training of the generator itself is responsible for minimizing the distance between the distribution of the training and generated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The possible solutions are to use distribution distance measures, like Kullback-Leibler (KL) divergence, Jensen-Shannon (JS) divergence, and the Earth-Mover (EM) distance (also called Wasserstein distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main advantage of using Wasserstein distance is due to its differentiability and having a continuous linear gradient [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A GAN that uses a Wasserstein loss, known as a WGAN, does not discriminate between real and generated distributions in the same way as other GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Instead, the WGAN discriminator is called a "critic," and it scores each instance with a real-valued score rather than predicting the probability that it is fake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This score is calculated so that the distance between scores for real and fake data is maximised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The advantage of the WGAN is that the training procedure is more stable and less sensitive to model architecture and selection of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The two loss functions can be written as: L𝑐𝑟𝑖𝑡𝑖𝑐 = 𝐷({x0, · · · , x𝑁 }) − 𝐷(𝐺({z0, · · · , z𝑁 })), L𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟 = 𝐷(𝐺({z0, · · · , z𝑁 })) (37) The discriminator tries to maximize L𝑐𝑟𝑖𝑡𝑖𝑐.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In other words, it tries to maximize the difference between its output on real instances and its output on fake instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generator tries to maximize L𝐺𝑒𝑛𝑒𝑟𝑎𝑡𝑜𝑟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In other words, It tries to maximize the discriminator’s output for its fake instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The benefit of Wasserstein loss is that it provides a useful gradient almost everywhere, allowing for the continued training of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It also means that a lower Wasserstein loss correlates with better generator image quality, meaning that it explicitly seeks a minimization of generator loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, it is less vulnerable to getting stuck in a local minimum than minimax-based GANs [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, accurately estimating the Wasserstein distance using batches requires unaffordable batch size, which significantly increases the amount of data needed [104].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 17 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Diffusion models Diffusion Models are generative models that rely on probabilistic likelihood estimation to generate data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' They were originally inspired by the physical phenomena called diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Diffusion Modelling is a learning procedure in which a model can learn the systematic decay of information due to adding a slight noise factor iteratively [100, 101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The learning procedure enables the possibility of recovering the decayed information from the noise again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' They model a series of noise distributions in a Markov Chain and they decode the original data by systematically removing the noises hierarchically [20, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The diffusion process consists of two steps, the forward process and the reconstruction (reverse) process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Gaussian noise is gradually added during the forward diffusion process until the data sample loses its distinguishable features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The reverse process removes the noise and restores the original data by utilizing a neural network model to learn the conditional probability densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Forward diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Given a data point x0 sampled from a real data distribution 𝑞(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The forward process aims to add a small noise iteratively 𝑇 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For every data point x0 the process produces a sequence of noisy samples x1, · · · , xT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The noise distributions is usually chosen to be Gaussian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Because the forecast of probability density at time 𝑡 is only dependent on the immediate predecessor at time 𝑡 − 1, the conditional probability density can be calculated as: 𝑞(x𝑡 |x𝑡−1) = N (x𝑡;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' √︁ 1 − 𝛽𝑡x𝑡−1, 𝛽𝑡I) 𝑞(x1:𝑇 |x0) = 𝑇 � 𝑡=1 𝑞(x𝑡 |x𝑡−1) (38) With a 𝛽𝑡 ∈ (0, 1) is a hyperparameter that can be taken as constant or variable and 𝑡 ∈ [1,𝑇].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As 𝑇 → ∞, xT becomes a pure Gaussian distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We can only sample the new states of the system using the gradient of the density function in Markov Chain updates, according to stochastic gradient Langevin dynamics [120].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The following formula can then be employed to calculate the sampling of a new data point at time 𝑡 with a step size 𝛿 dependent on the prior point at time 𝑡 − 1: x𝑡 = x𝑡−1 + 𝛿 2∇x log𝑞(x𝑡−1) + √ 𝛿𝝐𝑡, where 𝝐𝑡 ∼ N (0, I) (39) when 𝑇 → ∞, 𝝐 → 0 equals to the true probability density 𝑝(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The forward step does not require training a neural network;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' it only requires an iterative process for adding random Gaussian noises to the original data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Reverse diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Given the system’s current state, the reverse process requires the calculation of probability density at a previous time step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Calculating the 𝑞(x𝑡−1|x𝑡) at the time 𝑡 = 𝑇 is equal to producing data from an isotropic Gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, contrary to the forward process, the estimation of the past state from the present state requires the knowledge of every previous gradient, which is not feasible without the aid of a learning model that can forecast such estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This problem is resolved by training a neural network model that, using the learnt weights Θ and the current state at time 𝑡, estimates the value of 𝑝Θ(x𝑡−1|x𝑡) as formalized by the following equation: 𝑝Θ(x0:𝑇 ) = 𝑝(x𝑇 ) 𝑇 � 𝑡=1 𝑝Θ(x𝑡−1|x𝑡) 𝑝Θ(x𝑡−1|x𝑡) = N (x𝑡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 𝝁Θ(x𝑡,𝑡), 𝚺Θ(x𝑡,𝑡)) (40) where 𝝁Θ(x𝑡,𝑡) is the mean function proposed in [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The sample at time 𝑡 − 1 can then be computed as: x𝑡−1 = N (x𝑡−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1 √𝛼𝑡 � x𝑡 − 1 − 𝛼𝑡 √1 − ¯𝛼𝑡 𝝐Θ(x𝑡,𝑡) � , 𝚺Θ(x𝑡,𝑡)) (41) 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3 Diffusion model loss function (CONT, DIFF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The main task of the network model is to minimize the following loss: 𝐿𝑑𝑖𝑓 𝑓 𝑢𝑠𝑖𝑜𝑛 = E𝑡 � log𝑝(x𝑇 ) − ∑︁ 𝑡 ≥1 log 𝑝Θ(x𝑡−1|x𝑡) 𝑞(x𝑡 |x𝑡−1) � (42) In [44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 100],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' it is shown that a simplified version of 𝐿𝑑𝑖𝑓 𝑓 𝑢𝑠𝑖𝑜𝑛 is able to achieve better results: 𝐿simple 𝑑𝑖𝑓 𝑓 𝑢𝑠𝑖𝑜𝑛 = E𝑡∼[1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='𝑇 ],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='x0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='𝝐𝑡 � ∥𝝐𝑡 − 𝝐Θ(√ ¯𝛼𝑡x0 + √1 − ¯𝛼𝑡𝝐𝑡,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='𝑡)∥2� (43) It is important to notice that diffusion Models perform significantly better in some applications despite being computationally more expensive than alternative deep network structures [25,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 45,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 89,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 92,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 94,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 102].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 7 RANKING LOSSES Machine learning can be employed to solve ranking problems, which have important industrial applications, especially in information retrieval systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' These problems can be typically solved by employing supervised, semi-supervised, or reinforcement learning [15, 47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The goal of ranking losses, in contrast to other loss functions like the cross-entropy loss or MSE loss, is to anticipate the relative distances between inputs rather than learning to predict a label, a value, or a set of values given an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This is also sometimes called metric learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Nevertheless, the cross-entropy loss can be used in the top-one probability ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In this scenario, given the scores of all the objects, the top-one probability of an object in this model indicates the likelihood that it will be ranked first [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Ranking loss functions for training data can be highly customizable because they require only a method to measure the similarity between two data points, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', similarity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For example, consider a face verification dataset, pairs of photographs which belong to the same person will have a high similarity score, whereas those that don’t will have a low score [118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 In general, ranking loss functions require a feature extraction for two (or three) data instances, which returns an embedded representation for each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A metric function can then be defined to measure the similarity between those representations, such as the euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, the feature extractors are trained to produce similar representations for both inputs in case the inputs are similar or distant representations in case of dissimilarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Similar to Section 6, both pairwise and triplet ranking losses are presented in a general form, as shown in the Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The properties of the loss function (CONT, DIFF, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=') are identified based on the metric function chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Pairwise Ranking Triplet Ranking Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Overview of the ranking losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2Different tasks, applications, and neural network configurations use ranking losses (like Siamese Nets or Triplet Nets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Because of this, there are various losses can be used, including Contrastive loss, Margin loss, Hinge loss, and Triplet loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 19 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Pairwise Ranking loss In the context of Pairwise Ranking loss, positive and negative pairs of training data points are used [15, 21, 42, 69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Positive pairs are composed of an anchor sample x𝑎 and a positive sample x𝑝, which is similar to x𝑎 in the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Negative pairs are composed of an anchor sample x𝑎 and a negative sample x𝑛, which is dissimilar to x𝑎 in that metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The objective is to learn representations with a small distance 𝑑 between them for positive pairs and a greater distance than some margin value 𝑚 for negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Pairwise Ranking loss forces representations to have a 0 distance for positive pairs and a distance greater than a margin for negative pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Given r𝑎, r𝑝, and r𝑛 the embedded representations (the output of a feature extractor) of the input samples x𝑎, x𝑝, x𝑛 respectively and 𝑑 as a distance function the loss function can be written as: 𝐿𝑝𝑎𝑖𝑟𝑤𝑖𝑠𝑒 = � 𝑑(r𝑎, r𝑏) if positive pair max(0,𝑚 − 𝑑(r𝑎, r𝑛)) if negative pair (44) For positive pairs, the loss will vanish if the distance between the embedding representations of the two elements in the pair is 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' instead, the loss will increase as the distance between the two representations increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For negative pairs, the loss will vanish if the distance between the embedding representations of the two elements is greater than the margin 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, if the distance is less than 𝑚, the loss will be positive, and the model parameters will be updated to provide representations for the two items that are farther apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' When the distance between r𝑎 and r𝑛 is 0, the loss value will be at most 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The purpose of the margin is to create representations for negative pairs that are far enough, thus implicitly stopping the training on these pairs and allowing the model to focus on more challenging ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' If r0 and r1 are the pair elements representations, 𝑦 is a binary flag equal to 0 for a negative pair and to 1 for a positive pair, and the distance 𝑑 is the euclidean distance: 𝐿𝑝𝑎𝑖𝑟𝑤𝑖𝑠𝑒 (r0, r1,𝑦) = 𝑦||r0 − r1|| + (1 − 𝑦) max(0,𝑚 − ||r0 − r1||) (45) Unlike typical classification learning, this loss requires more training data and time because it requires access to all the data of all potential pairs during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Additionally, because training involves pairwise learning, it will output the binary distance from each class, which is more computationally expensive if there is incorrect classification [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Triplet Ranking loss Employing triplets of training data instances instead of pairs can produce better performance [18, 47, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The resultant loss is called Triplet Ranking loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A triplet consists of an anchor sample x𝑎, a positive sample x𝑝, and a negative sample x𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The objective is that the distance between the anchor sample and the negative sample representations 𝑑(r𝑎, r𝑛) is greater (and bigger than a margin 𝑚) than the distance between the anchor and positive representations 𝑑(r𝑎, r𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The same notation applies: 𝐿𝑡𝑟𝑖𝑝𝑙𝑒𝑡 (r𝑎, r𝑝, r𝑛) = max(0,𝑚 + 𝑑(r𝑎, r𝑝) − 𝑑(r𝑎, r𝑛)) (46) This loss is characterized by three different scenarios based on the values of r𝑎, r𝑝, r𝑛, and 𝑚: Easy Triplets: 𝑑(r𝑎, r𝑛) > 𝑑(r𝑎, r𝑝) + 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In the embedding space, the distance between the negative sample and the anchor sample is already large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The model parameters are not changed, and the loss is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 20 Hard Triplets: 𝑑(r𝑎, r𝑛) < 𝑑(r𝑎, r𝑝).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='Compared to the positive sample, the negative sample is closer to the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The loss is positive (and > 𝑚).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The model’s parameters are subject to change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Semi-Hard Triplets: 𝑑(r𝑎, r𝑝) < 𝑑(r𝑎, r𝑛) < 𝑑(r𝑎, r𝑝) + 𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The loss is still positive (and < 𝑚) nevertheless, the negative sample is further away from the anchor than the positive sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The model’s parameters are subject to change, and the loss is not 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss is sensitive to small changes in the input samples, so it cannot be generalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This means that once the model has been trained on a specific dataset, it cannot be applied to other datasets [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8 ENERGY-BASED LOSSES An Energy-Based Model (EBM) is a probabilistic model that uses a scalar energy function to describe the dependencies of the model variables [28, 31, 33, 34, 40, 41, 64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' An EBM can be formalised as 𝐹 : X × Y → R, where 𝐹 (x, y) stands for the relationship between the (x, y) pairings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Given an energy function and the input x, the best fitting value of y is computed with the following inference procedure: ˜y = 𝑎𝑟𝑔𝑚𝑖𝑛y{𝐹 (x, {y0, · · · , y𝑁 })} (47) Energy-based models provide fully generative models that can be used as an alternative to probabilistic estimation for prediction, classification, or decision-making tasks [29, 40, 41, 64, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The energy function 𝐸(x, y) ≡ 𝐹 (x, y) can be explicitly defined for all the values of y ∈ Y = {y0, · · · , y𝑁 } if and only if the size of the set Y is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In contrast, when the space of Y is sufficiently large, a specific strategy, known as the inference procedure, must be employed to find the y that minimizes 𝐸(x, {y0, · · · , y𝑁 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In many real situations, the inference procedure can produce an approximate result, which may or may not be the global minimum of 𝐸(x, {y0, · · · , y𝑁 }) for a given x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Moreover, it is possible that 𝐸(x, {y0, · · · , y𝑁 }) has several equivalent minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The best inference procedure to use often de- pends on the internal structure of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' For example, if Y is continuous and 𝐸(x, {y0, · · · , y𝑁 }) is smooth and differentiable everywhere concerning y, a gradient-based optimization algorithm can be employed [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In general, any probability density function 𝑝(x) for x ∈ R𝐷 can be rewritten as an EBM: 𝑝𝜽 (x) = exp(−𝐸𝜽 (x)) ∫ x′ exp(−𝐸𝜽 (x′))𝑑x′, (48) where the energy function (𝐸𝜽) can be any function parameterised by 𝜽 ∈ Θ (such as a neural network).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In these models, a prediction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' finding 𝑝(x0|x1, x2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=')) is done by fixing the values of the conditional variables, and estimating the remaining variables, (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' x0), by minimizing the energy function [106, 108, 123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' An EBM is trained by finding an energy function that associates low energies to values of x drawn from the underlying data distribution, 𝑝𝜃 (x) ∼ 𝑝𝐷 (x), and high energies for values of x not close to the underlying distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Training Given the aforementioned conceptual framework, the training can be thought as finding the model parameters that define the good match between the output (y ∈ Y) and the input (x ∈ X) for every step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This is done by estimating the best energy function from the set of energy functions (E) by scanning all the model parameters Θ [62, 103], where E = {𝐹 (Θ, X, Y) : Θ ∈ W = Θ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A loss function should behave similarly to the energy function described in Equation 47, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', lower 21 energy for a correct answer must be modeled by low losses, instead, higher energy, to all incorrect answers, by a higher loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Considering a set of training samples S = {(x𝑖, y𝑖) : 𝑖 = 1, · · · , 𝑁 }, during the training procedure, the loss function should have the effect of pushing-down 𝐸(Θ, y𝑖, x𝑖) and pulling-up 𝐸(Θ, ˜y𝑖, x𝑖), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', finding the parameters Θ that minimize the loss: Θ∗ = 𝑚𝑖𝑛Θ∈W𝐿𝑒𝑏𝑚(Θ, S) (49) The general form of the loss function L𝑒𝑏𝑚 is defined as: L𝑒𝑏𝑚(Θ, S) = 1 𝑁 𝑁 ∑︁ 𝑖=1 𝐿𝑒𝑏𝑚(y𝑖, 𝐸(Θ, Y, x𝑖)) + regularizer term (50) Where: 𝐿𝑒𝑏𝑚(y𝑖, 𝐸(Θ, Y, x𝑖)) is the per-sample loss y𝑖 is the desired output 𝐸(Θ, Y, x𝑖)) is energy surface for a given x𝑖 as y ∈ Y varies This is an average of a per-sample loss functional, denoted 𝐿𝑒𝑏𝑚(y𝑖, 𝐸(Θ, Y, x𝑖)), over the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This function depends on the desired output y𝑖 and on the energies derived by holding the input sample x𝑖 constant and changing the output scanning over the sample Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' With this definition, the loss remains unchanged when the training samples are mixed up and when the training set is repeated numerous times [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' As the size of the training set grows, the model is less likely to overfit [114].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Loss Functions for EBMs Following Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6, in this section, we introduce the energy loss first since it is the most straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Then we present common losses in machine learning that can be adapted to the energy based models, such as the Negative Log-Likelihood loss, the Hinge loss, and the Log loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Subsequently, we introduce more sophisticated losses like the Generalized Perceptron loss and the Generalized Margin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Finally, the Minimum classification error loss, the Square-square loss, and its variation Square-exponential loss are presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Energy loss Generalized Perceptron Generalized Margin Log loss Minimum classification error Square square Square exponential Negative Log-Likelihood Hinge Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Schematic overview of the energy based losses with their connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1 Energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The so-called energy loss is the most straightforward loss due to its simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' It can be simply defined by using the energy function as the per-sample loss: 𝐿𝑒𝑛𝑒𝑟𝑔𝑦(y𝑖, 𝐸(Θ, Y, x𝑖)) = 𝐸(Θ, x𝑖, y𝑖) (51) This loss is often used in regression tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Accordingly to its definition, it pulls the energy function down for values that are close to the correct data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' However, the energy 22 function is not pulled up for incorrect values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The assumption is that, by lowering the energy function in the correct location, the energy for incorrect values is left higher as a result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Due to this assumption, the training is sensitive to the model design and may result in energy collapse, leading to a largely flat energy function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2 Generalized Perceptron loss (L-CONT, CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Generalized Perceptron loss is defined as: 𝐿𝑝𝑒𝑟𝑐𝑒𝑝𝑡𝑟𝑜𝑛(y𝑖, 𝐸(Θ, Y, x𝑖)) = 𝐸(Θ, y𝑖, x𝑖) − [min y∈Y]{𝐸(Θ, {y0, · · · , y𝑁 }, x𝑖)} (52) This loss is positive definite as the second term is the lower bound of the first one, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', 𝐸(Θ, y𝑖, x𝑖) − [𝑚𝑖𝑛y∈Y]{𝐸(Θ, {y0, · · · , y𝑁 }, x𝑖)} ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' By minimizing this loss, the first term is pushed down and the energy of the model prediction is raised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Although it is widely used [23, 63], this loss is suboptimal as it does not detect the gap between the correct output and the incorrect ones, and it doesn’t restrict the function from assigning the same value to each wrong output y𝑖 and it may produce flat energy distributions [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3 Negative Log-Likelihood loss (CONT,DIFF,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In analogy with the description in Sec- tion 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1, the Negative Log-Likelihood loss (NLL) in the energy-based context is defined as: L𝑁𝐿𝐿(Θ, S) = 1 𝑁 𝑁 ∑︁ 𝑖=1 � 𝐸(Θ, y𝑖, x𝑖) + 1 𝛽 log ∫ y∈Y 𝑒𝛽𝐸(Θ,y,x𝑖) � (53) where S is the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss reduces to the perceptron loss when 𝛽 → ∞ and to the log loss in case Y has only two labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=', binary classification).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Since the integral above is intractable, considerable efforts have been devoted to finding approximation methods, including Monte-Carlo sampling methods [96], and variational methods [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' While these methods have been devised as approximate ways of minimizing the NLL loss, they can be viewed in the energy-based framework as different strategies for choosing the y’s whose energies will be pulled up [64, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The NLL is also known as the cross- entropy [67] loss and is widely used in many applications, including energy-based models [9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss function formulation is subject to the same limitations listed in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='4 Generalized Margin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The generalized margin loss is a more reliable version of the generalized perceptron loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The general form of the generalized margin loss in the context of energy-based training is defined as: 𝐿𝑚𝑎𝑟𝑔𝑖𝑛(Θ, x𝑖, y𝑖) = 𝑄𝑚(𝐸(Θ, x𝑖, y𝑖), 𝐸(Θ, x𝑖, ¯y𝑖)) (54) Where ¯y𝑖 is the so-called "most-offending incorrect output" which is the output that has the lowest energy among all possible outputs that are incorrect [64], 𝑚 is a positive margin parameter, and 𝑄𝑚 is a convex function which ensures that the loss receives low values for 𝐸(Θ, x𝑖, y𝑖) and high values for 𝐸(Θ, x𝑖, ¯y𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In other words, the loss function can ensure that the energy of the most offending incorrect output is greater by some arbitrary margin than the energy of the correct output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss function is written in the general form and a wide variety of losses that use specific margin function 𝑄𝑚 to produce a gap between the correct output and the wrong output are formalised in the following part of the section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Hinge loss (L-CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Already explained in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2, the hinge loss can be rewritten as: 𝐿ℎ𝑖𝑛𝑔𝑒 (Θ, x𝑖, y𝑖) = 𝑚𝑎𝑥(0,𝑚 + 𝐸(Θ, x𝑖, y𝑖) − 𝐸(Θ, x𝑖, ¯y𝑖)) (55) This loss enforces that the difference between the correct answer and the most offending incorrect answer be at least 𝑚 [1, 107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Individual energies are not required to take a specific value because 23 the hinge loss depends on energy differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss function shares limitations with the original Hinge loss defined in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Log loss (DIFF,CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss is similar to the hinge loss, but it sets a softer margin between the correct output and the most offending outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The log loss is defined as: 𝐿𝑙𝑜𝑔(Θ, x𝑖, y𝑖) = log(1 + 𝑒𝐸(Θ,x𝑖,y𝑖)−𝐸(Θ,x𝑖,¯y𝑖)) (56) This loss is also called soft hinge and it may produce overfitting on high dimensional datasets [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Minimum classification error loss (CONT, DIFF, CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A straightforward function that roughly counts the total number of classification errors while being smooth and differentiable is known as the Minimum Classification Error (MCE) loss [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The MCE is written as a sigmoid function: 𝐿𝑚𝑐𝑒 (Θ, x𝑖, y𝑖) = 𝜎(𝐸(Θ, x𝑖, y𝑖) − 𝐸(Θ, x𝑖, ¯y𝑖)) (57) Where 𝜎 is defined as 𝜎(𝑥) = (1+𝑒−𝑥)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' While this function lacks an explicit margin, it nevertheless produces an energy difference between the most offending incorrect output and the correct output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Square-square loss (CONT,CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Square-square loss deals differently with the energy of the correct output 𝐸(Θ, x𝑖, y𝑖) and the energy of the most offensive output 𝐸(Θ, x𝑖, ¯y𝑖) as: 𝐿𝑠𝑞−𝑠𝑞(Θ, x𝑖, y𝑖) = 𝐸(Θ, x𝑖, y𝑖)2 + (max(0,𝑚 − 𝐸(Θ, x𝑖, ¯y𝑖)))2 (58) The combination aims to minimize the energy of the correct output while enforcing a margin of at least 𝑚 on the most offending incorrect outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss is a modified version of the margin loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss can be only used when there is a lower bound on the energy function [42, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Square-exponential loss (CONT, DIFF, CONV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss is similar to the square-square loss function, and it only differs in the second term: 𝐿𝑠𝑞−𝑒𝑥𝑝 (Θ, x𝑖, y𝑖) = 𝐸(Θ, x𝑖, y𝑖)2 + 𝛾𝑒−𝐸(Θ,x𝑖,¯y𝑖) (59) While 𝛾 is a positive constant, the combination aims to minimize the energy of the correct output while pushing the energy of the most offending incorrect output to an infinite margin [21, 65, 82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss is considered a regularized version of the aforementioned square-square loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This loss, as for the Square-square loss, can be only used when there is a lower bound on the energy function [42, 65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 9 CONCLUSION The definition of an appropriate loss function is a critical part of solving many machine learning problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In this survey we have described 33 of the most commonly used loss functions from across the machine learning literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' These functions are appropriate for solving a wide range of problems, including classification, regression, sample generation and ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Overall, we made an effort to provide a useful resource for newcomers to the machine learning literature and advanced practitioners alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Each of the loss functions we describe have also been put into context and compared in a novel taxonomy, introduced in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' This gives practitioners a quick view of the different techniques and how they fit together, allowing them to decide and choose the most appropriate approach for their technique quickly and efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' We also hope this framing helps researchers to contextualise newly developed loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 24 REFERENCES [1] Yasemin Altun, Ioannis Tsochantaridis, and Thomas Hofmann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Hidden markov support vector machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 20th international conference on machine learning (ICML-03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [2] Jinwon An and Sungzoon Cho.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Variational autoencoder based anomaly detection using reconstruction probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Special Lecture on IE 2, 1 (2015), 1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [3] Brenna D Argall, Sonia Chernova, Manuela Veloso, and Brett Browning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A survey of robot learning from demonstration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Robotics and autonomous systems 57, 5 (2009), 469–483.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [4] Martin Arjovsky, Soumith Chintala, and Léon Bottou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Wasserstein GAN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 07875 [5] Peter Bartlett, Stéphane Boucheron, and Gábor Lugosi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Model Selection and Error Estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine Learning 48 (01 2002), 85–113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1023/A:1013999503812 [6] Peter L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Bartlett, Michael I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Jordan, and Jon D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' McAuliffe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Convexity, Classification, and Risk Bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 101 (2006), 138 – 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [7] Sebahattin Bektaş and Yasemin Şişman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The comparison of L1 and L2-norm minimization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' International Journal of the Physical Sciences 5, 11 (2010), 1721–1727.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [8] Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Gradient-based optimization of hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Neural computation 12, 8 (2000), 1889–1900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [9] Yoshua Bengio, Renato De Mori, Giovanni Flammia, and Ralf Kompe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Global optimization of a neural network- hidden Markov model hybrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE transactions on Neural Networks 3, 2 (1992), 252–259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [10] Yoshua Bengio, Réjean Ducharme, and Pascal Vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A neural probabilistic language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in neural information processing systems 13 (2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [11] Christopher M Bishop et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Neural networks for pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Oxford university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [12] Christopher M Bishop and Nasser M Nasrabadi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Pattern recognition and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [13] Gianluca Bontempi, Souhaib Ben Taieb, and Yann-Aël Le Borgne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning strategies for time series forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In European business intelligence summer school.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer, 62–77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [14] Bernhard E Boser, Isabelle M Guyon, and Vladimir N Vapnik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A training algorithm for optimal margin classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 5th Annual ACM Workshop on Computational Learning Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 144–152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [15] Jane Bromley, Isabelle Guyon, Yann LeCun, Eduard Säckinger, and Roopak Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1993.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Signature verification using a" siamese" time delay neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in neural information processing systems 6 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [16] Zhe Cao, Tao Qin, Tie-Yan Liu, Ming-Feng Tsai, and Hang Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Learning to rank: from pairwise approach to listwise approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 24th international conference on Machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 129–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [17] Varun Chandola, Arindam Banerjee, and Vipin Kumar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Anomaly detection: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ACM computing surveys (CSUR) 41, 3 (2009), 1–58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [18] Gal Chechik, Varun Sharma, Uri Shalit, and Samy Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Large Scale Online Learning of Image Similarity Through Ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Machine Learning Research 11, 3 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [19] Pin-Yu Chen, Huan Zhang, Yash Sharma, Jinfeng Yi, and Cho-Jui Hsieh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Zoo: Zeroth order optimization based black-box attacks to deep neural networks without training substitute models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 10th ACM workshop on artificial intelligence and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 15–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [20] Wai-Ki Ching and Michael K Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Models, algorithms and applications (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [21] Sumit Chopra, Raia Hadsell, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Learning a similarity metric discriminatively, with application to face verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In 2005 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’05), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE, 539–546.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [22] KR1442 Chowdhary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Fundamentals of artificial intelligence (2020), 603–649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [23] Michael Collins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Discriminative training methods for hidden markov models: Theory and experiments with perceptron algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 2002 conference on empirical methods in natural language processing (EMNLP 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1–8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [24] Andrew R Conn, Katya Scheinberg, and Luis N Vicente.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Introduction to derivative-free optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' SIAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [25] Prafulla Dhariwal and Alexander Nichol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Diffusion models beat gans on image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 34 (2021), 8780–8794.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [26] Amit Dhurandhar, Tejaswini Pedapati, Avinash Balakrishnan, Pin-Yu Chen, Karthikeyan Shanmugam, and Ruchir Puri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Model agnostic contrastive explanations for structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='00117 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [27] Laurent Dinh, Jascha Sohl-Dickstein, and Samy Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Density estimation using real nvp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1605.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='08803 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [28] Yilun Du, Toru Lin, and Igor Mordatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Model Based Planning with Energy Based Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ArXiv abs/1909.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='06878 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [29] Yilun Du and Igor Mordatch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Implicit generation and modeling with energy based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 25 [30] Bradley Efron and Trevor Hastie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Computer Age Statistical Inference: Algorithms, Evidence, and Data Science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Cambridge University Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1017/CBO9781316576533 [31] Chelsea Finn, Paul Christiano, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A connection between generative adversarial networks, inverse reinforcement learning, and energy-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='03852 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [32] William J Frawley, Gregory Piatetsky-Shapiro, and Christopher J Matheus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1992.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Knowledge discovery in databases: An overview.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' AI magazine 13, 3 (1992), 57–57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [33] Karl Friston.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The free-energy principle: a rough guide to the brain?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Trends in Cognitive Sciences 13, 7 (2022/08/04 2009), 293–301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='005 [34] Karl Friston, James Kilner, and Lee Harrison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A free energy principle for the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Physiology-Paris 100, 1 (2006), 70–87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='jphysparis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='001 Theoretical and Computational Neuroscience: Understanding Brain Functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [35] Claudio Gentile and Manfred K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' K Warmuth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Linear Hinge Loss and Average Margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Kearns, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Solla, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Cohn (Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' MIT Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' neurips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='cc/paper/1998/file/a14ac55a4f27472c5d894ec1c3c743d2-Paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='pdf [36] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ACM 63, 11 (2020), 139–144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [37] Ian J Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2661 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [38] Ian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Generative Adversarial Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2661 [39] Henry Gouk, Eibe Frank, Bernhard Pfahringer, and Michael J Cree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Regularisation of neural networks by enforcing lipschitz continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine Learning 110, 2 (2021), 393–416.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [40] Will Grathwohl, Kuan-Chieh Wang, Jörn-Henrik Jacobsen, David Duvenaud, Mohammad Norouzi, and Kevin Swersky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Your classifier is secretly an energy based model and you should treat it like one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='03263 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [41] Tuomas Haarnoja, Haoran Tang, Pieter Abbeel, and Sergey Levine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Reinforcement learning with deep energy- based policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' PMLR, 1352–1361.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [42] Raia Hadsell, Sumit Chopra, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Dimensionality reduction by learning an invariant mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR’06), Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE, 1735–1742.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [43] GM Harshvardhan, Mahendra Kumar Gourisaria, Manjusha Pandey, and Siddharth Swarup Rautaray.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A comprehensive survey and analysis of generative models in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Computer Science Review 38 (2020), 100285.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [44] Jonathan Ho, Ajay Jain, and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Denoising diffusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 33 (2020), 6840–6851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [45] Jonathan Ho, Chitwan Saharia, William Chan, David J Fleet, Mohammad Norouzi, and Tim Salimans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Cascaded Diffusion Models for High Fidelity Image Generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 23 (2022), 47–1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [46] Arthur E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Hoerl and Robert W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Kennard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Ridge Regression: Biased Estimation for Nonorthogonal Problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Technometrics 42, 1 (2000), 80–86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/stable/1271436 [47] Elad Hoffer and Nir Ailon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Deep metric learning using triplet network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International workshop on similarity- based pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer, 84–92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [48] Peter J Huber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1965.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A robust version of the probability ratio test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Annals of Mathematical Statistics (1965), 1753–1758.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [49] Shruti Jadon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A survey of loss functions for semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In 2020 IEEE Conference on Computational Intelligence in Bioinformatics and Computational Biology (CIBCB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE, 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [50] Wenxin Jiang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Process consistency for AdaBoost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Annals of Statistics 32, 1 (Feb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2004), 13–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1214/aos/1079120128 [51] Michael I Jordan, Zoubin Ghahramani, Tommi S Jaakkola, and Lawrence K Saul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' An introduction to variational methods for graphical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning 37, 2 (1999), 183–233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [52] Michael I Jordan and Tom M Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning: Trends, perspectives, and prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Science 349, 6245 (2015), 255–260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [53] James M Joyce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Kullback-leibler divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International encyclopedia of statistical science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer, 720–722.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [54] Biing-Hwang Juang, Wu Hou, and Chin-Hui Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Minimum classification error rate methods for speech recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Transactions on Speech and Audio processing 5, 3 (1997), 257–265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [55] Diederik P Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Auto-encoding variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='6114 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [56] Krzysztof C Kiwiel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Methods of descent for nondifferentiable optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [57] David G Kleinbaum, K Dietz, M Gail, Mitchel Klein, and Mitchell Klein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Logistic regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 26 [58] Gregory Koch, Richard Zemel, Ruslan Salakhutdinov, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Siamese neural networks for one-shot image recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In ICML deep learning workshop, Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Lille, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [59] Konstantina Kourou, Themis P Exarchos, Konstantinos P Exarchos, Michalis V Karamouzis, and Dimitrios I Fotiadis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning applications in cancer prognosis and prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Computational and structural biotechnology journal 13 (2015), 8–17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [60] T Krishnaiah, S Srinivasa Rao, K Madhumurthy, and KS Reddy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Neural network approach for modelling global solar radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Applied Sciences Research 3, 10 (2007), 1105–1111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [61] Jan Kukačka, Vladimir Golkov, and Daniel Cremers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='48550/ARXIV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1710.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='10686 [62] Rithesh Kumar, Sherjil Ozair, Anirudh Goyal, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Maximum entropy generators for energy-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1901.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='08508 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [63] Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Gradient-based learning applied to document recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE 86, 11 (1998), 2278–2324.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [64] Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, and F Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A tutorial on energy-based learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Predicting structured data 1, 0 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [65] Yann LeCun and Fu Jie Huang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Loss functions for discriminative training of energy-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International workshop on artificial intelligence and statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' PMLR, 206–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [66] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Lee, Yoonkyung Lee, Yoonkyung Lee, Yi Lin, Yi Lin, Grace Wahba, and Grace Wahba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Multicategory Support Vector Machines, Theory, and Application to the Classification of Microarray Data and Satellite Radiance Data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [67] Esther Levin and Michael Fleisher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1988.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Accelerated learning in layered neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Complex systems 2, 625-640 (1988), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [68] Gong Li and Jing Shi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' On comparing three artificial neural networks for wind speed forecasting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Applied Energy 87, 7 (2010), 2313–2320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [69] Yuncheng Li, Yale Song, and Jiebo Luo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Improving pairwise ranking for multi-label image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3617–3625.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [70] Sijia Liu, Pin-Yu Chen, Bhavya Kailkhura, Gaoyuan Zhang, Alfred O Hero III, and Pramod K Varshney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A primer on zeroth-order optimization in signal processing and machine learning: Principals, recent advances, and applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Signal Processing Magazine 37, 5 (2020), 43–54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [71] Dengsheng Lu and Qihao Weng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A survey of image classification methods and techniques for improving classification performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' International journal of Remote sensing 28, 5 (2007), 823–870.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [72] Gábor Lugosi and Nicolas Vayatis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' On the Bayes-risk consistency of regularized boosting methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Annals of Statistics 32 (2003), 30–55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [73] Batta Mahesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning algorithms-a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' International Journal of Science and Research (IJSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [Internet] 9 (2020), 381–386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [74] Shie Mannor, Ron Meir, and Tong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Greedy Algorithms for Classification – Consistency, Convergence Rates, and Adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Mach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 4 (2003), 713–741.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [75] Ajay Mathur and Giles M Foody.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Multiclass and binary SVM classification: Implications for training and classification users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Geoscience and remote sensing letters 5, 2 (2008), 241–245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [76] T Mitchell, B Buchanan, G DeJong, T Dietterich, P Rosenbloom, and A Waibel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1990.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Annual Review of Computer Science 4, 1 (1990), 417–433.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1146/annurev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='04.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='060190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='002221 [77] Tom M Mitchell and Tom M Mitchell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' McGraw-hill New York.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [78] Klaus-Robert Müller, Matthias Krauledat, Guido Dornhege, Gabriel Curio, and Benjamin Blankertz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning techniques for brain-computer interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Tech 49, 1 (2004), 11–22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [79] In Jae Myung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The Importance of Complexity in Model Selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Mathematical Psychology 44, 1 (2000), 190–204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1006/jmps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1283 [80] In Jae Myung.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Tutorial on maximum likelihood estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of mathematical Psychology 47, 1 (2003), 90–100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [81] Andrew Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Ng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Feature Selection, L1 vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' L2 Regularization, and Rotational Invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the Twenty-First International Conference on Machine Learning (Banff, Alberta, Canada) (ICML ’04).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Association for Computing Machinery, New York, NY, USA, 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1145/1015330.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1015435 [82] Margarita Osadchy, Matthew Miller, and Yann Cun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Synergistic face detection and pose estimation with energy-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in neural information processing systems 17 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [83] Daniel W Otter, Julian R Medina, and Jugal K Kalita.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A survey of the usages of deep learning for natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE transactions on neural networks and learning systems 32, 2 (2020), 604–624.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [84] Shashikant Patil, Kalpesh R Patil, Chandragouda R Patil, and Smit S Patil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Performance overview of an artificial intelligence in biomedics: a systematic approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' International Journal of Information Technology 12, 3 (2020), 963–973.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [85] Patricia Pauli, Anne Koch, Julian Berberich, Paul Kohler, and Frank Allgöwer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Training robust neural networks using Lipschitz bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Control Systems Letters 6 (2021), 121–126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 27 [86] Claudia Perlich, Brian Dalessandro, Troy Raeder, Ori Stitelman, and Foster Provost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning for targeted display advertising: Transfer learning in action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning 95, 1 (2014), 103–127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [87] Yunchen Pu, Zhe Gan, Ricardo Henao, Xin Yuan, Chunyuan Li, Andrew Stevens, and Lawrence Carin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Variational autoencoder for deep learning of images, labels and captions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in neural information processing systems 29 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [88] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Ross Quinlan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1986.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Induction of decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Machine learning 1, 1 (1986), 81–106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [89] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Hierarchical text-conditional image generation with clip latents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='06125 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [90] Jason D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Rennie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Smooth Hinge Classication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [91] Luis Miguel Rios and Nikolaos V Sahinidis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Derivative-free optimization: a review of algorithms and comparison of software implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Global Optimization 56, 3 (2013), 1247–1293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [92] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Björn Ommer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' High-resolution image synthesis with latent diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 10684–10695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [93] Frank Rosenblatt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1958.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The perceptron: a probabilistic model for information storage and organization in the brain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Psychological review 65, 6 (1958), 386.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [94] Chitwan Saharia, William Chan, Saurabh Saxena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Photorealistic Text-to- Image diffusion models with deep language understanding (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' URL https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' org/abs/2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='11487 ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [95] HR Shally and Rejimol Robinson RR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Survey for Mining Biomedic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' ([n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=']).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [96] Alexander Shapiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Monte Carlo sampling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Handbooks in operations research and management science 10 (2003), 353–425.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [97] Kamran Shaukat, Suhuai Luo, Vijay Varadharajan, Ibrahim A Hameed, and Min Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A survey on machine learning techniques for cyber security in the last decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Access 8 (2020), 222310–222354.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [98] Pramila P Shinde and Seema Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A review of machine learning and deep learning applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In 2018 Fourth international conference on computing communication control and automation (ICCUBEA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE, 1–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [99] Naum Zuselevich Shor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Minimization methods for non-differentiable functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer Science & Business Media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [100] Jascha Sohl-Dickstein, Eric Weiss, Niru Maheswaranathan, and Surya Ganguli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Deep unsupervised learning using nonequilibrium thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International Conference on Machine Learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' PMLR, 2256–2265.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [101] Yang Song and Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Generative modeling by estimating gradients of the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 32 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [102] Yang Song and Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Improved techniques for training score-based generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in neural information processing systems 33 (2020), 12438–12448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [103] Yang Song and Diederik P Kingma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' How to train your energy-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='03288 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [104] Jan Stanczuk, Christian Etmann, Lisa Maria Kreusser, and Carola-Bibiane Schönlieb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Wasserstein GANs work because they fail (to approximate the Wasserstein distance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:2103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='01678 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [105] I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Steinwart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Consistency of support vector machines and other regularized kernel classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Transactions on Information Theory 51, 1 (2005), 128–142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1109/TIT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='839514 [106] Kevin Swersky, Marc’Aurelio Ranzato, David Buchman, Nando D Freitas, and Benjamin M Marlin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' On autoen- coders and score matching for energy based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 28th international conference on machine learning (ICML-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1201–1208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [107] Ben Taskar, Carlos Guestrin, and Daphne Koller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Max-margin Markov networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in neural information processing systems 16 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [108] Yee Whye Teh, Max Welling, Simon Osindero, and Geoffrey E Hinton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Energy-based models for sparse overcomplete representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Machine Learning Research 4, Dec (2003), 1235–1260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [109] Robert Tibshirani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1996.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Regression Shrinkage and Selection via the Lasso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of the Royal Statistical Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Series B (Methodological) 58, 1 (1996), 267–288.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='jstor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/stable/2346178 [110] Yoshimasa Tsuruoka, Jun’ichi Tsujii, and Sophia Ananiadou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Stochastic Gradient Descent Training for L1- regularized Log-linear Models with Cumulative Penalty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 477–485.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='3115/1687878.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='1687946 [111] Fath U Min Ullah, Amin Ullah, Ijaz Ul Haq, Seungmin Rho, and Sung Wook Baik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Short-term prediction of residential power energy consumption via CNN and multi-layer bi-directional LSTM networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' IEEE Access 8 (2019), 123369–123380.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [112] Mohammad Valipour, Mohammad Ebrahim Banihabib, and Seyyed Mahmood Reza Behbahani.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Comparison of the ARMA, ARIMA, and the autoregressive artificial neural network models in forecasting the monthly inflow of Dez dam reservoir.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of hydrology 476 (2013), 433–441.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 28 [113] Aaron Van Oord, Nal Kalchbrenner, and Koray Kavukcuoglu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Pixel recurrent neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' PMLR, 1747–1756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [114] Vladimir Vapnik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1999.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' The nature of statistical learning theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Springer science & business media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [115] Aladin Virmaux and Kevin Scaman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Lipschitz regularity of deep neural networks: analysis and efficient estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advances in Neural Information Processing Systems 31 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [116] John Von Neumann and Oskar Morgenstern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Theory of games and economic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Princeton university press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [117] Jun Wang, Suncheng Feng, Yong Cheng, and Najla Al-Nabhan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Survey on the Loss Function of Deep Learning in Face Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Journal of Information Hiding and Privacy Protection 3, 1 (2021), 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [118] Jiang Wang, Yang Song, Thomas Leung, Chuck Rosenberg, Jingbin Wang, James Philbin, Bo Chen, and Ying Wu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Learning fine-grained image similarity with deep ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the IEEE conference on computer vision and pattern recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 1386–1393.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [119] Qi Wang, Yue Ma, Kun Zhao, and Yingjie Tian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A comprehensive survey of loss functions in machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Annals of Data Science (2020), 1–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [120] Max Welling and Yee W Teh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Bayesian learning via stochastic gradient Langevin dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In Proceedings of the 28th international conference on machine learning (ICML-11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Citeseer, 681–688.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [121] Cort J Willmott and Kenji Matsuura.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Advantages of the mean absolute error (MAE) over the root mean square error (RMSE) in assessing average model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Climate research 30, 1 (2005), 79–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [122] Yichao Wu and Yufeng Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Robust Truncated Hinge Loss Support Vector Machines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Statist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Assoc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 102 (2007), 974 – 983.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [123] Shuangfei Zhai, Yu Cheng, Weining Lu, and Zhongfei Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Deep structured energy based models for anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In International conference on machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' PMLR, 1100–1109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [124] Tong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Statistical Behavior and Consistency of Classification Methods based on Convex Risk Minimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [125] Tong Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' Solving Large Scale Linear Prediction Problems Using Stochastic Gradient Descent Algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' In ICML 2004: PROCEEDINGS OF THE TWENTY-FIRST INTERNATIONAL CONFERENCE ON MACHINE LEARNING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' OMNIPRESS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 919–926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' [126] Xiang Zhang, Lina Yao, Xianzhi Wang, Jessica Monaghan, David Mcalpine, and Yu Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' A survey on deep learning based brain computer interface: Recent advances and new frontiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' arXiv preprint arXiv:1905.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content='04149 66 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/QdE5T4oBgHgl3EQfZQ86/content/2301.05579v1.pdf'} diff --git a/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf b/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..68cacbf7584b8343c27039ed2b9d18169a55a0c8 --- /dev/null +++ b/T9FIT4oBgHgl3EQffis3/content/2301.11279v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bd4937ff78bac3b40e49029e7ed7323940299c02051a587651fa5c2219377369 +size 2382996 diff --git a/T9FIT4oBgHgl3EQffis3/vector_store/index.faiss b/T9FIT4oBgHgl3EQffis3/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..a331fdbd1f1a3a265f33f5823469112b28da795c --- /dev/null +++ b/T9FIT4oBgHgl3EQffis3/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:fcd5dc4a2469101f8472acba08758353522a1366a779ae30e5115f34d8220c4b +size 3145773 diff --git a/T9FIT4oBgHgl3EQffis3/vector_store/index.pkl b/T9FIT4oBgHgl3EQffis3/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..753ccf8ad13983ec3f889c21ad13483906480d54 --- /dev/null +++ b/T9FIT4oBgHgl3EQffis3/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a1cc67ef79cc3e4ec9531eb5b2170f92da16209a6f17b8cc07723f5ebdeb1639 +size 122071 diff --git a/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf b/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..84f965d894d7296ceeba1bcbcf3a571c517492a2 --- /dev/null +++ b/UNAyT4oBgHgl3EQfVvd8/content/2301.00149v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:bd072946f7f33c6ef430ac08edd24604ddd115db1ba47b01fad6ce8115786bdf +size 12505297 diff --git a/UNAyT4oBgHgl3EQfVvd8/vector_store/index.faiss b/UNAyT4oBgHgl3EQfVvd8/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5de9d8b1c233d32b01d6b00fd402b0fc7bcb6214 --- /dev/null +++ b/UNAyT4oBgHgl3EQfVvd8/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:74262adb646e652b12ec3d2c6182f654682358572010feeb9b39a1e4fcb642a7 +size 8912941 diff --git a/UNAyT4oBgHgl3EQfVvd8/vector_store/index.pkl b/UNAyT4oBgHgl3EQfVvd8/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..e4ae9fdd8b0400e7a6486b6813b545964a4cc5e1 --- /dev/null +++ b/UNAyT4oBgHgl3EQfVvd8/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:49257a06e9f56297aa0efa4802199c47279fb88fb55fea7395861a0d3ef99d37 +size 269023 diff --git a/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf b/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..82057aea237fe4466d97415ea564fb9f82facb49 --- /dev/null +++ b/W9E1T4oBgHgl3EQfvgXD/content/2301.03401v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:1e4551b218d2e2e0dcd86bf46e3480dcbcbc19d1f9fc4cde0368380f6b57d32f +size 1166419 diff --git a/W9E1T4oBgHgl3EQfvgXD/vector_store/index.faiss b/W9E1T4oBgHgl3EQfvgXD/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..60b77a40015edc53cdb53742bca6c61240592c76 --- /dev/null +++ b/W9E1T4oBgHgl3EQfvgXD/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:259c486eedc97be191a3a2e1f2185ee274a42c163c266d6e44dc230803c2ee70 +size 4522029 diff --git a/XNE0T4oBgHgl3EQf3QLX/vector_store/index.faiss b/XNE0T4oBgHgl3EQf3QLX/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..57836a2dd086c138381ebc91a419ea0bfd23b93a --- /dev/null +++ b/XNE0T4oBgHgl3EQf3QLX/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ac8d121330db0e52df60e32d92c7e7ffe882f1723c740a3db189681769e735e9 +size 4587565 diff --git a/ZNE3T4oBgHgl3EQfcgp3/content/tmp_files/2301.04526v1.pdf.txt b/ZNE3T4oBgHgl3EQfcgp3/content/tmp_files/2301.04526v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fd58156327ab4f28881af5e9c2b22ad793367eed --- /dev/null +++ b/ZNE3T4oBgHgl3EQfcgp3/content/tmp_files/2301.04526v1.pdf.txt @@ -0,0 +1,338 @@ +Sensitivity of DUNE to low energy physics searches +C. Cuesta𝑎,∗ on behalf of DUNE collaboration +𝑎Centro de Investigaciones Energé𝑡𝑖𝑐𝑎𝑠, 𝑀𝑒𝑑𝑖𝑜𝑎𝑚𝑏𝑖𝑒𝑛𝑡𝑎𝑙𝑒𝑠𝑦𝑇𝑒𝑐𝑛𝑜𝑙ógicas, CIEMAT, +28040, Madrid, Spain +E-mail: clara.cuesta@ciemat.es +The Deep Underground Neutrino Experiment (DUNE), a next-generation long-baseline neutrino +oscillation experiment, is a powerful tool to perform low energy physics searches. DUNE will be +uniquely sensitive to the electron-neutrino-flavour component of the burst of neutrinos expected +from the next Galactic core-collapse supernova, and also capable of detecting solar neutrinos. +DUNE will have four modules of 70-kton liquid argon mass in total, placed 1.5 km underground +at the Sanford Underground Research Facility in the USA. These modules are being designed +exploiting different liquid argon time projection chamber technologies and based on the physics +requirements that take into account the particularities of the low energy physics searches. +41st International Conference on High Energy physics - ICHEP2022 +July 6 - 13, 2022 +Bologna, Italy +∗Speaker +© Copyright owned by the author(s) under the terms of the Creative Commons +Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). +https://pos.sissa.it/ +arXiv:2301.04526v1 [hep-ex] 11 Jan 2023 + +Sensitivity of DUNE to low energy physics searches +C. Cuesta +1. +The Deep Underground Neutrino Experiment +The Deep Underground Neutrino Experiment (DUNE) is a next-generation long-baseline neu- +trino oscillation experiment with a primary physics goal of observing neutrino and antineutrino +oscillation patterns to precisely measure the parameters governing long-baseline neutrino oscillation +in a single experiment, and to test the three-flavor paradigm [1]. DUNE is also uniquely sensitive +to low energy neutrinos such as electron neutrinos from a galactic supernova burst (SNB) [2], and +to a broad range of physics beyond the Standard Model (BSM) [3], including nucleon decays. +DUNE will consist of a near detector placed at Fermilab close to the production point of the +muon neutrino beam of the Long-Baseline Neutrino Facility (LBNF), and four 17 kt liquid argon +time projection chambers (LArTPCs) as far detector (FD) in the Sanford Underground Research +Facility (SURF) at 4300 m.w.e. depth at 1300 km from Fermilab. +DUNE is anticipated to begin collecting physics data with Phase I, an initial experiment +configuration consisting of two FD modules and a minimal suite of near detector components, with +a 1.2 MW proton beam. The Phase II upgrades necessary to achieve DUNE’s physics goals are: +addition of FD modules three and four for a total FD fiducial mass of at least 40 kt, upgrade of the +proton beam power to 2.4 MW and of the near detector. +2. +The DUNE Far Detector +The DUNE FD will be sensitive to low energy neutrinos from astrophysical sources such as +a SNB and the Sun. The FD LArTPC technology provides good energy resolution, full particle +reconstruction with very high quality tracking, and energy thresholds as low as a few MeV may be +possible. +In these detectors, the ionization charge is drifted by an electric field towards the anode where +the charge is collected. Using the time arrival of the charge at the readout planes, a three-dimensional +track reconstruction is possible. Particles are identified by the rate of energy loss along the track. +The Ar scintillation light is also detected enabling fast timing of signals and event localization inside +the detector. +Different LAr technologies are being considered for the DUNE FD: the first module will employ +the single-phase horizontal-drift (HD) LArTPC technology [4], where the drift of 3.5 m is horizontal +with wrapped-wire readout including two induction and one charge collection anode planes [4]; and +the second module will employ the single-phase vertical-drift (VD) LArTPC technology where the +drift is vertical over 6 m. For low-energy signals, such as solar and SNB neutrinos, the VD design +provides opportunities to improve the physics performance, primarily due to improved photon +detection. +3. +Low energy events in DUNE +The few-MeV low energy regime is of particular interest for the detection of the burst of +neutrinos from a galactic core-collapse supernova, which has been the primary focus of DUNE low- +energy sensitivity studies, but DUNE will also have sensitivity to neutrinos from other astrophysical +sources, including solar neutrinos. +2 + +Sensitivity of DUNE to low energy physics searches +C. Cuesta +Among the different detection channels in LAr for the neutrino interactions, the dominant +interaction is the charged-current (CC) absorption of 𝜈𝑒 on 40𝐴𝑟, for which the observable are short +electron tracks plus deexcitation products from the excited 40𝐾∗ final state. Then, CC interactions of +𝜈𝑒 at MeV energies create short electron tracks in liquid argon, potentially accompanied by gamma +ray and other secondary particle signatures. Additional channels include a ¯𝜈𝑒 CC interaction and +electron scattering. Cross sections for the most relevant interactions are shown in Fig. 1. + Neutrino Energy (MeV) +10 +20 +30 +40 +50 +60 +70 +80 +90 +100 +) +2 + cm +-38 + Cross section (10 +-7 +10 +-6 +10 +-5 +10 +-4 +10 +-3 +10 +-2 +10 +-1 +10 +1 +10 +2 +10 + e +e +ν + e +e +ν + e +x +ν + e +x +ν +Ar +40 + +e +ν +Ar +40 + +e +ν + +Figure 1: Cross sections for supernova-relevant interactions in argon as a function of neutrino energy. +The predicted event rate from a supernova burst or the solar neutrinos may be calculated +by folding expected neutrino flux differential energy spectra with cross sections for the relevant +channels, and with detector response; this is done using SNOwGLoBES [5]. +We use the MARLEY (Model of Argon Reaction Low Energy Yields) generator [6] to simulate +tens-of-MeV neutrino-nucleus interactions in liquid argon. The LArSoft [7] Geant4-based software +package is used to simulate the final-state products from MARLEY in the DUNE LArTPC. SNB and +solar neutrino events due to their low energies manifest as spatially small events of few centimeters. +Understanding of cosmogenic and radiological backgrounds, dominated by 39Ar, is necessary, +although we expect a minor impact on reconstruction, the triggering efficiency for SNB neutrinos +could be affected. Background generation follows the BxDecay0 package1, a C++ library providing +simulated nuclear decays that is integrated into LArSoft. A radiological model is being developed +for the DUNE FD considering the HD and VD technologies. It includes radioactive decays in the +LAr bulk (39Ar, 42Ar, 85Kr, 222Rn and their decay chains), the cathode (drifted 42K from LArSoft’s +42Ar decays, 40K and the 238U decay chain), the charge readout planes (60Co and the 238U decay +chain), the photon detection system (222Rn decay chain), and external sources (gammas and neutrons +from surrounding rocks). +1https://github.com/BxCppDev/bxdecay0 +3 + +Sensitivity of DUNE to low energy physics searches +C. Cuesta + ergs/s) +52 +L (10 +e +ν +e +ν +x +ν +Infall +Neutronization Accretion +Cooling +0.1 +1 +10 + (MeV) +6 +8 +10 +12 +14 + Time (seconds) +2 +− +10 +1 +− +10 +1 +Alpha +2.5 +3 +3.5 +4 +4.5 +Figure 2: Expected time-dependent flux parameters for a specific model for an electron-capture supernova [8]. +No flavor transitions are assumed. The top plot shows the luminosity as a function of time, and the bottom +plot shows average neutrino energy. +4. +Supernova neutrinos in DUNE +Each supernova releases an intense source of neutrinos of all flavors. During a supernova +explosion, 99% of the gravitational binding energy of the star (∼ 1053 ergs) is released as neutrinos +and antineutrinos of all flavors, which play the role of astrophysical messengers, escaping from the +SN core. In the event of a galactic supernova explosion, DUNE data will probe the inner evolution +of the core-collapse mechanism by studying the time and energy spectra of neutrinos arriving at +DUNE. SNB neutrinos are emitted in a burst of a few tens of seconds duration [8]. Within DUNE, +three qualitative stages of the collapse can be distinguished, as shown in Figure ??: +1. The neutronization burst – a large pulse of 𝜈𝑒 emission takes place in the first 10’s of ms as +electrons capture on protons in the stellar core during the formation of a proto-neutron star. +A neutrino sphere forms around the proto-neutron star, inside which the density is so large +neutrinos become trapped at which point the neutronization burst of 𝜈𝑒 emission quenches. +2. The accretion phase – during accretion, lasting from tens to hundreds of ms, neutrino emission +is dominated by infall of gas from the outer layers of the progenitor onto the outer extant of +the proto-neutron star. +3. The cooling phase – after infall, the proto-neutron star cools over several seconds. The +neutrino opacity drops, allowing neutrinos to escape the core. While trapped within the core, +neutrino species thermalize so that there is nearly luminosity equipartition between neutrino +species during the cooling phase. +The predicted event rate from a SNB is calculated by folding together expected neutrino +differential energy spectra, cross sections for the relevant channels, and detector response using +SNOwGLoBES. Monte Carlo simulated events are generated using the time and energy of incident +4 + +Sensitivity of DUNE to low energy physics searches +C. Cuesta +neutrinos for a particular SN model using the MARLEY interaction model to simulate the 𝜈𝑒 CC +neutrino interaction and using the standard Geant4-based detector models to simulate the DUNE +far detector. Standard LArTPC algorithms are applied to reconstruct electron tracks. All visible +energy from the event is used to calculate the incident neutrino energy calorimetrically. Photon +detectors are typically used to determine the time of events, but DUNE is exploring how photon +detector calorimetry can expand its low-energy physics reach. +In DUNE, the trigger on a SNB can be done using either TPC or photon detection system +information. In both cases, the trigger scheme exploits the time coincidence of multiple signals +over a timescale matching the supernova luminosity evolution. A redundant and highly efficient +triggering scheme is under development. +A number of astrophysical phenomena associated withsupernovae areexpected to beobservable +in the supernova neutrino signal, providing a remarkable window into the event. In particular, the +supernova explosion mechanism, which in the current paradigm involves energy deposition into the +stellar envelope via neutrino interactions, is still not well understood, and the neutrinos themselves +will bring the insight needed to confirm or refute the paradigm. There are many other examples of +astrophysical observables, more details can be found in [2]. +Detecting neutrinos from a SNB, we can also learn a lot about neutrinos and particle physics. +A SNB can be thought as an extremely hermetic system, which can be used to search for new new +physics like Goldstone bosons, neutrino magnetic moments, "dark photons", "unparticles", extra- +dimensional gauge bosons, and sterile neutrinos. Also, self-interactions of neutrinos, neutrino +instability, and light gauge bosons can be studied. Such energy-loss-based analysis will make use +of two types of information. First, the total energy of the emitted neutrinos compared with the +expected release in the gravitational collapse. Second, the rate of cooling should be measured and +compared with what is expected from diffusion of the standard neutrinos. As DUNE is mostly +sensitive to 𝜈𝑒, complementary data of ¯𝜈𝑒 from water Cherenkov and scintillator experiments for +careful analysis of the flavor transition will be very useful. The flavor oscillation neutrino physics +and its signatures are a major part of the physics program in the different periods. +5. +Solar neutrinos in DUNE +Detection of solar and other low-energy neutrinos is challenging in a LArTPC because of +relatively high intrinsic detection energy thresholds for the charged-current interaction on argon of +about 5 MeV and because radioactive backgrounds in the same energy regime can affect triggering +capability. However, compared with other technologies, a LArTPC offers a large cross section +and unique potential channel-tagging signatures from deexcitation photons. Furthermore, observed +energy from the final state CC interaction is much more tightly correlated with the incident neutrino +energy on an event-by-event basis than the electron recoil spectrum from the ES channel that has been +used for past solar neutrino observations such as in Super-Kamiokande [9]. Due to this, DUNE will +make more precise spectral measurements. Though background rates are large, LArTPC detector +allows for background reduction using fiduacialization techniques and the solar neutrino event rate +is also substantial in the DUNE far detector, ∼100 per day, allowing samples of a few 105 events +after 10 years of data collection. Initial studies suggest DUNE could improve the measurement of +Δ𝑚2 +21 as well as observing the ℎ𝑒𝑝 and 8B solar neutrino flux, as shown in Figure ??. +5 + +Sensitivity of DUNE to low energy physics searches +C. Cuesta +Figure 3: Simulated solar neutrino spectrum with background for the DUNE Far Detector. +Detailed studies of solar neutrino detection capability are underway in DUNE along with sub- +sequent physics sensitivity studies. Similarly, DUNE can search for the Diffuse Supernova Neutrino +Background (DSNB) [10] at energies just above the endpoint of the solar neutrino spectrum. As +DUNE is primarily sensitive to the 𝜈𝑒 component, DUNE will be the only running experiment with +sensitivity to the neutrino component of the DSNB. +6. +Conclusions +The DUNE experiment will be sensitive to neutrinos with about 5 MeV up to several tens of +MeV, the regime of relevance for core-collapse supernova burst and solar neutrinos. This low-energy +regime presents particular challenges for triggering and reconstruction. The combined information +from DUNE’s TPC and PDS systems will provide a good reconstruction of these events. Software +tools that enable preliminary physics and astrophysics sensitivity studies have been developed. The +observation of a burst will also enable sensitivity to neutrino mass ordering, and potentially many +other topics. In addition, there is discovery potential for ℎ𝑒𝑝 neutrinos in DUNE and perform a +precision measurement of neutrino mixing and fluxes. +Acknowledgments +This project has received funding from the European Union Horizon 2020 Research and +Innovation programme under Grant no. 101004761, from Spanish Ministerio de Economia y Com- +petitividad (SEIDI-MINECO) under Grant no. FPA2016-77347-C2-1-P; and from the Comunidad +de Madrid. +6 + +DUNE Preliminary +10 +"BV。CC +107 +hep VeCC +Lm +10 +Neutron Capture +222 Rn +10 +42 Ar +400 +10 +10 +10 +10 +10° +0 +5 +10 +15 +20 +25 +30 +ReconstructedE(MeV)Sensitivity of DUNE to low energy physics searches +C. Cuesta +References +[1] DUNE collaboration, B. Abi et al., Long-baseline neutrino oscillation physics potential of +the DUNE experiment, Eur. Phys. J. C 80 (2020) 978. +[2] DUNE collaboration, B. Abi et al., Supernova Neutrino Burst Detection with the Deep +Underground Neutrino Experiment, Eur. Phys. J. C 81 (2021) 423, [2008.06647]. +[3] DUNE collaboration, B. Abi et al., Prospects for beyond the Standard Model physics +searches at the Deep Underground Neutrino Experiment, Eur. Phys. J. C 81 (2021) 322. +[4] DUNE collaboration, B. Abi et al., Deep Underground Neutrino Experiment (DUNE), Far +Detector Technical Design Report, Volume IV Far Detector Single-phase Technology, JINST +15 (2020) T08010, [2002.03010]. +[5] SNOwGLoBES, http://www.phy.duke.edu/ schol/snowglobes. +[6] S. Gardiner et al., MARLEY (Model of Argon Reaction Low Energy Yields), +http://www.marleygen.org. +[7] LArSoft, http://larsoft.org. +[8] L. Hudepohl, B. Muller, H.-T. Janka, A. Marek and G. Raffelt, Neutrino Signal of +Electron-Capture Supernovae from Core Collapse to Cooling, Phys. Rev. Lett. 104 (2010) +251101, [0912.0260]. +[9] Super-Kamiokande collaboration, K. Abe et al., Solar Neutrino Measurements in +Super-Kamiokande-IV, Phys. Rev. D 94 (2016) 052010, [1606.07538]. +[10] J. F. Beacom, The Diffuse Supernova Neutrino Background, Ann. Rev. Nucl. Part. Sci. 60 +(2010) 439–462, [1004.3311]. +7 + diff --git a/ZNE3T4oBgHgl3EQfcgp3/content/tmp_files/load_file.txt b/ZNE3T4oBgHgl3EQfcgp3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..fafd73ec1b032642bdf1902473fe931db50597cf --- /dev/null +++ b/ZNE3T4oBgHgl3EQfcgp3/content/tmp_files/load_file.txt @@ -0,0 +1,188 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf,len=187 +page_content='Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta𝑎,∗ on behalf of DUNE collaboration 𝑎Centro de Investigaciones Energé𝑡𝑖𝑐𝑎𝑠, 𝑀𝑒𝑑𝑖𝑜𝑎𝑚𝑏𝑖𝑒𝑛𝑡𝑎𝑙𝑒𝑠𝑦𝑇𝑒𝑐𝑛𝑜𝑙ógicas, CIEMAT, 28040, Madrid, Spain E-mail: clara.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='cuesta@ciemat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='es The Deep Underground Neutrino Experiment (DUNE), a next-generation long-baseline neutrino oscillation experiment, is a powerful tool to perform low energy physics searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' DUNE will be uniquely sensitive to the electron-neutrino-flavour component of the burst of neutrinos expected from the next Galactic core-collapse supernova, and also capable of detecting solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' DUNE will have four modules of 70-kton liquid argon mass in total, placed 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='5 km underground at the Sanford Underground Research Facility in the USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' These modules are being designed exploiting different liquid argon time projection chamber technologies and based on the physics requirements that take into account the particularities of the low energy physics searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 41st International Conference on High Energy physics - ICHEP2022 July 6 - 13, 2022 Bologna, Italy ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='04526v1 [hep-ex] 11 Jan 2023 Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The Deep Underground Neutrino Experiment The Deep Underground Neutrino Experiment (DUNE) is a next-generation long-baseline neu- trino oscillation experiment with a primary physics goal of observing neutrino and antineutrino oscillation patterns to precisely measure the parameters governing long-baseline neutrino oscillation in a single experiment, and to test the three-flavor paradigm [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' DUNE is also uniquely sensitive to low energy neutrinos such as electron neutrinos from a galactic supernova burst (SNB) [2], and to a broad range of physics beyond the Standard Model (BSM) [3], including nucleon decays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' DUNE will consist of a near detector placed at Fermilab close to the production point of the muon neutrino beam of the Long-Baseline Neutrino Facility (LBNF), and four 17 kt liquid argon time projection chambers (LArTPCs) as far detector (FD) in the Sanford Underground Research Facility (SURF) at 4300 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' depth at 1300 km from Fermilab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' DUNE is anticipated to begin collecting physics data with Phase I, an initial experiment configuration consisting of two FD modules and a minimal suite of near detector components, with a 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='2 MW proton beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The Phase II upgrades necessary to achieve DUNE’s physics goals are: addition of FD modules three and four for a total FD fiducial mass of at least 40 kt, upgrade of the proton beam power to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='4 MW and of the near detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The DUNE Far Detector The DUNE FD will be sensitive to low energy neutrinos from astrophysical sources such as a SNB and the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The FD LArTPC technology provides good energy resolution, full particle reconstruction with very high quality tracking, and energy thresholds as low as a few MeV may be possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' In these detectors, the ionization charge is drifted by an electric field towards the anode where the charge is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Using the time arrival of the charge at the readout planes, a three-dimensional track reconstruction is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Particles are identified by the rate of energy loss along the track.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The Ar scintillation light is also detected enabling fast timing of signals and event localization inside the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Different LAr technologies are being considered for the DUNE FD: the first module will employ the single-phase horizontal-drift (HD) LArTPC technology [4], where the drift of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='5 m is horizontal with wrapped-wire readout including two induction and one charge collection anode planes [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' and the second module will employ the single-phase vertical-drift (VD) LArTPC technology where the drift is vertical over 6 m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' For low-energy signals, such as solar and SNB neutrinos, the VD design provides opportunities to improve the physics performance, primarily due to improved photon detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Low energy events in DUNE The few-MeV low energy regime is of particular interest for the detection of the burst of neutrinos from a galactic core-collapse supernova, which has been the primary focus of DUNE low- energy sensitivity studies, but DUNE will also have sensitivity to neutrinos from other astrophysical sources, including solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 2 Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta Among the different detection channels in LAr for the neutrino interactions, the dominant interaction is the charged-current (CC) absorption of 𝜈𝑒 on 40𝐴𝑟, for which the observable are short electron tracks plus deexcitation products from the excited 40𝐾∗ final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Then, CC interactions of 𝜈𝑒 at MeV energies create short electron tracks in liquid argon, potentially accompanied by gamma ray and other secondary particle signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Additional channels include a ¯𝜈𝑒 CC interaction and electron scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cross sections for the most relevant interactions are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Neutrino Energy (MeV) 10 20 30 40 50 60 70 80 90 100 ) 2 cm 38 Cross section (10 7 10 6 10 5 10 4 10 3 10 2 10 1 10 1 10 2 10 e e ν e e ν e x ν e x ν Ar 40 e ν Ar 40 e ν Figure 1: Cross sections for supernova-relevant interactions in argon as a function of neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The predicted event rate from a supernova burst or the solar neutrinos may be calculated by folding expected neutrino flux differential energy spectra with cross sections for the relevant channels, and with detector response;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' this is done using SNOwGLoBES [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' We use the MARLEY (Model of Argon Reaction Low Energy Yields) generator [6] to simulate tens-of-MeV neutrino-nucleus interactions in liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The LArSoft [7] Geant4-based software package is used to simulate the final-state products from MARLEY in the DUNE LArTPC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' SNB and solar neutrino events due to their low energies manifest as spatially small events of few centimeters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Understanding of cosmogenic and radiological backgrounds, dominated by 39Ar, is necessary, although we expect a minor impact on reconstruction, the triggering efficiency for SNB neutrinos could be affected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Background generation follows the BxDecay0 package1, a C++ library providing simulated nuclear decays that is integrated into LArSoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' A radiological model is being developed for the DUNE FD considering the HD and VD technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' It includes radioactive decays in the LAr bulk (39Ar, 42Ar, 85Kr, 222Rn and their decay chains), the cathode (drifted 42K from LArSoft’s 42Ar decays, 40K and the 238U decay chain), the charge readout planes (60Co and the 238U decay chain), the photon detection system (222Rn decay chain), and external sources (gammas and neutrons from surrounding rocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 1https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='com/BxCppDev/bxdecay0 3 Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta ergs/s) 52 L (10 e ν e ν x ν Infall Neutronization Accretion Cooling 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='1 1 10 (MeV) 6 8 10 12 14 Time (seconds) 2 − 10 1 − 10 1 Alpha 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='5 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='5 Figure 2: Expected time-dependent flux parameters for a specific model for an electron-capture supernova [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' No flavor transitions are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The top plot shows the luminosity as a function of time, and the bottom plot shows average neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Supernova neutrinos in DUNE Each supernova releases an intense source of neutrinos of all flavors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' During a supernova explosion, 99% of the gravitational binding energy of the star (∼ 1053 ergs) is released as neutrinos and antineutrinos of all flavors, which play the role of astrophysical messengers, escaping from the SN core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' In the event of a galactic supernova explosion, DUNE data will probe the inner evolution of the core-collapse mechanism by studying the time and energy spectra of neutrinos arriving at DUNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' SNB neutrinos are emitted in a burst of a few tens of seconds duration [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Within DUNE, three qualitative stages of the collapse can be distinguished, as shown in Figure ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' : 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The neutronization burst – a large pulse of 𝜈𝑒 emission takes place in the first 10’s of ms as electrons capture on protons in the stellar core during the formation of a proto-neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' A neutrino sphere forms around the proto-neutron star, inside which the density is so large neutrinos become trapped at which point the neutronization burst of 𝜈𝑒 emission quenches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The accretion phase – during accretion, lasting from tens to hundreds of ms, neutrino emission is dominated by infall of gas from the outer layers of the progenitor onto the outer extant of the proto-neutron star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The cooling phase – after infall, the proto-neutron star cools over several seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The neutrino opacity drops, allowing neutrinos to escape the core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' While trapped within the core, neutrino species thermalize so that there is nearly luminosity equipartition between neutrino species during the cooling phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The predicted event rate from a SNB is calculated by folding together expected neutrino differential energy spectra, cross sections for the relevant channels, and detector response using SNOwGLoBES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Monte Carlo simulated events are generated using the time and energy of incident 4 Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta neutrinos for a particular SN model using the MARLEY interaction model to simulate the 𝜈𝑒 CC neutrino interaction and using the standard Geant4-based detector models to simulate the DUNE far detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Standard LArTPC algorithms are applied to reconstruct electron tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' All visible energy from the event is used to calculate the incident neutrino energy calorimetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Photon detectors are typically used to determine the time of events, but DUNE is exploring how photon detector calorimetry can expand its low-energy physics reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' In DUNE, the trigger on a SNB can be done using either TPC or photon detection system information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' In both cases, the trigger scheme exploits the time coincidence of multiple signals over a timescale matching the supernova luminosity evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' A redundant and highly efficient triggering scheme is under development.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' A number of astrophysical phenomena associated withsupernovae areexpected to beobservable in the supernova neutrino signal, providing a remarkable window into the event.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' In particular, the supernova explosion mechanism, which in the current paradigm involves energy deposition into the stellar envelope via neutrino interactions, is still not well understood, and the neutrinos themselves will bring the insight needed to confirm or refute the paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' There are many other examples of astrophysical observables, more details can be found in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Detecting neutrinos from a SNB, we can also learn a lot about neutrinos and particle physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' A SNB can be thought as an extremely hermetic system, which can be used to search for new new physics like Goldstone bosons, neutrino magnetic moments, "dark photons", "unparticles", extra- dimensional gauge bosons, and sterile neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Also, self-interactions of neutrinos, neutrino instability, and light gauge bosons can be studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Such energy-loss-based analysis will make use of two types of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' First, the total energy of the emitted neutrinos compared with the expected release in the gravitational collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Second, the rate of cooling should be measured and compared with what is expected from diffusion of the standard neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' As DUNE is mostly sensitive to 𝜈𝑒, complementary data of ¯𝜈𝑒 from water Cherenkov and scintillator experiments for careful analysis of the flavor transition will be very useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The flavor oscillation neutrino physics and its signatures are a major part of the physics program in the different periods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Solar neutrinos in DUNE Detection of solar and other low-energy neutrinos is challenging in a LArTPC because of relatively high intrinsic detection energy thresholds for the charged-current interaction on argon of about 5 MeV and because radioactive backgrounds in the same energy regime can affect triggering capability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' However, compared with other technologies, a LArTPC offers a large cross section and unique potential channel-tagging signatures from deexcitation photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Furthermore, observed energy from the final state CC interaction is much more tightly correlated with the incident neutrino energy on an event-by-event basis than the electron recoil spectrum from the ES channel that has been used for past solar neutrino observations such as in Super-Kamiokande [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Due to this, DUNE will make more precise spectral measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Though background rates are large, LArTPC detector allows for background reduction using fiduacialization techniques and the solar neutrino event rate is also substantial in the DUNE far detector, ∼100 per day, allowing samples of a few 105 events after 10 years of data collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Initial studies suggest DUNE could improve the measurement of Δ𝑚2 21 as well as observing the ℎ𝑒𝑝 and 8B solar neutrino flux, as shown in Figure ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='. 5 Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta Figure 3: Simulated solar neutrino spectrum with background for the DUNE Far Detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Detailed studies of solar neutrino detection capability are underway in DUNE along with sub- sequent physics sensitivity studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Similarly, DUNE can search for the Diffuse Supernova Neutrino Background (DSNB) [10] at energies just above the endpoint of the solar neutrino spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' As DUNE is primarily sensitive to the 𝜈𝑒 component, DUNE will be the only running experiment with sensitivity to the neutrino component of the DSNB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Conclusions The DUNE experiment will be sensitive to neutrinos with about 5 MeV up to several tens of MeV, the regime of relevance for core-collapse supernova burst and solar neutrinos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' This low-energy regime presents particular challenges for triggering and reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The combined information from DUNE’s TPC and PDS systems will provide a good reconstruction of these events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Software tools that enable preliminary physics and astrophysics sensitivity studies have been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' The observation of a burst will also enable sensitivity to neutrino mass ordering, and potentially many other topics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' In addition, there is discovery potential for ℎ𝑒𝑝 neutrinos in DUNE and perform a precision measurement of neutrino mixing and fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Acknowledgments This project has received funding from the European Union Horizon 2020 Research and Innovation programme under Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 101004761, from Spanish Ministerio de Economia y Com- petitividad (SEIDI-MINECO) under Grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' FPA2016-77347-C2-1-P;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' and from the Comunidad de Madrid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 6 DUNE Preliminary 10 "BV。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='CC 107 hep VeCC Lm 10 Neutron Capture 222 Rn 10 42 Ar 400 10 10 10 10 10° 0 5 10 15 20 25 30 ReconstructedE(MeV)Sensitivity of DUNE to low energy physics searches C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Cuesta References [1] DUNE collaboration, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=', Long-baseline neutrino oscillation physics potential of the DUNE experiment, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' C 80 (2020) 978.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [2] DUNE collaboration, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=', Supernova Neutrino Burst Detection with the Deep Underground Neutrino Experiment, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' C 81 (2021) 423, [2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='06647].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [3] DUNE collaboration, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=', Prospects for beyond the Standard Model physics searches at the Deep Underground Neutrino Experiment, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' C 81 (2021) 322.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [4] DUNE collaboration, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=', Deep Underground Neutrino Experiment (DUNE), Far Detector Technical Design Report, Volume IV Far Detector Single-phase Technology, JINST 15 (2020) T08010, [2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='03010].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [5] SNOwGLoBES, http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='phy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='duke.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='edu/ schol/snowglobes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [6] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Gardiner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=', MARLEY (Model of Argon Reaction Low Energy Yields), http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='marleygen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [7] LArSoft, http://larsoft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [8] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Hudepohl, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Muller, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Janka, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Marek and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Raffelt, Neutrino Signal of Electron-Capture Supernovae from Core Collapse to Cooling, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 104 (2010) 251101, [0912.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='0260].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [9] Super-Kamiokande collaboration, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=', Solar Neutrino Measurements in Super-Kamiokande-IV, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' D 94 (2016) 052010, [1606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='07538].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' [10] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Beacom, The Diffuse Supernova Neutrino Background, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 60 (2010) 439–462, [1004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content='3311].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZNE3T4oBgHgl3EQfcgp3/content/2301.04526v1.pdf'} diff --git a/aNE1T4oBgHgl3EQfcwSy/vector_store/index.pkl b/aNE1T4oBgHgl3EQfcwSy/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..99906897900995fa01b51439942045607aba0f95 --- /dev/null +++ b/aNE1T4oBgHgl3EQfcwSy/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:c3bd6de9c30cd448fb6266f07eb3d277aa3d50b37a19351cbebad13dc6c25446 +size 120199 diff --git a/atE1T4oBgHgl3EQfxAUb/content/tmp_files/2301.03416v1.pdf.txt b/atE1T4oBgHgl3EQfxAUb/content/tmp_files/2301.03416v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b17db8f2596d436ded19a5c6b730c2e08023427 --- /dev/null +++ b/atE1T4oBgHgl3EQfxAUb/content/tmp_files/2301.03416v1.pdf.txt @@ -0,0 +1,1592 @@ +ERNIE 3.0 Tiny: Frustratingly Simple Method to Improve Task-Agnostic +Distillation Generalization +Weixin Liu, Xuyi Chen, Jiaxiang Liu, Shikun Feng, Yu Sun, Hao Tian, Hua Wu +Baidu Inc. +{liuweixin, chenxuyi, liujiaxiang, fengshikun01, sunyu02, +tianhao, wu_hua}@baidu.com +Abstract +Task-agnostic knowledge distillation attempts +to address the problem of deploying large +pretrained +language +model +in +resource- +constrained scenarios by compressing a large +pretrained model called teacher into a smaller +one called student such that the student +can be directly finetuned on downstream +tasks and retains comparable performance. +However, we empirically find that there is +a generalization gap between the student +and the teacher in existing methods. In this +work, we show that we can leverage multi- +task learning in task-agnostic distillation to +advance the generalization of the resulted +student. In particular, we propose Multi-task +Infused Task-agnostic Knowledge Distillation +(MITKD). We first enhance the teacher by +multi-task training it on multiple downstream +tasks and then perform distillation to produce +the student. Experimental results demonstrate +that our method yields a student with much +better +generalization, +significantly +outper- +forms existing baselines, +and establishes +a new state-of-the-art result on in-domain, +out-domain, and low-resource datasets in the +setting of task-agnostic distillation. +More- +over, our method even exceeds an 8x larger +BERTBase on SQuAD and four GLUE tasks. +In addition, by combining ERNIE 3.0, our +method achieves state-of-the-art results on 10 +Chinese datasets. +1 +Introduction +Pretrained language models (PLMs) (Devlin et al., +2019; Liu et al., 2019d; Clark et al., 2020; He et al., +2021) have achieved great success in a wide range +of natural language processing tasks, however, their +enormous parameters often bring challenges to +serving them in real-life applications where com- +putational resources are limited. +Knowledge distillation (KD) (Hinton et al., +2015) has been widely utilized to tackle this prob- +lem. KD aims to compress a large PLM called +GLUE +SQuAD +76 +77.75 +79.5 +81.25 +83 +MITKD (5.3x Speedup) +Previous Best (5.3x Speedup) +BERT Base (1.0x Speedup) +Figure 1: Performance on GLUE and SQuAD. +teacher into a smaller one called student by trans- +ferring knowledge from teacher to student. In the +context of PLM compression, KD is usually ap- +plied in two different settings: task-specific (Sun +et al., 2019; Tang et al., 2019; Jiao et al., 2020; +Su et al., 2021) and task-agnostic (Sanh et al., +2019; Wang et al., 2020). The former transfers +task-specific knowledge from teacher to student for +a given task and often yields student with better +performance than the latter, but poses one disad- +vantage: task-specific KD needs to be performed +for every downstream task. Task-agnostic KD, on +the other hand, eliminates the need of distillation +for every single task by transferring general knowl- +edge to the student in a once-for-all fashion that +the student needs to be distilled only once and can +be directly finetuned on downstream tasks as sim- +ilar to the pretrain-finetune paradigm. A natural +research question is whether we can combine the +advantage of favorable downstream performance +and easy-to-deploy from these two types of distilla- +tion. +Previous attempt (Mukherjee et al., 2021) injects +downstream task knowledge into task-agnostic dis- +tillation by performing task-agnostic distillation on +a teacher finetuned in a downstream task. Although +this approach can improve the performance of the +student, knowledge from a single task may not be +sufficient to yield a generalizable student. In this +work, we show that the downstream generalization +of the student can be further improved by fusing +multi-task learning (MTL) into task-agnostic distil- +lation. +Existing works in MTL (Liu et al., 2019c; Agha- +arXiv:2301.03416v1 [cs.CL] 9 Jan 2023 + +janyan et al., 2021) point out that the model learns +a representation that generalizes better on new +tasks and domains through finetuned on multiple +tasks. We propose a distillation method Multi- +task Infused Task-agnostic Knowledge Distillation +(MITKD) to show that the generalizable representa- +tion brought by MTL can also benefit task-agnostic +distillation. +In particular, we first finetune the +teacher on multiple downstream tasks under the +setting of MTL to learn a generalizable representa- +tion and then perform task-agnostic distillation on +it. Specifically, our contribution includes: +1. We present a novel and simple method to com- +bine the advantage of task-agnostic and task- +specific distillation by fusing MTL into task- +agnostic distillation to improve the generaliza- +tion of the student. +2. We conduct extensive experiments to ver- +ify the effectiveness of MITKD on in- +domain datasets, out-domain datasets, and +low-resource datasets. Empirical results show +that MITKD consistently outperforms state- +of-the-art baselines in all three foregoing +scenarios, even outperforming an 8x larger +BERTBase on SQuAD and four GLUE tasks +shown in Figure 1 and Table 2. Moreover, by +applying MITKD on ERNIE 3.0, we obtain +ERNIE 3.0 Tiny which achieves state-of-the- +art results on 10 Chinese datasets. +3. Our empirical results bring out a message that +we should also pay attention on the knowledge +embedded in the teacher apart from pursuing a +stronger teacher, in order to improve student’s +downstream performance. +2 +Related Work +Knowledge Distillation +Knowledge distillation +(Hinton et al., 2015) mimics the output representa- +tion of the teacher and the student to guide the stu- +dent’s training. In the context of PLM, task-specific +methods aim to compress task-specific teacher over +task-specific data (Sun et al., 2019; Tang et al., +2019). (Liu et al., 2019b) strengthens the teacher +through multi-task learning. (Clark et al., 2019b) +compresses multiple task-specific models into one +student model. +On the other hand, task-agnostic methods aim to +compress pretrained teacher in a way such that the +resulted student can be easily adapted to down- +stream tasks via finetuning (Sanh et al., 2019). +In addition, task-agnostic method typically per- +forms distillation on pretraining data (i.e. the un- +supervised corpora on which the teacher is pre- +trained). (Wang et al., 2020, 2021) proposes to +mimic the self-attention of student and teacher. +(Khanuja et al., 2021) compresses multiple teachers +trained in different languages into a single student. +XtremeDistilTrans (Mukherjee et al., 2021) distills +a single-task finetuned teacher on augmented trans- +fer data generated from unsupervised data. +Multi-Task Learning +Multi-task learning learns +multiple tasks jointly so that the knowledge learned +from one task can benefit the others (Caruana, +1997; Luong et al., 2016). (Liu et al., 2019c) shows +that adding MTL at finetuning stage can boost the +performance of PLM. (Aghajanyan et al., 2021) +proposes a MTL stage named pre-finetuning stage +before the traditional finetuning stage. (Wei et al., +2022) combines MTL and prompting to improve +zero-shot performance. +3 +Method +In this section, we first categorize the existing +works in task-agnostic distillation into two differ- +ent types, then we propose MITKD and show the +difference between it and these existing works. +Vanilla Task-agnostic Distillation +Vanilla task- +agnostic distillation transfers general knowledge +from pretrained teacher to student without leverag- +ing any downstream knowledge. +Single-task Enhanced Task-agnostic Distilla- +tion +Single-task enhanced task-agnostic distilla- +tion exploits downstream knowledge by finetuning +the teacher on a single task and performing dis- +tillation on it. For example, XtremeDistilTrans +(Mukherjee et al., 2021) develops a cumbersome +distillation method that first intensively searches for +a source task inducing better transferability, then +finetunes the pretrained teacher in the source task. +After that, it performs task-agnostic distillation on +the finetuned teacher with augmented transfer data +generated from unsupervised data and a previously +task-agnostic distilled student as a warm start. +Multi-task Infused Task-agnostic Distillation +(MITKD) +While previous work leverages task +knowledge through carefully hunting for a task +inducing better transferability, we argue that we +can simply utilize the power of MTL to infuse +task knowledge into task-agnostic distillation to +improve student’s generalizability. In particular, +we propose a two-stage distillation method: we +first finetune a pretrained teacher on multiple tasks +and then perform vanilla task-agnostic distillation + +Method +Teacher +CHEMPROT +ANLI +ACL-ARC +SCIERC +PARTISAN +HELPFUL +SCOTUS +LEDGAR +FinBank +Avg. +Domain +- +Biology +General +Computer Science +News +Review +Law +Finance +- +RoBERTaLarge +- +86.9 +57.8 +82.5 +90.6 +87.5 +87.7 +77.6 +88.6 +90.1 +83.3 +RoBERTaBase +- +83.9 +53.4 +81.2 +89.9 +84.2 +87.7 +74.6 +87.4 +88.8 +81.2 +RoBERTaBase w/ MTL +- +84.0 +54.9 +81.6 +90.4 +85.8 +87.8 +74.6 +87.9 +89.0 +81.8 +MiniLMv2-L +RoBERTaLarge +74.5 +48.7 +70.0 +80.3 +77.0 +87.3 +66.6 +86.6 +85.1 +75.1 +MiniLMv2-B +RoBERTaBase +72.2 +48.0 +68.8 +76.7 +74.7 +87.1 +66.2 +86.5 +83.7 +73.8 +XtremeDistilTrans +ELECTRABase* +73.8 +50.5 +72.8 +81.4 +77.7 +87.1 +66.4 +86.5 +85.7 +75.8 +MITKD +RoBERTaBase w/MTL +79.0 +50.0 +74.4 +82.4 +82.0 +87.5 +71.2 +86.8 +85.9 +77.7 +Table 1: Out-domain results on the development sets. All results are produced by us using their publically available +checkpoints. For ANLI, we merge the train sets and dev sets at all three rounds into one train set and dev set, and +use them for training and evaluation. * means that ELECTRABase is finetuned on MNLI. +on it. Unlike previous work which requires search- +ing for the best transferable task and distilling on +augmented transfer data, our method simply ap- +plies MTL to the teacher and only needs to distill +on pretraining data as what vanilla task-agnostic +distillation does. Moreover, our empirical results +illustrate that MITKD does not only inject task +knowledge to the distillation, but more importantly, +brings in the generalization to improve the down- +stream performance of the student dramatically. +4 +Experiment +Experiment Setup +First, we finetune the teacher +with MTL. In particular, we adopt the base version +of Muppet’s (Aghajanyan et al., 2021) released +checkpoint 1 as our finetuned teacher. It is obtained +by finetuning a pretrained RoBERTaBase model +(Liu et al., 2019d) on around 50 tasks jointly. The +datasets used for the the MTL finetuning are listed +in Appendix A.4. +Then, we perform task-agnostic distillation on +the multi-task finetuned teacher using a classic task- +agnostic distillation method MiniLMv2 (Wang +et al., 2021) which mimics a more fine-grained +version of self-attention between the teacher and +the student. Following MiniLMv2, we utilizes the +pretraining datasets used in RoBERTa for the distil- +lation data. All the students in the experiment are +6-layer transformers with hidden size of 384, inter- +mediate size of 1536, and 12 attention heads. All +the results in the experiment section are reported +on the development set and are an average of 4 runs. +We also listed all the hyperparameters used in the +experiment in Appendix A.2. +Baselines +We select the classic task-agnostic dis- +tillation MiniLMv2 (Wang et al., 2021) as our base- +line for vanilla task-agnostic distillation. It utilizes +the same task-agnostic distillation algorithm as us +but is distilled from a larger teacher RoBERTaLarge. +In order to ablate the effectiveness of our method, +we also reproduce a MiniLMv2 with RoBERTaBase +1https://huggingface.co/facebook/ +muppet-roberta-base +which is essentially the same teacher as ours but +without MTL training. We denote the one dis- +tilled from RoBERTaLarge as MiniLMv2-L and the +other as MiniLMv2-B. As for single-task enhanced +task-agnostic distillation, we select the state-of-the- +art method XtremeDistilTrans (Mukherjee et al., +2021) as the baseline. In particular, it utilizes a +ELECTRABase (Clark et al., 2020) finetuned on +MNLI (Williams et al., 2018) as the teacher and +performs distillation from it on a student previously +task-agnostic distilled. +Tasks and Datasets +To demonstrate how MTL +can benefit task-agnostic distillation, we evaluate +the student on those tasks utilized in the MTL stage +of the teacher and those tasks which are not used in +MTL. For simplicity, we call the former in-domain +tasks and the latter out-domain tasks. In order to +verify the generalization improvement brought by +MITKD on out-domain datasets, we experiment +with nine datasets from seven domains. Refer to +Appendix A.1 for the dataset details. As for the +evaluation on in-domain tasks, we select GLUE +benchmark (Wang et al., 2018) and SQuAD 2.0 +(Rajpurkar et al., 2018) among the 50 tasks trained +in the MTL stage of the teacher. +Out-domain Generalization +As MTL brings +better generalization for new tasks and domains +to the teacher (Aghajanyan et al., 2021), we em- +pirically show that it also boosts the generaliza- +tion of the student through distillation by evaluat- +ing the downstream performance of the student on +out-domain tasks. In particular, we compare our +method with the mentioned baselines in Table 1. +Experimental results illustrate that our method +significantly outperforms other baselines, demon- +strating the generalization brought by our method. +Comparing XtremeDistilTrans and MiniLMv2-B, +we can see that introducing task knowledge to dis- +tillation can improve the performance of the stu- +dent. Moreover, MITKD further brings in multi- +2https://github.com/facebookresearch/fairseq/ +tree/main/examples/roberta + +Method +Teacher +MNLI +MRPC +QNLI +QQP +RTE +CoLA +SST +SQuADv2 +Avg. +BERTBase +- +84.5 +87.3 +91.7 +91.3 +68.6 +58.9 +93.2 +76.8 +81.5 +RoBERTaLarge +- +90.2 +90.9 +94.7 +92.2 +86.6 +68.0 +96.4 +89.4 +88.6 +RoBERTaBase +- +87.6 +90.2 +92.8 +91.9 +78.7 +63.6 +94.8 +83.7 +85.4 +RoBERTaBase w/ MTL +- +88.1 +91.7 +93.3 +91.9 +87.8 +64.6† +96.7 +84.7† +87.4 +MiniLMv2-L +RoBERTaLarge +84.4 +88.7 +90.9 +90.8 +69.9 +42.6 +92.0 +76.4 +79.5 +MiniLMv2-B +RoBERTaBase +83.4 +86.4 +90.2 +90.7 +59.6 +36.6 +91.9 +75.5 +76.8 +XtremeDistilTrans +ELECTRABase* +84.5 +89.0 +90.2 +90.4 +77.3 +40.6† +91.6 +74.4 +79.8 +MITKD +RoBERTaBase w/ MTL +85.1 +89.6 +91.7 +91.0 +77.6 +41.8 +93.6 +77.9 +81.0 +Table 2: In-Domain results on the development sets of GLUE and SQuAD 2.0 . +We quote the result for +RoBERTaBase w/ MTL, MiniLMv2-L and XtremeDistillTrans from their papers. The result for BERT is taken +from (Wang et al., 2020). The results for RoBERTa are taken from their github2. The number with † indicates that +the number is missing and we report our reproduced number using the publicly available checkpoint. We report F1 +for SQuAD. +task knowlegde and generalization led by MTL, +pushing the improvement by 2.0 average points. +Together, we observe that adding MTL into task- +agnostic distillation results in an impressive im- +provement of 3.9 points on out-domain tasks, com- +pared to the vanilla task-agnostic distillation base- +line MiniLMv2-B. In addition, it even exceeds +MiniLMv2-L by 2.6 points. +In-domain Performance +Besides the general- +ization improvement on out-domain datasets, +MITKD also improves the performance on in- +domain tasks. Table 2 shows the dev set result on +GLUE and SQuAD 2.0. MITKD achieves state-of- +the-art results on most tasks in GLUE and SQuAD +2.0, exceeding the best baseline by a margin of 1.2 +points. It even outperforms an 8x larger BERTBase +on four GLUE tasks and SQuAD 2.0 while being +5.3x faster than BERTBase.3 +Method +QNLI +CHEMPROT +LEDGAR +SCIREC +Domain +In +Biology +Law +Comp. Sci. +# of train data (1%) +1047 +41 +600 +32 +MiniLMv2-L +80.2 +33.5 +21.2 +47.0 +MiniLMv2-B +78.6 +33.5 +20.5 +47.0 +XtremeDsitilTran +84.1 +33.5 +18.3 +47.0 +MITKD +85.2 +33.5 +36.6 +47.0 +# of train data (10%) +10474 +416 +6000 +321 +MiniLMv2-L +86.6 +36.9 +74.2 +50.0 +MiniLMv2-B +84.7 +36.3 +73.9 +47.6 +XtremeDistilTran +87.7 +40.0 +66.7 +52.8 +MITKD +89.3 +50.2 +77.8 +54.8 +# of train data (50%) +52371 +2084 +30000 +1609 +MiniLMv2-L +89.8 +63.5 +83.8 +71.7 +MiniLMv2-B +88.8 +61.7 +83.4 +69.8 +XtremeDsitilTran +90.1 +66.3 +83.7 +75.0 +MITKD +91.0 +74.8 +84.8 +78.5 +Table 3: Transferability over different dataset scale set- +tings. +Performance on Low-resource Tasks +MITKD +also brings improvement on low-resource tasks +in both in-domain and out-domain. To demon- +strate that, we select datasets from various do- +mains and vary their training data size to 1%, +10%, and 50%. In particular, we use QNLI for +3Refer to Appendix A.3 for the details of speedup and +model size calculation. +in-domain, CHEMPROT for biology, LEDGAR +for legal, SCIREC for computer science. Note +that only QNLI has been used for MTL training +among these tasks. The results are presented in +Table 3, from which we can see that MITKD con- +sistently outperforms other baselines. Recall that +XtremeDistilTrans is obtained by distilling from +an MNLI-finetuned teacher. It performs relatively +well in QNLI which is similar to MNLI, but fails +to transfer well and even underperforms the vanilla +task-agnostic distillation baseline MiniLMv2-B in +the 1% and 10% settings of legal (LEDGAR). On +the other hand, MITKD consistently performs the +best, demonstrating that MTL brings significantly +better generalization to the student on low-resource +tasks than single task finetuning. +Method +Teacher +In-Domain +Out-Domain +RoBERTaLarge +- +88.6 +83.3 +RoBERTaBase +- +85.4 +81.2 +RoBERTaBase w/ MTL +- +87.4 +81.8 +MiniLMv2-L +RoBERTaLarge +79.5 +75.1 +MiniLMv2-B +RoBERTaBase +76.8 +73.8 +MITKD +RoBERTaBase w/ MTL +81.0 +77.7 +Table 4: Summary of average scores on in-domain and +out-domain tasks. +Larger Teacher or Better Knowledge? +There +are many ways to improve the performance of +the student. One straightforward way is to en- +large the teacher to obtain a stronger teacher +with better downstream performance: for exam- +ple, MiniLMv2-L outperforms MiniLMv2-B by +simply switching to a larger teacher, shown in Ta- +ble 4. However, several works (Mirzadeh et al., +2020; Li et al., 2021) witness failure when the size +gap between student and teacher are too large. On +the other hand, we propose to enhance the teacher +with more generalizable knowledge through MTL. +Table 4 shows that although our teacher has infe- +rior performance compared to RoBERTaLarge, the +produced student shows better performance on in- +domain and out-domain tasks. This suggests that +besides chasing after a larger and stronger teacher, + +we should also pay attention to the knowledge em- +bedded in the teacher. +Performance on Chinese Datasets +To verify +the effectiveness of MITKD on Chinese datasets, +we apply MITKD on ERNIE 3.0 (Sun et al., 2021) +to produce ERNIE 3.0 Tiny. +In particular, we +follow ERNIE 3.0 training process to reproduce +a Large version of ERNIE 3.0 and use it as the +teacher. +It is a 24-layer transformer with hid- +den size of 1024, intermediate size of 4096, and +16 attention heads. 28 datasets are selected for +MTL finetuning the teacher and are listed in Ap- +pendix A.5. All the students in this experiment are +4-layer transformers with hidden size of 312 and 12 +attention heads. The finetuning hyperparameters +for the students are listed in Appendix A.7. We +compare ERNIE 3.0 Tiny with the Chinese version +of TinyBERT 4 (Jiao et al., 2020) on the in-domain, +out-domain and low-resource datasets and list out +the result in Table 5. Refer to Appendix A.6 for the +dataset details. ERNIE 3.0 Tiny outperforms the +baseline by a margin of 4.3 average points, estab- +lishing a new state-of-the-art result. +5 +Conclusion +In this work, we propose a simple method to im- +prove task-agnostic distillation generalization by +leveraging MTL. The teacher is first augmented +by MTL and then distilled. Empirical results show +that our method outperforms several baselines on +in-domain, out-domain and low-resource tasks. +6 +Limitation +Compared with vanilla task-agnostic distillation, +MITKD has an additional multi-task finetuning +stage which may require additional computation +resources. +References +Armen Aghajanyan, Anchit Gupta, Akshat Shrivastava, +Xilun Chen, Luke Zettlemoyer, and Sonal Gupta. +2021. Muppet: Massive multi-task representations +with pre-finetuning. +In Proceedings of the 2021 +Conference on Empirical Methods in Natural Lan- +guage Processing, pages 5799–5811, Online and +Punta Cana, Dominican Republic. Association for +Computational Linguistics. +Aida Amini, Saadia Gabriel, Shanchuan Lin, Rik +Koncel-Kedziorski, Yejin Choi, and Hannaneh Ha- +jishirzi. 2019. +MathQA: Towards interpretable +4https://github.com/huawei-noah/ +Pretrained-Language-Model/tree/master/TinyBERT +math word problem solving with operation-based +formalisms. In Proceedings of the 2019 Conference +of the North American Chapter of the Association +for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), +pages 2357–2367, Minneapolis, Minnesota. Associ- +ation for Computational Linguistics. +Luisa Bentivogli, Peter Clark, Ido Dagan, and Danilo +Giampiccolo. 2009. The fifth pascal recognizing tex- +tual entailment challenge. In TAC. +Samuel R. Bowman, Gabor Angeli, Christopher Potts, +and Christopher D. Manning. 2015. A large anno- +tated corpus for learning natural language inference. +In Proceedings of the 2015 Conference on Empiri- +cal Methods in Natural Language Processing, pages +632–642, Lisbon, Portugal. Association for Compu- +tational Linguistics. +Rich Caruana. 1997. +Multitask learning. +Machine +learning, 28(1):41–75. +Ilias Chalkidis, Abhik Jana, Dirk Hartung, Michael +Bommarito, Ion Androutsopoulos, Daniel Katz, and +Nikolaos Aletras. 2022. LexGLUE: A benchmark +dataset for legal language understanding in English. +In Proceedings of the 60th Annual Meeting of the +Association for Computational Linguistics (Volume +1: Long Papers), pages 4310–4330, Dublin, Ireland. +Association for Computational Linguistics. +Jing Chen, Qingcai Chen, Xin Liu, Haijun Yang, +Daohe Lu, and Buzhou Tang. 2018. The BQ cor- +pus: +A large-scale domain-specific Chinese cor- +pus for sentence semantic equivalence identifica- +tion. +In Proceedings of the 2018 Conference on +Empirical Methods in Natural Language Processing, +pages 4946–4951, Brussels, Belgium. Association +for Computational Linguistics. +Christopher Clark, Kenton Lee, Ming-Wei Chang, +Tom Kwiatkowski, Michael Collins, and Kristina +Toutanova. 2019a. BoolQ: Exploring the surprising +difficulty of natural yes/no questions. In Proceed- +ings of NAACL-HLT 2019. +Kevin Clark, Minh-Thang Luong, Urvashi Khandelwal, +Christopher D. Manning, and Quoc V. Le. 2019b. +BAM! born-again multi-task networks for natural +language understanding. In Proceedings of the 57th +Annual Meeting of the Association for Computa- +tional Linguistics, pages 5931–5937, Florence, Italy. +Association for Computational Linguistics. +Kevin Clark, Minh-Thang Luong, Quoc V. Le, and +Christopher D. Manning. 2020. +Electra: +Pre- +training text encoders as discriminators rather than +generators. In International Conference on Learn- +ing Representations. +Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, +Ashish Sabharwal, Carissa Schoenick, and Oyvind +Tafjord. 2018. +Think you have solved question +answering? +try arc, the AI2 reasoning challenge. +CoRR, abs/1803.05457. + +Method +AFQMC +TNEWS +IFLYTEK +OCNLI +CLUEWSC +CSL +CANLI +SHOPPING10 +BUSTM +EPRSTMT +CSLDCP +Avg. +Domain +In-Domain +Out-Domain +Low-Resource +All +TinyBERT, Chinese +69.1 +54.0 +39.7 +69.6 +70.1 +75.1 +93.1 +93.6 +71.3 +78.6 +34.1 +68.0 +ERNIE 3.0 Tiny +72.4 +55.4 +50.6 +74.4 +68.4 +76.5 +98.2 +94.5 +72.3 +87.0 +45.1 +72.3 +Table 5: Performance on Chinese Datasets. All results listed are reported on dev set. +Alexis Conneau, Ruty Rinott, Guillaume Lample, Ad- +ina Williams, Samuel R. Bowman, Holger Schwenk, +and Veselin Stoyanov. 2018. Xnli: Evaluating cross- +lingual sentence representations. In Proceedings of +the 2018 Conference on Empirical Methods in Natu- +ral Language Processing. Association for Computa- +tional Linguistics. +Marie-Catherine de Marneffe, Mandy Simons, and Ju- +dith Tonhauser. 2019. The commitmentbank: Inves- +tigating projection in naturally occurring discourse. +Proceedings of Sinn und Bedeutung, 23(2):107–124. +Jacob Devlin, Ming-Wei Chang, Kenton Lee, and +Kristina Toutanova. 2019. +BERT: Pre-training of +deep bidirectional transformers for language under- +standing. +In Proceedings of the 2019 Conference +of the North American Chapter of the Association +for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), +pages 4171–4186, Minneapolis, Minnesota. Associ- +ation for Computational Linguistics. +Jay DeYoung, Sarthak Jain, Nazneen Fatema Rajani, +Eric Lehman, Caiming Xiong, Richard Socher, and +Byron C. Wallace. 2020. ERASER: A benchmark to +evaluate rationalized NLP models. In Proceedings +of the 58th Annual Meeting of the Association for +Computational Linguistics, pages 4443–4458, On- +line. Association for Computational Linguistics. +William B Dolan and Chris Brockett. 2005. Automati- +cally constructing a corpus of sentential paraphrases. +In Proceedings of the Third International Workshop +on Paraphrasing (IWP2005). +Dheeru Dua, Yizhong Wang, Pradeep Dasigi, Gabriel +Stanovsky, Sameer Singh, and Matt Gardner. 2019. +Drop: A reading comprehension benchmark requir- +ing discrete reasoning over paragraphs. In Proc. of +NAACL. +Vladimir Eidelman. 2019. Billsum: A corpus for auto- +matic summarization of us legislation. In Proceed- +ings of the 2nd Workshop on New Frontiers in Sum- +marization, pages 48–56. +Alexander Fabbri, Irene Li, Tianwei She, Suyi Li, and +Dragomir Radev. 2019. Multi-news: A large-scale +multi-document summarization dataset and abstrac- +tive hierarchical model. In Proceedings of the 57th +Annual Meeting of the Association for Computa- +tional Linguistics, pages 1074–1084, Florence, Italy. +Association for Computational Linguistics. +Pengcheng He, Xiaodong Liu, Jianfeng Gao, and +Weizhu Chen. 2021. Deberta: Decoding-enhanced +bert with disentangled attention. +In International +Conference on Learning Representations. +Karl Moritz Hermann, Tomas Kocisky, Edward Grefen- +stette, Lasse Espeholt, Will Kay, Mustafa Suleyman, +and Phil Blunsom. 2015. Teaching machines to read +and comprehend. In Advances in neural information +processing systems, pages 1693–1701. +Geoffrey E. Hinton, Oriol Vinyals, and Jeffrey Dean. +2015. Distilling the knowledge in a neural network. +CoRR, abs/1503.02531. +Eduard Hovy, Laurie Gerber, Ulf Hermjakob, Chin- +Yew Lin, and Deepak Ravichandran. 2001. Toward +semantics-based answer pinpointing. +In Proceed- +ings of the First International Conference on Human +Language Technology Research. +Hai Hu, Kyle Richardson, Liang Xu, Lu Li, Sandra +Kuebler, and Larry Moss. 2020. +Ocnli: Original +chinese natural language inference. In Findings of +EMNLP. +Lifu Huang, Ronan Le Bras, Chandra Bhagavatula, and +Yejin Choi. 2019. +Cosmos QA: Machine reading +comprehension with contextual commonsense rea- +soning. In Proceedings of the 2019 Conference on +Empirical Methods in Natural Language Processing +and the 9th International Joint Conference on Natu- +ral Language Processing (EMNLP-IJCNLP), pages +2391–2401, Hong Kong, China. Association for +Computational Linguistics. +Shankar Iyer, Nikhil Dandekar, and Kornel Csernai. +2017. First quora dataset release: Question pairs. +Xiaoqi Jiao, Yichun Yin, Lifeng Shang, Xin Jiang, +Xiao Chen, Linlin Li, Fang Wang, and Qun Liu. +2020. TinyBERT: Distilling BERT for natural lan- +guage understanding. In Findings of the Association +for Computational Linguistics: EMNLP 2020, pages +4163–4174, Online. Association for Computational +Linguistics. +Mandar Joshi, Eunsol Choi, Daniel Weld, and Luke +Zettlemoyer. 2017. +TriviaQA: A large scale dis- +tantly supervised challenge dataset for reading com- +prehension. In Proceedings of the 55th Annual Meet- +ing of the Association for Computational Linguistics +(Volume 1: Long Papers), pages 1601–1611, Van- +couver, Canada. Association for Computational Lin- +guistics. +David Jurgens, Srijan Kumar, Raine Hoover, Dan Mc- +Farland, and Dan Jurafsky. 2018. Measuring the evo- +lution of a scientific field through citation frames. +Transactions of the Association for Computational +Linguistics, 6:391–406. +Simran Khanuja, Melvin Johnson, and Partha Talukdar. +2021. Mergedistill: Merging language models using +pre-trained distillation. +In Findings of the Associ- +ation for Computational Linguistics: ACL-IJCNLP +2021, pages 2874–2887. + +Daniel Khashabi, Snigdha Chaturvedi, Michael Roth, +Shyam Upadhyay, and Dan Roth. 2018. +Looking +beyond the surface:a challenge set for reading com- +prehension over multiple sentences. In NAACL. +Tushar Khot, Ashish Sabharwal, and Peter Clark. 2018. +SciTail: A textual entailment dataset from science +question answering. In AAAI. +Johannes Kiesel, Maria Mestre, Rishabh Shukla, Em- +manuel Vincent, Payam Adineh, David Corney, +Benno Stein, and Martin Potthast. 2019. SemEval- +2019 task 4: +Hyperpartisan news detection. +In +Proceedings of the 13th International Workshop on +Semantic Evaluation, pages 829–839, Minneapo- +lis, Minnesota, USA. Association for Computational +Linguistics. +Diederik P. Kingma and Jimmy Ba. 2015. Adam: A +method for stochastic optimization. +In 3rd Inter- +national Conference on Learning Representations, +ICLR 2015, San Diego, CA, USA, May 7-9, 2015, +Conference Track Proceedings. +Jens Kringelum, Sonny Kim Kjaerulff, Søren Brunak, +Ole Lund, Tudor I Oprea, and Olivier Taboureau. +2016. ChemProt-3.0: a global chemical biology dis- +eases mapping. Database : the journal of biological +databases and curation, 2016:bav123. +Tom Kwiatkowski, Jennimaria Palomaki, Olivia Red- +field, Michael Collins, Ankur Parikh, Chris Alberti, +Danielle Epstein, Illia Polosukhin, Matthew Kelcey, +Jacob Devlin, Kenton Lee, Kristina N. Toutanova, +Llion Jones, Ming-Wei Chang, Andrew Dai, Jakob +Uszkoreit, Quoc Le, and Slav Petrov. 2019. Natu- +ral questions: a benchmark for question answering +research. Transactions of the Association of Compu- +tational Linguistics. +Guokun Lai, Qizhe Xie, Hanxiao Liu, Yiming Yang, +and Eduard Hovy. 2017. RACE: Large-scale ReAd- +ing comprehension dataset from examinations. In +Proceedings of the 2017 Conference on Empirical +Methods in Natural Language Processing, pages +785–794, Copenhagen, Denmark. Association for +Computational Linguistics. +Hector Levesque, Ernest Davis, and Leora Morgen- +stern. 2012. The winograd schema challenge. In +Thirteenth International Conference on the Princi- +ples of Knowledge Representation and Reasoning. +Citeseer. +Lei Li, Yankai Lin, Shuhuai Ren, Peng Li, Jie Zhou, +and Xu Sun. 2021. Dynamic knowledge distillation +for pre-trained language models. In Proceedings of +the 2021 Conference on Empirical Methods in Nat- +ural Language Processing, pages 379–389, Online +and Punta Cana, Dominican Republic. Association +for Computational Linguistics. +Xin Li and Dan Roth. 2002. Learning question clas- +sifiers. In COLING 2002: The 19th International +Conference on Computational Linguistics. +Yudong Li, Yuqing Zhang, Zhe Zhao, Linlin Shen, Wei- +jie Liu, Weiquan Mao, and Hui Zhang. 2022. CSL: +A large-scale Chinese scientific literature dataset. In +Proceedings of the 29th International Conference +on Computational Linguistics, pages 3917–3923, +Gyeongju, Republic of Korea. International Com- +mittee on Computational Linguistics. +Bang Liu, Di Niu, Haojie Wei, Jinghong Lin, Yancheng +He, Kunfeng Lai, and Yu Xu. 2019a. Matching ar- +ticle pairs with graphical decomposition and convo- +lutions. +In Proceedings of the 57th Annual Meet- +ing of the Association for Computational Linguis- +tics, pages 6284–6294, Florence, Italy. Association +for Computational Linguistics. +Linqing Liu, Huan Wang, Jimmy Lin, Richard Socher, +and Caiming Xiong. 2019b. +Mkd: a multi-task +knowledge distillation approach for pretrained lan- +guage models. arXiv preprint arXiv:1911.03588. +Xiaodong Liu, Pengcheng He, Weizhu Chen, and Jian- +feng Gao. 2019c. Multi-task deep neural networks +for natural language understanding. In Proceedings +of the 57th Annual Meeting of the Association for +Computational Linguistics, pages 4487–4496, Flo- +rence, Italy. Association for Computational Linguis- +tics. +Xin Liu, Qingcai Chen, Chong Deng, Huajun Zeng, +Jing Chen, Dongfang Li, and Buzhou Tang. 2018. +LCQMC:a large-scale Chinese question matching +corpus. +In Proceedings of the 27th International +Conference on Computational Linguistics, pages +1952–1962, Santa Fe, New Mexico, USA. Associ- +ation for Computational Linguistics. +Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- +dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, +Luke Zettlemoyer, and Veselin Stoyanov. 2019d. +Roberta: A robustly optimized bert pretraining ap- +proach. arXiv preprint arXiv:1907.11692. +Yi Luan, Luheng He, Mari Ostendorf, and Hannaneh +Hajishirzi. 2018. +Multi-task identification of enti- +ties, relations, and coreference for scientific knowl- +edge graph construction. In Proceedings of the 2018 +Conference on Empirical Methods in Natural Lan- +guage Processing, pages 3219–3232, Brussels, Bel- +gium. Association for Computational Linguistics. +Minh-Thang Luong, Quoc V. Le, Ilya Sutskever, +Oriol Vinyals, and Lukasz Kaiser. 2016. +Multi- +task sequence to sequence learning. +In 4th Inter- +national Conference on Learning Representations, +ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, +Conference Track Proceedings. +Andrew L. Maas, Raymond E. Daly, Peter T. Pham, +Dan Huang, Andrew Y. Ng, and Christopher Potts. +2011. Learning word vectors for sentiment analy- +sis. In Proceedings of the 49th Annual Meeting of +the Association for Computational Linguistics: Hu- +man Language Technologies, pages 142–150, Port- +land, Oregon, USA. Association for Computational +Linguistics. + +Pekka Malo, Ankur Sinha, Pekka Korhonen, Jyrki Wal- +lenius, and Pyry Takala. 2014. Good debt or bad +debt: Detecting semantic orientations in economic +texts. Journal of the Association for Information Sci- +ence and Technology, 65(4):782–796. +Julian J. McAuley, Christopher Targett, Qinfeng Shi, +and Anton van den Hengel. 2015. +Image-based +recommendations on styles and substitutes. CoRR, +abs/1506.04757. +R. Thomas McCoy, Ellie Pavlick, and Tal Linzen. 2019. +Right for the wrong reasons: Diagnosing syntac- +tic heuristics in natural language inference. CoRR, +abs/1902.01007. +Todor Mihaylov, Peter Clark, Tushar Khot, and Ashish +Sabharwal. 2018. Can a suit of armor conduct elec- +tricity? a new dataset for open book question an- +swering. In Proceedings of the 2018 Conference on +Empirical Methods in Natural Language Processing, +pages 2381–2391, Brussels, Belgium. Association +for Computational Linguistics. +Seyed Iman Mirzadeh, +Mehrdad Farajtabar, +Ang +Li, Nir Levine, Akihiro Matsukawa, and Hassan +Ghasemzadeh. 2020. Improved knowledge distilla- +tion via teacher assistant. In AAAI 2020 - 34th AAAI +Conference on Artificial Intelligence, AAAI 2020 +- 34th AAAI Conference on Artificial Intelligence, +pages 5191–5198. AAAI press. +Subhabrata Mukherjee, Ahmed Hassan Awadallah, and +Jianfeng Gao. 2021. Xtremedistiltransformers: Task +transfer for task-agnostic distillation. arXiv preprint +arXiv:2106.04563. +Shashi Narayan, Shay B. Cohen, and Mirella Lapata. +2018. Don’t give me the details, just the summary! +topic-aware convolutional neural networks for ex- +treme summarization. In Proceedings of the 2018 +Conference on Empirical Methods in Natural Lan- +guage Processing, pages 1797–1807, Brussels, Bel- +gium. Association for Computational Linguistics. +Yixin Nie, Adina Williams, Emily Dinan, Mohit +Bansal, Jason Weston, and Douwe Kiela. 2020. Ad- +versarial NLI: A new benchmark for natural lan- +guage understanding. In Proceedings of the 58th An- +nual Meeting of the Association for Computational +Linguistics, pages 4885–4901, Online. Association +for Computational Linguistics. +Bo Pang and Lillian Lee. 2005. +Seeing stars: Ex- +ploiting class relationships for sentiment categoriza- +tion with respect to rating scales. +In Proceed- +ings of the 43rd Annual Meeting of the Association +for Computational Linguistics (ACL’05), pages 115– +124, Ann Arbor, Michigan. Association for Compu- +tational Linguistics. +Mohammad +Taher +Pilehvar +and +Jose +Camacho- +Collados. 2019. WiC: the word-in-context dataset +for evaluating context-sensitive meaning represen- +tations. +In Proceedings of the 2019 Conference +of the North American Chapter of the Association +for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), +pages 1267–1273, Minneapolis, Minnesota. Associ- +ation for Computational Linguistics. +Maria Pontiki, Dimitris Galanis, Haris Papageorgiou, +Ion Androutsopoulos, Suresh Manandhar, Moham- +mad AL-Smadi, Mahmoud Al-Ayyoub, Yanyan +Zhao, Bing Qin, Orphée De Clercq, Véronique +Hoste, Marianna Apidianaki, Xavier Tannier, Na- +talia Loukachevitch, Evgeniy Kotelnikov, Nuria Bel, +Salud María Jiménez-Zafra, and Gül¸sen Eryi˘git. +2016. +SemEval-2016 task 5: Aspect based senti- +ment analysis. In Proceedings of the 10th Interna- +tional Workshop on Semantic Evaluation (SemEval- +2016), pages 19–30, San Diego, California. Associa- +tion for Computational Linguistics. +Pranav Rajpurkar, Robin Jia, and Percy Liang. 2018. +Know what you don’t know: Unanswerable ques- +tions for SQuAD. In Proceedings of the 56th An- +nual Meeting of the Association for Computational +Linguistics (Volume 2: Short Papers), pages 784– +789, Melbourne, Australia. Association for Compu- +tational Linguistics. +Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and +Percy Liang. 2016. SQuAD: 100,000+ questions for +machine comprehension of text. In Proceedings of +the 2016 Conference on Empirical Methods in Natu- +ral Language Processing, pages 2383–2392, Austin, +Texas. Association for Computational Linguistics. +Melissa Roemmele, Cosmin Adrian Bejan, and An- +drew S. Gordon. 2011. Choice of plausible alterna- +tives: An evaluation of commonsense causal reason- +ing. In 2011 AAAI Spring Symposium Series. +Victor Sanh, Lysandre Debut, Julien Chaumond, and +Thomas Wolf. 2019. Distilbert, a distilled version +of BERT: smaller, faster, cheaper and lighter. CoRR, +abs/1910.01108. +Richard Socher, Alex Perelygin, Jean Wu, Jason +Chuang, Christopher D Manning, Andrew Ng, and +Christopher Potts. 2013. +Recursive deep models +for semantic compositionality over a sentiment tree- +bank. +In Proceedings of the 2013 conference on +empirical methods in natural language processing, +pages 1631–1642. +Weiyue Su, Xuyi Chen, Shikun Feng, Jiaxiang Liu, +Weixin Liu, Yu Sun, Hao Tian, Hua Wu, and +Haifeng Wang. 2021. Ernie-tiny: A progressive dis- +tillation framework for pretrained transformer com- +pression. arXiv preprint arXiv:2106.02241. +Siqi Sun, Yu Cheng, Zhe Gan, and Jingjing Liu. 2019. +Patient knowledge distillation for BERT model com- +pression. In Proceedings of the 2019 Conference on +Empirical Methods in Natural Language Processing +and the 9th International Joint Conference on Natu- +ral Language Processing (EMNLP-IJCNLP), pages +4323–4332, Hong Kong, China. Association for +Computational Linguistics. + +Yu Sun, Shuohuan Wang, Shikun Feng, Siyu Ding, +Chao Pang, Junyuan Shang, Jiaxiang Liu, Xuyi +Chen, Yanbin Zhao, Yuxiang Lu, et al. 2021. Ernie +3.0: Large-scale knowledge enhanced pre-training +for language understanding and generation. arXiv +preprint arXiv:2107.02137. +Alon Talmor, Jonathan Herzig, Nicholas Lourie, and +Jonathan Berant. 2019. CommonsenseQA: A ques- +tion answering challenge targeting commonsense +knowledge. In Proceedings of the 2019 Conference +of the North American Chapter of the Association +for Computational Linguistics: Human Language +Technologies, Volume 1 (Long and Short Papers), +pages 4149–4158, Minneapolis, Minnesota. Associ- +ation for Computational Linguistics. +Raphael Tang, Yao Lu, Linqing Liu, Lili Mou, Olga +Vechtomova, and Jimmy Lin. 2019. Distilling task- +specific knowledge from bert into simple neural net- +works. arXiv preprint arXiv:1903.12136. +Don Tuggener, Pius von Däniken, Thomas Peetz, and +Mark Cieliebak. 2020. +LEDGAR: A large-scale +multi-label corpus for text classification of legal pro- +visions in contracts. In Proceedings of the 12th Lan- +guage Resources and Evaluation Conference, pages +1235–1241, Marseille, France. European Language +Resources Association. +Alex Wang, Amanpreet Singh, Julian Michael, Fe- +lix Hill, Omer Levy, and Samuel Bowman. 2018. +GLUE: A multi-task benchmark and analysis plat- +form for natural language understanding. +In Pro- +ceedings of the 2018 EMNLP Workshop Black- +boxNLP: Analyzing and Interpreting Neural Net- +works for NLP, pages 353–355, Brussels, Belgium. +Association for Computational Linguistics. +Wenhui Wang, Hangbo Bao, Shaohan Huang, Li Dong, +and Furu Wei. 2021. MiniLMv2: Multi-head self- +attention relation distillation for compressing pre- +trained transformers. In Findings of the Association +for Computational Linguistics: ACL-IJCNLP 2021, +pages 2140–2151, Online. Association for Computa- +tional Linguistics. +Wenhui Wang, Furu Wei, Li Dong, Hangbo Bao, Nan +Yang, and Ming Zhou. 2020. Minilm: Deep self- +attention distillation for task-agnostic compression +of pre-trained transformers. In Advances in Neural +Information Processing Systems 33: Annual Con- +ference on Neural Information Processing Systems +2020, NeurIPS 2020, December 6-12, 2020, virtual. +Alex Warstadt, Amanpreet Singh, and Samuel R. Bow- +man. 2019. Neural network acceptability judgments. +Transactions of the Association for Computational +Linguistics, 7:625–641. +Jason Wei, Maarten Bosma, Vincent Zhao, Kelvin Guu, +Adams Wei Yu, Brian Lester, Nan Du, Andrew M. +Dai, and Quoc V Le. 2022. +Finetuned language +models are zero-shot learners. In International Con- +ference on Learning Representations. +Johannes Welbl, Nelson F. Liu, and Matt Gardner. 2017. +Crowdsourcing multiple choice science questions. +In Proceedings of the 3rd Workshop on Noisy User- +generated Text, pages 94–106, Copenhagen, Den- +mark. Association for Computational Linguistics. +Adina Williams, Nikita Nangia, and Samuel Bowman. +2018. A broad-coverage challenge corpus for sen- +tence understanding through inference. In Proceed- +ings of the 2018 Conference of the North American +Chapter of the Association for Computational Lin- +guistics: Human Language Technologies, Volume +1 (Long Papers), pages 1112–1122. Association for +Computational Linguistics. +Liang Xu, Hai Hu, Xuanwei Zhang, Lu Li, Chenjie +Cao, Yudong Li, Yechen Xu, Kai Sun, Dian Yu, +Cong Yu, Yin Tian, Qianqian Dong, Weitang Liu, +Bo Shi, Yiming Cui, Junyi Li, Jun Zeng, Rongzhao +Wang, Weijian Xie, Yanting Li, Yina Patterson, +Zuoyu Tian, Yiwen Zhang, He Zhou, Shaoweihua +Liu, Zhe Zhao, Qipeng Zhao, Cong Yue, Xinrui +Zhang, Zhengliang Yang, Kyle Richardson, and +Zhenzhong Lan. 2020. CLUE: A Chinese language +understanding evaluation benchmark. +In Proceed- +ings of the 28th International Conference on Com- +putational Linguistics, pages 4762–4772, Barcelona, +Spain (Online). International Committee on Compu- +tational Linguistics. +Liang Xu, Xiaojing Lu, Chenyang Yuan, Xuanwei +Zhang, Huilin Xu, Hu Yuan, Guoao Wei, Xiang Pan, +Xin Tian, Libo Qin, et al. 2021. Fewclue: A chi- +nese few-shot learning evaluation benchmark. arXiv +preprint arXiv:2107.07498. +Shanshan Xu and Katja Markert. 2022. The Chinese +causative-passive homonymy disambiguation: an ad- +versarial dataset for NLI and a probing task. +In +Proceedings of the Thirteenth Language Resources +and Evaluation Conference, pages 4316–4323, Mar- +seille, France. European Language Resources Asso- +ciation. +Yi Yang, Wen-tau Yih, and Christopher Meek. 2015. +WikiQA: A challenge dataset for open-domain ques- +tion answering. +In Proceedings of the 2015 Con- +ference on Empirical Methods in Natural Language +Processing, pages 2013–2018, Lisbon, Portugal. As- +sociation for Computational Linguistics. +Yinfei Yang, Yuan Zhang, Chris Tar, and Jason +Baldridge. 2019. +PAWS-X: A cross-lingual ad- +versarial dataset for paraphrase identification. +In +Proceedings of the 2019 Conference on Empirical +Methods in Natural Language Processing and the +9th International Joint Conference on Natural Lan- +guage Processing (EMNLP-IJCNLP), pages 3687– +3692, Hong Kong, China. Association for Computa- +tional Linguistics. +Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Ben- +gio, William W. Cohen, Ruslan Salakhutdinov, and +Christopher D. Manning. 2018. +HotpotQA: A +dataset for diverse, explainable multi-hop question + +answering. In Conference on Empirical Methods in +Natural Language Processing (EMNLP). +Rowan Zellers, Yonatan Bisk, Roy Schwartz, and +Yejin Choi. 2018. SWAG: A large-scale adversar- +ial dataset for grounded commonsense inference. In +Proceedings of the 2018 Conference on Empirical +Methods in Natural Language Processing, pages 93– +104, Brussels, Belgium. Association for Computa- +tional Linguistics. +Rowan Zellers, Ari Holtzman, Yonatan Bisk, Ali +Farhadi, and Yejin Choi. 2019. +HellaSwag: Can +a machine really finish your sentence? +In Pro- +ceedings of the 57th Annual Meeting of the Asso- +ciation for Computational Linguistics, pages 4791– +4800, Florence, Italy. Association for Computational +Linguistics. +Ningyu Zhang, Mosha Chen, Zhen Bi, Xiaozhuan +Liang, Lei Li, Xin Shang, Kangping Yin, Chuanqi +Tan, Jian Xu, Fei Huang, Luo Si, Yuan Ni, Guo- +tong Xie, Zhifang Sui, Baobao Chang, Hui Zong, +Zheng Yuan, Linfeng Li, Jun Yan, Hongying Zan, +Kunli Zhang, Buzhou Tang, and Qingcai Chen. +2022. CBLUE: A Chinese biomedical language un- +derstanding evaluation benchmark. In Proceedings +of the 60th Annual Meeting of the Association for +Computational Linguistics (Volume 1: Long Papers), +pages 7888–7915, Dublin, Ireland. Association for +Computational Linguistics. +Rui Zhang and Joel Tetreault. 2019. This email could +save your life: Introducing the task of email subject +line generation. In Proceedings of the 57th Annual +Meeting of the Association for Computational Lin- +guistics, pages 446–456, Florence, Italy. Association +for Computational Linguistics. +Sheng Zhang, Xiaodong Liu, Jingjing Liu, Jianfeng +Gao, Kevin Duh, and Benjamin Van Durme. 2018. +Record: Bridging the gap between human and ma- +chine commonsense reading comprehension. arXiv +preprint arXiv:1810.12885. +Xiang Zhang, Junbo Zhao, and Yann LeCun. 2015. +Character-level convolutional networks for text clas- +sification. In Advances in Neural Information Pro- +cessing Systems, volume 28. Curran Associates, Inc. +A +Appendix +A.1 +Out-domain Datasets +We +evaluate +our +model +on +CHEMPROT +(Kringelum et al., 2016) for Biology, ANLI +(Nie et al., 2020) for web, ACL-ARC (Jurgens +et al., 2018) and SCIERC (Luan et al., 2018) +for computer science, PARTISAN (Kiesel et al., +2019) for news, HELPFUL (McAuley et al., 2015) +for review, SCOTUS(Chalkidis et al., 2022) and +LEDGAR (Tuggener et al., 2020) for law, and +FinBank (Malo et al., 2014) for finance. +A.2 +Hyperparameters +We use Adam (Kingma and Ba, 2015) optimizer +with β1 of 0.9 and β2 of 0.999 for all tasks. We use +linear warm up over the first 10% of training steps +and linear decay for the rest of training steps. The +dropout rate is set to 0.1. The weight decay is 0.01. +SQuAD 2.0 +The learning rate ranges from {3e-5, +6e-5, 8e-5, 9e-5}. The batch size is set to 32. The +training epoch is set to 3. +Others +For all other tasks, the epoch ranges from +{3, 5, 10}, the batch size ranges from {16, 32, 48}, +learning rate ranges from {1e-5, 2e-5, 5e-5}. For +CoLA, we finetune it with training epoch of 25. +A.3 +Model size and Speed Calculation +When calculating the size of a model, we count the +number of all the learnable parameters in the model +except the parameters of the embeddings. As for +the speedup, we quote the number from (Wang +et al., 2021). +A.4 +Datasets Used for Fintuning Teacher +1. MNLI (Williams et al., 2018) +2. CoLA (Warstadt et al., 2019) +3. SST (Socher et al., 2013) +4. MRPC (Dolan and Brockett, 2005) +5. QQP (Iyer et al., 2017) +6. QNLI (Rajpurkar et al., 2016) +7. RTE (Bentivogli et al., 2009) +8. WNLI (Levesque et al., 2012) +9. Bool Q (Clark et al., 2019a) +10. MultiRC (Khashabi et al., 2018) + +11. ReCoRD (Zhang et al., 2018) +12. WIC (Pilehvar and Camacho-Collados, 2019) +13. WSC (Levesque et al., 2012) +14. CB (de Marneffe et al., 2019) +15. COPA (Roemmele et al., 2011) +16. AG News (Zhang et al., 2015) +17. IMDB (Maas et al., 2011) +18. SNLI (Bowman et al., 2015) +19. HANS (McCoy et al., 2019) +20. Rotten Tomatoes (Pang and Lee, 2005) +21. Yelp Polarity (Zhang et al., 2015) +22. Eraser Multi RC (DeYoung et al., 2020) +23. Wiki QA (Yang et al., 2015) +24. Trec (Li and Roth, 2002; Hovy et al., 2001) +25. SciTail (Khot et al., 2018) +26. CNN Daily Mail (Hermann et al., 2015) +27. Billsum (Eidelman, 2019) +28. XSUM (Narayan et al., 2018) +29. Aeslc (Zhang and Tetreault, 2019) +30. Multinews (Fabbri et al., 2019) +31. Math QA (Amini et al., 2019) +32. Openbook QA (Mihaylov et al., 2018) +33. SWAG (Zellers et al., 2018) +34. HellaSWAG (Zellers et al., 2019) +35. RACE (Lai et al., 2017) +36. CommonSense QA (Talmor et al., 2019) +37. Cosmos QA (Huang et al., 2019) +38. AI2 ARC - Easy (Clark et al., 2018) +39. AI2 ARC - Challenge (Clark et al., 2018) +40. SCIQ (Welbl et al., 2017) +41. SQUAD (Rajpurkar et al., 2016) +42. NQ (Kwiatkowski et al., 2019) +43. DROP (Dua et al., 2019) +44. Hotpot (Yang et al., 2018) +45. Trivia QA (Joshi et al., 2017) +A.5 +Datasets Used for Fintuning ERNIE 3.0 +Titan +1. XNLI (Conneau et al., 2018) +2. OCNLI (Hu et al., 2020) +3. KUAKE-QQR (Zhang et al., 2022) +4. KUAKE-QTR (Zhang et al., 2022) +5. CMNLI (Xu et al., 2020) +6. PAWS-X (Yang et al., 2019) +7. CHIP-STS (Zhang et al., 2022) +8. CSL (Li et al., 2022) +9. CCKS2018-Task3 5 +10. CHIP2019-Task2 6 +11. ZHONGCE: It is an internal semantic textual +similarity dataset, consisting of 1025327 train- +ing examples and 8803 evaluation examples. +12. QBQTC 7 +13. LCQMC (Liu et al., 2018) +14. BQ (Chen et al., 2018) +15. AFQMC (Xu et al., 2020) +16. WAIMAI 8 +17. WEIBO 9 +18. NLPC2014-Task2 10 +19. SemEval2016-Task5-PHNS (Pontiki et al., +2016) +5http://www.sigkg.cn/ccks2018/?page_id=16 +6http://www.cips-chip.org.cn:8000/evaluation +7https://github.com/CLUEbenchmark/QBQTC +8https://github.com/SophonPlus/ +ChineseNlpCorpus/tree/master/datasets/waimai_10k +9https://github.com/SophonPlus/ +ChineseNlpCorpus/tree/master/datasets/weibo_ +senti_100k +10http://tcci.ccf.org.cn/conference/2014/pages/ +page04_dg.html + +Task +AFQMC +TNEWS +IFLYTEK +OCNLI +CLUEWSC +CSL +Epoch +3 +3 +3 +5 +50 +5 +Dropout Rate +0.1 +0.1 +{0.0, 0.1} +0.1 +{0.0, 0.1} +0.1 +Table 6: Epoch and dropout rate settings for chinese datasets. +20. SemEval2016-Task5-CAME (Pontiki et al., +2016) +21. THUCNEWS 11 +22. TNEWS (Xu et al., 2020) +23. IFLYTEK (Xu et al., 2020) +24. SOGOUNEWS (Zhang et al., 2015) +25. CHIP-CTC (Zhang et al., 2022) +26. CNSE (Liu et al., 2019a) +27. CNSS (Liu et al., 2019a) +28. CLUEWSC (Xu et al., 2020) +A.6 +Chinese Evaluation Datasets +We select the classification tasks in CLUE Bench- +mark (Xu et al., 2020) for the evaluation on +in-domain datasets, and CANLI (Xu and Mark- +ert, 2022) and SHOPPING1012 for the out- +domain datasets. In addition, we select BUSTM, +EPRSTMT and CSLDCP from FewCLUE (Xu +et al., 2021), a Chinese few-shot learning bench- +mark, for the low-resource datasets. For the low- +resource datasets, we merge the train sets and dev +sets from all sub-parts into one train set and dev +set, and use them for training and evaluation. +A.7 +Hypaerparameters for Chinese Datasets +For the in-domain datasets, we follow the hyper- +parameters settings in PaddleNLP GitHub repos- +itory13. In particular, the batch size ranges from +{16, 32, 64}, learning rate ranges from {1e-5, 2e-5, +3e-5, 5e-5}. The epoch and dropout rate for each +task are listed in Figure 6. All other settings such as +optimizer, and learning rate are the same as those +in Appendix A.2. For all other tasks, we use the +same hyperparameters in Appendix A.2. +11http://thuctc.thunlp.org +12https://github.com/SophonPlus/ +ChineseNlpCorpus/tree/master/datasets/online_ +shopping_10_cats +13https://github.com/PaddlePaddle/PaddleNLP/ +tree/develop/examples/benchmark/clue + diff --git a/atE1T4oBgHgl3EQfxAUb/content/tmp_files/load_file.txt b/atE1T4oBgHgl3EQfxAUb/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9c2a6f6438373d2e8ffb1b0681d86ccfc224c10 --- /dev/null +++ b/atE1T4oBgHgl3EQfxAUb/content/tmp_files/load_file.txt @@ -0,0 +1,1117 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf,len=1116 +page_content='ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Tiny: Frustratingly Simple Method to Improve Task-Agnostic Distillation Generalization Weixin Liu, Xuyi Chen, Jiaxiang Liu, Shikun Feng, Yu Sun, Hao Tian, Hua Wu Baidu Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' {liuweixin, chenxuyi, liujiaxiang, fengshikun01, sunyu02, tianhao, wu_hua}@baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com Abstract Task-agnostic knowledge distillation attempts to address the problem of deploying large pretrained language model in resource- constrained scenarios by compressing a large pretrained model called teacher into a smaller one called student such that the student can be directly finetuned on downstream tasks and retains comparable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' However, we empirically find that there is a generalization gap between the student and the teacher in existing methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In this work, we show that we can leverage multi- task learning in task-agnostic distillation to advance the generalization of the resulted student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we propose Multi-task Infused Task-agnostic Knowledge Distillation (MITKD).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We first enhance the teacher by multi-task training it on multiple downstream tasks and then perform distillation to produce the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Experimental results demonstrate that our method yields a student with much better generalization, significantly outper- forms existing baselines, and establishes a new state-of-the-art result on in-domain, out-domain, and low-resource datasets in the setting of task-agnostic distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' More- over, our method even exceeds an 8x larger BERTBase on SQuAD and four GLUE tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In addition, by combining ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0, our method achieves state-of-the-art results on 10 Chinese datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 1 Introduction Pretrained language models (PLMs) (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) have achieved great success in a wide range of natural language processing tasks, however, their enormous parameters often bring challenges to serving them in real-life applications where com- putational resources are limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Knowledge distillation (KD) (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) has been widely utilized to tackle this prob- lem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' KD aims to compress a large PLM called GLUE SQuAD 76 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='75 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='25 83 MITKD (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3x Speedup) Previous Best (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3x Speedup) BERT Base (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0x Speedup) Figure 1: Performance on GLUE and SQuAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' teacher into a smaller one called student by trans- ferring knowledge from teacher to student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In the context of PLM compression, KD is usually ap- plied in two different settings: task-specific (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Jiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Su et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) and task-agnostic (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The former transfers task-specific knowledge from teacher to student for a given task and often yields student with better performance than the latter, but poses one disad- vantage: task-specific KD needs to be performed for every downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Task-agnostic KD, on the other hand, eliminates the need of distillation for every single task by transferring general knowl- edge to the student in a once-for-all fashion that the student needs to be distilled only once and can be directly finetuned on downstream tasks as sim- ilar to the pretrain-finetune paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A natural research question is whether we can combine the advantage of favorable downstream performance and easy-to-deploy from these two types of distilla- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Previous attempt (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) injects downstream task knowledge into task-agnostic dis- tillation by performing task-agnostic distillation on a teacher finetuned in a downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Although this approach can improve the performance of the student, knowledge from a single task may not be sufficient to yield a generalizable student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In this work, we show that the downstream generalization of the student can be further improved by fusing multi-task learning (MTL) into task-agnostic distil- lation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Existing works in MTL (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Agha- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='03416v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='CL] 9 Jan 2023 janyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) point out that the model learns a representation that generalizes better on new tasks and domains through finetuned on multiple tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We propose a distillation method Multi- task Infused Task-agnostic Knowledge Distillation (MITKD) to show that the generalizable representa- tion brought by MTL can also benefit task-agnostic distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we first finetune the teacher on multiple downstream tasks under the setting of MTL to learn a generalizable representa- tion and then perform task-agnostic distillation on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Specifically, our contribution includes: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We present a novel and simple method to com- bine the advantage of task-agnostic and task- specific distillation by fusing MTL into task- agnostic distillation to improve the generaliza- tion of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We conduct extensive experiments to ver- ify the effectiveness of MITKD on in- domain datasets, out-domain datasets, and low-resource datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Empirical results show that MITKD consistently outperforms state- of-the-art baselines in all three foregoing scenarios, even outperforming an 8x larger BERTBase on SQuAD and four GLUE tasks shown in Figure 1 and Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Moreover, by applying MITKD on ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0, we obtain ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Tiny which achieves state-of-the- art results on 10 Chinese datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Our empirical results bring out a message that we should also pay attention on the knowledge embedded in the teacher apart from pursuing a stronger teacher, in order to improve student’s downstream performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2 Related Work Knowledge Distillation Knowledge distillation (Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) mimics the output representa- tion of the teacher and the student to guide the stu- dent’s training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In the context of PLM, task-specific methods aim to compress task-specific teacher over task-specific data (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Tang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019b) strengthens the teacher through multi-task learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019b) compresses multiple task-specific models into one student model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' On the other hand, task-agnostic methods aim to compress pretrained teacher in a way such that the resulted student can be easily adapted to down- stream tasks via finetuning (Sanh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In addition, task-agnostic method typically per- forms distillation on pretraining data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' the un- supervised corpora on which the teacher is pre- trained).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020, 2021) proposes to mimic the self-attention of student and teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Khanuja et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) compresses multiple teachers trained in different languages into a single student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' XtremeDistilTrans (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) distills a single-task finetuned teacher on augmented trans- fer data generated from unsupervised data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multi-Task Learning Multi-task learning learns multiple tasks jointly so that the knowledge learned from one task can benefit the others (Caruana, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Luong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019c) shows that adding MTL at finetuning stage can boost the performance of PLM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Aghajanyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) proposes a MTL stage named pre-finetuning stage before the traditional finetuning stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' (Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) combines MTL and prompting to improve zero-shot performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 3 Method In this section, we first categorize the existing works in task-agnostic distillation into two differ- ent types, then we propose MITKD and show the difference between it and these existing works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Vanilla Task-agnostic Distillation Vanilla task- agnostic distillation transfers general knowledge from pretrained teacher to student without leverag- ing any downstream knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Single-task Enhanced Task-agnostic Distilla- tion Single-task enhanced task-agnostic distilla- tion exploits downstream knowledge by finetuning the teacher on a single task and performing dis- tillation on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' For example, XtremeDistilTrans (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) develops a cumbersome distillation method that first intensively searches for a source task inducing better transferability, then finetunes the pretrained teacher in the source task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' After that, it performs task-agnostic distillation on the finetuned teacher with augmented transfer data generated from unsupervised data and a previously task-agnostic distilled student as a warm start.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multi-task Infused Task-agnostic Distillation (MITKD) While previous work leverages task knowledge through carefully hunting for a task inducing better transferability, we argue that we can simply utilize the power of MTL to infuse task knowledge into task-agnostic distillation to improve student’s generalizability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we propose a two-stage distillation method: we first finetune a pretrained teacher on multiple tasks and then perform vanilla task-agnostic distillation Method Teacher CHEMPROT ANLI ACL-ARC SCIERC PARTISAN HELPFUL SCOTUS LEDGAR FinBank Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Domain Biology General Computer Science News Review Law Finance RoBERTaLarge 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 RoBERTaBase 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 RoBERTaBase w/ MTL 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 MiniLMv2-L RoBERTaLarge 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 MiniLMv2-B RoBERTaBase 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 XtremeDistilTrans ELECTRABase* 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 MITKD RoBERTaBase w/MTL 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 Table 1: Out-domain results on the development sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' All results are produced by us using their publically available checkpoints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' For ANLI, we merge the train sets and dev sets at all three rounds into one train set and dev set, and use them for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' * means that ELECTRABase is finetuned on MNLI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Unlike previous work which requires search- ing for the best transferable task and distilling on augmented transfer data, our method simply ap- plies MTL to the teacher and only needs to distill on pretraining data as what vanilla task-agnostic distillation does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Moreover, our empirical results illustrate that MITKD does not only inject task knowledge to the distillation, but more importantly, brings in the generalization to improve the down- stream performance of the student dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 4 Experiment Experiment Setup First, we finetune the teacher with MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we adopt the base version of Muppet’s (Aghajanyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) released checkpoint 1 as our finetuned teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' It is obtained by finetuning a pretrained RoBERTaBase model (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019d) on around 50 tasks jointly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The datasets used for the the MTL finetuning are listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Then, we perform task-agnostic distillation on the multi-task finetuned teacher using a classic task- agnostic distillation method MiniLMv2 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) which mimics a more fine-grained version of self-attention between the teacher and the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Following MiniLMv2, we utilizes the pretraining datasets used in RoBERTa for the distil- lation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' All the students in the experiment are 6-layer transformers with hidden size of 384, inter- mediate size of 1536, and 12 attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' All the results in the experiment section are reported on the development set and are an average of 4 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We also listed all the hyperparameters used in the experiment in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Baselines We select the classic task-agnostic dis- tillation MiniLMv2 (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) as our base- line for vanilla task-agnostic distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' It utilizes the same task-agnostic distillation algorithm as us but is distilled from a larger teacher RoBERTaLarge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In order to ablate the effectiveness of our method, we also reproduce a MiniLMv2 with RoBERTaBase 1https://huggingface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='co/facebook/ muppet-roberta-base which is essentially the same teacher as ours but without MTL training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We denote the one dis- tilled from RoBERTaLarge as MiniLMv2-L and the other as MiniLMv2-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' As for single-task enhanced task-agnostic distillation, we select the state-of-the- art method XtremeDistilTrans (Mukherjee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) as the baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, it utilizes a ELECTRABase (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) finetuned on MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) as the teacher and performs distillation from it on a student previously task-agnostic distilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Tasks and Datasets To demonstrate how MTL can benefit task-agnostic distillation, we evaluate the student on those tasks utilized in the MTL stage of the teacher and those tasks which are not used in MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' For simplicity, we call the former in-domain tasks and the latter out-domain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In order to verify the generalization improvement brought by MITKD on out-domain datasets, we experiment with nine datasets from seven domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 for the dataset details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' As for the evaluation on in-domain tasks, we select GLUE benchmark (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) and SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) among the 50 tasks trained in the MTL stage of the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Out-domain Generalization As MTL brings better generalization for new tasks and domains to the teacher (Aghajanyan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021), we em- pirically show that it also boosts the generaliza- tion of the student through distillation by evaluat- ing the downstream performance of the student on out-domain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we compare our method with the mentioned baselines in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Experimental results illustrate that our method significantly outperforms other baselines, demon- strating the generalization brought by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Comparing XtremeDistilTrans and MiniLMv2-B, we can see that introducing task knowledge to dis- tillation can improve the performance of the stu- dent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Moreover, MITKD further brings in multi- 2https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/facebookresearch/fairseq/ tree/main/examples/roberta Method Teacher MNLI MRPC QNLI QQP RTE CoLA SST SQuADv2 Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' BERTBase 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 RoBERTaLarge 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 RoBERTaBase 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 RoBERTaBase w/ MTL 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6† 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7† 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 MiniLMv2-L RoBERTaLarge 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 MiniLMv2-B RoBERTaBase 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 XtremeDistilTrans ELECTRABase* 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6† 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 MITKD RoBERTaBase w/ MTL 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Table 2: In-Domain results on the development sets of GLUE and SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We quote the result for RoBERTaBase w/ MTL, MiniLMv2-L and XtremeDistillTrans from their papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The result for BERT is taken from (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The results for RoBERTa are taken from their github2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The number with † indicates that the number is missing and we report our reproduced number using the publicly available checkpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We report F1 for SQuAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' task knowlegde and generalization led by MTL, pushing the improvement by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 average points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Together, we observe that adding MTL into task- agnostic distillation results in an impressive im- provement of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 points on out-domain tasks, com- pared to the vanilla task-agnostic distillation base- line MiniLMv2-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In addition, it even exceeds MiniLMv2-L by 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In-domain Performance Besides the general- ization improvement on out-domain datasets, MITKD also improves the performance on in- domain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Table 2 shows the dev set result on GLUE and SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' MITKD achieves state-of- the-art results on most tasks in GLUE and SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0, exceeding the best baseline by a margin of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' It even outperforms an 8x larger BERTBase on four GLUE tasks and SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 while being 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3x faster than BERTBase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 Method QNLI CHEMPROT LEDGAR SCIREC Domain In Biology Law Comp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' # of train data (1%) 1047 41 600 32 MiniLMv2-L 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 MiniLMv2-B 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 XtremeDsitilTran 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 MITKD 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 # of train data (10%) 10474 416 6000 321 MiniLMv2-L 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 MiniLMv2-B 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 XtremeDistilTran 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 MITKD 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 # of train data (50%) 52371 2084 30000 1609 MiniLMv2-L 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 MiniLMv2-B 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 XtremeDsitilTran 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 MITKD 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 Table 3: Transferability over different dataset scale set- tings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Performance on Low-resource Tasks MITKD also brings improvement on low-resource tasks in both in-domain and out-domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' To demon- strate that, we select datasets from various do- mains and vary their training data size to 1%, 10%, and 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we use QNLI for 3Refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 for the details of speedup and model size calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' in-domain, CHEMPROT for biology, LEDGAR for legal, SCIREC for computer science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Note that only QNLI has been used for MTL training among these tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The results are presented in Table 3, from which we can see that MITKD con- sistently outperforms other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Recall that XtremeDistilTrans is obtained by distilling from an MNLI-finetuned teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' It performs relatively well in QNLI which is similar to MNLI, but fails to transfer well and even underperforms the vanilla task-agnostic distillation baseline MiniLMv2-B in the 1% and 10% settings of legal (LEDGAR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' On the other hand, MITKD consistently performs the best, demonstrating that MTL brings significantly better generalization to the student on low-resource tasks than single task finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Method Teacher In-Domain Out-Domain RoBERTaLarge 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 RoBERTaBase 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 RoBERTaBase w/ MTL 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 MiniLMv2-L RoBERTaLarge 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 MiniLMv2-B RoBERTaBase 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='8 MITKD RoBERTaBase w/ MTL 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 Table 4: Summary of average scores on in-domain and out-domain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Larger Teacher or Better Knowledge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' There are many ways to improve the performance of the student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' One straightforward way is to en- large the teacher to obtain a stronger teacher with better downstream performance: for exam- ple, MiniLMv2-L outperforms MiniLMv2-B by simply switching to a larger teacher, shown in Ta- ble 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' However, several works (Mirzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) witness failure when the size gap between student and teacher are too large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' On the other hand, we propose to enhance the teacher with more generalizable knowledge through MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Table 4 shows that although our teacher has infe- rior performance compared to RoBERTaLarge, the produced student shows better performance on in- domain and out-domain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' This suggests that besides chasing after a larger and stronger teacher, we should also pay attention to the knowledge em- bedded in the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Performance on Chinese Datasets To verify the effectiveness of MITKD on Chinese datasets, we apply MITKD on ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 (Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021) to produce ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Tiny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, we follow ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 training process to reproduce a Large version of ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 and use it as the teacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' It is a 24-layer transformer with hid- den size of 1024, intermediate size of 4096, and 16 attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 28 datasets are selected for MTL finetuning the teacher and are listed in Ap- pendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' All the students in this experiment are 4-layer transformers with hidden size of 312 and 12 attention heads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The finetuning hyperparameters for the students are listed in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We compare ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Tiny with the Chinese version of TinyBERT 4 (Jiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) on the in-domain, out-domain and low-resource datasets and list out the result in Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Refer to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 for the dataset details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Tiny outperforms the baseline by a margin of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 average points, estab- lishing a new state-of-the-art result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 5 Conclusion In this work, we propose a simple method to im- prove task-agnostic distillation generalization by leveraging MTL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The teacher is first augmented by MTL and then distilled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Empirical results show that our method outperforms several baselines on in-domain, out-domain and low-resource tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 6 Limitation Compared with vanilla task-agnostic distillation, MITKD has an additional multi-task finetuning stage which may require additional computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' References Armen Aghajanyan, Anchit Gupta, Akshat Shrivastava, Xilun Chen, Luke Zettlemoyer, and Sonal Gupta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Muppet: Massive multi-task representations with pre-finetuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Natural Lan- guage Processing, pages 5799–5811, Online and Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Aida Amini, Saadia Gabriel, Shanchuan Lin, Rik Koncel-Kedziorski, Yejin Choi, and Hannaneh Ha- jishirzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' MathQA: Towards interpretable 4https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/huawei-noah/ Pretrained-Language-Model/tree/master/TinyBERT math word problem solving with operation-based formalisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 2357–2367, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Luisa Bentivogli, Peter Clark, Ido Dagan, and Danilo Giampiccolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The fifth pascal recognizing tex- tual entailment challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In TAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Bowman, Gabor Angeli, Christopher Potts, and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A large anno- tated corpus for learning natural language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2015 Conference on Empiri- cal Methods in Natural Language Processing, pages 632–642, Lisbon, Portugal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Rich Caruana.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 1997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multitask learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Machine learning, 28(1):41–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ilias Chalkidis, Abhik Jana, Dirk Hartung, Michael Bommarito, Ion Androutsopoulos, Daniel Katz, and Nikolaos Aletras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' LexGLUE: A benchmark dataset for legal language understanding in English.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 4310–4330, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Jing Chen, Qingcai Chen, Xin Liu, Haijun Yang, Daohe Lu, and Buzhou Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The BQ cor- pus: A large-scale domain-specific Chinese cor- pus for sentence semantic equivalence identifica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 4946–4951, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Christopher Clark, Kenton Lee, Ming-Wei Chang, Tom Kwiatkowski, Michael Collins, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' BoolQ: Exploring the surprising difficulty of natural yes/no questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceed- ings of NAACL-HLT 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Kevin Clark, Minh-Thang Luong, Urvashi Khandelwal, Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Manning, and Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' BAM!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' born-again multi-task networks for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computa- tional Linguistics, pages 5931–5937, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Kevin Clark, Minh-Thang Luong, Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Le, and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Electra: Pre- training text encoders as discriminators rather than generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In International Conference on Learn- ing Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Peter Clark, Isaac Cowhey, Oren Etzioni, Tushar Khot, Ashish Sabharwal, Carissa Schoenick, and Oyvind Tafjord.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Think you have solved question answering?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' try arc, the AI2 reasoning challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CoRR, abs/1803.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='05457.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Method AFQMC TNEWS IFLYTEK OCNLI CLUEWSC CSL CANLI SHOPPING10 BUSTM EPRSTMT CSLDCP Avg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Domain In-Domain Out-Domain Low-Resource All TinyBERT, Chinese 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Tiny 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 Table 5: Performance on Chinese Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' All results listed are reported on dev set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Alexis Conneau, Ruty Rinott, Guillaume Lample, Ad- ina Williams, Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Bowman, Holger Schwenk, and Veselin Stoyanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xnli: Evaluating cross- lingual sentence representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natu- ral Language Processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Marie-Catherine de Marneffe, Mandy Simons, and Ju- dith Tonhauser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The commitmentbank: Inves- tigating projection in naturally occurring discourse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Proceedings of Sinn und Bedeutung, 23(2):107–124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' BERT: Pre-training of deep bidirectional transformers for language under- standing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4171–4186, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Jay DeYoung, Sarthak Jain, Nazneen Fatema Rajani, Eric Lehman, Caiming Xiong, Richard Socher, and Byron C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Wallace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' ERASER: A benchmark to evaluate rationalized NLP models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 58th Annual Meeting of the Association for Computational Linguistics, pages 4443–4458, On- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' William B Dolan and Chris Brockett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Automati- cally constructing a corpus of sentential paraphrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the Third International Workshop on Paraphrasing (IWP2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Dheeru Dua, Yizhong Wang, Pradeep Dasigi, Gabriel Stanovsky, Sameer Singh, and Matt Gardner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Drop: A reading comprehension benchmark requir- ing discrete reasoning over paragraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' of NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Vladimir Eidelman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Billsum: A corpus for auto- matic summarization of us legislation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceed- ings of the 2nd Workshop on New Frontiers in Sum- marization, pages 48–56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Alexander Fabbri, Irene Li, Tianwei She, Suyi Li, and Dragomir Radev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multi-news: A large-scale multi-document summarization dataset and abstrac- tive hierarchical model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computa- tional Linguistics, pages 1074–1084, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Pengcheng He, Xiaodong Liu, Jianfeng Gao, and Weizhu Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Deberta: Decoding-enhanced bert with disentangled attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In International Conference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Karl Moritz Hermann, Tomas Kocisky, Edward Grefen- stette, Lasse Espeholt, Will Kay, Mustafa Suleyman, and Phil Blunsom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Teaching machines to read and comprehend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Advances in neural information processing systems, pages 1693–1701.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Geoffrey E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Hinton, Oriol Vinyals, and Jeffrey Dean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Distilling the knowledge in a neural network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CoRR, abs/1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='02531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Eduard Hovy, Laurie Gerber, Ulf Hermjakob, Chin- Yew Lin, and Deepak Ravichandran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Toward semantics-based answer pinpointing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceed- ings of the First International Conference on Human Language Technology Research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Hai Hu, Kyle Richardson, Liang Xu, Lu Li, Sandra Kuebler, and Larry Moss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ocnli: Original chinese natural language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Findings of EMNLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Lifu Huang, Ronan Le Bras, Chandra Bhagavatula, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Cosmos QA: Machine reading comprehension with contextual commonsense rea- soning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natu- ral Language Processing (EMNLP-IJCNLP), pages 2391–2401, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Shankar Iyer, Nikhil Dandekar, and Kornel Csernai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' First quora dataset release: Question pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xiaoqi Jiao, Yichun Yin, Lifeng Shang, Xin Jiang, Xiao Chen, Linlin Li, Fang Wang, and Qun Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' TinyBERT: Distilling BERT for natural lan- guage understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: EMNLP 2020, pages 4163–4174, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Mandar Joshi, Eunsol Choi, Daniel Weld, and Luke Zettlemoyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' TriviaQA: A large scale dis- tantly supervised challenge dataset for reading com- prehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 55th Annual Meet- ing of the Association for Computational Linguistics (Volume 1: Long Papers), pages 1601–1611, Van- couver, Canada.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Lin- guistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' David Jurgens, Srijan Kumar, Raine Hoover, Dan Mc- Farland, and Dan Jurafsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Measuring the evo- lution of a scientific field through citation frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Transactions of the Association for Computational Linguistics, 6:391–406.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Simran Khanuja, Melvin Johnson, and Partha Talukdar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Mergedistill: Merging language models using pre-trained distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Findings of the Associ- ation for Computational Linguistics: ACL-IJCNLP 2021, pages 2874–2887.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Daniel Khashabi, Snigdha Chaturvedi, Michael Roth, Shyam Upadhyay, and Dan Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Looking beyond the surface:a challenge set for reading com- prehension over multiple sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In NAACL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Tushar Khot, Ashish Sabharwal, and Peter Clark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SciTail: A textual entailment dataset from science question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In AAAI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Johannes Kiesel, Maria Mestre, Rishabh Shukla, Em- manuel Vincent, Payam Adineh, David Corney, Benno Stein, and Martin Potthast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SemEval- 2019 task 4: Hyperpartisan news detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 13th International Workshop on Semantic Evaluation, pages 829–839, Minneapo- lis, Minnesota, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Diederik P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Kingma and Jimmy Ba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Adam: A method for stochastic optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In 3rd Inter- national Conference on Learning Representations, ICLR 2015, San Diego, CA, USA, May 7-9, 2015, Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Jens Kringelum, Sonny Kim Kjaerulff, Søren Brunak, Ole Lund, Tudor I Oprea, and Olivier Taboureau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' ChemProt-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0: a global chemical biology dis- eases mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Database : the journal of biological databases and curation, 2016:bav123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Tom Kwiatkowski, Jennimaria Palomaki, Olivia Red- field, Michael Collins, Ankur Parikh, Chris Alberti, Danielle Epstein, Illia Polosukhin, Matthew Kelcey, Jacob Devlin, Kenton Lee, Kristina N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Toutanova, Llion Jones, Ming-Wei Chang, Andrew Dai, Jakob Uszkoreit, Quoc Le, and Slav Petrov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Natu- ral questions: a benchmark for question answering research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Transactions of the Association of Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Guokun Lai, Qizhe Xie, Hanxiao Liu, Yiming Yang, and Eduard Hovy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' RACE: Large-scale ReAd- ing comprehension dataset from examinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2017 Conference on Empirical Methods in Natural Language Processing, pages 785–794, Copenhagen, Denmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Hector Levesque, Ernest Davis, and Leora Morgen- stern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The winograd schema challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Thirteenth International Conference on the Princi- ples of Knowledge Representation and Reasoning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Citeseer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Lei Li, Yankai Lin, Shuhuai Ren, Peng Li, Jie Zhou, and Xu Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Dynamic knowledge distillation for pre-trained language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2021 Conference on Empirical Methods in Nat- ural Language Processing, pages 379–389, Online and Punta Cana, Dominican Republic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xin Li and Dan Roth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Learning question clas- sifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In COLING 2002: The 19th International Conference on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yudong Li, Yuqing Zhang, Zhe Zhao, Linlin Shen, Wei- jie Liu, Weiquan Mao, and Hui Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CSL: A large-scale Chinese scientific literature dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 29th International Conference on Computational Linguistics, pages 3917–3923, Gyeongju, Republic of Korea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' International Com- mittee on Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Bang Liu, Di Niu, Haojie Wei, Jinghong Lin, Yancheng He, Kunfeng Lai, and Yu Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Matching ar- ticle pairs with graphical decomposition and convo- lutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meet- ing of the Association for Computational Linguis- tics, pages 6284–6294, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Linqing Liu, Huan Wang, Jimmy Lin, Richard Socher, and Caiming Xiong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Mkd: a multi-task knowledge distillation approach for pretrained lan- guage models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:1911.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='03588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xiaodong Liu, Pengcheng He, Weizhu Chen, and Jian- feng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multi-task deep neural networks for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, pages 4487–4496, Flo- rence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguis- tics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xin Liu, Qingcai Chen, Chong Deng, Huajun Zeng, Jing Chen, Dongfang Li, and Buzhou Tang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' LCQMC:a large-scale Chinese question matching corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 27th International Conference on Computational Linguistics, pages 1952–1962, Santa Fe, New Mexico, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yinhan Liu, Myle Ott, Naman Goyal, Jingfei Du, Man- dar Joshi, Danqi Chen, Omer Levy, Mike Lewis, Luke Zettlemoyer, and Veselin Stoyanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Roberta: A robustly optimized bert pretraining ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:1907.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='11692.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yi Luan, Luheng He, Mari Ostendorf, and Hannaneh Hajishirzi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multi-task identification of enti- ties, relations, and coreference for scientific knowl- edge graph construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Lan- guage Processing, pages 3219–3232, Brussels, Bel- gium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Minh-Thang Luong, Quoc V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Le, Ilya Sutskever, Oriol Vinyals, and Lukasz Kaiser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multi- task sequence to sequence learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In 4th Inter- national Conference on Learning Representations, ICLR 2016, San Juan, Puerto Rico, May 2-4, 2016, Conference Track Proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Andrew L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Maas, Raymond E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Daly, Peter T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Pham, Dan Huang, Andrew Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ng, and Christopher Potts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Learning word vectors for sentiment analy- sis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 49th Annual Meeting of the Association for Computational Linguistics: Hu- man Language Technologies, pages 142–150, Port- land, Oregon, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Pekka Malo, Ankur Sinha, Pekka Korhonen, Jyrki Wal- lenius, and Pyry Takala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Good debt or bad debt: Detecting semantic orientations in economic texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Journal of the Association for Information Sci- ence and Technology, 65(4):782–796.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Julian J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' McAuley, Christopher Targett, Qinfeng Shi, and Anton van den Hengel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Image-based recommendations on styles and substitutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CoRR, abs/1506.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='04757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Thomas McCoy, Ellie Pavlick, and Tal Linzen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Right for the wrong reasons: Diagnosing syntac- tic heuristics in natural language inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CoRR, abs/1902.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='01007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Todor Mihaylov, Peter Clark, Tushar Khot, and Ashish Sabharwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Can a suit of armor conduct elec- tricity?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' a new dataset for open book question an- swering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 2381–2391, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Seyed Iman Mirzadeh, Mehrdad Farajtabar, Ang Li, Nir Levine, Akihiro Matsukawa, and Hassan Ghasemzadeh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Improved knowledge distilla- tion via teacher assistant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In AAAI 2020 - 34th AAAI Conference on Artificial Intelligence, AAAI 2020 34th AAAI Conference on Artificial Intelligence, pages 5191–5198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' AAAI press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Subhabrata Mukherjee, Ahmed Hassan Awadallah, and Jianfeng Gao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xtremedistiltransformers: Task transfer for task-agnostic distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='04563.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Shashi Narayan, Shay B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Cohen, and Mirella Lapata.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Don’t give me the details, just the summary!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' topic-aware convolutional neural networks for ex- treme summarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Lan- guage Processing, pages 1797–1807, Brussels, Bel- gium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yixin Nie, Adina Williams, Emily Dinan, Mohit Bansal, Jason Weston, and Douwe Kiela.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ad- versarial NLI: A new benchmark for natural lan- guage understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 58th An- nual Meeting of the Association for Computational Linguistics, pages 4885–4901, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Bo Pang and Lillian Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Seeing stars: Ex- ploiting class relationships for sentiment categoriza- tion with respect to rating scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceed- ings of the 43rd Annual Meeting of the Association for Computational Linguistics (ACL’05), pages 115– 124, Ann Arbor, Michigan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Mohammad Taher Pilehvar and Jose Camacho- Collados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WiC: the word-in-context dataset for evaluating context-sensitive meaning represen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 1267–1273, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Maria Pontiki, Dimitris Galanis, Haris Papageorgiou, Ion Androutsopoulos, Suresh Manandhar, Moham- mad AL-Smadi, Mahmoud Al-Ayyoub, Yanyan Zhao, Bing Qin, Orphée De Clercq, Véronique Hoste, Marianna Apidianaki, Xavier Tannier, Na- talia Loukachevitch, Evgeniy Kotelnikov, Nuria Bel, Salud María Jiménez-Zafra, and Gül¸sen Eryi˘git.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SemEval-2016 task 5: Aspect based senti- ment analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 10th Interna- tional Workshop on Semantic Evaluation (SemEval- 2016), pages 19–30, San Diego, California.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Associa- tion for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Pranav Rajpurkar, Robin Jia, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Know what you don’t know: Unanswerable ques- tions for SQuAD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 56th An- nual Meeting of the Association for Computational Linguistics (Volume 2: Short Papers), pages 784– 789, Melbourne, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Pranav Rajpurkar, Jian Zhang, Konstantin Lopyrev, and Percy Liang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SQuAD: 100,000+ questions for machine comprehension of text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2016 Conference on Empirical Methods in Natu- ral Language Processing, pages 2383–2392, Austin, Texas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Melissa Roemmele, Cosmin Adrian Bejan, and An- drew S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Gordon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Choice of plausible alterna- tives: An evaluation of commonsense causal reason- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In 2011 AAAI Spring Symposium Series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Victor Sanh, Lysandre Debut, Julien Chaumond, and Thomas Wolf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Distilbert, a distilled version of BERT: smaller, faster, cheaper and lighter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CoRR, abs/1910.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='01108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Richard Socher, Alex Perelygin, Jean Wu, Jason Chuang, Christopher D Manning, Andrew Ng, and Christopher Potts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Recursive deep models for semantic compositionality over a sentiment tree- bank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2013 conference on empirical methods in natural language processing, pages 1631–1642.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Weiyue Su, Xuyi Chen, Shikun Feng, Jiaxiang Liu, Weixin Liu, Yu Sun, Hao Tian, Hua Wu, and Haifeng Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ernie-tiny: A progressive dis- tillation framework for pretrained transformer com- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:2106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='02241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Siqi Sun, Yu Cheng, Zhe Gan, and Jingjing Liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Patient knowledge distillation for BERT model com- pression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natu- ral Language Processing (EMNLP-IJCNLP), pages 4323–4332, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yu Sun, Shuohuan Wang, Shikun Feng, Siyu Ding, Chao Pang, Junyuan Shang, Jiaxiang Liu, Xuyi Chen, Yanbin Zhao, Yuxiang Lu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ernie 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0: Large-scale knowledge enhanced pre-training for language understanding and generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='02137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Alon Talmor, Jonathan Herzig, Nicholas Lourie, and Jonathan Berant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CommonsenseQA: A ques- tion answering challenge targeting commonsense knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Volume 1 (Long and Short Papers), pages 4149–4158, Minneapolis, Minnesota.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Associ- ation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Raphael Tang, Yao Lu, Linqing Liu, Lili Mou, Olga Vechtomova, and Jimmy Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Distilling task- specific knowledge from bert into simple neural net- works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:1903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='12136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Don Tuggener, Pius von Däniken, Thomas Peetz, and Mark Cieliebak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' LEDGAR: A large-scale multi-label corpus for text classification of legal pro- visions in contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 12th Lan- guage Resources and Evaluation Conference, pages 1235–1241, Marseille, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' European Language Resources Association.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Alex Wang, Amanpreet Singh, Julian Michael, Fe- lix Hill, Omer Levy, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' GLUE: A multi-task benchmark and analysis plat- form for natural language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Pro- ceedings of the 2018 EMNLP Workshop Black- boxNLP: Analyzing and Interpreting Neural Net- works for NLP, pages 353–355, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Wenhui Wang, Hangbo Bao, Shaohan Huang, Li Dong, and Furu Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' MiniLMv2: Multi-head self- attention relation distillation for compressing pre- trained transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Findings of the Association for Computational Linguistics: ACL-IJCNLP 2021, pages 2140–2151, Online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Wenhui Wang, Furu Wei, Li Dong, Hangbo Bao, Nan Yang, and Ming Zhou.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Minilm: Deep self- attention distillation for task-agnostic compression of pre-trained transformers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Advances in Neural Information Processing Systems 33: Annual Con- ference on Neural Information Processing Systems 2020, NeurIPS 2020, December 6-12, 2020, virtual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Alex Warstadt, Amanpreet Singh, and Samuel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Bow- man.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Neural network acceptability judgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Transactions of the Association for Computational Linguistics, 7:625–641.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Jason Wei, Maarten Bosma, Vincent Zhao, Kelvin Guu, Adams Wei Yu, Brian Lester, Nan Du, Andrew M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Dai, and Quoc V Le.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Finetuned language models are zero-shot learners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In International Con- ference on Learning Representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Johannes Welbl, Nelson F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Liu, and Matt Gardner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Crowdsourcing multiple choice science questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 3rd Workshop on Noisy User- generated Text, pages 94–106, Copenhagen, Den- mark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Adina Williams, Nikita Nangia, and Samuel Bowman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A broad-coverage challenge corpus for sen- tence understanding through inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceed- ings of the 2018 Conference of the North American Chapter of the Association for Computational Lin- guistics: Human Language Technologies, Volume 1 (Long Papers), pages 1112–1122.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Liang Xu, Hai Hu, Xuanwei Zhang, Lu Li, Chenjie Cao, Yudong Li, Yechen Xu, Kai Sun, Dian Yu, Cong Yu, Yin Tian, Qianqian Dong, Weitang Liu, Bo Shi, Yiming Cui, Junyi Li, Jun Zeng, Rongzhao Wang, Weijian Xie, Yanting Li, Yina Patterson, Zuoyu Tian, Yiwen Zhang, He Zhou, Shaoweihua Liu, Zhe Zhao, Qipeng Zhao, Cong Yue, Xinrui Zhang, Zhengliang Yang, Kyle Richardson, and Zhenzhong Lan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CLUE: A Chinese language understanding evaluation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceed- ings of the 28th International Conference on Com- putational Linguistics, pages 4762–4772, Barcelona, Spain (Online).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' International Committee on Compu- tational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Liang Xu, Xiaojing Lu, Chenyang Yuan, Xuanwei Zhang, Huilin Xu, Hu Yuan, Guoao Wei, Xiang Pan, Xin Tian, Libo Qin, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Fewclue: A chi- nese few-shot learning evaluation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='07498.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Shanshan Xu and Katja Markert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The Chinese causative-passive homonymy disambiguation: an ad- versarial dataset for NLI and a probing task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the Thirteenth Language Resources and Evaluation Conference, pages 4316–4323, Mar- seille, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' European Language Resources Asso- ciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yi Yang, Wen-tau Yih, and Christopher Meek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WikiQA: A challenge dataset for open-domain ques- tion answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2015 Con- ference on Empirical Methods in Natural Language Processing, pages 2013–2018, Lisbon, Portugal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' As- sociation for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yinfei Yang, Yuan Zhang, Chris Tar, and Jason Baldridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' PAWS-X: A cross-lingual ad- versarial dataset for paraphrase identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2019 Conference on Empirical Methods in Natural Language Processing and the 9th International Joint Conference on Natural Lan- guage Processing (EMNLP-IJCNLP), pages 3687– 3692, Hong Kong, China.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Zhilin Yang, Peng Qi, Saizheng Zhang, Yoshua Ben- gio, William W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Cohen, Ruslan Salakhutdinov, and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Manning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' HotpotQA: A dataset for diverse, explainable multi-hop question answering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Conference on Empirical Methods in Natural Language Processing (EMNLP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Rowan Zellers, Yonatan Bisk, Roy Schwartz, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SWAG: A large-scale adversar- ial dataset for grounded commonsense inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 2018 Conference on Empirical Methods in Natural Language Processing, pages 93– 104, Brussels, Belgium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computa- tional Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Rowan Zellers, Ari Holtzman, Yonatan Bisk, Ali Farhadi, and Yejin Choi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' HellaSwag: Can a machine really finish your sentence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Pro- ceedings of the 57th Annual Meeting of the Asso- ciation for Computational Linguistics, pages 4791– 4800, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Ningyu Zhang, Mosha Chen, Zhen Bi, Xiaozhuan Liang, Lei Li, Xin Shang, Kangping Yin, Chuanqi Tan, Jian Xu, Fei Huang, Luo Si, Yuan Ni, Guo- tong Xie, Zhifang Sui, Baobao Chang, Hui Zong, Zheng Yuan, Linfeng Li, Jun Yan, Hongying Zan, Kunli Zhang, Buzhou Tang, and Qingcai Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CBLUE: A Chinese biomedical language un- derstanding evaluation benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers), pages 7888–7915, Dublin, Ireland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Rui Zhang and Joel Tetreault.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' This email could save your life: Introducing the task of email subject line generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Proceedings of the 57th Annual Meeting of the Association for Computational Lin- guistics, pages 446–456, Florence, Italy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Sheng Zhang, Xiaodong Liu, Jingjing Liu, Jianfeng Gao, Kevin Duh, and Benjamin Van Durme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Record: Bridging the gap between human and ma- chine commonsense reading comprehension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' arXiv preprint arXiv:1810.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='12885.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Xiang Zhang, Junbo Zhao, and Yann LeCun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Character-level convolutional networks for text clas- sification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In Advances in Neural Information Pro- cessing Systems, volume 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 Out-domain Datasets We evaluate our model on CHEMPROT (Kringelum et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2016) for Biology, ANLI (Nie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) for web, ACL-ARC (Jurgens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) and SCIERC (Luan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) for computer science, PARTISAN (Kiesel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) for news, HELPFUL (McAuley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) for review, SCOTUS(Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) and LEDGAR (Tuggener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) for law, and FinBank (Malo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2014) for finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2 Hyperparameters We use Adam (Kingma and Ba, 2015) optimizer with β1 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='9 and β2 of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='999 for all tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' We use linear warm up over the first 10% of training steps and linear decay for the rest of training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The dropout rate is set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The weight decay is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 The learning rate ranges from {3e-5, 6e-5, 8e-5, 9e-5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The batch size is set to 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The training epoch is set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Others For all other tasks, the epoch ranges from {3, 5, 10}, the batch size ranges from {16, 32, 48}, learning rate ranges from {1e-5, 2e-5, 5e-5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' For CoLA, we finetune it with training epoch of 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='3 Model size and Speed Calculation When calculating the size of a model, we count the number of all the learnable parameters in the model except the parameters of the embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' As for the speedup, we quote the number from (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='4 Datasets Used for Fintuning Teacher 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' MNLI (Williams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CoLA (Warstadt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SST (Socher et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2013) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' MRPC (Dolan and Brockett, 2005) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' QQP (Iyer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2017) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' QNLI (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2016) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' RTE (Bentivogli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2009) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WNLI (Levesque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2012) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Bool Q (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019a) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' MultiRC (Khashabi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' ReCoRD (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WIC (Pilehvar and Camacho-Collados, 2019) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WSC (Levesque et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2012) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CB (de Marneffe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' COPA (Roemmele et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2011) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' AG News (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' IMDB (Maas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2011) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SNLI (Bowman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' HANS (McCoy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Rotten Tomatoes (Pang and Lee, 2005) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Yelp Polarity (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Eraser Multi RC (DeYoung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Wiki QA (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Trec (Li and Roth, 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Hovy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2001) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SciTail (Khot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CNN Daily Mail (Hermann et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Billsum (Eidelman, 2019) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' XSUM (Narayan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Aeslc (Zhang and Tetreault, 2019) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Multinews (Fabbri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Math QA (Amini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Openbook QA (Mihaylov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SWAG (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' HellaSWAG (Zellers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' RACE (Lai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2017) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CommonSense QA (Talmor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Cosmos QA (Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' AI2 ARC - Easy (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' AI2 ARC - Challenge (Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SCIQ (Welbl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2017) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SQUAD (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2016) 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' NQ (Kwiatkowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' DROP (Dua et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Hotpot (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' Trivia QA (Joshi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2017) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='5 Datasets Used for Fintuning ERNIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0 Titan 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' XNLI (Conneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' OCNLI (Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' KUAKE-QQR (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' KUAKE-QTR (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CMNLI (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' PAWS-X (Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CHIP-STS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CSL (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CCKS2018-Task3 5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CHIP2019-Task2 6 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' ZHONGCE: It is an internal semantic textual similarity dataset, consisting of 1025327 train- ing examples and 8803 evaluation examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' QBQTC 7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' LCQMC (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' BQ (Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2018) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' AFQMC (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WAIMAI 8 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' WEIBO 9 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' NLPC2014-Task2 10 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SemEval2016-Task5-PHNS (Pontiki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2016) 5http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='sigkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='cn/ccks2018/?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='page_id=16 6http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='cips-chip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='cn:8000/evaluation 7https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/CLUEbenchmark/QBQTC 8https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/SophonPlus/ ChineseNlpCorpus/tree/master/datasets/waimai_10k 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/SophonPlus/ ChineseNlpCorpus/tree/master/datasets/weibo_ senti_100k 10http://tcci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='ccf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='cn/conference/2014/pages/ page04_dg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='html Task AFQMC TNEWS IFLYTEK OCNLI CLUEWSC CSL Epoch 3 3 3 5 50 5 Dropout Rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 {0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1} 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='1 Table 6: Epoch and dropout rate settings for chinese datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SemEval2016-Task5-CAME (Pontiki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2016) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' THUCNEWS 11 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' TNEWS (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' IFLYTEK (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' SOGOUNEWS (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2015) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CHIP-CTC (Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2022) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CNSE (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019a) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CNSS (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2019a) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' CLUEWSC (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='6 Chinese Evaluation Datasets We select the classification tasks in CLUE Bench- mark (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2020) for the evaluation on in-domain datasets, and CANLI (Xu and Mark- ert, 2022) and SHOPPING1012 for the out- domain datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In addition, we select BUSTM, EPRSTMT and CSLDCP from FewCLUE (Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=', 2021), a Chinese few-shot learning bench- mark, for the low-resource datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' For the low- resource datasets, we merge the train sets and dev sets from all sub-parts into one train set and dev set, and use them for training and evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='7 Hypaerparameters for Chinese Datasets For the in-domain datasets, we follow the hyper- parameters settings in PaddleNLP GitHub repos- itory13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' In particular, the batch size ranges from {16, 32, 64}, learning rate ranges from {1e-5, 2e-5, 3e-5, 5e-5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' The epoch and dropout rate for each task are listed in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' All other settings such as optimizer, and learning rate are the same as those in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' For all other tasks, we use the same hyperparameters in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content=' 11http://thuctc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='thunlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='org 12https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/SophonPlus/ ChineseNlpCorpus/tree/master/datasets/online_ shopping_10_cats 13https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} +page_content='com/PaddlePaddle/PaddleNLP/ tree/develop/examples/benchmark/clue' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/atE1T4oBgHgl3EQfxAUb/content/2301.03416v1.pdf'} diff --git a/cdAyT4oBgHgl3EQf-foD/content/2301.00891v1.pdf b/cdAyT4oBgHgl3EQf-foD/content/2301.00891v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fa2d58f0f916c30a53a6a95fee53a75c47e38225 --- /dev/null +++ b/cdAyT4oBgHgl3EQf-foD/content/2301.00891v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:42ac3c8148621028e0dc61f53a5b77eb7b97599f015bd30fd84e7f7748c22e0a +size 1091497 diff --git a/cdAyT4oBgHgl3EQf-foD/vector_store/index.pkl b/cdAyT4oBgHgl3EQf-foD/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..bad7cd054ddab3123d5693cbe61911c46258f356 --- /dev/null +++ b/cdAyT4oBgHgl3EQf-foD/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:851bcaf5f4e63171e49d49f339aa8068fa6f7bc13a20569550d41f9bd006f3de +size 83212 diff --git a/cdE3T4oBgHgl3EQfGQmo/content/tmp_files/2301.04313v1.pdf.txt b/cdE3T4oBgHgl3EQfGQmo/content/tmp_files/2301.04313v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..f325d4d67c83bfd05f72340336a2a2906b40e922 --- /dev/null +++ b/cdE3T4oBgHgl3EQfGQmo/content/tmp_files/2301.04313v1.pdf.txt @@ -0,0 +1,2806 @@ +arXiv:2301.04313v1 [math.AT] 11 Jan 2023 +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +MORGAN OPIE +Abstract. We show that complex rank 3 topological vector bundles on CP 5 are determined +by their Chern classes, except in the case that c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3). To +address this case, we produce a universal class in the tmf(3)-cohomology of a Thom spectrum +related to BU(3), where tmf(3) denotes topological modular forms localized at 3. From this +class and orientation data, we construct a Z/3-valued invariant of the bundles of interest and +prove that our invariant separates distinct bundles with the same Chern classes. +Contents +1. +Introduction +1 +1.1. +Paper outline +5 +1.2. +Acknowledgements +5 +1.3. +Conventions +6 +2. +A count of rank 3 bundles on CP 5 +6 +2.1. +3-complete rank 3 vector bundles on CP 5 +7 +2.2. +Proof of technical claims +9 +2.3. +2-complete rank 3 vector bundles on CP 5 +12 +2.4. +The Schwarzenberger conditions +13 +3. +Defining a twisted tmf(3)-valued invariant +16 +3.1. +Proof outline: existence and uniqueness of a twisted tmf(3) invariant +18 +3.2. +Proof of Proposition 3.11 +20 +3.3. +The cohomology of Th(BU(3)c1≡0; −γ3) and related spectra +25 +3.4. +Proof of Proposition 3.13 +28 +4. +Untwisting the invariant for rank 3 bundles on CP 5 +30 +4.1. +Background on orientability and orientations +31 +4.2. +Selecting orientations and the definition of ρ +31 +4.3. +The invariant ρ separates Chern classes for rank 3 bundles on CP 5 +35 +4.4. +Computing ρ on certain sums of line bundles +38 +4.5. +A 3-torsion tmf(3)-valued invariant for rank 2 bundles +39 +References +39 +1. Introduction +Let X be a finite-dimensional CW-complex. From the perspective of homotopy theory, a +topological vector bundle of complex rank r over a space X is identified with a classifying +map X → BU(r). Topologically equivalent vector bundles over X correspond to homotopy +equivalent maps to BU(r). +The integral cohomology of BU(r) is generated by universal Chern classes c1, . . . cr, with +ci ∈ H2i(BU(r); Z). These give rise to important invariants of complex bundles: the Chern +1 + +2 +MORGAN OPIE +classes of a bundle, defined for V : X → BU(r) as the pullbacks +ci(V ) := V ∗(ci) ∈ H2i(X; Z). +In the case X = CP n, Chern classes are complete invariants of the stable equivalence class +of the bundle. Explicitly, this means that V : CP n → BU(r) and W : CP n → BU(r′) have the +same Chern classes if and only if there exist integers n, m greater than zero such that V ⊕ Cn +and W ⊕ Cm are topologically equivalent. Here, C is the trivial rank 1 bundle on X. +This leads to the following fundamental question: +Question 1.1. Are Chern classes sufficient to determine the (unstable) topological class of a +complex rank r vector bundle on CP n, up to topological equivalence? If not, what invariants +beyond Chern classes are needed to distinguish such bundles? +Rank 1 bundles on all spaces CP n are determined by their first Chern class [3, 14]. Rank +≥ n bundles on CP n are also determined by their Chern classes. For r strictly between 1 and +n, there is no uniform answer (although some patterns have been found when restricting to +bundles with all Chern classes zero, see [13]). +In [5], Atiyah and Rees answer Question 1.1 for complex rank 2 topological bundles on CP 3 +by producing a Z/2-valued invariant α, which can be viewed as a characteristic class in the +generalized cohomology of a classifying space. +Theorem 1.2 ([5, Theorem 2.8 and 3.3]). Given a1, a2 ∈ Z with a1a2 ≡ 0 (mod 2), the number +of rank 2 bundles on CP 3 with i-th Chern class ai is: +• equal to 2 if a1 ≡ 0 (mod 2); and +• equal to 1 otherwise. +In the first case, a rank 2 vector bundle on CP 3 is determined by c1, c2, and α. +Remark 1.3. The condition a1a2 ≡ 0 (mod 2) is necessary and sufficient for two integers to +be the Chern classes of a rank 2 bundle on CP 3. +Atiyah–Rees’ works shows that the classification of rank 2 bundles on CP 3 is a 2-primary +problem. Since there are similarities between the 2-primary homotopical structure of BU(2) +and the 3-primary homotopical structure of BU(3), one might hope for an analogy between the +classification of rank 2 bundles on CP 3 and of rank 3 bundles on CP 5. Our goal is to realize +this analogy and answer Question 1.1 for rank 3 bundles on CP 5. We do this by defining a +Z/3-valued invariant ρ of such bundles and proving the following: +Theorem 1.4. Given a1, a2, a3 ∈ Z satisfying the the Schwarzenberger condition S5 (see +Lemma 2.16), the number of bundles of rank 3 on CP 5 with i-th Chern class equal to ai is: +• equal to 3 if a1 ≡ 0 (mod 3) and a2 ≡ 0 (mod 3); and +• equal to 1 otherwise. +In the first case, a rank 3 bundle on CP 5 is determined by c1, c2, c3 and ρ. +Remark 1.5. In Subsection 2.4, we show that the Schwarzenberger condition S5 is necessary +and sufficient for three integers to be the Chern classes of a rank 3 bundle on CP 5. We also +give S5 explicitly in this section. +A priori, there is no simple geometric relationship between topologically distinct bundles +with the same Chern classes. However, in both the case of rank 2 bundles on CP 3 and the case +of rank 3 bundles on CP 5, any two bundles with the same Chern classes differ by an explicit +action defined as follows. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +3 +Construction 1.6. Associated to an inclusion D2n ֒→ CP n of a disk in the top cell of CP n, +we define +Q: CP n → S2n ∨ CP n +by collapsing the boundary of D2n to a point. Given vector bundles V : CP n → BU(n) and +σ: S2n → BU(r) we define +σV := (σ ∨ V ) ◦ Q: CP n → BU(r). +Diagrammatically: +S2n +CP n +S2n ∨ CP n +BU(r) +CP n +ι1 +σ +Q +σ∨V +ι2 +V +where ι1 and ι2 are the standard maps of the summands into the wedge. The association +(σ, V ) �→ σV +defines an action of π2nBU(r) on equivalence classes of rank r vector bundles over CP n. +This action preserves Chern classes provided that n > r. Therefore, if nontrivial, this action +gives topologically distinct bundles with the same Chern classes. +In the case that r = 2 and n = 3, the action of π6BU(2) ≃ Z/2 on rank 2 bundles on CP 3 +with fixed Chern data is transitive, and is free if and only if c1 ≡ 0 (mod 2). This aligns with +the enumeration of bundles with fixed Chern data in Theorem 1.2 above. The theorem below +shows that the role of Construction 1.6 in analyzing rank 3 bundles on CP 5 is analogous. +Theorem 1.7. Let a1, a2, and a3 be integers satisfying S5. Let Va1,a2,a3 be the set of homotopy +classes of rank 3 bundles on CP 5 with i-th Chern class equal to ai. Then: +(1) The action of π10BU(3) ≃ Z/3 on rank 3 bundles over CP 5, as given in Construc- +tion 1.6, induces a transitive action on Va1,a2,a3. +(2) If a1 or a2 is nonzero mod 3, then the action of π10BU(3) on Va1,a2,a3 is trivial. +(3) If a1 and a2 are zero mod 3, then the action of π10BU(3) on Va1,a2,a3 is free. +This refines the enumeration result in Theorem 1.4. +Theorem 1.7 says that, if a1, a2, a3 satisfy S5, a1 ≡ 0 (mod 3), and a2 ≡ 0 (mod 3), then +the set of complex rank 3 topological vector bundles on CP 5 with i-th Chern class ai is a torsor +for Z/3. The goal of the rest of the paper is to trivialize this torsor via a bundle invariant. To +explain our approach to defining such an invariant for rank 3 bundles on CP 5, we discuss the +α-invariant of rank 2 bundles on CP 3 in greater detail. +The Atiyah–Rees invariant α is initially defined for rank 2 bundles with c1 = 0. +Such +bundles are classified by maps to BSU(2), allowing an invariant to be defined via a universal +class in the generalized cohomology of BSU(2) rather than BU(2). Atiyah and Rees give a +class α ∈ KO4(BSU(2)), where KO denotes real K-theory. They define the α-invariant of +V : CP 3 → BSU(2) as +α(V ) := p∗V ∗(α) ∈ KO−2(point) ≃ Z/2, +where V ∗ is pullback with respect to V and p∗ : KO∗(CP 3) → KO∗−6(point) is the KO-theory +pushforward for the spin manifold CP 3. They extend α to bundles with c1(V ) ≡ 0 (mod 2) by +α(V ) := α +� +V ⊗ O(−c1(V ) +2 +) +� +. + +4 +MORGAN OPIE +Alternatively, the Atiyah–Rees invariant can be rephrased as a twisted characteristic class. +Recall that, given a virtual bundle W over a space X, the Thom spectrum of X with respect to +W, written Th(X; W), can be viewed as a twisted version of the suspension spectrum Σ∞ ++ X. +By a twisted characteristic class, we will mean a class in some generalized cohomology of a +Thom spectrum over a classifying space. +In this framework, one can show that there is a class ˜α ∈ KO∗(Th(BU(2); −γ2)) which +extends α in a precise sense. Given any rank 2 vector bundle on CP 3, the pullback of ˜α gives +a class +˜α(V ) := V ∗ ˜α ∈ KO4(Th(CP 3; −V )). +If c1(V ) ≡ 0 (mod 2), V is canonically KO-oriented, yielding a KO-Thom isomorphism +KO∗(Th(CP 3; −V )) ≃ KO∗(Σ∞ ++ CP 3). +We can thus define +α′(V ) = p∗(˜α(V )) ∈ KO−2(point) ≃ Z/2, +where as before p∗ is the KO-theory pushforward. The invariant α′ also distinguishes rank 2 +bundles on CP 3 with c1 ≡ 0 (mod 2) and agrees with the original α-invariant when c1(V ) = 0. +The insight here is that both BSU(2) and Th(BU(2); −γ2) stabilize π6BU(2), in the follow- +ing sense. While π6BU(2) ≃ Z/2, the stable homotopy group π6 (Σ∞BU(2)) is trivial, so bun- +dles differing by an element in the unstable group π6BU(2) cannot be distinguished by a char- +acteristic class in the generalized cohomology of BU(2) itself. However, both π6 (Σ∞BSU(2)) +and π6 Th(BU(2); −γ) are nontrivial and are canonically isomorphic to π6BU(2), permitting +their KO-cohomology to supply the classes α and α′, respectively. +Recalling Theorem 1.7, the relevant group for understanding rank 3 bundles on CP 5 is +π10BU(3). We also find that π10BU(3) ≃ Z/3 is stably trivial. By the above discussion, we +might attempt to classify rank 3 bundles on CP 5 by first stabilizing π10BU(3) and then detect- +ing it with some generalized cohomology theory. Indeed, our strategy to define an invariant of +rank 3 bundles on CP 5 is as follows: +• We identify a Thom spectrum related to BU(3) which stabilizes π10BU(3) (Introduction +to Section 3); +• We define a twisted characteristic class in an appropriate generalized cohomology of +this Thom spectrum, with certain key properties (Section 3); and +• We show that our twisted characteristic class can be resolved to an honest invariant ρ, +via orientation data, and that this invariant distinguishes vector bundles with the same +Chern data (Section 4). +The main result of Section 3 can be stated as follows: +Theorem 1.8. Let BU(3)c1≡0 be the homotopy fiber of c1 (mod 3): BU(3) → K(Z/3, 2). Let +tmf(3) denote the 3-localization of the spectrum of topological modular forms. There is a class +˜ρ ∈ tmf(3) +−3(sk26 Th(BU(3)c1≡0; −γ3)) +such that the pullback of ˜ρ with respect to the Thomificiation of a generator for π10BU(3)c1≡0 +induces an isomorphism +(1.1) +π10BU(3)c1≡0 ≃ π13 tmf(3) . +The class ˜ρ and isomorphism (1.1) tell us that the cohomology theory tmf(3) stably detects +π10BU(3) and therefore retains information about the bundles of interest. Under pullback, +the class ˜ρ together with Thom isomorphisms determined by orientation data give rise to the +invariant ρ of Theorem 1.4, as follows. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +5 +Theorem 1.8 gives an association +(1.2) +V �→ Th(V )∗(˜ρ) ∈ tmf(3) +−3(Th(CP 5; −V )), +where Th(V ) denotes the Thomification of the classifying map V : CP 5 → BU(3). +Equa- +tion (1.2) does not define an invariant of V because the target depends on V itself. However, +vector bundles with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) are tmf(3)-orientable and therefore +admit a tmf(3)-Thom isomorphisms tmf(3) +∗(Th(CP 5; −V )) ≃ tmf(3) +∗(Σ∞ ++ CP 5). The problem is +not quite solved: we need a consistent way of choosing Thom isomorphisms. This is the main +project of Section 4, which involves a detailed study of tmf(3)-orientations for the relevant bun- +dles. The problem cannot be reduced to a known orientation problem (e.g. using the celebrated +string orientation for topological modular forms [2]). +1.1. Paper outline. +The proof of Theorem 1.7 is the main project of Section 2 and proceeds via analyses of the +set of homotopy classes of maps from CP 5 to BU(3) localized at the primes 3 and 2. These +arguments are carried out in Subsections 2.1 and 2.3, respectively, and involve obstruction- +theoretic arguments. Subsection 2.2 proves claims used in Subsection 2.1. In Subsection 2.4 we +compute the Schwarzenberger condition explicitly and show that it is necessary and sufficient +for three integers to be the Chern classes of a rank 3 bundle on CP 5. +In Subsection 3.1, we outline our method to produce the class ˜ρ in the tmf(3)-cohomology of +sk26 Th(BU(3)c1≡0; −γ3). The remaining Subsections 3.2, 3.3, and 3.4 supply the details of the +proof, which includes a uniqueness result. This concludes the proof of Theorem 1.8. +In Subsection 4.1, we review the theory of Thom isomorphisms and orientations and also +establish notation. In Subsection 4.2, we study orientations of rank 3 bundles on CP 5 with +c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) and isolate a desirable set of tmf(3)-orientations. Using this +set of orientations, we are able to produce a well-defined invariant: in Section 4.3, we combine +orientation data with ˜ρ to define the invariant ρ of complex rank 3 topological bundles on CP 5 +with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3). We prove that ρ separates topological equivalence +classes of rank 3 bundles on CP 5 with the same Chern data. This completes the proof of +Theorem 1.4. +The remaining subsections offer examples and suggest future directions. In Subsection 4.4, +we show that ρ(L⊕3) = 0 for L a line bundle with c1(L) ≡ 0 (mod 3). +We also state an +additivity result for ρ on sums of line bundles. In Subsection 4.5, we show that the methods +discussed in this paper also produce a 3-local invariant of rank 2 bundles. +1.2. Acknowledgements. +First and foremost I want to thank my PhD advisor, Mike Hopkins, for suggesting this project +and for his immense support throughout my PhD program. I am also immensely grateful to +Haynes Miller for his mentorship during my time in graduate school; and to both Haynes and +Elden Elmanto for serving on my dissertation committee and offering feedback on my thesis +write-up. After my move to UCLA, Mike Hill’s guidance and encouragement – mathematical +and practical – were invaluable for me while improving and revising my thesis. Hood Chatham +and Jeremy Hahn were both extremely generous in offering specific suggestions for methods and +strategies used in this paper. This work benefited greatly from my conversations with Aravind +Asok, Lukas Brantner, Yang Hu, Dev Sinha, Alexander Smith, and Dylan Wilson. +While working on this project, the author was supported by the National Science Foundation +under Award No. 2202914. + +6 +MORGAN OPIE +1.3. Conventions. +• “Vector bundle” will refer to a complex, topological vector bundle. As such, we use +“vector bundle” and “map to BU(r)” interchangeably. “Rank” refers to complex rank. +• Given spaces X and Y , we write [X, Y ] for homotopy classes of maps from X to Y . +• H∗ will refer to ordinary cohomology with Z coefficients, except otherwise stated. We +write HF ∗ +p for cohomology with Fp = Z/p coefficients. +• If C is a space, then π∗C will refer to unstable homotopy groups. If X is a spectrum, +π∗X will refer to its stable homotopy groups. Thus, the stable homotopy groups of a +space C will be written as π∗(Σ∞ ++ C). +• Given a space or spectrum X, we write τnX for its n-th Postnikov section. +• Given virtual bundle W on a topological space Y , we write Th(Y ; W) for the Thom +spectrum of W. Given a map f : X → Y of spaces, f has a Thomification +Th(f) : Th(X; f ∗W) → Th(Y ; W). +When taking Thom spectra, we assume all bundles have virtual dimension zero. +• Given a space or spectrum X, we write Xˆ +p for its completion at a prime p, and X(p) +for its localization away from p. +• In Sections 3 and 4, all spaces and spectra are implicitly localized at the prime 3. +• Given an E∞-ring spectrum R and two R-modules X, Y , we write +MapsR(X, Y ) := MapsR- Mod(X, Y ) = R- Mod(X, Y ). +2. A count of rank 3 bundles on CP 5 +The primary goal of this section is to prove Theorem 1.7 by computing the set [CP 5, BU(3)]. +For this we need the homotopy of BU(3) through degree 10, which can be computed via the +fiber sequence +BSU(3) → BU(3) → BU(1) +and its associated homotopy long exact sequence. The homotopy of U(1) ≃ S1 is known and +enough of the homotopy of SU(3) is computed in [22]. We give the result in Figure 1. +π2 +π3 +π4 +π5 +π6 +π7 +π8 +π9 +π10 +BU(3) +Z +0 +Z +0 +Z +Z/6 +0 +Z/12 +Z/3 +Figure 1. Homotopy of BU(3) +Note that the torsion in πnBU(3) for n ≤ 10 is either 2- or 3- primary. Thus we may break +the computation into analyses at the primes 2 and 3. +The key tool which allows us to study the problem one prime at a time is the theory of +rationalization and completion of spaces. Given a space X, let Xˆ +p denote its p-completion. The +Fracture Theorem for completion, as stated in [18, Theorem 13.1.1], implies that an element in +[CP 5, BU(3)] is the same data as pairs of maps +f2 : CP 5 → BU(3)ˆ +2, +f3 : CP 5 → BU(3)ˆ +3 +such that the Zˆ +2- and Zˆ +3-valued Chern classes are both in the image of the canonical inclusion +Z ֒→ Zˆ +p and agree under this identification. +The in-depth analysis in the proof of Theorem 1.7 is calculating [CP 5, BU(3)ˆ +3]. This is +carried out in Subsection 2.1, with supporting technical results in Subsection 2.2. In Subsec- +tion 2.3, we compute [CP 5, BU(3)ˆ +2]. In Subsection 2.4, we show that the Schwarzenberger + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +7 +condition S5 is necessary and sufficient for three integers to be the Chern classes of a rank 3 +bundle on CP 5. This completes the proof of Theorem 1.7 and justifies Remark 1.5. +2.1. 3-complete rank 3 vector bundles on CP 5. +We give the first stages of a Postnikov-type tower for the 3-completion of BU(3) and analyze +maps from CP 5 into this tower. +Claim 2.1. There is a tower of principal fibrations given by the solid arrows below: +K(Z/3, 10) +P10 +K(Z/3, 7) × K(Z/3, 9) +P9 +K(Z/3, 11) +BU(3) +K(Z, 2) × K(Z, 4) × K(Z, 6) +K(Z/3, 8) × K(Z/3, 10) +τ10 +U +(c1,c2,c3) +τ9 +k7×k9 +where (c1, c2, c3) induces a 3-complete equivalence on τ6BU(3); τ9 induces a 3-complete equiv- +alence on τ9BU(3); and τ10 induces a 3-complete equivalence on τ10BU(3). +At present, the explicit forms of k7 × k9 and U are not needed. +Given Claim 2.1, we can calculate [CP 5, τ10BU(3)ˆ +3] ≃ [CP 5, BU(3)ˆ +3] by working up the +tower. We need the following standard lemma. +Lemma 2.2. Let X be a connected space. For any other space Y let Y X denote the mapping +space. Given a fiber sequence of connected spaces +(2.1) +F +E +B. +and a map f : X → E so that the composite map to B is nullhomotopic, the set of homotopy +classes of choices of lifts of f to F is a torsor for coker +� +π1(EX, f) → π1(BX, 0) +� +. +We apply Lemma 2.2 to the diagram in Claim 2.1. Candidate Chern data (a1, a2, a2) ∈ +H2(CP 5) × H4(CP 5) × H6(CP 5) lifts to P9 if and only if (k7 × k9) ◦ (a1, a2, a3) ≡ 0 (mod 3). +This is a mod 3 condition on Chern classes, which we do not compute since we recover the +condition via different methods in Subsection 2.4. +By Lemma 2.2, the number of lifts to P9 are a torsor for a quotient of +π1 +�� +K(Z/3, 8) × K(Z/3, 10) +�CP 3� +≃ HF 7 +3 (CP 3) × HF 9 +3 (CP 3) = 0, +so when a lift exists it is unique. +There are no obstructions to lifting from P9 to P10, since HF 11 +3 +(CP 3) = 0. Choices of lift +are a torsor for +coker +� +π1(P9 +CP 5) +U◦− +−−−→ π1(K(Z/3, 11)CP 5) +� +Since π1(K(Z/11)CP 5) ≃ π0(K(Z/3, 10)CP 5)) ≃ HF 10 +3 +(CP 5) ≃ Z/3, there are two possibilities +that will depend on a1, a2, a3: +• The map is surjective, the cokernel is trivial, and there is a unique lift; or +• The map is zero, the cokernel is Z/3 and there are three lifts. +To compute Im(U ◦ −), we consider a related problem. From the principal fibration +(2.2) +K(Z/3, 7) × K(Z/3, 9) → P9 → K(Z, 2) × K(Z, 4) × K(Z, 6), +we get an action of the fiber +� +K(Z/3, 7) × K(Z/3, 9) +� +× P9 → P9. This gives an action +(2.3) +π1 +� +K(Z/3, 7)CP 5 × K(Z/3, 9)CP 5� +× π1(P CP 5 +9 +) → π1(P CP 5 +9 +). + +8 +MORGAN OPIE +Claim 2.3. The action given in Equation (2.3) is transitive. +Assuming this claim too, fix a1, a2, a3 ∈ Z with (k7 × k9) ◦ (a1, a2, a3) = 0. +Consider the +diagram: +(2.4) +P9 +K(Z/3, 11) +∗ × CP 5 +S1 × CP 5 +K(Z/3, 7) × K(Z/3, 9) × P9, +U +[a1,a2,a3] +∗×1 +a +(xι1t3,yι1t4,a) +† +⋆ +m +m∗U +where only the triangles † and ⋆ commute. In the above: +(1) a: S1 × CP 3 → τ5BU(2) restricts to [a1, a2, a3] on ∗ × CP 3. +(2) x,y ∈ Z/3 are arbitrary coefficients of the classes ι1t3 and ι1t4, which are the natural +generators of +HF 7 +3 (S1 × CP 5) ≃ HF 1 +3 (S1 ⊗ HF 6 +3 (CP 5) +≃ Z/3{ι1} ⊗ Z/3{t3} +and +HF 9 +3 (S1 × CP 5) ≃ HF 1 +3 (S1) ⊗ HF 8 +3 (CP 5) +≃ Z/3{ι1} ⊗ Z/3{t4}. +Given Claim 2.3, to compute Im(U ◦ −), it suffices to compute m∗U ◦ (xι1t3, yι1t4, a) as (x, y) +ranges over Z/3 × Z/3. We will obtain formula for the difference +(2.5) +m∗U ◦ (xι1t3, yι1t5, a) − m∗U ◦ (0, 0, a). +Showing that Im(U ◦ −) is Z/3 is equivalent to finding x, y ∈ Z/3 so that the difference in +Equation (2.5) is nonzero. +Claim 2.4. The class m∗U ∈ HF 11 +3 +� +K(Z/3, 7) × K(Z/3, 9) × P9 +� +is given by +m∗U = U + P 1(ι′ +7) − ι2ι′ +9 + ι2 +2ι′ +7 − ι4ι′ +7 ∈ HF 11 +3 +� +K(Z/3, 7) × K(Z/3, 9) × P9 +� +, +where ι′ +7 and ι′ +9 generate HF 7 +3 +� +K(Z/3, 7) +� +and HF 9 +3 +� +K(Z/3, 9) +� +, respectively; and where ι2, ι4 +are the images in HF ∗ +3 (P9) of generators in HF i +3 (K(Z, 2)) and HF i +3 (K(Z, 4)), respectively. +Given Claim 2.4, since ι2 pulls back to c1 (mod 3) in HF ∗ +3 (CP 5) and ι4 pulls back to c2 +(mod 3), we see that +m∗U ◦ (xι1t3, yι1t4, a) − U ∗m ◦ (0, 0, a) = U + xP 1(ι1t3) − (a1t)yι1t4 ++ (a2 +1t2)xι1t3 − (a2t2)xι1t3 − U += −(ya1 − xa2 +1 + xa2)ι1t5. +The quantity ya1 − xa2 +1 + xa2 is zero mod 3 for all choices of x and y if and only if a1 and a2 +are both zero mod 3. +Predicated on Claims 2.1, 2.3, and 2.4, we have shown: +Proposition 2.5. Given integers (a1, a2, a3) with (k7 × k9) ◦ (a1, a2, a3) = 0, the following two +situations can occur: +• If either a1 or a2 are nonzero mod 3, then the map in U ◦ − is surjective and, up to +homotopy, there is a unique 3-local vector bundle with i-th Chern class ai. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +9 +• If a1 ≡ 0 (mod 3) and a2 ≡ 0 (mod 3), then the map U ◦ − is zero and there are three +distinct homotopy classes of 3-local vector bundles with i-th Chern class ai. +2.2. Proof of technical claims. +We now prove the claims 2.1, 2.3, and 2.4, completing the proof of Proposition 2.5. +Proof of Claim 2.1. The map pullback of the Chern class map +c := (c1, c2, c3): BU(3) → K(Z, 2) × K(Z, 4) × K(Z, 6) +on mod 3-cohomology is a 3-complete equivalence through degree 6. We correct the degree 8 +and degree 10 cohomology terms simultaneously via a map +K(Z, 2) × K(Z, 4) × K(Z, 6) +K(Z/3, 8) × K(Z/3, 10), +k7×k9 +and a factorization +P9 +BU(3) +K(Z, 2) × K(Z, 4) × K(Z, 6) +K(Z/3Z, 8) × K(Z/3Z, 10), +τ9 +c +k7×k9 +where P9 := hofib(k7 × k9), such that: +(1) The map (k7 × k9) ◦ c is nullhomotopic; and +(2) The lift τ9 : BU(3) → P9 is mod 3-cohomology isomorphism up to at least degree 10 +and therefore realizes the 9-truncation of BU(3), up to 3-completion. +We complete (1) in Construction 2.6 and (2) in Verification 2.7. +Construction 2.6. Let P i denote the mod 3 Steenrod operation of degree 4i. +The first +relation among Steenrod operations on Chern classes is P 1 on c2: P 1(c2) = c2 +1c2 + c2 +2 − c1c3. +Let ιj denote a generator for HF j +3 (K(Z, j)) and take +k7 := P 1ι4 − ι2 +2ι4 − ι2 +4 + ι2ι6 ∈ HF 8 +3 +� +K(Z, 2) × K(Z, 4) × K(Z, 6)Z +� +. +We identify a candidate for k9 by computing P 1 on c3. P 1c3 = c3(c2 +1 + c2) so let +k9 := P 1ι6 − ι6 +� +ι2 +2 + ι4 +� += P 1ι6 − ι6ι2 +2 − ι6ι4 ∈ HF 10 +3 +� +K(Z, 2) × K(Z, 4) × K(Z, 6)Z +� +. +Verification 2.7. One computes the HF3-cohomology of integral Eilenberg Mac Lane spaces, +which can be computed directly from the path-loop fibration. The main result we need is the +following: +Proposition 2.8. Let ιj generate HF j +3 K(Z, j) for j = 2, 4, 6. We can identify the multiplicative +structure of HF ∗ +3 (K(Z, 2) × K(Z, 4) × K(Z, 6)) through degree 11 as follows: +� +HF ∗ +3 +� +K(Z, 2) × K(Z, 4) × K(Z, 6) +�� +≤11 ≃ (Z/3Z[ι2, ι4, ι6, Y8, W10] ⊗ Λ[N9, S11])≤11 +where the subscript indicates the degree of the polynomial or exterior generator, the notation +(−)≤11 indicates that we quotient by all elements of degree at least 12, and +Y8 = P 1ι4 +W10 = P 1ι6 +N9 = βP 1ι4 +S11 = βP 1ι6. + +10 +MORGAN OPIE +From the above, we can compute the Serre spectral sequence for the fibration +K(Z/3, 7) × K(Z/3, 9) → P9 → K(Z, 2) × K(Z, 4) × K(Z, 6). +The E2-page is given in Figure 2. Moreover, if β denotes the Bockstein power operation: +L8 = βι7 +R10 = βι9 +M11 = P 1ι7. +To obtain the associated graded of HF ∗ +3 (P9), we compute all relevant differentials using a +combination of the following two facts (Kudo’s transgression theorem, see [15] or [20, Ch. 6]): +• Given a principal fibration F → E → K(Z/pZ, n), the fundamental class ιn+1 is +transgressive in the mod p Serre spectral sequence for ΩK(Z/pZ, n) → F → E. +• A power operation applied to a transgressive class is transgressive; transgressions com- +mute with power operations. +From the above items and the fact that L8 = β(ι7), we deduce that +d7(ι7) = Y8 − ι2 +2ι4 − ι2 +4 + ι6ι2, and +d8(L8) = β(d7(ι7)) = β(Y8 − ι2 +2ι4 − ι2 +4 + ι6ι2) = β(Y8) = N9. +Similarly, since β(ι9) = R10, we get that +d9(ι9) = W10 − ι6(ι2 +2 − 2ι4) and +d10(R10) = β +� +W10 − ι6(ι2 +2 + ι4) +� += S11. +All other terms strictly below the dotted line in Figure 2 are computed using the Liebniz rule. +Thus the images of ι2, ι4, and ι6 are polynomial generators for HF ∗ +3 (P9) up to degree 10; since +The E2-page of a spectral sequence computing HF ∗ +3 (P9). +11 M11 +10 +R10 +9 +ι9 +8 +L8 +7 +ι7 +6 +5 +4 +3 +2 +1 +0 +ι2 +ι4 +ι6 +Y8 +N9 +W10 +S11 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +Figure 2. Only multiplicative generators for the E2-page are indicated. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +11 +c∗ : HF 2j +3 +(K(Z, 2) × K(Z, 4) × K(Z, 6)) → HF ∗ +3 (BU(3)Z) satisfies ι2j �→ cj, this shows that a +lift of (c1, c2, c3) induces an equivalence through degree 9, completing Verification 2.7. +Evidently the next stage in the tower is given by a class U : P9 → K(Z/3, 11). +□ +Proof of Claim 2.3. Consider the π1 portion of the homotopy long exact sequence associated +to the fibration (2.2) +(2.6) +π1 +�� +K(Z/3, 7) × K(Z/3, 9) +�CP 5� +π1(P CP 5 +9 +) +π1 +�� +K(Z, 2) × K(Z, 4) × K(Z, 6) +�CP 5� +s +where the basepoint for (K(Z, 2) × K(Z, 4) × K(Z, 6))CP 5 +is (a1, a2, a3). The last term of (2.6) +is zero, so s is surjective. The action (2.3) is given on +(x, a) ∈ π1 +�� +K(Z/3, 7) × K(Z/3, 9) +�CP 5� +× π1(P CP 5 +9 +) +by +(2.7) +(x, a) �→ s(x)a ∈ π1(P CP 5 +9 +). +Thus, surjectivity of s implies the action is transitive. +□ +Proof of Claim 2.4. In order to understand U more explicitly, we study the spectral sequence +in Figure 2 up to and including the dotted line. Computing differentials, we see that the class +M11 = P 1(ι7) detects a nonzero class in HF 11 +3 +(P9). +The action that (2.7) gives rise to a map of spectral sequences, from the Serre spectral +sequence for +K(Z/3, 7) × K(Z/3, 9) → P9 → K(Z, 2) × K(Z, 4) × K(Z, 6) +to the Serre spectral sequence for +(2.8) +� +K(Z/3, 7) × K(Z/3, 9) +�×2 +→ K(Z/3, 7) × K(Z/3, 9) × P9 → +3 +� +i=1 +K(Z, 2i). +We compute this map of spectral sequences using the fiber-by-fiber action. +For i = 7 and i = 9, let ιi and ι′ +i generate the two copies of HF i +3 (K(Z/3, i)) in the fiber of +Equation 2.8. The comultiplication on the fiber implies that the coaction on the E2-page is: +ι7 �→ ι7 + ι′ +7, +ι9 �→ ι9 + ι′ +9. +We claim that, in the double complex of the source sequence, U should be represented by +(2.9) +M11 − ι4ι7 + ι2 +2ι7 − ι2ι9. + +12 +MORGAN OPIE +To see this, note that M11 is transgressive and +d11(M11) = d11(P 1ι7) += P 1(d7(ι7)) += P 1(P 1ι4 − ι2 +2ι4 − ι2 +4 + ι2ι6) += −P 2ι4 + ι4 +2ι4 − ι2 +2P 1ι4 + ι4P 1ι4 + ι3 +2ι6 + ι2P 1ι6 += −ι3 +4 + ι4 +2ι4 − ι2 +2P 1ι4 + ι4P 1ι4 + ι3 +2ι6 + ι2P 1ι6 += +� +ι4(d7ι7) + ι2 +2ι2 +4 − ι2ι4ι6 +� ++ ι4 +2ι4 − ι2 +2P 1ι4 + ι3 +2ι6 + ι2P 1ι6 += +� +ι4(d7ι7) − ι2 +2d7(ι7) + ι3 +2ι6 +� +− ι2ι4ι6 + ι3 +2ι6 + ι2P 1ι6 += ι4(d7ι7) − ι2 +2d7(ι7) − ι2ι4ι6 − ι3 +2ι6 + ι2P 1ι6 += ι4(d7ι7) − ι2 +2(d7ι7) + ι2(d9ι9). +This indicates that a cocycle representative for U in the double complex computing H∗(P9; Z/2) +is Equation (2.9) on the E2-page. +Therefore, on the E2-page, we see that the action on the class representing U is detected by +(M11 − ι4ι7 + ι2 +2ι7 − ι2ι9) +m∗ +�−−→ (M11 − ι4ι7 + ι2 +2ι7 − ι2ι9 + P 1ι′ +7 − ι4ι′ +7 + ι2 +2ι′ +7 − ι2ι′ +9). +Passing to the E∞-page of the Serre spectral sequence for Equation (2.8) we see that m∗U is +detected by +M11 + P 1ι′ +7 − ι4ι′ +7 + ι2 +2ι′ +7 − ι2ι′ +9 ∈ HF ∗ +3 +� +K(Z/3, 7) × K(Z/3, 9) × P9 +� +, +and therefore m∗U = U + P 1ι′ +7 − ι4ι′ +7 + ι2 +2ι′ +7 − ι2ι′ +9, completing the proof of the claim. +□ +Remark 2.9. Instead of analyzing the tower of Diagram 2.1, we could transpose over the +skeleton-truncation adjunction and instead lift a map sk0(CP 5) → BU(3)ˆ +3 up the higher skeleta +of CP 5. The action of coker(U ◦ −) corresponds to the action of π10BU(3)ˆ +3 on lifts of a given +map sk9 CP 5 → BU(3)ˆ +3 to the 10-skeleton. This shows that the action from Construction 1.6 +is the relevant one. +2.3. 2-complete rank 3 vector bundles on CP 5. +In this section we show there are no 2-complete bundles on CP 3 which were not already +detected by Chern classes. +Proposition 2.10. Consider the map c: BU(3)ˆ +2 → K(Zˆ +2, 2) × K(Zˆ +2, 4) × K(Zˆ +2, 6) given by +the product c1 × c2 × c3 of two-completed Chern classes. The induced 2-complete Chern class +map from [CP 5, BU(3)ˆ +2] to H2(CP 5, Zˆ +2) × H4(CP 5, Zˆ +2) × H6(CP 5, Zˆ +2) is injective. +Proof. To understand [CP 5, BU(3)ˆ +2], we build a map from CP 5 into BU(3)ˆ +2 cell-by-cell. First, +recall the 2-complete homotopy of BU(3), as in Figure 3, computed from Figure 1. +π2 +π3 +π4 +π5 +π6 +π7 +π8 +π9 +π10 +BU(3)ˆ +2 +Zˆ +2 +0 +Zˆ +2 +0 +Zˆ +2 +Z/2 +0 +Z/4 +0 +Figure 3. 2-complete homotopy of BU(3) + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +13 +Consider the dotted arrows (i) to (v) in Diagram (2.10) below. +(2.10) +CP 5 +sk8 CP 5 +sk6 CP 5 +BU(3)ˆ +2 +sk4 CP 5 +sk2 CP 5 +∗ +(v) +(iv) +(iii) +(ii) +(i) +An arrow (i) corresponds to a 2-complete first Chern class CP 2 → K(Zˆ +2, 2). The obstruction +to lifting further is in π3(BU(3)ˆ +2) = 0. +The choices of lifts to an arrow (ii) are acted on +transitively by π3(BU(3)ˆ +2) ≃ Zˆ +2 and correspond to c2. The obstructions to lifting to an arrow +(iii) lie in π5(BU(3)ˆ +2) = 0, and the choices of lift to (iii) correspond to c3. +The obstruction to a lift to a map (iv) lies in π7(BU(3)ˆ +2) ≃ Z/2. The choices of lifts are +acted on transitively by π8(BU(3)ˆ +2) ≃ 0. The obstruction to lifting from (iv) to (v) are in +π9(BU(3)ˆ +2) ≃ Z/4. The choices of lift are acted upon transitively by π10(BU(3)ˆ +2) = 0. +□ +Remark 2.11. We have shown that there are mod 2 and mod 4 conditions on the Chern +classes of a rank 3 vector bundle on CP 5, but no new 2-primary invariants. However, for a +general 10-skeletal space (one that is not even), there may be additional 2-complete bundles +not determined by Chern classes. +2.4. The Schwarzenberger conditions. +Finally, we discuss necessary conditions for a collection of integers a1, a2, a3 ∈ Z to be the +Chern classes of a topological vector bundle of rank 3 on CP 5. Following [25], let integers +c1, . . . , ck be given. Inductively, let +s1 := c1 +sk(c1, . . . , ck) := Σk−1 +i=1 (−1)i+1cisk−i +f1(s1) := Identity +fn(s1, . . . , sn) := fn−1(s2, . . . , sn) − (n − 1)fn−1(s1, . . . , sn−1). +Definition 2.12. The Schwarzenberger condition Sk on a set c1, . . . , ck of integers is the +requirement that, for each 1 ≤ n ≤ k, +fn(s1(⃗c), . . . , sn(⃗c)) ≡ 0 +(mod n!) , +where ⃗c := (c1, . . . , ck). +Remark 2.13. This condition has different forms, e.g. see [12, Appendix A]. +Theorem 2.14 ([25], Theorem A). Integers c1, . . . , ck ∈ Z are the Chern classes of a rank k +vector bundle on CP k if and only if c1, . . . , ck satisfy the condition Sk. +From this we can to prove: + +14 +MORGAN OPIE +Lemma 2.15. Let a1, a2, a3 ∈ Z. Then there exists a complex rank 3 topological vector bundle +V on CP 5 with ci(V ) = ai if and only if the 5-tuple (a1, a2, a3, 0, 0) satisfies the Schwarzenberger +condition S5. +Proof. By Theorem 2.14 above, the condition is necessary. +To show the condition S5 is sufficient, we prove that a rank 5 vector V ′ bundle on CP 5 with +c4 and c5 equal to zero is in fact is isomorphic to a bundle V ⊕ C2, i.e. its stable class has a +rank 3 representative. Consider [26, Proposition 5.7.5], which implies that any (complex) rank +7 vector bundle on CP 5 with top four Chern classes zero is a sum of a rank 3 bundle and two +trivial bundles. We apply this to V ′ ⊕ C2 to get the desired result. +□ +We now work out S5 explicitly. +Lemma 2.16. The condition S5 on (a1, a2, a3, 0, 0) is equivalent to the system of equations: +a3 + a1a2 ≡ 0 +(mod 2) +−a2 +1a2 + a1a3 − a2 +2 + a2 ≡ 0 +(mod 3) +a1a2 − a2 +1a3 − a1a2 +2 + a2a3 + a2 +1a2 − a1a3 + a2 +2 ≡ 0 +(mod 3) +−a1 +3a2 + a1 +2a3 + a1a2 +2 − a2a3 − a1a2 + a3 ≡ 0 +(mod 4) +Proof. Using the definitions preceding Theorem 2.14, we evaluate s1, . . . , s5 at (a1, a2, a3, 0, 0). +Let ⃗a := (a1, a2, a3, 0, 0). +s1(⃗a) = a1 +s2(⃗a) = a2 +1 − 2a2 +s3(⃗a) = a3 +1 − 3a1a2 + 3a3 +s4(⃗a) = a4 +1 − 4a2 +1a2 + 4a1a3 + 2a2 +2 +s5(⃗a) = a5 +1 − 5a3 +1a2 + 5a2 +1a3 + 5a1a2 +2 − 5a2a3. +We now compute fi(s1, . . . , si) for 1 ≤ i ≤ 5. +f1(s1) = s1 +f2(s1, s2) = s2 − s1 +f3(s1, s2, s3) = (s3 − s2) − 2(s2 − s1) += s3 − 3s2 + 2s1 +f4(s1, s2, s3, s4) = s4 − 3s3 + 2s2 − 3(s3 − 3s2 + 2s1) += s4 − 6s3 + 11s2 − 6s1 +f5(s1, s2, s3, s4, s5) = s5 − 6s4 + 11s3 − 6s2 − 4(s4 − 6s3 + 11s2 − 6s1) += s5 − 10s4 + 35s3 − 50s2 + 24s1. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +15 +From the above we get: +f2(si)|⃗a = a2 +1 − 2a2 − a1 += a1(a1 − 1) − 2a2 +f3(si)|⃗a = a3 +1 − 3a1a2 + 3a3 − 3(a2 +1 − 2a2) + 2a1 += a1(a1 − 1)(a1 − 2) − 3a1a2 + 3a3 − 6a2 +f4(si)⃗a = a4 +1 − 4a2 +1a2 + 4a1a3 + 2a2 +2 − 6(a3 +1 − 3a1a2 + 3a3) + 11(a2 +1 − 2a2) − 6(a1) += a1(a1 − 1)(a1 − 2)(a1 − 3) − 4a2 +1a2 + 4a1a3 + 2a2 +2 + 18a1a2 − 18a3 − 22a2 +f5(si)|⃗a = a5 +1 − 5a3 +1a2 + 5a2 +1a3 + 5a1a2 +2 − 5a2a3 +− 10(a4 +1 − 4a2 +1a2 + 4a1a3 + 2a2 +2) + 35(a3 +1 − 3a1a2 + 3a3) − 50(a2 +1 − 2a2) + 24a1 += +4 +� +i=0 +(a1 − i) + 5(−a3 +1a2 + a2 +1a3 + a1a2 +2 − a2a3) +− 10(−4a2 +1a2 + 4a1a3 + 2a2 +2) + 35(−3a1a2 + 3a3). +For simplicity, write fi(⃗a) for fi(s1, . . . , si)|⃗a. We now expand the equations fi(⃗a) ≡ 0 (mod i!) +for 2 ≤ i ≤ 5: +f2 (mod 2!) : a1(a1 − 1) − 2a2 ≡ 0 (mod 2) for any a1, a2, so this gives no condition. +f3 (mod 3!) : f3(⃗a) ≡ a3 + a1a2 (mod 2), so we get the condition +(2.11) +a3 + a1a2 ≡ 0 +(mod 2). +Since all terms of f3(⃗a) are divisible by 3, there is no 3-primary condition from f3. +f4 (mod 4!) : All terms of f4(⃗a) are divisible by 2, so this gives no condition. +Since f4(⃗a) ≡ −a2 +1a2 + a1a3 + 2a2 +2 − a2 (mod 3), this gives the condition +(2.12) +−a2 +1a2 + a1a3 − a2 +2 + a2 ≡ 0 +(mod 3). +Consider f4(⃗a) ≡ 2a2 +2 + 2a1a2 + 2a3 − 2a2 (mod 4). This quantity is zero +(mod 4) if +and only if a2 +2 + a1a2 − a3 − a2 ≡ 0 (mod 2). However, this condition is implied by +(2.11), so we get no new constraint. +f5 (mod 5!) : f5(⃗a) ≡ a3 +1a2 + a2 +1a3 + a1a2 +2 + a2a3 + a1a2 + a3 (mod 2) giving the condition +a1a3 + a1a2 + a2a3 + a3 ≡ 0 +(mod 2). +However, this condition is implied by (2.11) so we get no new constraint. +Next, we reduce (mod 3): +f5(⃗a) ≡ a3 +1a2 − a2 +1a3 − a1a2 +2 + a2a3 + a2 +1a2 − a1a3 + a2 +2 +(mod 3) +Thus f5(⃗a) ≡ 0 (mod 3) if and only if +(2.13) +a1a2 − a2 +1a3 − a1a2 +2 + a2a3 + a2 +1a2 − a1a3 + a2 +2 ≡ 0 +(mod 3). +Since f5(⃗a) ≡ 0 (mod 5), the last condition comes from +f5(⃗a) +(mod 4) ≡ −a13a2 + a12a3 + a1a22 − a2a3 − a1a2 + a3. +This gives the condition: +(2.14) +−a1 +3a2 + a1 +2a3 + a1a2 +2 − a2a3 − a1a2 + a3 ≡ 0 +(mod 4) +Equations (2.11), (2.12), (2.13), and (2.14) are precisely the conditions to be proved. +□ + +16 +MORGAN OPIE +3. Defining a twisted tmf(3)-valued invariant +By Theorem 1.7, rank 3 bundles on CP 5 that are not determined by their Chern data arise +from the action of π10BU(3) ≃ Z/3 given in Construction 1.6. To go from an action to an +invariant, we must study the unstable homotopy of BU(3) in greater detail. We outline the key +insights in Lemmas 3.2, 3.3, and 3.7 below. These are motivational but not logically necessary +for what follows, so we omit most proofs. +Convention 3.1. Throughout this section all spaces and spectra are implicitly localized at 3. +For any n, there is a fiber sequence +S2n+1 +δn+1 +−−−→ BU(n) → BU(n + 1) +(for example, see [21, Section 72]). For each n, δn+1 is the attaching map for a (2n + 2)-cell +corresponding to cn+1 ∈ H2n+2BU(n+1). The homotopy class of δn+1 is linked to the existence +of non-isomorphic vector bundles with the same Chern data in both the case of rank 2 bundles +on CP 3 and that of rank 3 bundles on CP 5, as shown by the next result. +Lemma 3.2. A generator for π6BU(2) is given by the composite +S6 +η−→ S5 +δ3 +−→ BU(2), +where the η is an unstable representative for the class by the same name in π∗S, which is the +first element of order 2. +A generator for π10BU(3) is given by the composite +S10 +α1 +−→ S7 +δ4 +−→ BU(3), +where the α1 is an unstable representative for the class by the same name in π∗S, which is the +first element of order 3. +Moreover, we can describe the generator for π10(BU(3)) as a further composite. +Lemma 3.3. Let x: S4 → BU(3) generate π4BU(3). Then there is a map ǫ: S7 → S4 such +that: +(1) ǫ ◦ x generates the three-torsion in π7BU(3); and +(2) Σ∞ǫ = α1. +Remark 3.4. Lemma 3.2 (for BU(3)) and Lemma 3.3 will follow from the the proof of Theo- +rem 3.9. +Combining the previous two lemmas, π10BU(3) is generated by +(3.1) +S10 +α1 +−→ S7 +ǫ−→ S4 +x−→ BU(3). +This shows that a generator σ for π10BU(3) is stably trivial every sense: first, the bundle +on BU(3) represented by σ is stably trivial as a map to BU; second, Σ∞σ = Σ∞α2 +1x = 0 since +α2 +1 = 0 in π∗S. In fact the composition in (3.1) is null after just one suspension. +Instead of applying the suspension spectrum functor to Diagram (3.1), we can take a Thom +spectrum with respect to one of the canonical bundles that are present. Let V be a bundle on +BU(3) (for example, the universal bundle γ3, its determinant, or −γ3). Equation (3.1) gives a +sequence of spectra + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +17 +(3.2) +S10 +S7 +Σ∞ ++ S10 +Σ∞ ++ S7 +Th(S4; V |S4) +Th(BU(3); V ). +˜v +α1 +Th(α1) +Th(ǫ) +Th(x) +For various choices of V , we can ask whether ˜v is null. +Remark 3.5. To obtain Diagram (3.2) by Thomifying, note that the bundles V |S7 and V |S10 +are trivial as maps to bu and fix a spherical orientation for V |S7. +The Thom spectrum Th(S4; V |S4) has two cells, one in degree four and one in degree zero. +Thomifying can either keep the cells split or introduce an α1-attaching map, in which case +the Thom spectrum is C(α1).1 Which option occurs depends on V . In order for the dotted +composite ˜v to be nonzero, the latter must occur. Moreover, given any spectrum X together +with a map C(α1) → X, if +(3.3) +� +S10 → S7 → C(α1) → X +� +̸≃ 0. +Then the image in X of the 4-cell of C(α1) cannot support a P 1 in X. +From experimentation, it seems that taking X = Th(BU(3); V ) with V a canonical construc- +tion on γ3 fails one condition or the other: either Th(S4; V |S4) ≃ Σ∞ ++ S4 or there is a nonzero +P 1 on the relevant 4-cell of Th(BU(3); V ). To resolve this issue, we modify our classifying +space. +Definition 3.6. Let BU(n)c1≡0 := hofib +� +c1 (mod 3): BU(n) → K(Z/3, 2) +� +. +Note that: +• The space BU(n)c1≡0 is even, so any rank n on CP k bundle with c1 ≡ 0 (mod 3) lifts +uniquely, up to homotopy, along the natural map BU(n)c1≡0 → BU(n). +• The natural map BU(n)c1≡0 → BU(n) is a homotopy equivalence above degree 2. +• The space BU(n)c1≡0 carries a universal bundle which we denote γn. +Thus our previous analyses of rank 3 bundles on CP 5 can be repeated after adding the con- +straint c1 ≡ 0 (mod 3) and substituting BU(3)c1≡0 in place of BU(3). In particular, we get a +modification of Diagram (3.2): +(3.4) +S10 +S7 +Σ∞ ++ S10 +Σ∞ ++ S7 +Th(S4; −γ3) +Th(BU(3)c1≡0; −γ3). +˜v +α1 +Th(α1) +Th(ǫ) +Th(x) +Lemma 3.7. In the diagram above, Th(S4; −γ3) = C(α1) and the element ˜v is nontrivial in +π10 +� +Th(BU(3)c1≡0; −γ3) +� +. +It will be useful to have better terminology for the Thomified homotopy classes such as ˜v. +Definition 3.8. Given a pointed map y: Sn → BU(r)c1≡0 representing a stably trivial bundle +on Sn, and a nullhomotopy u of the composite map to BGL1S, let Thu(y): Sn → BU(r)−γr +denote the composite +Sn +i2 +−→ Σ∞ ++ Sn +u≃ +−−→ Th(Sn; −y) → Th(BU(r); −γr), +1C(α1) is the cofiber of the map α1 : S3 → S0. + +18 +MORGAN OPIE +where the arrow i2 is the inclusion of the top cell determined by a base point and the map u≃ +is the spherical Thom isomorphism determined by u.2 +The main goal of this section is to prove the following: +Theorem 3.9. Given σ: S7 → BU(3) generating π7BU(3) and a Thom class u0 for σ, there +is a unique class +˜ρ ∈ tmf(3) +−3(sk26 Th(BU(3)c1≡0; −γ3)) +such that +Thu0(σ)∗ ˜ρ = α1β1 ∈ π13 tmf(3), +where sk26 Th(BU(3)c1≡0; −γ3) is the Thomification of a 26-skeleton of BU(3)c1≡0. +Remark 3.10. The reader may wonder why we look for a class in tmf(3) cohomology. The +spectrum X = Σ−3 tmf(3) carries a natural map C(α1) → Σ−3 tmf(3) induced by α1 : S0 → +Σ−3 tmf(3) such that Equation (3.3) holds. Moreover, tmf(3) is one of the simplest ring spectra +with this property. This makes tmf(3) is a natural candidate to detect π10BU(3)c1≡0. +In Subsection 3.1, we outline the proof strategy for Theorem 3.9. +We prove ˜ρ exists in +Subsection 3.2, predicated on cohomology calculations which are recorded in Subsection 3.3. +The proof of uniqueness of ˜ρ, given in Subsection 3.4, also uses these calculations. +3.1. Proof outline: existence and uniqueness of a twisted tmf(3) invariant. +In this subsection we will state a sequence of propositions and explain how they imply Theo- +rem 3.9. To begin, let u be the canonical Thom class in the HF3-cohomology of Th(BU(3)c1≡0; −γ3). +As a module over the mod 3 Steenrod algebra, +HF ∗ +3 Th(BU(3)c1≡0; −γ3) ≃ +� +HF ∗ +3 BU(3)c1≡0 +� +· u. +We find that P 1(u) = −c2 · u and P 1P 1(u) = 0 (see Proposition 3.17), so there is a map +k: C(α1) → Th(BU(3)c1≡0; −γ3) +that takes the 0-cell in C(α1) to the 0-cell in Th(BU(3)c1≡0; −γ3) and the 4-cell in C(α1) to the +cell dual to −c2 · u. +Proposition 3.11. The map k: C(α1) → sk26 Th(BU(3)c1≡0; −γ3) splits after tensoring with +tmf(3). More precisely, there is a map of tmf(3)-modules +r: (sk26 Th(BU(3)c1≡0; −γ3)) ⊗ tmf(3) → C(α1 ⊗ tmf(3)) +so that r ◦ (k ⊗ tmf(3)) is homotopic to the identity. +Fix u0 a spherical orientation for σ: S10 → BU(3)c1≡0 generating π10BU(3)c1≡0. Assuming the +previous proposition, we immediately obtain: +Corollary 3.12. There is a map ˜ρ: sk26 Th(BU(3)c1≡0; −γ3) → Σ−3 tmf(3) such that +� +S10 +Thu0 (σ) +−−−−−→ sk26 Th(BU(3)c1≡0; −γ3) +˜ρ−→ Σ−3 tmf(3) +� += α1β1 +in π13 tmf(3). +2We recall the basics of orientations and Thom isomorphisms in Subsection 4.1. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +19 +Proof of Corollary 3.12 assuming Proposition 3.11. Let r and k be as above. The map α1 : S0 → +Σ−3 tmf(3) extends over C(α1), since α2 +1 = 0. Let o : C(α1) → Σ−3 tmf(3) denote an extension +and let ¯o = o ⊗ tmf(3) be the associated map of tmf(3)-modules. Let 1: S → tmf(3) be the unit +map. We then define ˜ρ to be the composite +sk36 Th(BU(3)c1≡0; −γ3) ⊗ S +sk36(Th(BU(3)c1≡0; −γ3)) ⊗ tmf(3) +C(α1) ⊗ tmf(3) +Σ−3 tmf(3) . +1⊗ι +˜ρ +r +¯o +To show this map has the desired property, consider the homotopy commutative diagram: +sk36(Th(BU(3)c1≡0; −γ3)) ⊗ tmf(3) +C(α1) ⊗ tmf(3) +Σ−3 tmf(3) +sk36(Th(BU(3)c1≡0; −γ3)) ⊗ S +C(α1) ⊗ S +S10 +r +¯o +k⊗tmf(3) +1⊗ι +1⊗ι +o +k +Th(σ) +β1[0] +where β1[0] is the image of β1 ∈ π10S0 under S0 → C(α1). The fact that the lower triangle +commutes is a consequence of the fact that the map +S10 +α1 +−→ S7 +α1[4] +−−−→ C(α1) = ⟨α1, α1, α1⟩ = β1. +The map o ◦ β1[0] ∈ π13 tmf(3) is precisely α1β1. +□ +The splitting is not canonical. However, given an identification Th(S10; −σ) ≃ S0 ⊕ S10, the +class ˜ρ is uniquely determined by requiring it to restrict to α1β1 on S10. +Proposition 3.13. Let ǫ: S7 → BU(3) generate π10BU(3) and let u0 be a spherical orientation +for ǫ. Let Thu0 be as in Definition 3.8. +Any two classes +˜ρ, ˜ρ′ ∈ tmf(3) +−3(sk26 Th(BU(3)c1≡0; −γ3)) +such that Thu0(ǫ)∗(˜ρ) = Thu0(ǫ)∗(˜ρ′) = β1 ∈ π13 tmf(3) are in fact equal in tmf(3) +−3(Th(BU(3)c1≡0; −γ3)). +We will prove this in Subsection 3.4. This immediately implies: +Corollary 3.14. Let σ = α1 ◦ ǫ ∈ π10BU(3). Then: +• σ = α1 ◦ ǫ generates π10BU(3). +• If ˜ρ, ˜ρ′ ∈ tmf(3) +−3(sk26 Th(BU(3)c1≡0; −γ3)) satisfy +Thu0(σ)∗(˜ρ) = Thu0(σ)∗(˜ρ′) = α1β1 ∈ π13 tmf(3), +then ˜ρ = ˜ρ′. +Proof. Note that the orientation u0 for ǫ gives an orientation u0 for σ such that +α1 · Thu0(ǫ) = Thu0(σ). +For the second item, suppose that ρ′, ρ both satisfy +Th(σ)∗(˜ρ) = α1β1 = Th(σ)∗(˜ρ′) + +20 +MORGAN OPIE +in π ∗ tmf(3). This implies: +Th(ǫ)∗(˜ρ) = β1 = Th(ǫ)∗(˜ρ′), +so, by Proposition 3.13, ˜ρ = ˜ρ′ ∈ tmf(3) +−3 � +sk26 Th(BU(3)c1≡0; −γ3) +� +. +For the first item, note that Th(σ)∗(˜ρ) ̸= 0, which implies σ ̸= 0. Since π10BU(3) is cyclic, +this implies σ generates. +□ +Together, Corollary 3.12 and Corollary 3.14 imply Theorem 3.9. It remains to prove Propo- +sitions 3.11 and 3.13. These are the projects of the next subsections. +3.2. Proof of Proposition 3.11. +By a skeleton of Th(BU(3)c1≡0; −γ3) we mean a term in a filtration of Th(BU(3)c1≡0; −γ3) ob- +tained by Thomifying a skeletal filtration of BU(3)c1≡0. Precisely, BU(3)c1≡0 has a cell structure +filtering the space by a sequence of pushouts as in Diagram (3.5) below. +(3.5) +... +∨j∈Hi+2Si+3 +ski+2 BU(3)c1≡0 +∨j∈HiSi+1 +ski BU(3)c1≡0 +... +∨¯ci+2,j +∨¯cij +Each Hi above is a finite indexing set. Each skeleton carries a bundle pulled back from γ3 on +BU(3)c1≡0. We can Thomify all diagrams involved to obtain a filtration for Th(BU(3)c1≡0; −γ3): +each stage in Diagram (3.5) gives pushout in spectra +(3.6) +∗ +ski+2 Th(BU(3)c1≡0; −γ3) +⊕HiSi +ski Th(BU(3)c1≡0; −γ3) . +⊕Hicij +In Diagram 3.6, we define ski Th(BU(3)c1≡0; −γ3) := Th(ski BU(3)c1≡0; −γ3) and cij is the +Thomification of ¯cij restricted to Si. +Consider the cofiber +C := cof(C(α1) +k−→ sk26 Th(BU(3)c1≡0; −γ3). +Lemma 3.15. With notation as above, π0 Mapstmf(3)(Σ−1C ⊗ tmf(3), C(α1) ⊗ tmf(3)) = 0. +Given this lemma, the going around map Σ−1C ⊗ tmf(3) → C(α1) ⊗ tmf(3) is null and there is +an extension making Diagram 3.7 homotopy commutative, which is the desired section. +(3.7) +Σ−1C ⊗ tmf(3) +C(α1) ⊗ tmf(3) +sk26 Th(BSU(3); −γ3) ⊗ tmf(3) +C(α1) ⊗ tmf(3) +0 +k += +∃r +Proof of Lemma 3.15. Using the free-forgetful adjunction for tmf(3)-modules: +π0 Mapstmf(3)(Σ−1C ⊗ tmf(3), C(α1) ⊗ tmf(3)) ≃ π0 MapsS(Σ−1C, C(α1) ⊗ tmf(3)). + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +21 +To establish the result, we compute π0 MapsS(Σ−1C, C(α1) ⊗ tmf(3)) via an Atiyah–Hirzebruch +spectral sequence +(3.8) +Ep,q +2 += Hp � +Σ−1C; +� +tmf(3) +� +−q C(α1) +� +=⇒ πp+q MapsS +� +Σ−1C, C(α1) ⊗ tmf(3) +� +. +We use grading conventions indicated in Figure 4 and we depict the spectral sequence in Fig- +ure 6. +Schematic grading convention for the Atiyah–Hirzebruch spectral sequence. +p = +4 +3 +2 +1 +0 +q = +−4 +−3 +−2 +−1 +0 +d2 +d3 +d4 +Figure 4. Dotted lines indicate diagonals p + q = i which converge to an +associated graded of the i-th homotopy; the direction of the first few differential +are indicated. +We need to understand some aspects of HF ∗ +3 Σ−1(C). From Proposition 3.17, whose proof +we defer to the next subsection, we have that +HF ∗ +3 (Th(BU(3)c1≡0; −γ3)) ≃ Z/3[t, c3, c3] · u +as a module over the Steenrod algebra, where |t| = 2, |c2| = 4, |c3| = 6. We have drawn the +P 1-action on those classes which are not multiples of t in Diagram 5. Note that k∗ is surjective, +so HF ∗ +3 (C) can be identified with a submodule of HF ∗ +3 (Th(BU(3)c1≡0; −γ3)) consisting of all +elements except Z/3-multiples of u and c2 · u. Given a class tlci +2cj +3 · u ∈ HF ∗ +3 (C), we refer the +reader to Proposition 3.18 to deduce that +(3.9) +P 1(tlci +2cj +3 · u) = l(tl+2ci +2cj +3 · u) + (i + j − 1)tlci+1 +2 +cj +3 · u. +Since HF ∗ +3 (Σ−1C) has odd cohomology, terms on the diagonal p + q = 0 on the E2-page of +the Atiyah–Hirzebruch spectral sequence of Equation 3.8 arise from odd elements in π∗C(α1)⊗ +tmf(3). In the relevant range, we can describe π∗(C(α1) ⊗ tmf(3)) as follows: +π∗≤26(C(α1) ⊗ tmf(3)) ≃ Z/3 · {1, α1[4], α1β1[4], β1[0], β2[0]} ⊕ H, +where +• H has only even terms (arising from the classes c2 and c4 in π∗ tmf(3)) and these terms +support no α1 or β1 multiplications, so can be disregarded for our calculation. + +22 +MORGAN OPIE +P 1-module structure of Th(BU(3)c1≡0; −γ3) through degree 18, excluding generators +divisible by t. +22 +c2c3 +3 · u +20 +c2 +2c2 +3 · u +18 +c3 +2c3 · u +c3 +3 · u +16 +c4 +2 · u +c2c2 +3 · u +14 +c2 +2c3 · u +12 +c3 +2 · u +c2 +3 · u +10 +c2c3 · u +8 +c2 +2 · u +6 +c3 · u +4 +c2 · u +2 +0 +u +Figure 5. The left column is the degree of the cohomology class. Dotted lines +indicate ±α1 attaching maps detected by a P 1 in HF ∗ +3 Th(BU(3)c1≡0; −γ3). +We omit classes above degree 18 that do not attach to cells at or below 18. +• The S-module structure is given by: +α1 · (α1[4]) = β1[0] +β1 · 1 = β1[0] +α1 · (α1β1[4]) = β2 +1[0] +β1 · (β1[0]) = β2 +1[0] +β1 · (α1[4]) = α1β1[4], +and all other multiplications are zero. +• The degree of an indicated class is the degree of the corresponding class in π∗ tmf(3) +plus the number in the bracket.3 So, for example, α1[4] has degree 7. +The only odd classes in π∗≤26 +� +C(α1) ⊗ tmf(3) +� +are α1β1[4] and α1[4], so contributions on the +E2-page are in bidegrees (−17, 17) and (−7, 7). First consider the classes which are not multiples +of t +α1[4] ⊗ (c2 +2u), α1β1[4] ⊗ (c3 +2c3 · u), α1β1[4] ⊗ (c3 +3 · u). +3These elements are named by the classes that detect them on the E2-page of the Atiyah–Hirzebruch spectral +sequence computing π∗(C(α1) ⊗ tmf(3))). + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +23 +By inspection of Figure 5 and the fact that ⟨α1, α1, α1⟩ = β1 we see that +d4(α1[4] ⊗ c2 +2 · u) = β1[0] ⊗ c3 +2 · u +d4(α1β1[4] ⊗ c3 +3 · u) = −β2 +1[0] ⊗ c2c3 +3 · u +α1β1[4] ⊗ c3 +2c3 · u = d8(β1[0] ⊗ c2c3 · u) +In degree 7 we have +t4 · u +c2 +2. +By Equation (3.9) we have: +P 1(t4 · u) = t6 · u − t4c2 · u +P 1(c2 +2 · u) = c3 +2 · u +Therefore: +d4(α1[4] ⊗ c2 +2 · u) = β1[0] ⊗ c3 +2 · u +d4(α1[4] ⊗ t4 · u) = β1[0] ⊗ (t6 · u − t4c2 · u). +Thus the cell (−7, 7) does not contribute to the E∞-page. +In degree 17 we have generators: +t9 · u +t7c2 · u +t6c3 · u +t5c2 +2 · u +t4c2c3 · u +t3c3 +2 · u +t3c2 +3 · u +t2c2 +2c3 · u +tc4 +2 · u +tc2c2 +3 · u +c3 +2c3 · u +c3 +3 · u +Using Equation 3.9, we see that: +The image of P 1 on the generators for H17(Σ−1C) listed above is: +−t9c2 · u +t9c2 · u +0 +−t7c2 +2 · u + t7c3 +2 · u +t6c2c3 · u + t4c2 +2c3 · u +−t3c4 +2 · u +t3c2c2 +3 · u +−t4c2 +2c3 · u − t2c3 +2c3 · u +t3c4 +2 · u +t3c2c2 +3 · u − tc2 +2c2 +3 · u +0 +−c2c3 +3 · u +Since d4(α1β1[4] ⊗ x) = β2 +1 ⊗ P 1(x), we see that ker(d4): E4 +(−17,17) → E4 +(−20,21) is generated +by α1β1 tensored with: +t9 · u + t7c2 · u +t6c3 · u +tc4 +2 · u + t3c3 +2 · u +c3 +2c3 · u +The next differential involved is a d8 with source (-10,9), which is computed as +d8(β1[0] ⊗ y) = α1β1[4] ⊗ P 1P 1(y). + +24 +MORGAN OPIE +Using Equation (3.9), we get: +P 1 � +P 1(t5 · u) +� += P 1 � +P 1(t5) +� +· u − P 1(t5)P 1(u) + t5P 1 � +P 1(u) +� += (7 ∗ 5)t9 · u − (5t7)(−c2 · u) + t5 · 0 = −(t9 · u + t7c2 · u), +P 1 � +P 1(tc2 +2u) +� += P 1 � +P 1(t) +� +· u − P 1(t)P 1(c2 +2u) + tP 1 � +P 1(c2 +2u) +� += −(t3c3 +2 · u + tc4 +2 · u), +P 1 � +P 1(t2c3 · u) +� += P 1 � +P 1(t2) +� +· u − P 1(t2)P 1(c3 · u) + t2P 1 � +P 1(c3 · u) +� += −t6c3 · u, +since P 1(c2 · u) = 0. And similarly: +P 1 � +P 1(c2c3 · u) +� += P 1 � +P 1(c2) +� +· u − P 1(c2)P 1(c3 · u) + c2P 1 � +P 1(c3 · u) +� += −c3 +2c3 · u, +This shows that gr +� +π0(MapsS(Σ−1C, C(α1) ⊗ tmf(3)) +� += 0, which implies the group itself is +zero. +□ +The relevant part of the E2-page of the Atiyah–Hirzebruch spectral sequence +computing π∗ MapsS(Σ−1C, C(α1) ⊗ tmf(3)). +21 +19 +17 +15 +13 +11 +9 +7 +− q +20 +19 18 +17 +16 15 14 13 12 11 10 +9 +8 +7 +β2[0] +αβ[4] +β[0] +α[4] +d4 +d8 +d4 +Figure 6. The x-axis is graded by q with the generator of π−q(C(α1)⊗tmf(3)). +The y-axis is graded by the degree of generators for HF q +3 (Σ−1C). A circle in +degree (p, −q) indicates a non-zero three-torsion group. Rectangles mark the +bidegrees that can contribute to the p+q = 0 line. We indicate the differentials +that eliminate the terms on p + q = 0. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +25 +Remark 3.16. One might hope that C(α1) ⊗ tmf(3) split from ski Th(BU(3)c1≡0; −γ3) ⊗ tmf(3) +for i > 26. In addition to being a cleaner result, such an extension might allow us to extend +the ρ invariant to vector bundles on higher-dimensional spaces. Unfortunately, our approach +to splitting C(α1) ⊗ tmf(3) is computationally intensive and it is not at all clear how far up the +skeleton the splitting extends. However, from discussions with M.J. Hopkins, we believe ˜ρ can +be extended to all of Th(BU(3)c1≡0; −γ3) by a different method. We hope to work this out in +the future, although it is not necessary for our result. +3.3. The cohomology of Th(BU(3)c1≡0; −γ3) and related spectra. +This subsection includes the main calculations needed for the proof of Theorem 3.9. Some +of these results have already been used in the previous subsection, and some are necessary for +the next subsection. We begin with the cohomology of the relevant classifying space and the +Thom spectrum of interest. +Proposition 3.17. The P 1-module structure on the +(mod 3)-cohomology of BU(3)c1≡0 and +Th(BU(3)c1≡0; −γ3) is given as follows: +(1) HF ∗ +3 (BU(3)c1≡0) ≃ Z/3[t, c2, c3], where |t| = 2, |c2| = 4, and |c3| = 6; and +P 1c2 = c2 +2 +(P 1P 1)c2 = 2c3 +2 +P 1t = t3 +(P 1P 1)t = 0 +P 1c3 = c2c3 +(P 1P 1)c3 = −c2 +2c3. +(2) HF ∗ +3 (Th(BU(3)c1≡0; −γ3)) ≃ HF ∗ +3 (BU(3)c1≡0) · u, with +P 1u = −c2 · u +(P 1P 1)u = 0 +where we view HF ∗ +3 (BU(3)c1≡0) as a P 1-algebra and therefore the P 1-module structure +is determined by the action on u. +Proof. Part (1) follows from Steenrod operations on Chern classes in HF ∗ +3 (BU(3)), together +with the fact the natural map BU(3)c1≡0 → BU(3) induces c1 �→ 0, c2 �→ c2, and c3 �→ c3. +Part (2) follows from the HF3-Thom isomorphism together with the universal formula for +Steenrod operations on Thom classes, which we now explain. +However, we will later need +formulas for Steenrod operations on the Thom class for the bundle −γ4 on BU(4)c1≡0, so we +give these and then deduce those for −γ3 on BU(3)c1≡0. +We first compute P 1 on the Thom class uγ4 for γ4 on BU(4) via the universal example of +MU(1)×4. +P 1(wxyz) = w3xyz + wx3yz + wxy3z + wxyz3 += wxyz(w2 + x2 + y2 + z2) += wxyz((w + x + y + z)2 − 2(wx + wy + wz + xy + xz + yz)), +implying that +(3.10) +P 1(uγ4) = (c2 +1 + c2)uγ4. +The Chern classes for −γ4 are the coefficients of the power series inverse to +c(γ4) = 1 + c1t + c2t2 + c3t3 + c4t4. + +26 +MORGAN OPIE +1 +c(γ4) = 1 − (c1t + c2t2 + c3t3 + c4t4) + (c1t + c2t2 + c3t3 + c4t4)2 +− (c1t + c2t2 + c3t3 + c4t4)3 + (c1t + c2t2 + c3t3 + c4t4)4 + . . . += 1 − c1t + (c2 +1 − c2)t2 + (−c3 +1 + 2c1c2 − c3)t3 + (c4 +1 − 3c2 +1c2 + c2 +2 + 2c3c1 − c4)t4+, . . . +so that +c1(−γ4) = −c1 +c2(−γ4) = c2 +1 − c2 +c3(−γ4) = −c3 +1 + 2c1c2 − c3 +c4(−γ4) = c4 +1 + c2 +2 − c3c1 − c4. +The above implies +P 1(u−γ4) = +� +(−c1)2 + c2 +1 − c2 +� +· u−γ4 = −(c2 +1 + c2) · u−γ4. +(3.11) +We next calculate P 1c2 +1 and P 1c2 in H∗(BU(4)). For c1 we have: +P 1((w + x + y + z)2) = 2(w + x + y + z)P 1(w + x + y + z) += 2(w + x + y + z)(w + x + y + z)3 = 2(w + x + y + z)4, +and +P 1(c2 +1) = −c4 +1. +(3.12) +Let sn = sn(w, x, y, z) denote the n-th elementary symmetric polynomial in x, y, z, w. The +computation for c2 goes as follows: +P 1(wx + wy + wz + xy + xz + yz) = w3x + wx3 + w3y + wy3 + w3z + wz3 ++ x3y + xy3 + x3z + xz3 + y3z + yz3 += s2(w2 + x2 + y2 + z2) − s3s1 + xyzw += s2(s2 +1 + s2) − s3s1 + s4 +Therefore: +P 1(c2) = c2 +1c2 + c2 +2 − c1c3 + c4 +(3.13) +Combining Equations (3.11),(3.12), and (3.13) we get: +P 1P 1(u−γ4) = − +� +P 1(c2 +1 + c2) · u−γ4 + (c2 +1 + c2)P 1(u−γ4) +� += − +� +− c4 +1 + c2 +1c2 + c2 +2 − c1c3 + c4 − (c2 +1 + c2)2) +� +· u−γ4 += −(c4 +1 − c2 +1c2 − c1c3 + c4) · u−γ4. +Now let ˜u denote the Thom class of the negative of the universal bundle on BU(4)c1≡0 and +as before let u denote the Thom class of −γ3 on BU(3)c1≡0. Since the universal bundle has zero +first Chern class, we get: +P 1(˜u) = −c2 · ˜u +P 1P 1(˜u) = −c4 · ˜u +(3.14) +P 1(u) = −c2 · u +P 1P 1(u) = 0. +(3.15) +This completes the proof. +□ +From the above we can use the Liebniz rule for P 1 to derive: +Corollary 3.18. In HF ∗ +3 (Th(BU(3)c1≡0; −γ3)), +P 1(ti · u) = iti+2 · u − tic3 · u +P 1(ci +2cj +3 · u) = (i + j − 1)ci+1 +2 +cj +3 · u + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +27 +In order to prove that ˜ρ is unique in Subsection 3.4, we will need to analyze a certain cofiber. +To that end, the following calculation will be useful. +Proposition 3.19. Let ǫ: S7 → BU(3)c1≡0 generate π7BU(3)c1≡0, let u the Thom class for ǫ, +and let C(ǫ) denote the cofiber of Thu0(ǫ): S7 → Th(BU(3)c1≡0; −γ3). As a module over P 1, +HF ∗ +3 (C(ǫ)) ≃ HF ∗ +3 Th(BU(3)c1≡0; −γ3) ⊕ Z/3 · {y}, +where |y| = 8 and +P 1(−c2 · u) = y +P 1(y) = 0 +Proof. Let ι: BU(3) → BU(4) denote the natural map. Because the composite ι ◦ ǫ: S7 → +BU(4) is null, we get a homotopy commutative diagram +(3.16) +S7 +BU(3)c1≡0 +(BU(3)c1≡0)/S7 +BU(4)c1≡0, +0 +γ3⊕C +δ +where δ is any extension of the bundle γ3 ⊕ C to the cofiber. We get a homotopy pushout a by +taking Thom spectra: +(3.17) +S0 ⊕ S7 +S0 +Th(BU(3)c1≡0; −γ3) +Th((BU(3)c1≡0)/S7; −δ), +p1 +Th(ǫ) +where p1 is projection onto the first factor. Therefore +C(ǫ) ≃ Th((BU(3)c1≡0)/S7; −δ). +Let T := Th(BU(3)c1≡0; −γ3) and T4 := Th(BU(4)c1≡0; −γ4) below. From Equation (3.16) we +get a commuting diagram +(3.18) +Hi−1(T ) +Hi−1(S7) +Hi(C(ǫ)) +Hi(T ) +Hi(S7) +Hi(T4) +0 +a +0 +where Hi denotes either Z or Z/3 coefficients and the top row is exact. The map a induces a +ring isomorphism +(3.19) +H∗(C(ǫ))/H>8 ≃ H∗(Th(BU(4)c1≡0; −γ4))/H>8, +where H>8 is the ideal of elements of degree greater than 8. Equation (3.19) identifies the class +c4 · u with a class y ∈ H∗(C(ǫ)) and implies that +P 1(−c2 · u) = y +P 1(y) = 0, +as was to be shown. To complete the proof, note that Diagram (3.18) implies +H∗(Th(BU(3)c1≡0; −γ3)) ≃ H∗(C(ǫ))/⟨y⟩ +as modules over the Steenrod algebra. +□ + +28 +MORGAN OPIE +3.4. Proof of Proposition 3.13. +Consider the cofiber C := Cof(ǫ: S7 → sk26 BU(3)c1≡0). Let +C′(ǫ) = Cof(S7 → sk26 Th(BU(3)c1≡0; −γ3)) ≃ Th(C; −δ), +where δ: C → BU(4) is an extension of γ3 ⊕ C over C. We get a diagram +(3.20) +S10 +Σ−1C′(ǫ) +S7 +sk26 Th(BU(3)c1≡0; −γ3) +Σ−3 tmf(3), +α1 +˜v +φ +β1 +where ˜v is as in Diagram (3.4). A dotted arrow is an element ˜ρ ∈ tmf(3) +−3 Th(BU(3)c1≡0; −γ3) +satisfying Corollary 3.12. Choices of the dotted arrow in Diagram (3.20) up to homotopy are a +torsor for π0 MapsS(C′(ǫ), Σ−4 tmf(3)). We show that this group is zero via an Atiyah–Hirzebruch +spectral sequence +E2 +p,q = Hp(C′(ǫ); π−q+3 tmf(3)) =⇒ tmf(3) +p+q−3(C′(ǫ)), +depicted in Figure 7. +We computed the cohomology of C(ǫ) in Proposition 3.19 and H∗≤26C(ǫ) ≃ H∗≤26C′(ǫ). +The homotopy of tmf(3) is known [6, 17]. The terms along the line p + q = 0 on the E2-page +are u ⊗ α1 and α1β1 tensored with elements in H10(C′(ǫ)) ≃ H10(Th(BU(3)c1≡0; −γ3)/S7) · u. +Since π∗ tmf(3) has no other odd homotopy groups until degree 27, there are no further possible +contributions to consider. +First, consider u ⊗ α1. Since P 1P 1(u) = y and ⟨α1, α1, α1⟩ = β1, d8(u ⊗ α1) = β1 ⊗ y ̸= 0 +and the class α1 ⊗ u does not survive the spectral sequence. +On the other hand, there are many bidegree (−10, 10) classes to check: +E2 +(−10,10) ≃ α1β1 ⊗ ⟨t5 · u, t3c2 · u, t2c3 · u, tc2 +2 · u, c2c3 · u⟩. +We claim that all classes above support a differential or are the target of a nonzero differential. +Figure 7 shows the first interesting differentials in and out of this cell on the E2-page. +First, we check which of the above are the target of a differential in (A) below. Then we +check that the remaining classes support a nonzero differential in (B). +(A) The only possible differential is a d4 originating in bidegree (−7, 6) is computed as +follows: classes in this cell are β1 times classes in +(3.21) +H6(BU(3)c1≡0) · u ≃ ⟨t3 · u, tc2 · u, c3 · u⟩ +Proposition 3.19 implies: +P 1(t3u) = t3P 1(u) = −c2t3u +P 1(tc2 · u) = t3c2 · u + tc2 +2 · u − tc2 +2 · u = t3c2 · u +P 1(c3 · u) = c2c3 · u − c2c3 · u = 0 +Therefore: d4(β1 ⊗ t3 · u) = −α1β1 ⊗ c2t3 · u and the target dies on the E5-page. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +29 +A larger portion of the E2-page of the Atiyah–Hirzebruch spectral sequence +computing tmf(3) +∗ (C′(ǫ)). +p = +18 +17 +16 +15 +14 +13 +12 +11 +10 +9 +8 +7 +6 +q = −17 −16 −15 −14 −13 −12 −11 −10 −9 −8 −7 +π−q +β2 +1, +c2 +4 +αβ +c6 +β +c4c6 +d4 +d8 +Figure 7. The dashed line indicates the p + q = 0 line converging to +π0 MapsS(C′(ǫ), Σ−3 tmf(3)). A dot indicates a non-zero three-torsion group; +a square a non-zero torsion-free group. The diamond in bidegree (−10, 10) +indicate nonzero E2-terms which may contribute to the computation. +(B) A class β1α1 ⊗ (z · u) which survives to the E8-page supports a nonzero d8 if P 2z is +nonzero. Using Proposition 3.19, facts about Steenrod operations we get: +−(P 1P 1)(t5 · u) = −(P 1P 1)(t5) · u + P 1(t5)P 1(u) += (t9 + t7c2) · u +−(P 1P 1)(t2c3 · u) = −(P 1P 1)(t2)c3 · u + P 1(t2)P 1(c3 · u) − t2(P 1P 1)(c3 · u) += t6c3 · u +−(P 1P 1)(tc2 +2 · u) = −(P 1P 1)(tc2 · u)c2 + P 1(tc2 · u)P 1(c2) − P 1P 1(c2)(tc2 · u) += −P 1(t3c2 · u)c2 + t3c3 +2 · u + tc4 +2 · u += −t3P 1(c2 · u) + t3c2 +2 · u + tc4 +2u += t3c2 +2 · u + tc4 +2 · u +−(P 1P 1)(c2c3 · u) = −(P 1P 1)(c3 · u)c2 + P 1(c3 · u)P 1(c2) − (P 1P 1)(c2)(c3 · u) += −(P 1P 1)(c2)c3 · u += c3 +2c3 · u. + +30 +MORGAN OPIE +This shows that the classes t5 · u, t2c3 · u, tc2 +2 · u, c2c2 · u support nonzero differentials +whose joint span is 4-dimensional. +No terms on the p + q = 0 line survive the spectral sequence, so the group in question is zero. +4. Untwisting the invariant for rank 3 bundles on CP 5 +The work of the previous Section 3 provides an association +V �→ Th(V )∗˜ρ ∈ tmf(3) +−3(Th(CP 5; −V )). +Vector bundles on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) are tmf(3)-orientable, as we +now explain: Any complex bundle V carries an HZ-Thom class. Since τ0 tmf(3) = HZ, V is +tmf(3)-orientable if and only if this class lifts up the Postnikov tower for tmf(3). Since CP 5 +is finite-dimensional, this is a finite lifting problem. The Postnikov tower for tmf(3) through +degree 10 has only one stage that obstructs lifting the HZ-Thom class. This gives the condition +that c2 +1 − 2c2 ≡ 0 (mod 3). +Thus, for V over CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3), there exist isomorphisms +tmf(3) +∗(Th(CP 5; −V )) ≃ tmf(3) +∗(Σ∞ ++ CP 5). However, there are many choices of such an iso- +morphism and our choice cannot be dependent on V . +The ideal way to resolve this would be via a universal example: a space B together with a +bundle VB which carries a (canonical) tmf(3)-orientation, such that all bundles of interest are +canonically pullbacks of VB and therefore inherit a tmf(3)-orientation. Classical examples of this +phenomenon are numerous: BU(n) is canonically HZ oriented, giving the classical HZ-Thom +isomorphism for complex bundles; BSpin is canonically KO-oriented via the Atiyah–Bott– +Shapiro orientation, giving a canonical KO-Thom isomorphism for spin bundles [4]; BString +is canonically tmf-oriented, giving a canonical tmf-orientation for string bundles [2]. +Our bundles are not string, nor is there an obvious candidate for a universal example. Other, +more hands-on, approaches to resolving orientability problems can be found in the literature, +for example in [9, 7]4. The approach we take here is informed by discussions with H. Chatham. +In Theorem 4.9, we show that for V : CP 5 → BU(3) with c1 ≡ 0 (mod 3) and c2 ≡ 0 +(mod 3), there is a set of Thom isomorphisms subject to some concrete restrictions, with the +following property: there is a map i: S10 → Σ∞ ++ CP 5 such that, for any Thom isomorphism v +in this distinguished set, the image of Th(V )∗(˜ρ) under the restriction +(4.1) +tmf(3) +∗(Th(CP 5; −V )) +v−→ tmf(3) +∗(Σ∞ ++ CP 5) +i∗ +−→ tmf(3) +−3(S10) +does not depend on the choice of v. Thus, applying the composite in Equation (4.1) to the +class Th(V )∗(˜ρ) defines the desired invariant ρ(V ). +In Subsection 4.1, we briefly recapitulate the theory of orientations and establish notation. +In Subsection 4.2 we study orientation of bundles on CP 5 in detail, define the set of orientations +which make Equation (4.1) independent of choice within this set. We prove independence in +Theorem 4.9. We can then define the invariant ρ for V rank 3 on CP 5 with c1 ≡ 0 (mod 3) +and c2 ≡ 0 (mod 3) via the definition indicated in the previous paragraph. +The next Subsection 4.3 features the verification that ρ distinguishes bundles with the same +Chern classes, completing the proof of Theorem 1.4. +The final two subsections include some computations (Subsection 4.4) and the observation +that ρ can also be defined for rank 2 bundles (Subsection 4.5). In these subsections we also +discuss future research questions that we hope to address. +Convention 4.1. Throughout this section, all spaces and spectra are implicitly localized at +the prime 3. +4Since LK(2)TMF = EO2 at the prime 3, the orientability studied in [9, 7] is closely related to ours. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +31 +4.1. Background on orientability and orientations. +We present the relevant background on orientability, orientations, Thom classes, and Thom +isomorphisms. +The classical version is as follows. +Let V be a vector bundle over a space +X, with disc bundle dV and sphere bundle sV . +Let Sn → dV be the inclusion of a fiber +inducing i: Sn → dV/sV . Recall that, classically, a Thom class for a vector bundle V over +X in generalized cohomology theory E is a class v ∈ Edim V (dV/sV ) such that i∗v is a unit +in E0(S0) ≃ En(Sn). Orientability refers to the existence of such a class; an orientation is a +choice of such a class. Given an such orientation, pairing with the Thom class under the Thom +diagonal gives a Thom isomorphism +(−) · u: E∗(Σ∞ ++ X) ≃ E∗(Th(X; V )). +Convention 4.2. We will use the terms orientation, Thom class, and Thom isomorphism +synonymously. +Alternatively, a vector bundle V : X → BU(n) gives rise to a (stable) spherical bundle +(4.2) +VS := +� +X → BU(n) → BU +J−→ BGL1S +� +, +where J the complex J-homomorphism.5 The unit map 1: S → E induces a map BGL11: BGL1S → +BGL1E and obtain +(4.3) +VE := +� +X +VS +−→ BGL1S +BGL11 +−−−−−→ BGL1E +� +. +E-orientability is the condition that VE in Equation (4.3) is nullhomotopic; an E-orientation +is a choice of nullhomotopy of VE (see [1], building on [19]). +Remark 4.3. An orientation gives not just an isomorphism on cohomology, but a isomorphism +of E-modules: +Th(X; V ) ⊗ E ≃ Σ∞ ++ X ⊗ E. +Notation 4.4. In this section we will need to refer to many different maps associated to a +given V : X → BU(r). Our notation is slightly abusive, but clear from context. +• V will refer to any of the following associated maps: +– The definitional map X → BU(r), +– The composite X → BU(r) → BU, and +– The transpose of the previous item under the loops-suspension adjunction, which +is a map Σ∞ ++ X → bu. +• For E a commutative ring spectrum, VE will refer to both: +– The composite X +V−→ BU +J−→ BGL1S +BGL11 +−−−−−→ BGL1E, and +– The transpose Σ∞ ++ X +V−→ bu +j−→ bgl1S +bgl11 +−−−→ bgl1E. +4.2. Selecting orientations and the definition of ρ. +The 3-local spectrum Σ∞ ++ CP 5 has a splitting arising from the Adams splitting of Σ∞ ++ CP ∞: +(4.4) +Σ∞ ++ CP 5 ≃ X0 ⊕ X1, +where +X1 ≃ Σ2C(α1) ⊕ S10, +X0 ≃ Σ∞ ++ HP2 ≃ S0 ⊕ Σ4C(α1). +The summand Σ∞ ++ HP 2 is split via p = Σ∞ ++ p′ where p′ : CP 5 → HP 2 is the map of spaces given +by taking the 10-skeleton of the quotient map CP ∞ → HP ∞. +5The complex J homomorphism is commonly defined as a map J : U → GL1S. Our J is the delooping of +the former. + +32 +MORGAN OPIE +Remark 4.5. We fix isomorphisms of X0 and X1 with the sums above. We will write i for all +inclusions of summands and p for all projections onto summands. These choices can be made +once independent of any future bundles involved. +Recall the terminology from Notation 4.4. To study tmf(3)-orientations is to study nullho- +motopies of +Vtmf(3) : Σ∞ ++ CP 5 → bgl1 tmf(3) . +Our strategy is to restrict Vtmf(3) to each summand in the decomposition of Equation (4.4) and +separately study nullhomotopies on each summand. On the summand X1, we show that the +bundles of interest possess a certain canonical orientation arising from the image of j spectrum, +while the choice on the summand X0 will turn out not to matter. +At a prime p, the image of j spectrum bj can be defined as the cofiber of the map ψq−1: bu → +bu, where ψq is the q-th Adams operation and q is a topological generator for Z× +(p). By stable +Adams conjecture (proved6 in [8]) there is a factorization of the j homomorphism +j = +� +bu → bj +j′ +−→ bgl1S +� +, +where the map bu → bj is the natural map to the cofiber of ψq − 1. +Lemma 4.6. The map Vj := +� +X1 +V |X1 +−−−→ bu → bj +� +is nullhomotopic. Up to homotopy, there is +a unique such nullhomotopy. +Proof. The Atiyah–Hirzebruch spectral sequence for computing π∗ MapsS(X1, bj) shows that +both π0(MapsSp(X1, bj) and π0(MapsSp(X1, Σ−1bj) are trivial, since π≤10bj is concentrated in +degrees 0, 4, and 8 [16, Sec. 4] and HF ∗ +p X1 is concentrated in degrees 2, 6, and 10. +□ +Definition 4.7. Let X be a space and let W : X → bu be a given. Assume that the composite +Wj = +� +X +W +−→ bu → bj +� +is canonically null homotopic. Given any generalized cohomology theory E, the j-orientation +of WE will refer to the distinguished nullhomotopy obtained by whiskering the the canonical +nullhomotopy of Wj with the composite bj +j′ +−→ bgl1S +bgl11 +−−−→ bgl1E. +In particular, given a vector bundle V of rank 3 on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 +(mod 3), j-orientation of (V |X1)tmf(3) will refer to the nullhomotopy of the composite +X1 +V |X1 +−−−→ bu → bj → bgl1 tmf(3) +obtained from the unique nullhomotopy of (VX1)j given by Lemma 4.6. +We are interested in orientations which extend j-orientations. +Definition 4.8. Let Y be a space together with a splitting Σ∞ ++ Y = Z ⊕ X. Let E be a ring +spectrum with τ0E = HZ(3). Given a vector bundle V on Y is such that X +V |X +−−−→ bu → bj is +canonically null, we say that a E-orientation v of V satisfies condition (∗) if: +(∗1) v restricts to the j-orientation of (V |X)E; and +(∗2) v lifts the canonical HZ-orientation under 0-truncation bgl1E → bgl1HZ(3). +The main goal of this section is to prove the following: +6Notable previous attempts at the stable Adams conjecture are [10, 11]. The Adams conjecture for spaces +is proved in [24, 23]. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +33 +Theorem 4.9. Let V be a vector bundle over CP 5 with c1(V ) ≡ c2(V ) ≡ 0 (mod 3). Let v be +a tmf(3)-orientation for V satisfying condition (∗) with respect to the decomposition Σ∞ ++ CP ∞ = +X0 ⊕ X1. Then the composite +(4.5) +ρ(V ) := +� +S10 +i⊗1 +−−→ Σ∞ ++ CP 5 ⊗ tmf(3) +v−→ Th(CP 5; −V ) ⊗ tmf(3) +Th(V )∗(˜ρ) +−−−−−−−→ Σ−3 tmf(3) +� +is independent of v, as an element in π0 +� +MapsS(S10, Σ−3 tmf(3)) +� +≃ π13 tmf(3) ≃ Z/3. +Remark 4.10. Note that v satisfying the hypothesis of the theorem exists. Let +V : Σ∞ ++ HP 2 ⊕ X1 → bgl1S +be tmf(3)-orientable. Since π0 MapsS(X1, gl1HZ) = 0, there is a unique nullhomotopy of any null +homotopic map X1 → bgl1HZ. Summing the j-orientation of V |X1 with any tmf(3)-orientation +of V |Σ∞ ++ HP 2 lifting the canonical HZ orientation gives an appropriate orientation of V . +The rest of this section is devoted to the proof of Theorem 4.9. To begin, suppose that v and w +are two tmf(3)-orientations of a bundle V , both satisfying (∗). Consider the following diagram +tmf(3)-modules: +(4.6) +Σ∞ ++ CP 5 ⊗ tmf(3) +S10 ⊗ tmf(3) +Σ∞ ++ CP 5 ⊗ tmf(3), +i +i +w−1v +where w−1v is the automorphism of CP 5 ⊗ tmf(3) obtained from composing the Thom isomor- +phisms corresponding to v with the inverse of the Thom isomorphism corresponding to w (see +Remark 4.3). Note that, if Diagram 4.6 were to commute up to homotopy, the theorem would +be immediate. However, commutativity is stronger than necessary. More precisely, we only +need the diagram to commute after applying a certain tmf(3)-cohomology. We will return to +this after some preliminary calculations. +Let Auttmf(3) denote automorphisms in the category of tmf(3)-modules. +We study which +elements of π0 Auttmf(3)(Σ∞ ++ CP 5 ⊗tmf(3)) arise as ratios of orientations satisfying condition (∗). +Since CP ∞ ≃ X0 ⊕ X1, an automorphism a of Σ∞ ++ CP 5 ⊗ tmf(3) is represented by +(4.7) +a = +� +a00 +a01 +a10 +a11 +� +where the aij ∈ Mapstmf(3)(Xi ⊗ tmf(3), Xj ⊗ tmf(3)) for i ̸= j and aii ∈ Auttmf(3)(Xi ⊗ tmf(3)). +To study the failure of Diagram 4.6 to commute, we examine the possibilities for a10 and a11. +Proposition 4.11. Suppose that v, w are two tmf(3)-Thom classes for a tmf(3)-orientable bun- +dle of rank 3 on CP 5 and that both satisfy condition (∗). Let a = w−1v be the associated +automorphism of Σ∞ ++ CP 5 ⊗ tmf(3) . Then a10 : X1 ⊗ tmf(3) → Σ∞ ++ HP 2 ⊗ tmf(3) is null. + +34 +MORGAN OPIE +Proof. Consider the cofiber sequence of spectra X1 +i−→ Σ∞ ++ CP 5 +p−→ X0. Recall that p = Σ∞ ++ p′, +where p′ is the 10-skeleton of the natural map CP ∞ → HP ∞. We have a diagram +(4.8) +X1 +Σ∞ ++ CP 5 +bu +bj +bgl1S +bgl1 tmf(3) . +Σ∞ ++ HP 2 +=⇒ +(†) +V |X1 +0 +Σ∞ ++ p′ +V +j′ +bgl11 +q +In Diagram 4.8, the nullhomotopy marked (†) is the unique one. The dashed arrow q is de- +termined by the nullhomotopy (†) of the fiber. Given this, a choice v of nullhomotopy Vtmf(3) +extending the canonical j-orientation of V |X1 is equivalent to a choice v′ of nullhomotopy +bgl11 ◦ j′ ◦ q. +Thus, the map p′ of spaces gives rise to a map Th(p′): Th(CP 5; −V ) → Th(HP 2; −j′ ◦ q) +participating in a homotopy commutative diagram +Th(CP 5; −V ) ⊗ tmf(3) +Th(HP 2; −j′ ◦ p) ⊗ tmf(3) +Σ∞ ++ CP 5 ⊗ tmf(3) +Σ∞ ++ HP 2 ⊗ tmf(3) . +v +Th(p′)⊗tmf(3) +v′ +Σ∞ ++ p′⊗tmf(3) +Applying the same argument to obtain w and w′, we get a homotopy commutative diagram +(4.9) +Σ∞ ++ CP 5 ⊗ tmf(3) +Σ∞ ++ HP 2 ⊗ tmf(3) +Th(p′): Th(CP 5; −V ) ⊗ tmf(3) +Th(HP 2; −j′ ◦ q) ⊗ tmf(3) +Σ∞ ++ CP 5 ⊗ tmf(3) +Σ∞ ++ HP 2 ⊗ tmf(3) . +v +Σ∞ ++ p′ +v′ +w−1 +Th(p′)⊗tmf(3) +(w′)−1 +Σ∞ ++ p′ +Diagram 4.9 implies that a = w−1v has a10 ≃ 0. +□ +Proof of Theorem 4.9. Given v and w both tmf(3)-orientations for V satisfying condition (∗), +let a = w−1v and let a11 : X1 → X1 be as in Equation (4.7). By Proposition 4.11, all but the +left-most triangle in the following diagram commute: +(4.10) +X1 +Σ∞ ++ CP 5 ⊗ tmf(3) +S10 ⊗ tmf(3) +(CP 5)−V ⊗ tmf(3) +BU(3)−γ3 ⊗ tmf(3) . +X1 +Σ∞ ++ CP 5 ⊗ tmf(3) +w +i +i +Th(V ) +a11 +w−1v +v +To get that ρ(V ) as in Diagram (4.5) is well-defined, it suffices to prove that Diagram (4.10) +commutes after applying the functor +tmf(3) +−3(−) = π0 +� +Mapstmf(3) +� +(−) ⊗ tmf(3), Σ−3 tmf(3) +�� +. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +35 +Note that the map a11 ◦ i splits as a sum b0 ⊕ b1, where b0 ∈ Auttmf(3)(S10 ⊗ tmf(3)) and +b1 ∈ MapsS(S10, Σ2C(α1) ⊗ tmf(3)). Since tmf(3) +−3(Σ2C(α1)) = 0 by an Atiyah–Hirzebruch +spectral sequence, it suffices to show that b∗ +0 is the identity on tmf(3) +−3(−). +We have reduced to studying the tmf(3)-module automorphisms of S10 ⊗ tmf(3) which arise +from automorphisms of Σ∞ ++ CP 5 ⊗ tmf(3) of the form w−1v for v, w satisfying conditions (∗). +Next, we partially compute the set of all nullhomotopies of Vtmf(3) : Σ∞ ++ CP 5 → bgl1 tmf(3). +This set is a torsor for +G := π0 MapsS(Σ∞ ++ CP 5, Σ−1blg1 tmf(3)) = π0 MapsS(Σ∞ ++ CP 5, gl1 tmf(3)). +Fix one nullhomotopy v of Vtmf(3) which satisfies the hypotheses of Theorem 4.9. We can a +group homomorphism h: G → π0 Auttmf(3)(Σ∞ ++ CP 5 ⊗ tmf(3)) by +g �→ +� +Σ∞ ++ CP 5 ⊗ tmf(3) +gv +−→ Th(CP 5; −f) ⊗ tmf(3) +v−1 +−−→ Σ∞ ++ CP 5 ⊗ tmf(3) +� +and a subgroup G′ := {h ∈ G | hv satisfies conditions (∗) } ⊂ G. This induces a group homo- +morphisms +h′ := +� +G′ ֒→ G +h−→ π0 Auttmf(3)(Σ∞ ++ CP 5 ⊗ tmf(3)) +˜π−→ π0 Auttmf(3)(S10 ⊗ tmf(3)) +� +, +where ˜π takes an automorphism a of Σ∞ ++ CP 5 ⊗tmf(3) to the component b0 ∈ π0 Auttmf(3)(S10 ⊗ +tmf(3)). To show that Diagram 4.10 commutes after applying tmf(3) +−3(−), it suffices to show +that h′ is the zero map. We show that G′ is a subgroup of Z/3 ⊕ Z(3). Since π0 Auttmf(3)(S10 ⊗ +tmf(3)) ≃ Z× +(3) and Z/3 ⊕ Z(3) admits no nontrivial maps to Z× +(3), this suffices. +Given a basepoint for CP 5, we split the zero cell of Σ∞ ++ CP 5. Requiring a given orientation +to lift to the canonical HZ-orientation determines the nullhomotopy on S0. Therefore we have +that G′ ⊂ π0 MapsSp(Σ∞CP 5, gl1 tmf(3)). We compute this group via an Atiyah–Hirzebruch +spectral sequence shown in Figure 8 below. +Thus, there is an extension of Z(3)-modules +0 → Z/3 → π0 MapsSp(Σ∞CP 5, gl1 tmf(3)) → Z(3) → 0. +Since Ext1 +Z(3)(Z(3), Z/3) = 0, the group is Z(3) ⊕ Z/3. +□ +4.3. The invariant ρ separates Chern classes for rank 3 bundles on CP 5. +Our next goal is to show that the invariant ρ defined by Theorem 4.9 distinguishes vector +bundles of rank 3 on CP 5 with the same Chern classes. Recall that π10BU(3)c1≡0 ≃ Z/3 acts +on [CP 5, BU(3)c1≡0] as in Construction 1.6. Given (f, σ) ∈ [CP 5, BU(3)c1≡0] × π10BU(3)c1≡0, +(f, σ) �→ σV := +� +CP 5 +Q +−→ CP 5 ∨ S10 +f∨σ +−−−→ BU(3)c1≡0 +� +. +Fix a1, a2, and a3 with a1 ≡ 0 (mod 3). By Theorem 1.7, this action restricts to a transitive +one on each set +Va1,a2,a3 = {V : CP 5 → BU(3)c1≡0 | ci(V ) = ai}/ ∼, +which is free if only if a2 ≡ 0 (mod 3). +By Theorem 3.9 we have a Thomification isomorphism +t: π10(BU(3)c1≡0) → π10(Σ−3 tmf(3)), +defined relative to an orientation for a generator of π10 Th(BU(3)c1≡0; −γ3). More precisely, let +σ ∈ π10 Th(BU(3)c1≡0; −γ3) and an take an orientation v0 for σ satisfying condition (∗) from +Definition 4.8 with respect to the splitting Σ∞ ++ S10 ≃ S0 ⊕ S10. Then +(4.11) +t(σ) := ˜ρ ◦ Thv0(σ), + +36 +MORGAN OPIE +Nonzero terms on the E2-page of the Atiyah–Hirzebruch spectral sequence +Hp(Σ∞CP 5; π−qgl1 tmf(3)) =⇒ πp+q MapsS(Σ∞CP 5, gl1 tmf(3)). +p = +10 +9 +8 +7 +6 +5 +4 +3 +2 +q = −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 +Figure 8. A circle indicates a non-zero three-torsion group; a square a non- +zero torsion-free group; and a diamond a group Z× +(3). The stars in bidegree +(−10, 10) and (−8, 8) indicate terms along the p + q = 0 line which contribute. +with Thv0 as in Definition 3.8. +Convention 4.12. We write v0 for both the nullhomotopy of σ: Σ∞ ++ S10 → bgl1S and the +associated nullhomotopy of bgl11 ◦ σ. +Our main goal in this subsection is to prove: +Proposition 4.13. Let V be a rank 3 vector bundle on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 +(mod 3). The ρ(σV ) = ρ(V ) + t(σ). +This implies that the invariant ρ separates Chern classes as follows. +Corollary 4.14. If V1, V2 are rank 3 vector bundles on CP 5 that have the same Chern classes, +and such that +c1(V1) = c1(V2) ≡ 0 +(mod 3), +c2(V1) = c2(V2) ≡ 0 +(mod 3). +Then ρ(V1) = ρ(V2) if and only if V1 and V2 are represented by homotopic maps to BU(3), i.e. +if and only if V1 and V2 are topologically equivalent. +Proof of Corollary 4.14 assuming Proposition 4.13. Since π10(BU(3)c1≡0) acts transitively, there +is some element σ ∈ π10(BU(3)c1≡0) such that +V1 = σV2 and ρ(V2) = ρ(V1) + t(σ). +Since t is an isomorphism, ρ(V1) = ρ(V2) if and only if σ = 0 if and only if V1 ≃ V2. +□ +Proof of Proposition 4.13. Consider the diagram +(4.12) +Σ∞ ++ CP 5 +bgl1 tmf(3) . +Σ∞ ++ CP 5 ⊕ S10, +σV +Σ∞ ++ Q +V ⊕σ + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +37 +where Q: CP 5 → CP 5 ∨ S10 is as in Construction 1.6. By Equation 4.4, Σ∞ ++ CP 5 ≃ Σ∞ ++ HP 2 ⊕ +Σ2C(α2) ⊕ S10. Under this identification, +Σ∞ ++ Q = 1 ⊕ 1 ⊕ ∆: Σ∞ ++ HP 2 ⊕ Σ2C(α2) ⊕ S10 → Σ∞ ++ HP 2 ⊕ Σ2C(α2) ⊕ (S10 ⊕ S10). +Let X′ +1 := Σ2C(α2) ⊕ (S10 ⊕ S10). +We obtain an orientation v′ of V ⊕ σ by summing +the j-orientation of (V ⊕ σ|X′ +1)tmf(3) with any nullhomotopy of (V |Σ∞ ++ HP 2)tmf(3) that lifts the +canonical HZ(3)-orientation. By construction, v′ satisfies (∗). This induces a nullhomotopy v +of (σV )tmf(3) and a nullhomotopy ¯v of V , both of which satisfy condition (∗). Thus we get a +commuting diagram of tmf(3)-modules: +Σ−3 tmf(3) +(CP 5)−σV ⊗ tmf(3) +(CP 5 ⊕ S10)−(V ⊕σ) ⊗ tmf(3) +Σ∞ ++ CP 5 ⊗ tmf(3) +(Σ∞ ++ CP 5 ⊕ S10) ⊗ tmf(3), +Th(Q) +(σV )∗ ˜ρ +(V ⊕σ)∗ ˜ρ +Q +v−1 +(v′)−1 +where we suppress tensoring with tmf(3) from the horizontal arrows. Below, the pushout of +spaces on the left induces the diagram of Thom spectra on the right: +∗ +S10 +S0 +(S10)−σ +CP 5 +CP 5 ∨ S10 +(CP 5)−V +(CP 5 ∨ S10)−V ⊕σ. +The j-orientation for σS gives an equivalence +(4.13) +(CP5 ∨ S10)−V ⊕σ ≃ (CP 5)−V ⊕ S10. +The nullhomotopy v′|S10 of σtmf(3) extends the j-orientation. So, using the identification (4.13), +we have that Th(V ⊕ σ)∗ ˜ρ = Th(V )∗˜ρ ⊕ t(σ). Thus we get the homotopy commutative Dia- +gram (4.14) of tmf(3)-modules below: +(4.14) +(CP 5)−σV ⊗ tmf(3) +� +(CP 5)−V ⊕ S10� +⊗ tmf(3) +Σ−3 tmf(3) +Σ∞ ++ CP 5 ⊗ tmf(3) +� +Σ∞ ++ CP 5 ⊕ S10� +⊗ tmf(3) +S10 ⊗ tmf(3) +� +S10 ⊕ S10� +⊗ tmf(3) +. +Th(σV )∗ ˜ρ +Th(V )∗ ˜ρ⊕t(σ) +Q +v−1 +¯v−1⊕1 +ρ(V )+t(σ) +i +∆⊗tmf(3) +i⊕1 + +38 +MORGAN OPIE +Comparing the two outer paths from the lower left-hand corner to Σ−3 tmf(3) gives the result. +□ +4.4. Computing ρ on certain sums of line bundles. +Let ρ be as defined by Theorem 4.9. For L a line bundle, we define ρ(L) := ρ(L ⊕ C2). +Lemma 4.15. Suppose that O(a1), O(a2) and O(a3) are line bundles on CP 5 with ai ≡ 0 +(mod 3). Then ρ (O(a1) ⊕ O(a2) ⊕ O(a3)) = ρ(O(a1)) + ρ(O(a2)) + ρ(O(a3)) ∈ Z/3. +This immediately implies: +Corollary 4.16. Let a be an integer divisible by 3. Then ρ(O(a)⊕3) = 0. +Proof of Lemma 4.15. Let V := ⊕3 +i=1O(ai). Let ˜V be the bundle on (CP 5)×3 given by +˜V := ⊕3 +i=1p∗ +i O(ai), +where pi is the i-th projection. We can factor V : CP 5 → BU(3)c1≡0 as follows +V : CP 5 +∆ +−→ CP 5×3 +˜V−→ BU(3)c1≡0 +and naturally identify Th((CP 5)×3; − ˜V ) ≃ ⊗i Th(CP 5; −O(ai)). Using tmf(3)-orientations sat- +isfying (∗) 7 for each O(ai), we get a diagram of tmf(3)-modules: +Th(CP 5; −V ) +⊗i Th(CP 5; −O(ai)) +Th(BU(3)c1≡0; −γ3) +Σ−3 tmf(3) +Σ∞ ++ CP 5 +(Σ∞ ++ CP 5)⊗3 +S10 +(S10)⊗3 +S10 +S10 ⊕ S10 ⊕ S10 +Th(∆) +Th( ˜V ) +˜ρ +Σ∞ ++ ∆ +≃tmf(3) +≃tmf(3) +Σ∞∆ +ρ(V ) +1⊕1⊕1 +⊕iρ(O(ai)) +where all terms in the diagram are implicitly tensored with tmf(3) and the maps marked ≃tmf(3) +are tmf(3)-Thom isomorphisms. The diagram is homotopy commutative and comparing the +dashed arrows proves the Lemma. +□ +While this section provides some computations of ρ, we do not have a general formula. +Indeed, it is unclear what a formula for ρ should look like, since ρ cannot be computed from +Chern classes. Some inspiration can be drawn from [5], where Atiyah and Rees show that the +α invariant of a rank 2 bundles on CP 3 can be computed as a holomorphic semi-characteristic +[5, Theorem 4.2], provided we choose an holomorphic representative for the topological class of +the bundle. This leads to the following question: +Question 4.17. For V an algebraic vector bundle on CP 5, is there some description of the +invariant ρ(V ) in terms of sheaf cohomology of V ? +7See Definition 4.8. + +A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 +39 +4.5. A 3-torsion tmf(3)-valued invariant for rank 2 bundles. +The homotopy groups of BU(3) through degree 10 are fairly sparse, so a complete analysis +of [CP 5, BU(3)] was possible. This allowed us to classify rank 3 bundles on CP 5 by first enu- +merating such bundles and second defining an invariant to distinguish them. While unraveling +the structure of π∗BU(3), we began to suspect that ρ may also be interesting in the case of +rank 2 bundles on CP 5. +Proposition 4.18. The class ˜ρ: sk26 Th(BU(3)c1≡0; −γ3) → Σ−3 tmf(3) factors through sk26 +Th(BU(2)c1≡0; −γ2). Moreover: +a. The map induced map π10BU(2)c1≡0 → π10Σ−3 tmf(3) given by Thomifying with respect +to −γ2 followed by ˜ρ is a bijection. +b. The invariant V �→ ρ(V ) distinguishes 3-local equivalence classes of rank 2 vector bun- +dles on CP 5 with fixed c1, c2 where additionally c1 ≡ 0 (mod 3). +Proof. For (a), recall Diagram 3.1. +Note that the unstable generator for π4BU(3) factors +through BU(2), so Theorem 3.9 shows that the image of a generator for π10BU(2)c1≡0 is nonzero. +Therefore it suffices to check that π10BU(2) ≃ Z/3. This is classical: since BSU(2) ≃ BS3, +the homotopy in the relevant range is given in Figure 9. +π2 +π3 +π4 +π5 +π6 +π7 +π8 +π9 +π10 +BU(2) +Z +0 +Z +Z/2 +Z/2 +Z/12 +Z/2 +Z/2 +Z/3 +Figure 9. Homotopy of BU(3) +Since CP 5 is even, only even 3-local homotopy gives rise to 3-local invariants (odd 3-local +homotopy contributes to constraints on possible Chern classes). Therefore a argument as in +Remark 2.9 shows that the π10BU(3)-action as in Construction 1.6 is the only source of 3-local +invariants beyond Chern classes; for rank 2 bundles on CP 5 with c1 ≡ 0 (mod 3), these bundles +detected by ρ. +□ +Questions 4.19. Proposition 4.18 opens up several avenues of inquiry: +• For which c1, c2 ∈ Z is the action of π10BU(2) nontrivial? In other words, when is the +invariant ρ of rank 2 bundles on CP 5 is occupied? +• What is the 2-local enumeration of rank 2 bundles on CP 5 with fixed Chern data? +Figure 9 shows that there is significant two-local data to analyze. +References +[1] M. Ando, A. J. Blumberg, D. Gepner, M. J. Hopkins, and C. Rezk. Units of ring spectra, orientations and +thom spectra via rigid infinite loop space theory. Journal of Topology, 7(4):1077–1117, 2014. +[2] M. Ando, M. J. Hopkins, and C. Rezk. Multiplicative orientations of ko-theory and the spectrum of topo- +logical modular forms. Available at faculty.math.illinois.edu/∼mando/papers/koandtmf.pdf, 2010. +[3] B. Antieau and E. Elmanto. A primer for unstable motivic homotopy theory. Surveys on recent developments +in algebraic geometry, 95:305–370, 2017. +[4] M. Atiyah, R. Bott, and A. Shapiro. Clifford modules. Topology, 3(1):3–38, 1964. +[5] M. Atiyah and E. Rees. Vector bundles on projective 3-space. Inventiones Math., 35:131–153, 1976. +[6] T. Bauer. Computation of the homotopy of the spectrum tmf. Geom. Topolo. Monogr., 13:11–40, 2008. +[7] P. +Bhattacharya +and +H. +Chatham. +On +the +EO-orientability +of +vector +bundles. +Available +at +arXiv:2003.03795v3, 2020. +[8] P. Bhattacharya and N. Kitchloo. The stable Adams conjecture and higher associative structures on Moore +spectra. Ann. of Math., 195:375–420, 2022. +[9] H. Chatham. An orientation map for height p − 1 real e theory. 2019. + +40 +MORGAN OPIE +[10] E. M. Friedlander. The infinite loop adams conjecture via classification theorems for F-spaces. Math. Proc. +Cambridge Philos. Soc., 87(1):109–150, 1980. +[11] E. M. Friedlander and R. M. Seymour. Two proofs of the stable adams conjecture. Bull. Amer. Math. Soc., +83(6):1300–1302, 1977. +[12] F. Hirzebruch. Topological methods in algebraic geometry. Springer, 1995. +[13] Y. Hu. Metastable complex vector bundles over complex projective spaces. arXiv:2202.11800, 2022. +[14] K. Kodaira and D. C. Spencer. Divisor class groups on algebraic varieties. Proc. Nat. Acad. Sci. U.S.A., +39:872–877, 1953. +[15] T. Kudo. A transgression theorem. Mem. Fac. Sci. Kyusyu Univ., A(9):79–81, 1956. +[16] M. Mahowald and D. Ravenel. Towards a global understanding of the stable homotopy groups of spheres. In +S. Gitler, editor, The Leftschets Centennial Conference: Proceedings on Algebraic Topology II, volume 58 +of Contemporary Mathematics, pages 109–118. AMS, 1987. +[17] A. Mathew. The homotopy groups of TMF. Available at math.uchicago.edu/∼amathew/tmfhomotopy.pdf. +[18] J. P. May and K. Ponto. More Concise Algebraic Topology: Localization, Completion, and Model Cate- +gories. Chicago Lectures in Mathematics. The Unviersity of Chicago Press, Chicago and London, 2012. +[19] J. P. May and F. Quinn. e∞ ring space and e∞ ring spectra. Lecture Notes in Mathematics, 577, 1977. +[20] J. McCleary. A User’s Guide to Spectral Sequences. Cambridge Studies in Advanced Mathematics. Cam- +bridge University Press, 2 edition, 2000. +[21] H. Miller. Lectures on algebraic topology ii. Available at http://math.mit.edu/∼hrm/papers/lectures-905- +906.pdf, 2020. +[22] M. Mimura and H. Toda. Homotopy groups of SU(3), SU(4), and Sp(2). J. Math. Kyoto Univ., 3(2):217– +250, 1963. +[23] D. Quillen. The Adams conjecture. Topology, 10:67–80, 1971. +[24] D. Sullivan. Genetics of homotopy theory and the Adams conjecture. Ann. of Math., 100:1–79, 1974. +[25] A. Thomas. Almost complex structures on complex projective spaces. Trans. Amer. Math. Soc., 193:123– +132, 1974. +[26] A. Zabrodsky. Hopf Spaces. North-Holland Publishing Company, Amsterdam, New York, Oxford, 1976. + diff --git a/cdE3T4oBgHgl3EQfGQmo/content/tmp_files/load_file.txt b/cdE3T4oBgHgl3EQfGQmo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6f66d2a5979d2d676835fabe65c93ab8b629d4c0 --- /dev/null +++ b/cdE3T4oBgHgl3EQfGQmo/content/tmp_files/load_file.txt @@ -0,0 +1,1565 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf,len=1564 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='04313v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='AT] 11 Jan 2023 A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 MORGAN OPIE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We show that complex rank 3 topological vector bundles on CP 5 are determined by their Chern classes, except in the case that c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To address this case, we produce a universal class in the tmf(3)-cohomology of a Thom spectrum related to BU(3), where tmf(3) denotes topological modular forms localized at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From this class and orientation data, we construct a Z/3-valued invariant of the bundles of interest and prove that our invariant separates distinct bundles with the same Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Introduction 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Paper outline 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Acknowledgements 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Conventions 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A count of rank 3 bundles on CP 5 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 3-complete rank 3 vector bundles on CP 5 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of technical claims 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2-complete rank 3 vector bundles on CP 5 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Schwarzenberger conditions 13 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Defining a twisted tmf(3)-valued invariant 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof outline: existence and uniqueness of a twisted tmf(3) invariant 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The cohomology of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) and related spectra 25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13 28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Untwisting the invariant for rank 3 bundles on CP 5 30 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Background on orientability and orientations 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Selecting orientations and the definition of ρ 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The invariant ρ separates Chern classes for rank 3 bundles on CP 5 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Computing ρ on certain sums of line bundles 38 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A 3-torsion tmf(3)-valued invariant for rank 2 bundles 39 References 39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Introduction Let X be a finite-dimensional CW-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From the perspective of homotopy theory, a topological vector bundle of complex rank r over a space X is identified with a classifying map X → BU(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Topologically equivalent vector bundles over X correspond to homotopy equivalent maps to BU(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The integral cohomology of BU(r) is generated by universal Chern classes c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' cr, with ci ∈ H2i(BU(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' These give rise to important invariants of complex bundles: the Chern 1 2 MORGAN OPIE classes of a bundle, defined for V : X → BU(r) as the pullbacks ci(V ) := V ∗(ci) ∈ H2i(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the case X = CP n, Chern classes are complete invariants of the stable equivalence class of the bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Explicitly, this means that V : CP n → BU(r) and W : CP n → BU(r′) have the same Chern classes if and only if there exist integers n, m greater than zero such that V ⊕ Cn and W ⊕ Cm are topologically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Here, C is the trivial rank 1 bundle on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This leads to the following fundamental question: Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Are Chern classes sufficient to determine the (unstable) topological class of a complex rank r vector bundle on CP n, up to topological equivalence?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' If not, what invariants beyond Chern classes are needed to distinguish such bundles?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Rank 1 bundles on all spaces CP n are determined by their first Chern class [3, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Rank ≥ n bundles on CP n are also determined by their Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For r strictly between 1 and n, there is no uniform answer (although some patterns have been found when restricting to bundles with all Chern classes zero, see [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In [5], Atiyah and Rees answer Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 for complex rank 2 topological bundles on CP 3 by producing a Z/2-valued invariant α, which can be viewed as a characteristic class in the generalized cohomology of a classifying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 ([5, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a1, a2 ∈ Z with a1a2 ≡ 0 (mod 2), the number of rank 2 bundles on CP 3 with i-th Chern class ai is: equal to 2 if a1 ≡ 0 (mod 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and equal to 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the first case, a rank 2 vector bundle on CP 3 is determined by c1, c2, and α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The condition a1a2 ≡ 0 (mod 2) is necessary and sufficient for two integers to be the Chern classes of a rank 2 bundle on CP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Atiyah–Rees’ works shows that the classification of rank 2 bundles on CP 3 is a 2-primary problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since there are similarities between the 2-primary homotopical structure of BU(2) and the 3-primary homotopical structure of BU(3), one might hope for an analogy between the classification of rank 2 bundles on CP 3 and of rank 3 bundles on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our goal is to realize this analogy and answer Question 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 for rank 3 bundles on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We do this by defining a Z/3-valued invariant ρ of such bundles and proving the following: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a1, a2, a3 ∈ Z satisfying the the Schwarzenberger condition S5 (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='16), the number of bundles of rank 3 on CP 5 with i-th Chern class equal to ai is: equal to 3 if a1 ≡ 0 (mod 3) and a2 ≡ 0 (mod 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and equal to 1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the first case, a rank 3 bundle on CP 5 is determined by c1, c2, c3 and ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, we show that the Schwarzenberger condition S5 is necessary and sufficient for three integers to be the Chern classes of a rank 3 bundle on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We also give S5 explicitly in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A priori, there is no simple geometric relationship between topologically distinct bundles with the same Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, in both the case of rank 2 bundles on CP 3 and the case of rank 3 bundles on CP 5, any two bundles with the same Chern classes differ by an explicit action defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 3 Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Associated to an inclusion D2n ֒→ CP n of a disk in the top cell of CP n, we define Q: CP n → S2n ∨ CP n by collapsing the boundary of D2n to a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given vector bundles V : CP n → BU(n) and σ: S2n → BU(r) we define σV := (σ ∨ V ) ◦ Q: CP n → BU(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Diagrammatically: S2n CP n S2n ∨ CP n BU(r) CP n ι1 σ Q σ∨V ι2 V where ι1 and ι2 are the standard maps of the summands into the wedge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The association (σ, V ) �→ σV defines an action of π2nBU(r) on equivalence classes of rank r vector bundles over CP n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This action preserves Chern classes provided that n > r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Therefore, if nontrivial, this action gives topologically distinct bundles with the same Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the case that r = 2 and n = 3, the action of π6BU(2) ≃ Z/2 on rank 2 bundles on CP 3 with fixed Chern data is transitive, and is free if and only if c1 ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This aligns with the enumeration of bundles with fixed Chern data in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The theorem below shows that the role of Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6 in analyzing rank 3 bundles on CP 5 is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let a1, a2, and a3 be integers satisfying S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let Va1,a2,a3 be the set of homotopy classes of rank 3 bundles on CP 5 with i-th Chern class equal to ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then: (1) The action of π10BU(3) ≃ Z/3 on rank 3 bundles over CP 5, as given in Construc- tion 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6, induces a transitive action on Va1,a2,a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (2) If a1 or a2 is nonzero mod 3, then the action of π10BU(3) on Va1,a2,a3 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (3) If a1 and a2 are zero mod 3, then the action of π10BU(3) on Va1,a2,a3 is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This refines the enumeration result in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 says that, if a1, a2, a3 satisfy S5, a1 ≡ 0 (mod 3), and a2 ≡ 0 (mod 3), then the set of complex rank 3 topological vector bundles on CP 5 with i-th Chern class ai is a torsor for Z/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The goal of the rest of the paper is to trivialize this torsor via a bundle invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To explain our approach to defining such an invariant for rank 3 bundles on CP 5, we discuss the α-invariant of rank 2 bundles on CP 3 in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Atiyah–Rees invariant α is initially defined for rank 2 bundles with c1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Such bundles are classified by maps to BSU(2), allowing an invariant to be defined via a universal class in the generalized cohomology of BSU(2) rather than BU(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Atiyah and Rees give a class α ∈ KO4(BSU(2)), where KO denotes real K-theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' They define the α-invariant of V : CP 3 → BSU(2) as α(V ) := p∗V ∗(α) ∈ KO−2(point) ≃ Z/2, where V ∗ is pullback with respect to V and p∗ : KO∗(CP 3) → KO∗−6(point) is the KO-theory pushforward for the spin manifold CP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' They extend α to bundles with c1(V ) ≡ 0 (mod 2) by α(V ) := α � V ⊗ O(−c1(V ) 2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 4 MORGAN OPIE Alternatively, the Atiyah–Rees invariant can be rephrased as a twisted characteristic class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Recall that, given a virtual bundle W over a space X, the Thom spectrum of X with respect to W, written Th(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' W), can be viewed as a twisted version of the suspension spectrum Σ∞ + X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By a twisted characteristic class, we will mean a class in some generalized cohomology of a Thom spectrum over a classifying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In this framework, one can show that there is a class ˜α ∈ KO∗(Th(BU(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ2)) which extends α in a precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given any rank 2 vector bundle on CP 3, the pullback of ˜α gives a class ˜α(V ) := V ∗ ˜α ∈ KO4(Th(CP 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' If c1(V ) ≡ 0 (mod 2), V is canonically KO-oriented, yielding a KO-Thom isomorphism KO∗(Th(CP 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )) ≃ KO∗(Σ∞ + CP 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We can thus define α′(V ) = p∗(˜α(V )) ∈ KO−2(point) ≃ Z/2, where as before p∗ is the KO-theory pushforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The invariant α′ also distinguishes rank 2 bundles on CP 3 with c1 ≡ 0 (mod 2) and agrees with the original α-invariant when c1(V ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The insight here is that both BSU(2) and Th(BU(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ2) stabilize π6BU(2), in the follow- ing sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' While π6BU(2) ≃ Z/2, the stable homotopy group π6 (Σ∞BU(2)) is trivial, so bun- dles differing by an element in the unstable group π6BU(2) cannot be distinguished by a char- acteristic class in the generalized cohomology of BU(2) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, both π6 (Σ∞BSU(2)) and π6 Th(BU(2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ) are nontrivial and are canonically isomorphic to π6BU(2), permitting their KO-cohomology to supply the classes α and α′, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Recalling Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7, the relevant group for understanding rank 3 bundles on CP 5 is π10BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We also find that π10BU(3) ≃ Z/3 is stably trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By the above discussion, we might attempt to classify rank 3 bundles on CP 5 by first stabilizing π10BU(3) and then detect- ing it with some generalized cohomology theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Indeed, our strategy to define an invariant of rank 3 bundles on CP 5 is as follows: We identify a Thom spectrum related to BU(3) which stabilizes π10BU(3) (Introduction to Section 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We define a twisted characteristic class in an appropriate generalized cohomology of this Thom spectrum, with certain key properties (Section 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and We show that our twisted characteristic class can be resolved to an honest invariant ρ, via orientation data, and that this invariant distinguishes vector bundles with the same Chern data (Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The main result of Section 3 can be stated as follows: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let BU(3)c1≡0 be the homotopy fiber of c1 (mod 3): BU(3) → K(Z/3, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let tmf(3) denote the 3-localization of the spectrum of topological modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' There is a class ˜ρ ∈ tmf(3) −3(sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) such that the pullback of ˜ρ with respect to the Thomificiation of a generator for π10BU(3)c1≡0 induces an isomorphism (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) π10BU(3)c1≡0 ≃ π13 tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The class ˜ρ and isomorphism (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) tell us that the cohomology theory tmf(3) stably detects π10BU(3) and therefore retains information about the bundles of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Under pullback, the class ˜ρ together with Thom isomorphisms determined by orientation data give rise to the invariant ρ of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 5 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8 gives an association (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) V �→ Th(V )∗(˜ρ) ∈ tmf(3) −3(Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )), where Th(V ) denotes the Thomification of the classifying map V : CP 5 → BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Equa- tion (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) does not define an invariant of V because the target depends on V itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, vector bundles with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) are tmf(3)-orientable and therefore admit a tmf(3)-Thom isomorphisms tmf(3) ∗(Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )) ≃ tmf(3) ∗(Σ∞ + CP 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The problem is not quite solved: we need a consistent way of choosing Thom isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This is the main project of Section 4, which involves a detailed study of tmf(3)-orientations for the relevant bun- dles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The problem cannot be reduced to a known orientation problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' using the celebrated string orientation for topological modular forms [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Paper outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 is the main project of Section 2 and proceeds via analyses of the set of homotopy classes of maps from CP 5 to BU(3) localized at the primes 3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' These arguments are carried out in Subsections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, respectively, and involve obstruction- theoretic arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 proves claims used in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4 we compute the Schwarzenberger condition explicitly and show that it is necessary and sufficient for three integers to be the Chern classes of a rank 3 bundle on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, we outline our method to produce the class ˜ρ in the tmf(3)-cohomology of sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The remaining Subsections 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4 supply the details of the proof, which includes a uniqueness result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This concludes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, we review the theory of Thom isomorphisms and orientations and also establish notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2, we study orientations of rank 3 bundles on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) and isolate a desirable set of tmf(3)-orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Using this set of orientations, we are able to produce a well-defined invariant: in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, we combine orientation data with ˜ρ to define the invariant ρ of complex rank 3 topological bundles on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We prove that ρ separates topological equivalence classes of rank 3 bundles on CP 5 with the same Chern data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The remaining subsections offer examples and suggest future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, we show that ρ(L⊕3) = 0 for L a line bundle with c1(L) ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We also state an additivity result for ρ on sums of line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5, we show that the methods discussed in this paper also produce a 3-local invariant of rank 2 bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' First and foremost I want to thank my PhD advisor, Mike Hopkins, for suggesting this project and for his immense support throughout my PhD program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' I am also immensely grateful to Haynes Miller for his mentorship during my time in graduate school;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and to both Haynes and Elden Elmanto for serving on my dissertation committee and offering feedback on my thesis write-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' After my move to UCLA, Mike Hill’s guidance and encouragement – mathematical and practical – were invaluable for me while improving and revising my thesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hood Chatham and Jeremy Hahn were both extremely generous in offering specific suggestions for methods and strategies used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This work benefited greatly from my conversations with Aravind Asok, Lukas Brantner, Yang Hu, Dev Sinha, Alexander Smith, and Dylan Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' While working on this project, the author was supported by the National Science Foundation under Award No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2202914.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 6 MORGAN OPIE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' “Vector bundle” will refer to a complex, topological vector bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' As such, we use “vector bundle” and “map to BU(r)” interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' “Rank” refers to complex rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given spaces X and Y , we write [X, Y ] for homotopy classes of maps from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' H∗ will refer to ordinary cohomology with Z coefficients, except otherwise stated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We write HF ∗ p for cohomology with Fp = Z/p coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' If C is a space, then π∗C will refer to unstable homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' If X is a spectrum, π∗X will refer to its stable homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus, the stable homotopy groups of a space C will be written as π∗(Σ∞ + C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a space or spectrum X, we write τnX for its n-th Postnikov section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given virtual bundle W on a topological space Y , we write Th(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' W) for the Thom spectrum of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a map f : X → Y of spaces, f has a Thomification Th(f) : Th(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' f ∗W) → Th(Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' When taking Thom spectra, we assume all bundles have virtual dimension zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a space or spectrum X, we write Xˆ p for its completion at a prime p, and X(p) for its localization away from p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Sections 3 and 4, all spaces and spectra are implicitly localized at the prime 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given an E∞-ring spectrum R and two R-modules X, Y , we write MapsR(X, Y ) := MapsR- Mod(X, Y ) = R- Mod(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A count of rank 3 bundles on CP 5 The primary goal of this section is to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 by computing the set [CP 5, BU(3)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For this we need the homotopy of BU(3) through degree 10, which can be computed via the fiber sequence BSU(3) → BU(3) → BU(1) and its associated homotopy long exact sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The homotopy of U(1) ≃ S1 is known and enough of the homotopy of SU(3) is computed in [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We give the result in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' π2 π3 π4 π5 π6 π7 π8 π9 π10 BU(3) Z 0 Z 0 Z Z/6 0 Z/12 Z/3 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Homotopy of BU(3) Note that the torsion in πnBU(3) for n ≤ 10 is either 2- or 3- primary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus we may break the computation into analyses at the primes 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The key tool which allows us to study the problem one prime at a time is the theory of rationalization and completion of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a space X, let Xˆ p denote its p-completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Fracture Theorem for completion, as stated in [18, Theorem 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1], implies that an element in [CP 5, BU(3)] is the same data as pairs of maps f2 : CP 5 → BU(3)ˆ 2, f3 : CP 5 → BU(3)ˆ 3 such that the Zˆ 2- and Zˆ 3-valued Chern classes are both in the image of the canonical inclusion Z ֒→ Zˆ p and agree under this identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The in-depth analysis in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 is calculating [CP 5, BU(3)ˆ 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This is carried out in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, with supporting technical results in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, we compute [CP 5, BU(3)ˆ 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, we show that the Schwarzenberger A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 7 condition S5 is necessary and sufficient for three integers to be the Chern classes of a rank 3 bundle on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 and justifies Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 3-complete rank 3 vector bundles on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We give the first stages of a Postnikov-type tower for the 3-completion of BU(3) and analyze maps from CP 5 into this tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' There is a tower of principal fibrations given by the solid arrows below: K(Z/3, 10) P10 K(Z/3, 7) × K(Z/3, 9) P9 K(Z/3, 11) BU(3) K(Z, 2) × K(Z, 4) × K(Z, 6) K(Z/3, 8) × K(Z/3, 10) τ10 U (c1,c2,c3) τ9 k7×k9 where (c1, c2, c3) induces a 3-complete equivalence on τ6BU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' τ9 induces a 3-complete equiv- alence on τ9BU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and τ10 induces a 3-complete equivalence on τ10BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' At present, the explicit forms of k7 × k9 and U are not needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, we can calculate [CP 5, τ10BU(3)ˆ 3] ≃ [CP 5, BU(3)ˆ 3] by working up the tower.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We need the following standard lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let X be a connected space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For any other space Y let Y X denote the mapping space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a fiber sequence of connected spaces (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) F E B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and a map f : X → E so that the composite map to B is nullhomotopic, the set of homotopy classes of choices of lifts of f to F is a torsor for coker � π1(EX, f) → π1(BX, 0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 to the diagram in Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Candidate Chern data (a1, a2, a2) ∈ H2(CP 5) × H4(CP 5) × H6(CP 5) lifts to P9 if and only if (k7 × k9) ◦ (a1, a2, a3) ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This is a mod 3 condition on Chern classes, which we do not compute since we recover the condition via different methods in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2, the number of lifts to P9 are a torsor for a quotient of π1 �� K(Z/3, 8) × K(Z/3, 10) �CP 3� ≃ HF 7 3 (CP 3) × HF 9 3 (CP 3) = 0, so when a lift exists it is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' There are no obstructions to lifting from P9 to P10, since HF 11 3 (CP 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Choices of lift are a torsor for coker � π1(P9 CP 5) U◦− −−−→ π1(K(Z/3, 11)CP 5) � Since π1(K(Z/11)CP 5) ≃ π0(K(Z/3, 10)CP 5)) ≃ HF 10 3 (CP 5) ≃ Z/3, there are two possibilities that will depend on a1, a2, a3: The map is surjective, the cokernel is trivial, and there is a unique lift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' or The map is zero, the cokernel is Z/3 and there are three lifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To compute Im(U ◦ −), we consider a related problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From the principal fibration (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) K(Z/3, 7) × K(Z/3, 9) → P9 → K(Z, 2) × K(Z, 4) × K(Z, 6), we get an action of the fiber � K(Z/3, 7) × K(Z/3, 9) � × P9 → P9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This gives an action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) π1 � K(Z/3, 7)CP 5 × K(Z/3, 9)CP 5� × π1(P CP 5 9 ) → π1(P CP 5 9 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 8 MORGAN OPIE Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The action given in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Assuming this claim too, fix a1, a2, a3 ∈ Z with (k7 × k9) ◦ (a1, a2, a3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the diagram: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4) P9 K(Z/3, 11) ∗ × CP 5 S1 × CP 5 K(Z/3, 7) × K(Z/3, 9) × P9, U [a1,a2,a3] ∗×1 a (xι1t3,yι1t4,a) † ⋆ m m∗U where only the triangles † and ⋆ commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the above: (1) a: S1 × CP 3 → τ5BU(2) restricts to [a1, a2, a3] on ∗ × CP 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (2) x,y ∈ Z/3 are arbitrary coefficients of the classes ι1t3 and ι1t4, which are the natural generators of HF 7 3 (S1 × CP 5) ≃ HF 1 3 (S1 ⊗ HF 6 3 (CP 5) ≃ Z/3{ι1} ⊗ Z/3{t3} and HF 9 3 (S1 × CP 5) ≃ HF 1 3 (S1) ⊗ HF 8 3 (CP 5) ≃ Z/3{ι1} ⊗ Z/3{t4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, to compute Im(U ◦ −), it suffices to compute m∗U ◦ (xι1t3, yι1t4, a) as (x, y) ranges over Z/3 × Z/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We will obtain formula for the difference (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) m∗U ◦ (xι1t3, yι1t5, a) − m∗U ◦ (0, 0, a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Showing that Im(U ◦ −) is Z/3 is equivalent to finding x, y ∈ Z/3 so that the difference in Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The class m∗U ∈ HF 11 3 � K(Z/3, 7) × K(Z/3, 9) × P9 � is given by m∗U = U + P 1(ι′ 7) − ι2ι′ 9 + ι2 2ι′ 7 − ι4ι′ 7 ∈ HF 11 3 � K(Z/3, 7) × K(Z/3, 9) × P9 � , where ι′ 7 and ι′ 9 generate HF 7 3 � K(Z/3, 7) � and HF 9 3 � K(Z/3, 9) � , respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and where ι2, ι4 are the images in HF ∗ 3 (P9) of generators in HF i 3 (K(Z, 2)) and HF i 3 (K(Z, 4)), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, since ι2 pulls back to c1 (mod 3) in HF ∗ 3 (CP 5) and ι4 pulls back to c2 (mod 3), we see that m∗U ◦ (xι1t3, yι1t4, a) − U ∗m ◦ (0, 0, a) = U + xP 1(ι1t3) − (a1t)yι1t4 + (a2 1t2)xι1t3 − (a2t2)xι1t3 − U = −(ya1 − xa2 1 + xa2)ι1t5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The quantity ya1 − xa2 1 + xa2 is zero mod 3 for all choices of x and y if and only if a1 and a2 are both zero mod 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Predicated on Claims 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, we have shown: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given integers (a1, a2, a3) with (k7 × k9) ◦ (a1, a2, a3) = 0, the following two situations can occur: If either a1 or a2 are nonzero mod 3, then the map in U ◦ − is surjective and, up to homotopy, there is a unique 3-local vector bundle with i-th Chern class ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 9 If a1 ≡ 0 (mod 3) and a2 ≡ 0 (mod 3), then the map U ◦ − is zero and there are three distinct homotopy classes of 3-local vector bundles with i-th Chern class ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of technical claims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We now prove the claims 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, completing the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map pullback of the Chern class map c := (c1, c2, c3): BU(3) → K(Z, 2) × K(Z, 4) × K(Z, 6) on mod 3-cohomology is a 3-complete equivalence through degree 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We correct the degree 8 and degree 10 cohomology terms simultaneously via a map K(Z, 2) × K(Z, 4) × K(Z, 6) K(Z/3, 8) × K(Z/3, 10), k7×k9 and a factorization P9 BU(3) K(Z, 2) × K(Z, 4) × K(Z, 6) K(Z/3Z, 8) × K(Z/3Z, 10), τ9 c k7×k9 where P9 := hofib(k7 × k9), such that: (1) The map (k7 × k9) ◦ c is nullhomotopic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and (2) The lift τ9 : BU(3) → P9 is mod 3-cohomology isomorphism up to at least degree 10 and therefore realizes the 9-truncation of BU(3), up to 3-completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We complete (1) in Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6 and (2) in Verification 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Construction 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let P i denote the mod 3 Steenrod operation of degree 4i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The first relation among Steenrod operations on Chern classes is P 1 on c2: P 1(c2) = c2 1c2 + c2 2 − c1c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ιj denote a generator for HF j 3 (K(Z, j)) and take k7 := P 1ι4 − ι2 2ι4 − ι2 4 + ι2ι6 ∈ HF 8 3 � K(Z, 2) × K(Z, 4) × K(Z, 6)Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We identify a candidate for k9 by computing P 1 on c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' P 1c3 = c3(c2 1 + c2) so let k9 := P 1ι6 − ι6 � ι2 2 + ι4 � = P 1ι6 − ι6ι2 2 − ι6ι4 ∈ HF 10 3 � K(Z, 2) × K(Z, 4) × K(Z, 6)Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Verification 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' One computes the HF3-cohomology of integral Eilenberg Mac Lane spaces, which can be computed directly from the path-loop fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The main result we need is the following: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ιj generate HF j 3 K(Z, j) for j = 2, 4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We can identify the multiplicative structure of HF ∗ 3 (K(Z, 2) × K(Z, 4) × K(Z, 6)) through degree 11 as follows: � HF ∗ 3 � K(Z, 2) × K(Z, 4) × K(Z, 6) �� ≤11 ≃ (Z/3Z[ι2, ι4, ι6, Y8, W10] ⊗ Λ[N9, S11])≤11 where the subscript indicates the degree of the polynomial or exterior generator, the notation (−)≤11 indicates that we quotient by all elements of degree at least 12, and Y8 = P 1ι4 W10 = P 1ι6 N9 = βP 1ι4 S11 = βP 1ι6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 10 MORGAN OPIE From the above, we can compute the Serre spectral sequence for the fibration K(Z/3, 7) × K(Z/3, 9) → P9 → K(Z, 2) × K(Z, 4) × K(Z, 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The E2-page is given in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Moreover, if β denotes the Bockstein power operation: L8 = βι7 R10 = βι9 M11 = P 1ι7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To obtain the associated graded of HF ∗ 3 (P9), we compute all relevant differentials using a combination of the following two facts (Kudo’s transgression theorem, see [15] or [20, Ch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 6]): Given a principal fibration F → E → K(Z/pZ, n), the fundamental class ιn+1 is transgressive in the mod p Serre spectral sequence for ΩK(Z/pZ, n) → F → E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A power operation applied to a transgressive class is transgressive;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' transgressions com- mute with power operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From the above items and the fact that L8 = β(ι7), we deduce that d7(ι7) = Y8 − ι2 2ι4 − ι2 4 + ι6ι2, and d8(L8) = β(d7(ι7)) = β(Y8 − ι2 2ι4 − ι2 4 + ι6ι2) = β(Y8) = N9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Similarly, since β(ι9) = R10, we get that d9(ι9) = W10 − ι6(ι2 2 − 2ι4) and d10(R10) = β � W10 − ι6(ι2 2 + ι4) � = S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' All other terms strictly below the dotted line in Figure 2 are computed using the Liebniz rule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus the images of ι2, ι4, and ι6 are polynomial generators for HF ∗ 3 (P9) up to degree 10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' since The E2-page of a spectral sequence computing HF ∗ 3 (P9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 11 M11 10 R10 9 ι9 8 L8 7 ι7 6 5 4 3 2 1 0 ι2 ι4 ι6 Y8 N9 W10 S11 0 1 2 3 4 5 6 7 8 9 10 11 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Only multiplicative generators for the E2-page are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 11 c∗ : HF 2j 3 (K(Z, 2) × K(Z, 4) × K(Z, 6)) → HF ∗ 3 (BU(3)Z) satisfies ι2j �→ cj, this shows that a lift of (c1, c2, c3) induces an equivalence through degree 9, completing Verification 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Evidently the next stage in the tower is given by a class U : P9 → K(Z/3, 11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the π1 portion of the homotopy long exact sequence associated to the fibration (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6) π1 �� K(Z/3, 7) × K(Z/3, 9) �CP 5� π1(P CP 5 9 ) π1 �� K(Z, 2) × K(Z, 4) × K(Z, 6) �CP 5� s where the basepoint for (K(Z, 2) × K(Z, 4) × K(Z, 6))CP 5 is (a1, a2, a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The last term of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6) is zero, so s is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The action (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) is given on (x, a) ∈ π1 �� K(Z/3, 7) × K(Z/3, 9) �CP 5� × π1(P CP 5 9 ) by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7) (x, a) �→ s(x)a ∈ π1(P CP 5 9 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus, surjectivity of s implies the action is transitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Proof of Claim 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In order to understand U more explicitly, we study the spectral sequence in Figure 2 up to and including the dotted line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Computing differentials, we see that the class M11 = P 1(ι7) detects a nonzero class in HF 11 3 (P9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The action that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7) gives rise to a map of spectral sequences, from the Serre spectral sequence for K(Z/3, 7) × K(Z/3, 9) → P9 → K(Z, 2) × K(Z, 4) × K(Z, 6) to the Serre spectral sequence for (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8) � K(Z/3, 7) × K(Z/3, 9) �×2 → K(Z/3, 7) × K(Z/3, 9) × P9 → 3 � i=1 K(Z, 2i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We compute this map of spectral sequences using the fiber-by-fiber action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For i = 7 and i = 9, let ιi and ι′ i generate the two copies of HF i 3 (K(Z/3, i)) in the fiber of Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The comultiplication on the fiber implies that the coaction on the E2-page is: ι7 �→ ι7 + ι′ 7, ι9 �→ ι9 + ι′ 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We claim that, in the double complex of the source sequence, U should be represented by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9) M11 − ι4ι7 + ι2 2ι7 − ι2ι9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 12 MORGAN OPIE To see this,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' note that M11 is transgressive and d11(M11) = d11(P 1ι7) = P 1(d7(ι7)) = P 1(P 1ι4 − ι2 2ι4 − ι2 4 + ι2ι6) = −P 2ι4 + ι4 2ι4 − ι2 2P 1ι4 + ι4P 1ι4 + ι3 2ι6 + ι2P 1ι6 = −ι3 4 + ι4 2ι4 − ι2 2P 1ι4 + ι4P 1ι4 + ι3 2ι6 + ι2P 1ι6 = � ι4(d7ι7) + ι2 2ι2 4 − ι2ι4ι6 � + ι4 2ι4 − ι2 2P 1ι4 + ι3 2ι6 + ι2P 1ι6 = � ι4(d7ι7) − ι2 2d7(ι7) + ι3 2ι6 � − ι2ι4ι6 + ι3 2ι6 + ι2P 1ι6 = ι4(d7ι7) − ι2 2d7(ι7) − ι2ι4ι6 − ι3 2ι6 + ι2P 1ι6 = ι4(d7ι7) − ι2 2(d7ι7) + ι2(d9ι9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This indicates that a cocycle representative for U in the double complex computing H∗(P9;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Z/2) is Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9) on the E2-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Therefore, on the E2-page, we see that the action on the class representing U is detected by (M11 − ι4ι7 + ι2 2ι7 − ι2ι9) m∗ �−−→ (M11 − ι4ι7 + ι2 2ι7 − ι2ι9 + P 1ι′ 7 − ι4ι′ 7 + ι2 2ι′ 7 − ι2ι′ 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Passing to the E∞-page of the Serre spectral sequence for Equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8) we see that m∗U is detected by M11 + P 1ι′ 7 − ι4ι′ 7 + ι2 2ι′ 7 − ι2ι′ 9 ∈ HF ∗ 3 � K(Z/3, 7) × K(Z/3, 9) × P9 � , and therefore m∗U = U + P 1ι′ 7 − ι4ι′ 7 + ι2 2ι′ 7 − ι2ι′ 9, completing the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Instead of analyzing the tower of Diagram 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, we could transpose over the skeleton-truncation adjunction and instead lift a map sk0(CP 5) → BU(3)ˆ 3 up the higher skeleta of CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The action of coker(U ◦ −) corresponds to the action of π10BU(3)ˆ 3 on lifts of a given map sk9 CP 5 → BU(3)ˆ 3 to the 10-skeleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This shows that the action from Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6 is the relevant one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2-complete rank 3 vector bundles on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In this section we show there are no 2-complete bundles on CP 3 which were not already detected by Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the map c: BU(3)ˆ 2 → K(Zˆ 2, 2) × K(Zˆ 2, 4) × K(Zˆ 2, 6) given by the product c1 × c2 × c3 of two-completed Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The induced 2-complete Chern class map from [CP 5, BU(3)ˆ 2] to H2(CP 5, Zˆ 2) × H4(CP 5, Zˆ 2) × H6(CP 5, Zˆ 2) is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To understand [CP 5, BU(3)ˆ 2], we build a map from CP 5 into BU(3)ˆ 2 cell-by-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' First, recall the 2-complete homotopy of BU(3), as in Figure 3, computed from Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' π2 π3 π4 π5 π6 π7 π8 π9 π10 BU(3)ˆ 2 Zˆ 2 0 Zˆ 2 0 Zˆ 2 Z/2 0 Z/4 0 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2-complete homotopy of BU(3) A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 13 Consider the dotted arrows (i) to (v) in Diagram (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10) CP 5 sk8 CP 5 sk6 CP 5 BU(3)ˆ 2 sk4 CP 5 sk2 CP 5 ∗ (v) (iv) (iii) (ii) (i) An arrow (i) corresponds to a 2-complete first Chern class CP 2 → K(Zˆ 2, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The obstruction to lifting further is in π3(BU(3)ˆ 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The choices of lifts to an arrow (ii) are acted on transitively by π3(BU(3)ˆ 2) ≃ Zˆ 2 and correspond to c2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The obstructions to lifting to an arrow (iii) lie in π5(BU(3)ˆ 2) = 0, and the choices of lift to (iii) correspond to c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The obstruction to a lift to a map (iv) lies in π7(BU(3)ˆ 2) ≃ Z/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The choices of lifts are acted on transitively by π8(BU(3)ˆ 2) ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The obstruction to lifting from (iv) to (v) are in π9(BU(3)ˆ 2) ≃ Z/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The choices of lift are acted upon transitively by π10(BU(3)ˆ 2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We have shown that there are mod 2 and mod 4 conditions on the Chern classes of a rank 3 vector bundle on CP 5, but no new 2-primary invariants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, for a general 10-skeletal space (one that is not even), there may be additional 2-complete bundles not determined by Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Schwarzenberger conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Finally, we discuss necessary conditions for a collection of integers a1, a2, a3 ∈ Z to be the Chern classes of a topological vector bundle of rank 3 on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Following [25], let integers c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , ck be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Inductively, let s1 := c1 sk(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , ck) := Σk−1 i=1 (−1)i+1cisk−i f1(s1) := Identity fn(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , sn) := fn−1(s2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , sn) − (n − 1)fn−1(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , sn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Schwarzenberger condition Sk on a set c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , ck of integers is the requirement that, for each 1 ≤ n ≤ k, fn(s1(⃗c), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , sn(⃗c)) ≡ 0 (mod n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=') , where ⃗c := (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This condition has different forms, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' see [12, Appendix A].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14 ([25], Theorem A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Integers c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , ck ∈ Z are the Chern classes of a rank k vector bundle on CP k if and only if c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , ck satisfy the condition Sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From this we can to prove: 14 MORGAN OPIE Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let a1, a2, a3 ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then there exists a complex rank 3 topological vector bundle V on CP 5 with ci(V ) = ai if and only if the 5-tuple (a1, a2, a3, 0, 0) satisfies the Schwarzenberger condition S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14 above, the condition is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To show the condition S5 is sufficient, we prove that a rank 5 vector V ′ bundle on CP 5 with c4 and c5 equal to zero is in fact is isomorphic to a bundle V ⊕ C2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' its stable class has a rank 3 representative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider [26, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5], which implies that any (complex) rank 7 vector bundle on CP 5 with top four Chern classes zero is a sum of a rank 3 bundle and two trivial bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We apply this to V ′ ⊕ C2 to get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ We now work out S5 explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The condition S5 on (a1, a2, a3, 0, 0) is equivalent to the system of equations: a3 + a1a2 ≡ 0 (mod 2) −a2 1a2 + a1a3 − a2 2 + a2 ≡ 0 (mod 3) a1a2 − a2 1a3 − a1a2 2 + a2a3 + a2 1a2 − a1a3 + a2 2 ≡ 0 (mod 3) −a1 3a2 + a1 2a3 + a1a2 2 − a2a3 − a1a2 + a3 ≡ 0 (mod 4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Using the definitions preceding Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14, we evaluate s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , s5 at (a1, a2, a3, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ⃗a := (a1, a2, a3, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' s1(⃗a) = a1 s2(⃗a) = a2 1 − 2a2 s3(⃗a) = a3 1 − 3a1a2 + 3a3 s4(⃗a) = a4 1 − 4a2 1a2 + 4a1a3 + 2a2 2 s5(⃗a) = a5 1 − 5a3 1a2 + 5a2 1a3 + 5a1a2 2 − 5a2a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We now compute fi(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , si) for 1 ≤ i ≤ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' f1(s1) = s1 f2(s1, s2) = s2 − s1 f3(s1, s2, s3) = (s3 − s2) − 2(s2 − s1) = s3 − 3s2 + 2s1 f4(s1, s2, s3, s4) = s4 − 3s3 + 2s2 − 3(s3 − 3s2 + 2s1) = s4 − 6s3 + 11s2 − 6s1 f5(s1, s2, s3, s4, s5) = s5 − 6s4 + 11s3 − 6s2 − 4(s4 − 6s3 + 11s2 − 6s1) = s5 − 10s4 + 35s3 − 50s2 + 24s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='From the above we get: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='f2(si)|⃗a = a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 2a2 − a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= a1(a1 − 1) − 2a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='f3(si)|⃗a = a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 3a1a2 + 3a3 − 3(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 2a2) + 2a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= a1(a1 − 1)(a1 − 2) − 3a1a2 + 3a3 − 6a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='f4(si)⃗a = a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 4a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a2 + 4a1a3 + 2a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 − 6(a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 3a1a2 + 3a3) + 11(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 2a2) − 6(a1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= a1(a1 − 1)(a1 − 2)(a1 − 3) − 4a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a2 + 4a1a3 + 2a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 + 18a1a2 − 18a3 − 22a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='f5(si)|⃗a = a5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 5a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a2 + 5a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a3 + 5a1a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 − 5a2a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='− 10(a4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 4a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a2 + 4a1a3 + 2a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) + 35(a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 3a1a2 + 3a3) − 50(a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 − 2a2) + 24a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='i=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='(a1 − i) + 5(−a3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a2 + a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a3 + a1a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 − a2a3) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='− 10(−4a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1a2 + 4a1a3 + 2a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) + 35(−3a1a2 + 3a3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For simplicity, write fi(⃗a) for fi(s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' , si)|⃗a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We now expand the equations fi(⃗a) ≡ 0 (mod i!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=') for 2 ≤ i ≤ 5: f2 (mod 2!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=') : a1(a1 − 1) − 2a2 ≡ 0 (mod 2) for any a1, a2, so this gives no condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' f3 (mod 3!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=') : f3(⃗a) ≡ a3 + a1a2 (mod 2), so we get the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11) a3 + a1a2 ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since all terms of f3(⃗a) are divisible by 3, there is no 3-primary condition from f3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' f4 (mod 4!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=') : All terms of f4(⃗a) are divisible by 2, so this gives no condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since f4(⃗a) ≡ −a2 1a2 + a1a3 + 2a2 2 − a2 (mod 3), this gives the condition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12) −a2 1a2 + a1a3 − a2 2 + a2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider f4(⃗a) ≡ 2a2 2 + 2a1a2 + 2a3 − 2a2 (mod 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This quantity is zero (mod 4) if and only if a2 2 + a1a2 − a3 − a2 ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, this condition is implied by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11), so we get no new constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' f5 (mod 5!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=') : f5(⃗a) ≡ a3 1a2 + a2 1a3 + a1a2 2 + a2a3 + a1a2 + a3 (mod 2) giving the condition a1a3 + a1a2 + a2a3 + a3 ≡ 0 (mod 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, this condition is implied by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11) so we get no new constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Next, we reduce (mod 3): f5(⃗a) ≡ a3 1a2 − a2 1a3 − a1a2 2 + a2a3 + a2 1a2 − a1a3 + a2 2 (mod 3) Thus f5(⃗a) ≡ 0 (mod 3) if and only if (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13) a1a2 − a2 1a3 − a1a2 2 + a2a3 + a2 1a2 − a1a3 + a2 2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since f5(⃗a) ≡ 0 (mod 5), the last condition comes from f5(⃗a) (mod 4) ≡ −a13a2 + a12a3 + a1a22 − a2a3 − a1a2 + a3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This gives the condition: (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14) −a1 3a2 + a1 2a3 + a1a2 2 − a2a3 − a1a2 + a3 ≡ 0 (mod 4) Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14) are precisely the conditions to be proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ 16 MORGAN OPIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Defining a twisted tmf(3)-valued invariant By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7, rank 3 bundles on CP 5 that are not determined by their Chern data arise from the action of π10BU(3) ≃ Z/3 given in Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To go from an action to an invariant, we must study the unstable homotopy of BU(3) in greater detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We outline the key insights in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' These are motivational but not logically necessary for what follows, so we omit most proofs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Convention 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Throughout this section all spaces and spectra are implicitly localized at 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For any n, there is a fiber sequence S2n+1 δn+1 −−−→ BU(n) → BU(n + 1) (for example, see [21, Section 72]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For each n, δn+1 is the attaching map for a (2n + 2)-cell corresponding to cn+1 ∈ H2n+2BU(n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The homotopy class of δn+1 is linked to the existence of non-isomorphic vector bundles with the same Chern data in both the case of rank 2 bundles on CP 3 and that of rank 3 bundles on CP 5, as shown by the next result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A generator for π6BU(2) is given by the composite S6 η−→ S5 δ3 −→ BU(2), where the η is an unstable representative for the class by the same name in π∗S, which is the first element of order 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A generator for π10BU(3) is given by the composite S10 α1 −→ S7 δ4 −→ BU(3), where the α1 is an unstable representative for the class by the same name in π∗S, which is the first element of order 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Moreover, we can describe the generator for π10(BU(3)) as a further composite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let x: S4 → BU(3) generate π4BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then there is a map ǫ: S7 → S4 such that: (1) ǫ ◦ x generates the three-torsion in π7BU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and (2) Σ∞ǫ = α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 (for BU(3)) and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3 will follow from the the proof of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Combining the previous two lemmas, π10BU(3) is generated by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) S10 α1 −→ S7 ǫ−→ S4 x−→ BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This shows that a generator σ for π10BU(3) is stably trivial every sense: first, the bundle on BU(3) represented by σ is stably trivial as a map to BU;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' second, Σ∞σ = Σ∞α2 1x = 0 since α2 1 = 0 in π∗S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In fact the composition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) is null after just one suspension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Instead of applying the suspension spectrum functor to Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1), we can take a Thom spectrum with respect to one of the canonical bundles that are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let V be a bundle on BU(3) (for example, the universal bundle γ3, its determinant, or −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) gives a sequence of spectra A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 17 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) S10 S7 Σ∞ + S10 Σ∞ + S7 Th(S4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V |S4) Th(BU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' ˜v α1 Th(α1) Th(ǫ) Th(x) For various choices of V , we can ask whether ˜v is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To obtain Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) by Thomifying, note that the bundles V |S7 and V |S10 are trivial as maps to bu and fix a spherical orientation for V |S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Thom spectrum Th(S4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V |S4) has two cells, one in degree four and one in degree zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thomifying can either keep the cells split or introduce an α1-attaching map, in which case the Thom spectrum is C(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1 Which option occurs depends on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In order for the dotted composite ˜v to be nonzero, the latter must occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Moreover, given any spectrum X together with a map C(α1) → X, if (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) � S10 → S7 → C(α1) → X � ̸≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then the image in X of the 4-cell of C(α1) cannot support a P 1 in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From experimentation, it seems that taking X = Th(BU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V ) with V a canonical construc- tion on γ3 fails one condition or the other: either Th(S4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V |S4) ≃ Σ∞ + S4 or there is a nonzero P 1 on the relevant 4-cell of Th(BU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To resolve this issue, we modify our classifying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let BU(n)c1≡0 := hofib � c1 (mod 3): BU(n) → K(Z/3, 2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Note that: The space BU(n)c1≡0 is even, so any rank n on CP k bundle with c1 ≡ 0 (mod 3) lifts uniquely, up to homotopy, along the natural map BU(n)c1≡0 → BU(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The natural map BU(n)c1≡0 → BU(n) is a homotopy equivalence above degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The space BU(n)c1≡0 carries a universal bundle which we denote γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus our previous analyses of rank 3 bundles on CP 5 can be repeated after adding the con- straint c1 ≡ 0 (mod 3) and substituting BU(3)c1≡0 in place of BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In particular, we get a modification of Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4) S10 S7 Σ∞ + S10 Σ∞ + S7 Th(S4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' ˜v α1 Th(α1) Th(ǫ) Th(x) Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the diagram above, Th(S4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) = C(α1) and the element ˜v is nontrivial in π10 � Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' It will be useful to have better terminology for the Thomified homotopy classes such as ˜v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a pointed map y: Sn → BU(r)c1≡0 representing a stably trivial bundle on Sn, and a nullhomotopy u of the composite map to BGL1S, let Thu(y): Sn → BU(r)−γr denote the composite Sn i2 −→ Σ∞ + Sn u≃ −−→ Th(Sn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −y) → Th(BU(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γr), 1C(α1) is the cofiber of the map α1 : S3 → S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 18 MORGAN OPIE where the arrow i2 is the inclusion of the top cell determined by a base point and the map u≃ is the spherical Thom isomorphism determined by u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 The main goal of this section is to prove the following: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given σ: S7 → BU(3) generating π7BU(3) and a Thom class u0 for σ, there is a unique class ˜ρ ∈ tmf(3) −3(sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) such that Thu0(σ)∗ ˜ρ = α1β1 ∈ π13 tmf(3), where sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) is the Thomification of a 26-skeleton of BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The reader may wonder why we look for a class in tmf(3) cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The spectrum X = Σ−3 tmf(3) carries a natural map C(α1) → Σ−3 tmf(3) induced by α1 : S0 → Σ−3 tmf(3) such that Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Moreover, tmf(3) is one of the simplest ring spectra with this property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This makes tmf(3) is a natural candidate to detect π10BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, we outline the proof strategy for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We prove ˜ρ exists in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2, predicated on cohomology calculations which are recorded in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The proof of uniqueness of ˜ρ, given in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, also uses these calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof outline: existence and uniqueness of a twisted tmf(3) invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In this subsection we will state a sequence of propositions and explain how they imply Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To begin, let u be the canonical Thom class in the HF3-cohomology of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' As a module over the mod 3 Steenrod algebra, HF ∗ 3 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ≃ � HF ∗ 3 BU(3)c1≡0 � u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We find that P 1(u) = −c2 · u and P 1P 1(u) = 0 (see Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='17), so there is a map k: C(α1) → Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) that takes the 0-cell in C(α1) to the 0-cell in Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) and the 4-cell in C(α1) to the cell dual to −c2 · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map k: C(α1) → sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) splits after tensoring with tmf(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' More precisely, there is a map of tmf(3)-modules r: (sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ⊗ tmf(3) → C(α1 ⊗ tmf(3)) so that r ◦ (k ⊗ tmf(3)) is homotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Fix u0 a spherical orientation for σ: S10 → BU(3)c1≡0 generating π10BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Assuming the previous proposition, we immediately obtain: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' There is a map ˜ρ: sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) → Σ−3 tmf(3) such that � S10 Thu0 (σ) −−−−−→ sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ˜ρ−→ Σ−3 tmf(3) � = α1β1 in π13 tmf(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2We recall the basics of orientations and Thom isomorphisms in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 19 Proof of Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12 assuming Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let r and k be as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map α1 : S0 → Σ−3 tmf(3) extends over C(α1), since α2 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let o : C(α1) → Σ−3 tmf(3) denote an extension and let ¯o = o ⊗ tmf(3) be the associated map of tmf(3)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let 1: S → tmf(3) be the unit map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We then define ˜ρ to be the composite sk36 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ⊗ S sk36(Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ⊗ tmf(3) C(α1) ⊗ tmf(3) Σ−3 tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 1⊗ι ˜ρ r ¯o To show this map has the desired property, consider the homotopy commutative diagram: sk36(Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ⊗ tmf(3) C(α1) ⊗ tmf(3) Σ−3 tmf(3) sk36(Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ⊗ S C(α1) ⊗ S S10 r ¯o k⊗tmf(3) 1⊗ι 1⊗ι o k Th(σ) β1[0] where β1[0] is the image of β1 ∈ π10S0 under S0 → C(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The fact that the lower triangle commutes is a consequence of the fact that the map S10 α1 −→ S7 α1[4] −−−→ C(α1) = ⟨α1, α1, α1⟩ = β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map o ◦ β1[0] ∈ π13 tmf(3) is precisely α1β1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ The splitting is not canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, given an identification Th(S10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −σ) ≃ S0 ⊕ S10, the class ˜ρ is uniquely determined by requiring it to restrict to α1β1 on S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ǫ: S7 → BU(3) generate π10BU(3) and let u0 be a spherical orientation for ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let Thu0 be as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Any two classes ˜ρ, ˜ρ′ ∈ tmf(3) −3(sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) such that Thu0(ǫ)∗(˜ρ) = Thu0(ǫ)∗(˜ρ′) = β1 ∈ π13 tmf(3) are in fact equal in tmf(3) −3(Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We will prove this in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This immediately implies: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let σ = α1 ◦ ǫ ∈ π10BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then: σ = α1 ◦ ǫ generates π10BU(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' If ˜ρ, ˜ρ′ ∈ tmf(3) −3(sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) satisfy Thu0(σ)∗(˜ρ) = Thu0(σ)∗(˜ρ′) = α1β1 ∈ π13 tmf(3), then ˜ρ = ˜ρ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Note that the orientation u0 for ǫ gives an orientation u0 for σ such that α1 · Thu0(ǫ) = Thu0(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For the second item, suppose that ρ′, ρ both satisfy Th(σ)∗(˜ρ) = α1β1 = Th(σ)∗(˜ρ′) 20 MORGAN OPIE in π ∗ tmf(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This implies: Th(ǫ)∗(˜ρ) = β1 = Th(ǫ)∗(˜ρ′), so, by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13, ˜ρ = ˜ρ′ ∈ tmf(3) −3 � sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For the first item, note that Th(σ)∗(˜ρ) ̸= 0, which implies σ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since π10BU(3) is cyclic, this implies σ generates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Together, Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12 and Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14 imply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' It remains to prove Propo- sitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' These are the projects of the next subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By a skeleton of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) we mean a term in a filtration of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ob- tained by Thomifying a skeletal filtration of BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Precisely, BU(3)c1≡0 has a cell structure filtering the space by a sequence of pushouts as in Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' ∨j∈Hi+2Si+3 ski+2 BU(3)c1≡0 ∨j∈HiSi+1 ski BU(3)c1≡0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' ∨¯ci+2,j ∨¯cij Each Hi above is a finite indexing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Each skeleton carries a bundle pulled back from γ3 on BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We can Thomify all diagrams involved to obtain a filtration for Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3): each stage in Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) gives pushout in spectra (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6) ∗ ski+2 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ⊕HiSi ski Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' ⊕Hicij In Diagram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6, we define ski Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) := Th(ski BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) and cij is the Thomification of ¯cij restricted to Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the cofiber C := cof(C(α1) k−→ sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' With notation as above, π0 Mapstmf(3)(Σ−1C ⊗ tmf(3), C(α1) ⊗ tmf(3)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given this lemma, the going around map Σ−1C ⊗ tmf(3) → C(α1) ⊗ tmf(3) is null and there is an extension making Diagram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7 homotopy commutative, which is the desired section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7) Σ−1C ⊗ tmf(3) C(α1) ⊗ tmf(3) sk26 Th(BSU(3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ⊗ tmf(3) C(α1) ⊗ tmf(3) 0 k = ∃r Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Using the free-forgetful adjunction for tmf(3)-modules: π0 Mapstmf(3)(Σ−1C ⊗ tmf(3), C(α1) ⊗ tmf(3)) ≃ π0 MapsS(Σ−1C, C(α1) ⊗ tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 21 To establish the result, we compute π0 MapsS(Σ−1C, C(α1) ⊗ tmf(3)) via an Atiyah–Hirzebruch spectral sequence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8) Ep,q 2 = Hp � Σ−1C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' � tmf(3) � −q C(α1) � =⇒ πp+q MapsS � Σ−1C, C(α1) ⊗ tmf(3) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We use grading conventions indicated in Figure 4 and we depict the spectral sequence in Fig- ure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Schematic grading convention for the Atiyah–Hirzebruch spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' p = 4 3 2 1 0 q = −4 −3 −2 −1 0 d2 d3 d4 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Dotted lines indicate diagonals p + q = i which converge to an associated graded of the i-th homotopy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' the direction of the first few differential are indicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We need to understand some aspects of HF ∗ 3 Σ−1(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='17, whose proof we defer to the next subsection, we have that HF ∗ 3 (Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ≃ Z/3[t, c3, c3] · u as a module over the Steenrod algebra, where |t| = 2, |c2| = 4, |c3| = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We have drawn the P 1-action on those classes which are not multiples of t in Diagram 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Note that k∗ is surjective, so HF ∗ 3 (C) can be identified with a submodule of HF ∗ 3 (Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) consisting of all elements except Z/3-multiples of u and c2 · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a class tlci 2cj 3 · u ∈ HF ∗ 3 (C), we refer the reader to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='18 to deduce that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9) P 1(tlci 2cj 3 · u) = l(tl+2ci 2cj 3 · u) + (i + j − 1)tlci+1 2 cj 3 · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since HF ∗ 3 (Σ−1C) has odd cohomology, terms on the diagonal p + q = 0 on the E2-page of the Atiyah–Hirzebruch spectral sequence of Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8 arise from odd elements in π∗C(α1)⊗ tmf(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In the relevant range, we can describe π∗(C(α1) ⊗ tmf(3)) as follows: π∗≤26(C(α1) ⊗ tmf(3)) ≃ Z/3 · {1, α1[4], α1β1[4], β1[0], β2[0]} ⊕ H, where H has only even terms (arising from the classes c2 and c4 in π∗ tmf(3)) and these terms support no α1 or β1 multiplications, so can be disregarded for our calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 22 MORGAN OPIE P 1-module structure of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) through degree 18, excluding generators divisible by t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 22 c2c3 3 · u 20 c2 2c2 3 · u 18 c3 2c3 · u c3 3 · u 16 c4 2 · u c2c2 3 · u 14 c2 2c3 · u 12 c3 2 · u c2 3 · u 10 c2c3 · u 8 c2 2 · u 6 c3 · u 4 c2 · u 2 0 u Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The left column is the degree of the cohomology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Dotted lines indicate ±α1 attaching maps detected by a P 1 in HF ∗ 3 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We omit classes above degree 18 that do not attach to cells at or below 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The S-module structure is given by: α1 · (α1[4]) = β1[0] β1 · 1 = β1[0] α1 · (α1β1[4]) = β2 1[0] β1 · (β1[0]) = β2 1[0] β1 · (α1[4]) = α1β1[4], and all other multiplications are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The degree of an indicated class is the degree of the corresponding class in π∗ tmf(3) plus the number in the bracket.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3 So, for example, α1[4] has degree 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The only odd classes in π∗≤26 � C(α1) ⊗ tmf(3) � are α1β1[4] and α1[4], so contributions on the E2-page are in bidegrees (−17, 17) and (−7, 7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' First consider the classes which are not multiples of t α1[4] ⊗ (c2 2u), α1β1[4] ⊗ (c3 2c3 · u), α1β1[4] ⊗ (c3 3 · u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 3These elements are named by the classes that detect them on the E2-page of the Atiyah–Hirzebruch spectral sequence computing π∗(C(α1) ⊗ tmf(3))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 23 By inspection of Figure 5 and the fact that ⟨α1, α1, α1⟩ = β1 we see that d4(α1[4] ⊗ c2 2 · u) = β1[0] ⊗ c3 2 · u d4(α1β1[4] ⊗ c3 3 · u) = −β2 1[0] ⊗ c2c3 3 · u α1β1[4] ⊗ c3 2c3 · u = d8(β1[0] ⊗ c2c3 · u) In degree 7 we have t4 · u c2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9) we have: P 1(t4 · u) = t6 · u − t4c2 · u P 1(c2 2 · u) = c3 2 · u Therefore: d4(α1[4] ⊗ c2 2 · u) = β1[0] ⊗ c3 2 · u d4(α1[4] ⊗ t4 · u) = β1[0] ⊗ (t6 · u − t4c2 · u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus the cell (−7, 7) does not contribute to the E∞-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In degree 17 we have generators: t9 · u t7c2 · u t6c3 · u t5c2 2 · u t4c2c3 · u t3c3 2 · u t3c2 3 · u t2c2 2c3 · u tc4 2 · u tc2c2 3 · u c3 2c3 · u c3 3 · u Using Equation 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' we see that: The image of P 1 on the generators for H17(Σ−1C) listed above is: −t9c2 · u t9c2 · u 0 −t7c2 2 · u + t7c3 2 · u t6c2c3 · u + t4c2 2c3 · u −t3c4 2 · u t3c2c2 3 · u −t4c2 2c3 · u − t2c3 2c3 · u t3c4 2 · u t3c2c2 3 · u − tc2 2c2 3 · u 0 −c2c3 3 · u Since d4(α1β1[4] ⊗ x) = β2 1 ⊗ P 1(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' we see that ker(d4): E4 (−17,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='17) → E4 (−20,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='21) is generated by α1β1 tensored with: t9 · u + t7c2 · u t6c3 · u tc4 2 · u + t3c3 2 · u c3 2c3 · u The next differential involved is a d8 with source (-10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' which is computed as d8(β1[0] ⊗ y) = α1β1[4] ⊗ P 1P 1(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 24 MORGAN OPIE Using Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9), we get: P 1 � P 1(t5 · u) � = P 1 � P 1(t5) � u − P 1(t5)P 1(u) + t5P 1 � P 1(u) � = (7 ∗ 5)t9 · u − (5t7)(−c2 · u) + t5 · 0 = −(t9 · u + t7c2 · u), P 1 � P 1(tc2 2u) � = P 1 � P 1(t) � u − P 1(t)P 1(c2 2u) + tP 1 � P 1(c2 2u) � = −(t3c3 2 · u + tc4 2 · u), P 1 � P 1(t2c3 · u) � = P 1 � P 1(t2) � u − P 1(t2)P 1(c3 · u) + t2P 1 � P 1(c3 · u) � = −t6c3 · u, since P 1(c2 · u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' And similarly: P 1 � P 1(c2c3 · u) � = P 1 � P 1(c2) � u − P 1(c2)P 1(c3 · u) + c2P 1 � P 1(c3 · u) � = −c3 2c3 · u, This shows that gr � π0(MapsS(Σ−1C, C(α1) ⊗ tmf(3)) � = 0, which implies the group itself is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ The relevant part of the E2-page of the Atiyah–Hirzebruch spectral sequence computing π∗ MapsS(Σ−1C, C(α1) ⊗ tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 21 19 17 15 13 11 9 7 − q 20 19 18 17 16 15 14 13 12 11 10 9 8 7 β2[0] αβ[4] β[0] α[4] d4 d8 d4 Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The x-axis is graded by q with the generator of π−q(C(α1)⊗tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The y-axis is graded by the degree of generators for HF q 3 (Σ−1C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A circle in degree (p, −q) indicates a non-zero three-torsion group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Rectangles mark the bidegrees that can contribute to the p+q = 0 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We indicate the differentials that eliminate the terms on p + q = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 25 Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' One might hope that C(α1) ⊗ tmf(3) split from ski Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ⊗ tmf(3) for i > 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In addition to being a cleaner result, such an extension might allow us to extend the ρ invariant to vector bundles on higher-dimensional spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Unfortunately, our approach to splitting C(α1) ⊗ tmf(3) is computationally intensive and it is not at all clear how far up the skeleton the splitting extends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, from discussions with M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hopkins, we believe ˜ρ can be extended to all of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) by a different method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We hope to work this out in the future, although it is not necessary for our result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The cohomology of Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) and related spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This subsection includes the main calculations needed for the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Some of these results have already been used in the previous subsection, and some are necessary for the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We begin with the cohomology of the relevant classifying space and the Thom spectrum of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The P 1-module structure on the (mod 3)-cohomology of BU(3)c1≡0 and Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) is given as follows: (1) HF ∗ 3 (BU(3)c1≡0) ≃ Z/3[t, c2, c3], where |t| = 2, |c2| = 4, and |c3| = 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and P 1c2 = c2 2 (P 1P 1)c2 = 2c3 2 P 1t = t3 (P 1P 1)t = 0 P 1c3 = c2c3 (P 1P 1)c3 = −c2 2c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (2) HF ∗ 3 (Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ≃ HF ∗ 3 (BU(3)c1≡0) · u, with P 1u = −c2 · u (P 1P 1)u = 0 where we view HF ∗ 3 (BU(3)c1≡0) as a P 1-algebra and therefore the P 1-module structure is determined by the action on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Part (1) follows from Steenrod operations on Chern classes in HF ∗ 3 (BU(3)), together with the fact the natural map BU(3)c1≡0 → BU(3) induces c1 �→ 0, c2 �→ c2, and c3 �→ c3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Part (2) follows from the HF3-Thom isomorphism together with the universal formula for Steenrod operations on Thom classes, which we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, we will later need formulas for Steenrod operations on the Thom class for the bundle −γ4 on BU(4)c1≡0, so we give these and then deduce those for −γ3 on BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We first compute P 1 on the Thom class uγ4 for γ4 on BU(4) via the universal example of MU(1)×4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' P 1(wxyz) = w3xyz + wx3yz + wxy3z + wxyz3 = wxyz(w2 + x2 + y2 + z2) = wxyz((w + x + y + z)2 − 2(wx + wy + wz + xy + xz + yz)), implying that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10) P 1(uγ4) = (c2 1 + c2)uγ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Chern classes for −γ4 are the coefficients of the power series inverse to c(γ4) = 1 + c1t + c2t2 + c3t3 + c4t4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 26 MORGAN OPIE 1 c(γ4) = 1 − (c1t + c2t2 + c3t3 + c4t4) + (c1t + c2t2 + c3t3 + c4t4)2 − (c1t + c2t2 + c3t3 + c4t4)3 + (c1t + c2t2 + c3t3 + c4t4)4 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' = 1 − c1t + (c2 1 − c2)t2 + (−c3 1 + 2c1c2 − c3)t3 + (c4 1 − 3c2 1c2 + c2 2 + 2c3c1 − c4)t4+, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' so that c1(−γ4) = −c1 c2(−γ4) = c2 1 − c2 c3(−γ4) = −c3 1 + 2c1c2 − c3 c4(−γ4) = c4 1 + c2 2 − c3c1 − c4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The above implies P 1(u−γ4) = � (−c1)2 + c2 1 − c2 � u−γ4 = −(c2 1 + c2) · u−γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11) We next calculate P 1c2 1 and P 1c2 in H∗(BU(4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For c1 we have: P 1((w + x + y + z)2) = 2(w + x + y + z)P 1(w + x + y + z) = 2(w + x + y + z)(w + x + y + z)3 = 2(w + x + y + z)4, and P 1(c2 1) = −c4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12) Let sn = sn(w, x, y, z) denote the n-th elementary symmetric polynomial in x, y, z, w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The computation for c2 goes as follows: P 1(wx + wy + wz + xy + xz + yz) = w3x + wx3 + w3y + wy3 + w3z + wz3 + x3y + xy3 + x3z + xz3 + y3z + yz3 = s2(w2 + x2 + y2 + z2) − s3s1 + xyzw = s2(s2 1 + s2) − s3s1 + s4 Therefore: P 1(c2) = c2 1c2 + c2 2 − c1c3 + c4 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13) Combining Equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11),(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13) we get: P 1P 1(u−γ4) = − � P 1(c2 1 + c2) · u−γ4 + (c2 1 + c2)P 1(u−γ4) � = − � − c4 1 + c2 1c2 + c2 2 − c1c3 + c4 − (c2 1 + c2)2) � u−γ4 = −(c4 1 − c2 1c2 − c1c3 + c4) · u−γ4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Now let ˜u denote the Thom class of the negative of the universal bundle on BU(4)c1≡0 and as before let u denote the Thom class of −γ3 on BU(3)c1≡0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since the universal bundle has zero first Chern class, we get: P 1(˜u) = −c2 · ˜u P 1P 1(˜u) = −c4 · ˜u (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14) P 1(u) = −c2 · u P 1P 1(u) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ From the above we can use the Liebniz rule for P 1 to derive: Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In HF ∗ 3 (Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)), P 1(ti · u) = iti+2 · u − tic3 · u P 1(ci 2cj 3 · u) = (i + j − 1)ci+1 2 cj 3 · u A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 27 In order to prove that ˜ρ is unique in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, we will need to analyze a certain cofiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To that end, the following calculation will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ǫ: S7 → BU(3)c1≡0 generate π7BU(3)c1≡0, let u the Thom class for ǫ, and let C(ǫ) denote the cofiber of Thu0(ǫ): S7 → Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' As a module over P 1, HF ∗ 3 (C(ǫ)) ≃ HF ∗ 3 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) ⊕ Z/3 · {y}, where |y| = 8 and P 1(−c2 · u) = y P 1(y) = 0 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ι: BU(3) → BU(4) denote the natural map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Because the composite ι ◦ ǫ: S7 → BU(4) is null, we get a homotopy commutative diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='16) S7 BU(3)c1≡0 (BU(3)c1≡0)/S7 BU(4)c1≡0, 0 γ3⊕C δ where δ is any extension of the bundle γ3 ⊕ C to the cofiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We get a homotopy pushout a by taking Thom spectra: (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='17) S0 ⊕ S7 S0 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) Th((BU(3)c1≡0)/S7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −δ), p1 Th(ǫ) where p1 is projection onto the first factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Therefore C(ǫ) ≃ Th((BU(3)c1≡0)/S7;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let T := Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) and T4 := Th(BU(4)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ4) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' From Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='16) we get a commuting diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='18) Hi−1(T ) Hi−1(S7) Hi(C(ǫ)) Hi(T ) Hi(S7) Hi(T4) 0 a 0 where Hi denotes either Z or Z/3 coefficients and the top row is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map a induces a ring isomorphism (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19) H∗(C(ǫ))/H>8 ≃ H∗(Th(BU(4)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ4))/H>8, where H>8 is the ideal of elements of degree greater than 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19) identifies the class c4 · u with a class y ∈ H∗(C(ǫ)) and implies that P 1(−c2 · u) = y P 1(y) = 0, as was to be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To complete the proof, note that Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='18) implies H∗(Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ≃ H∗(C(ǫ))/⟨y⟩ as modules over the Steenrod algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ 28 MORGAN OPIE 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the cofiber C := Cof(ǫ: S7 → sk26 BU(3)c1≡0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let C′(ǫ) = Cof(S7 → sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)) ≃ Th(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −δ), where δ: C → BU(4) is an extension of γ3 ⊕ C over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We get a diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='20) S10 Σ−1C′(ǫ) S7 sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) Σ−3 tmf(3), α1 ˜v φ β1 where ˜v is as in Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A dotted arrow is an element ˜ρ ∈ tmf(3) −3 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) satisfying Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Choices of the dotted arrow in Diagram (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='20) up to homotopy are a torsor for π0 MapsS(C′(ǫ), Σ−4 tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We show that this group is zero via an Atiyah–Hirzebruch spectral sequence E2 p,q = Hp(C′(ǫ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' π−q+3 tmf(3)) =⇒ tmf(3) p+q−3(C′(ǫ)), depicted in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We computed the cohomology of C(ǫ) in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19 and H∗≤26C(ǫ) ≃ H∗≤26C′(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The homotopy of tmf(3) is known [6, 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The terms along the line p + q = 0 on the E2-page are u ⊗ α1 and α1β1 tensored with elements in H10(C′(ǫ)) ≃ H10(Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3)/S7) · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since π∗ tmf(3) has no other odd homotopy groups until degree 27, there are no further possible contributions to consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' First, consider u ⊗ α1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since P 1P 1(u) = y and ⟨α1, α1, α1⟩ = β1, d8(u ⊗ α1) = β1 ⊗ y ̸= 0 and the class α1 ⊗ u does not survive the spectral sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' On the other hand, there are many bidegree (−10, 10) classes to check: E2 (−10,10) ≃ α1β1 ⊗ ⟨t5 · u, t3c2 · u, t2c3 · u, tc2 2 · u, c2c3 · u⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We claim that all classes above support a differential or are the target of a nonzero differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Figure 7 shows the first interesting differentials in and out of this cell on the E2-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' First, we check which of the above are the target of a differential in (A) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then we check that the remaining classes support a nonzero differential in (B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (A) The only possible differential is a d4 originating in bidegree (−7, 6) is computed as follows: classes in this cell are β1 times classes in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='21) H6(BU(3)c1≡0) · u ≃ ⟨t3 · u, tc2 · u, c3 · u⟩ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19 implies: P 1(t3u) = t3P 1(u) = −c2t3u P 1(tc2 · u) = t3c2 · u + tc2 2 · u − tc2 2 · u = t3c2 · u P 1(c3 · u) = c2c3 · u − c2c3 · u = 0 Therefore: d4(β1 ⊗ t3 · u) = −α1β1 ⊗ c2t3 · u and the target dies on the E5-page.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 29 A larger portion of the E2-page of the Atiyah–Hirzebruch spectral sequence computing tmf(3) ∗ (C′(ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' p = 18 17 16 15 14 13 12 11 10 9 8 7 6 q = −17 −16 −15 −14 −13 −12 −11 −10 −9 −8 −7 π−q β2 1, c2 4 αβ c6 β c4c6 d4 d8 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The dashed line indicates the p + q = 0 line converging to π0 MapsS(C′(ǫ), Σ−3 tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A dot indicates a non-zero three-torsion group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' a square a non-zero torsion-free group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The diamond in bidegree (−10, 10) indicate nonzero E2-terms which may contribute to the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' (B) A class β1α1 ⊗ (z · u) which survives to the E8-page supports a nonzero d8 if P 2z is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Using Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' facts about Steenrod operations we get: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='−(P 1P 1)(t5 · u) = −(P 1P 1)(t5) · u + P 1(t5)P 1(u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= (t9 + t7c2) · u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='−(P 1P 1)(t2c3 · u) = −(P 1P 1)(t2)c3 · u + P 1(t2)P 1(c3 · u) − t2(P 1P 1)(c3 · u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= t6c3 · u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='−(P 1P 1)(tc2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 · u) = −(P 1P 1)(tc2 · u)c2 + P 1(tc2 · u)P 1(c2) − P 1P 1(c2)(tc2 · u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= −P 1(t3c2 · u)c2 + t3c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 · u + tc4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 · u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= −t3P 1(c2 · u) + t3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 · u + tc4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= t3c2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 · u + tc4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 · u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='−(P 1P 1)(c2c3 · u) = −(P 1P 1)(c3 · u)c2 + P 1(c3 · u)P 1(c2) − (P 1P 1)(c2)(c3 · u) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= −(P 1P 1)(c2)c3 · u ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='= c3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2c3 · u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 30 MORGAN OPIE This shows that the classes t5 · u, t2c3 · u, tc2 2 · u, c2c2 · u support nonzero differentials whose joint span is 4-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' No terms on the p + q = 0 line survive the spectral sequence, so the group in question is zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Untwisting the invariant for rank 3 bundles on CP 5 The work of the previous Section 3 provides an association V �→ Th(V )∗˜ρ ∈ tmf(3) −3(Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Vector bundles on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) are tmf(3)-orientable, as we now explain: Any complex bundle V carries an HZ-Thom class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since τ0 tmf(3) = HZ, V is tmf(3)-orientable if and only if this class lifts up the Postnikov tower for tmf(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since CP 5 is finite-dimensional, this is a finite lifting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Postnikov tower for tmf(3) through degree 10 has only one stage that obstructs lifting the HZ-Thom class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This gives the condition that c2 1 − 2c2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus, for V over CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3), there exist isomorphisms tmf(3) ∗(Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )) ≃ tmf(3) ∗(Σ∞ + CP 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, there are many choices of such an iso- morphism and our choice cannot be dependent on V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The ideal way to resolve this would be via a universal example: a space B together with a bundle VB which carries a (canonical) tmf(3)-orientation, such that all bundles of interest are canonically pullbacks of VB and therefore inherit a tmf(3)-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Classical examples of this phenomenon are numerous: BU(n) is canonically HZ oriented, giving the classical HZ-Thom isomorphism for complex bundles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' BSpin is canonically KO-oriented via the Atiyah–Bott– Shapiro orientation, giving a canonical KO-Thom isomorphism for spin bundles [4];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' BString is canonically tmf-oriented, giving a canonical tmf-orientation for string bundles [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our bundles are not string, nor is there an obvious candidate for a universal example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Other, more hands-on, approaches to resolving orientability problems can be found in the literature, for example in [9, 7]4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The approach we take here is informed by discussions with H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Chatham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9, we show that for V : CP 5 → BU(3) with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3), there is a set of Thom isomorphisms subject to some concrete restrictions, with the following property: there is a map i: S10 → Σ∞ + CP 5 such that, for any Thom isomorphism v in this distinguished set, the image of Th(V )∗(˜ρ) under the restriction (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) tmf(3) ∗(Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V )) v−→ tmf(3) ∗(Σ∞ + CP 5) i∗ −→ tmf(3) −3(S10) does not depend on the choice of v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus, applying the composite in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) to the class Th(V )∗(˜ρ) defines the desired invariant ρ(V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1, we briefly recapitulate the theory of orientations and establish notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2 we study orientation of bundles on CP 5 in detail, define the set of orientations which make Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1) independent of choice within this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We prove independence in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We can then define the invariant ρ for V rank 3 on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3) via the definition indicated in the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The next Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3 features the verification that ρ distinguishes bundles with the same Chern classes, completing the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The final two subsections include some computations (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4) and the observation that ρ can also be defined for rank 2 bundles (Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In these subsections we also discuss future research questions that we hope to address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Throughout this section, all spaces and spectra are implicitly localized at the prime 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 4Since LK(2)TMF = EO2 at the prime 3, the orientability studied in [9, 7] is closely related to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Background on orientability and orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We present the relevant background on orientability, orientations, Thom classes, and Thom isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The classical version is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let V be a vector bundle over a space X, with disc bundle dV and sphere bundle sV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let Sn → dV be the inclusion of a fiber inducing i: Sn → dV/sV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Recall that, classically, a Thom class for a vector bundle V over X in generalized cohomology theory E is a class v ∈ Edim V (dV/sV ) such that i∗v is a unit in E0(S0) ≃ En(Sn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Orientability refers to the existence of such a class;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' an orientation is a choice of such a class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given an such orientation, pairing with the Thom class under the Thom diagonal gives a Thom isomorphism (−) · u: E∗(Σ∞ + X) ≃ E∗(Th(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We will use the terms orientation, Thom class, and Thom isomorphism synonymously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Alternatively, a vector bundle V : X → BU(n) gives rise to a (stable) spherical bundle (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2) VS := � X → BU(n) → BU J−→ BGL1S � , where J the complex J-homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5 The unit map 1: S → E induces a map BGL11: BGL1S → BGL1E and obtain (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) VE := � X VS −→ BGL1S BGL11 −−−−−→ BGL1E � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' E-orientability is the condition that VE in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3) is nullhomotopic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' an E-orientation is a choice of nullhomotopy of VE (see [1], building on [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' An orientation gives not just an isomorphism on cohomology, but a isomorphism of E-modules: Th(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V ) ⊗ E ≃ Σ∞ + X ⊗ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In this section we will need to refer to many different maps associated to a given V : X → BU(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our notation is slightly abusive, but clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' V will refer to any of the following associated maps: – The definitional map X → BU(r), – The composite X → BU(r) → BU, and – The transpose of the previous item under the loops-suspension adjunction, which is a map Σ∞ + X → bu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For E a commutative ring spectrum, VE will refer to both: – The composite X V−→ BU J−→ BGL1S BGL11 −−−−−→ BGL1E, and – The transpose Σ∞ + X V−→ bu j−→ bgl1S bgl11 −−−→ bgl1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Selecting orientations and the definition of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The 3-local spectrum Σ∞ + CP 5 has a splitting arising from the Adams splitting of Σ∞ + CP ∞: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4) Σ∞ + CP 5 ≃ X0 ⊕ X1, where X1 ≃ Σ2C(α1) ⊕ S10, X0 ≃ Σ∞ + HP2 ≃ S0 ⊕ Σ4C(α1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The summand Σ∞ + HP 2 is split via p = Σ∞ + p′ where p′ : CP 5 → HP 2 is the map of spaces given by taking the 10-skeleton of the quotient map CP ∞ → HP ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 5The complex J homomorphism is commonly defined as a map J : U → GL1S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our J is the delooping of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 32 MORGAN OPIE Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We fix isomorphisms of X0 and X1 with the sums above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We will write i for all inclusions of summands and p for all projections onto summands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' These choices can be made once independent of any future bundles involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Recall the terminology from Notation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To study tmf(3)-orientations is to study nullho- motopies of Vtmf(3) : Σ∞ + CP 5 → bgl1 tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our strategy is to restrict Vtmf(3) to each summand in the decomposition of Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4) and separately study nullhomotopies on each summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' On the summand X1, we show that the bundles of interest possess a certain canonical orientation arising from the image of j spectrum, while the choice on the summand X0 will turn out not to matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' At a prime p, the image of j spectrum bj can be defined as the cofiber of the map ψq−1: bu → bu, where ψq is the q-th Adams operation and q is a topological generator for Z× (p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By stable Adams conjecture (proved6 in [8]) there is a factorization of the j homomorphism j = � bu → bj j′ −→ bgl1S � , where the map bu → bj is the natural map to the cofiber of ψq − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map Vj := � X1 V |X1 −−−→ bu → bj � is nullhomotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Up to homotopy, there is a unique such nullhomotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Atiyah–Hirzebruch spectral sequence for computing π∗ MapsS(X1, bj) shows that both π0(MapsSp(X1, bj) and π0(MapsSp(X1, Σ−1bj) are trivial, since π≤10bj is concentrated in degrees 0, 4, and 8 [16, Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 4] and HF ∗ p X1 is concentrated in degrees 2, 6, and 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let X be a space and let W : X → bu be a given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Assume that the composite Wj = � X W −→ bu → bj � is canonically null homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given any generalized cohomology theory E, the j-orientation of WE will refer to the distinguished nullhomotopy obtained by whiskering the the canonical nullhomotopy of Wj with the composite bj j′ −→ bgl1S bgl11 −−−→ bgl1E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In particular, given a vector bundle V of rank 3 on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3), j-orientation of (V |X1)tmf(3) will refer to the nullhomotopy of the composite X1 V |X1 −−−→ bu → bj → bgl1 tmf(3) obtained from the unique nullhomotopy of (VX1)j given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We are interested in orientations which extend j-orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let Y be a space together with a splitting Σ∞ + Y = Z ⊕ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let E be a ring spectrum with τ0E = HZ(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a vector bundle V on Y is such that X V |X −−−→ bu → bj is canonically null, we say that a E-orientation v of V satisfies condition (∗) if: (∗1) v restricts to the j-orientation of (V |X)E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and (∗2) v lifts the canonical HZ-orientation under 0-truncation bgl1E → bgl1HZ(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The main goal of this section is to prove the following: 6Notable previous attempts at the stable Adams conjecture are [10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Adams conjecture for spaces is proved in [24, 23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 33 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let V be a vector bundle over CP 5 with c1(V ) ≡ c2(V ) ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let v be a tmf(3)-orientation for V satisfying condition (∗) with respect to the decomposition Σ∞ + CP ∞ = X0 ⊕ X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then the composite (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) ρ(V ) := � S10 i⊗1 −−→ Σ∞ + CP 5 ⊗ tmf(3) v−→ Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V ) ⊗ tmf(3) Th(V )∗(˜ρ) −−−−−−−→ Σ−3 tmf(3) � is independent of v, as an element in π0 � MapsS(S10, Σ−3 tmf(3)) � ≃ π13 tmf(3) ≃ Z/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Note that v satisfying the hypothesis of the theorem exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let V : Σ∞ + HP 2 ⊕ X1 → bgl1S be tmf(3)-orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since π0 MapsS(X1, gl1HZ) = 0, there is a unique nullhomotopy of any null homotopic map X1 → bgl1HZ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Summing the j-orientation of V |X1 with any tmf(3)-orientation of V |Σ∞ + HP 2 lifting the canonical HZ orientation gives an appropriate orientation of V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The rest of this section is devoted to the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To begin, suppose that v and w are two tmf(3)-orientations of a bundle V , both satisfying (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the following diagram tmf(3)-modules: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6) Σ∞ + CP 5 ⊗ tmf(3) S10 ⊗ tmf(3) Σ∞ + CP 5 ⊗ tmf(3), i i w−1v where w−1v is the automorphism of CP 5 ⊗ tmf(3) obtained from composing the Thom isomor- phisms corresponding to v with the inverse of the Thom isomorphism corresponding to w (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Note that, if Diagram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6 were to commute up to homotopy, the theorem would be immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' However, commutativity is stronger than necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' More precisely, we only need the diagram to commute after applying a certain tmf(3)-cohomology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We will return to this after some preliminary calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let Auttmf(3) denote automorphisms in the category of tmf(3)-modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We study which elements of π0 Auttmf(3)(Σ∞ + CP 5 ⊗tmf(3)) arise as ratios of orientations satisfying condition (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since CP ∞ ≃ X0 ⊕ X1, an automorphism a of Σ∞ + CP 5 ⊗ tmf(3) is represented by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7) a = � a00 a01 a10 a11 � where the aij ∈ Mapstmf(3)(Xi ⊗ tmf(3), Xj ⊗ tmf(3)) for i ̸= j and aii ∈ Auttmf(3)(Xi ⊗ tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To study the failure of Diagram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6 to commute, we examine the possibilities for a10 and a11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Suppose that v, w are two tmf(3)-Thom classes for a tmf(3)-orientable bun- dle of rank 3 on CP 5 and that both satisfy condition (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let a = w−1v be the associated automorphism of Σ∞ + CP 5 ⊗ tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then a10 : X1 ⊗ tmf(3) → Σ∞ + HP 2 ⊗ tmf(3) is null.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 34 MORGAN OPIE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the cofiber sequence of spectra X1 i−→ Σ∞ + CP 5 p−→ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Recall that p = Σ∞ + p′, where p′ is the 10-skeleton of the natural map CP ∞ → HP ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We have a diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8) X1 Σ∞ + CP 5 bu bj bgl1S bgl1 tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Σ∞ + HP 2 =⇒ (†) V |X1 0 Σ∞ + p′ V j′ bgl11 q In Diagram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8, the nullhomotopy marked (†) is the unique one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The dashed arrow q is de- termined by the nullhomotopy (†) of the fiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given this, a choice v of nullhomotopy Vtmf(3) extending the canonical j-orientation of V |X1 is equivalent to a choice v′ of nullhomotopy bgl11 ◦ j′ ◦ q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus, the map p′ of spaces gives rise to a map Th(p′): Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V ) → Th(HP 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −j′ ◦ q) participating in a homotopy commutative diagram Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V ) ⊗ tmf(3) Th(HP 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −j′ ◦ p) ⊗ tmf(3) Σ∞ + CP 5 ⊗ tmf(3) Σ∞ + HP 2 ⊗ tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' v Th(p′)⊗tmf(3) v′ Σ∞ + p′⊗tmf(3) Applying the same argument to obtain w and w′, we get a homotopy commutative diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9) Σ∞ + CP 5 ⊗ tmf(3) Σ∞ + HP 2 ⊗ tmf(3) Th(p′): Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V ) ⊗ tmf(3) Th(HP 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −j′ ◦ q) ⊗ tmf(3) Σ∞ + CP 5 ⊗ tmf(3) Σ∞ + HP 2 ⊗ tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' v Σ∞ + p′ v′ w−1 Th(p′)⊗tmf(3) (w′)−1 Σ∞ + p′ Diagram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9 implies that a = w−1v has a10 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given v and w both tmf(3)-orientations for V satisfying condition (∗), let a = w−1v and let a11 : X1 → X1 be as in Equation (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11, all but the left-most triangle in the following diagram commute: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10) X1 Σ∞ + CP 5 ⊗ tmf(3) S10 ⊗ tmf(3) (CP 5)−V ⊗ tmf(3) BU(3)−γ3 ⊗ tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' X1 Σ∞ + CP 5 ⊗ tmf(3) w i i Th(V ) a11 w−1v v To get that ρ(V ) as in Diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5) is well-defined, it suffices to prove that Diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10) commutes after applying the functor tmf(3) −3(−) = π0 � Mapstmf(3) � (−) ⊗ tmf(3), Σ−3 tmf(3) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 35 Note that the map a11 ◦ i splits as a sum b0 ⊕ b1, where b0 ∈ Auttmf(3)(S10 ⊗ tmf(3)) and b1 ∈ MapsS(S10, Σ2C(α1) ⊗ tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since tmf(3) −3(Σ2C(α1)) = 0 by an Atiyah–Hirzebruch spectral sequence, it suffices to show that b∗ 0 is the identity on tmf(3) −3(−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We have reduced to studying the tmf(3)-module automorphisms of S10 ⊗ tmf(3) which arise from automorphisms of Σ∞ + CP 5 ⊗ tmf(3) of the form w−1v for v, w satisfying conditions (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Next, we partially compute the set of all nullhomotopies of Vtmf(3) : Σ∞ + CP 5 → bgl1 tmf(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This set is a torsor for G := π0 MapsS(Σ∞ + CP 5, Σ−1blg1 tmf(3)) = π0 MapsS(Σ∞ + CP 5, gl1 tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Fix one nullhomotopy v of Vtmf(3) which satisfies the hypotheses of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We can a group homomorphism h: G → π0 Auttmf(3)(Σ∞ + CP 5 ⊗ tmf(3)) by g �→ � Σ∞ + CP 5 ⊗ tmf(3) gv −→ Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −f) ⊗ tmf(3) v−1 −−→ Σ∞ + CP 5 ⊗ tmf(3) � and a subgroup G′ := {h ∈ G | hv satisfies conditions (∗) } ⊂ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This induces a group homo- morphisms h′ := � G′ ֒→ G h−→ π0 Auttmf(3)(Σ∞ + CP 5 ⊗ tmf(3)) ˜π−→ π0 Auttmf(3)(S10 ⊗ tmf(3)) � , where ˜π takes an automorphism a of Σ∞ + CP 5 ⊗tmf(3) to the component b0 ∈ π0 Auttmf(3)(S10 ⊗ tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' To show that Diagram 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='10 commutes after applying tmf(3) −3(−), it suffices to show that h′ is the zero map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We show that G′ is a subgroup of Z/3 ⊕ Z(3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since π0 Auttmf(3)(S10 ⊗ tmf(3)) ≃ Z× (3) and Z/3 ⊕ Z(3) admits no nontrivial maps to Z× (3), this suffices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given a basepoint for CP 5, we split the zero cell of Σ∞ + CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Requiring a given orientation to lift to the canonical HZ-orientation determines the nullhomotopy on S0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Therefore we have that G′ ⊂ π0 MapsSp(Σ∞CP 5, gl1 tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We compute this group via an Atiyah–Hirzebruch spectral sequence shown in Figure 8 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus, there is an extension of Z(3)-modules 0 → Z/3 → π0 MapsSp(Σ∞CP 5, gl1 tmf(3)) → Z(3) → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since Ext1 Z(3)(Z(3), Z/3) = 0, the group is Z(3) ⊕ Z/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The invariant ρ separates Chern classes for rank 3 bundles on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our next goal is to show that the invariant ρ defined by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9 distinguishes vector bundles of rank 3 on CP 5 with the same Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Recall that π10BU(3)c1≡0 ≃ Z/3 acts on [CP 5, BU(3)c1≡0] as in Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Given (f, σ) ∈ [CP 5, BU(3)c1≡0] × π10BU(3)c1≡0, (f, σ) �→ σV := � CP 5 Q −→ CP 5 ∨ S10 f∨σ −−−→ BU(3)c1≡0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Fix a1, a2, and a3 with a1 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='7, this action restricts to a transitive one on each set Va1,a2,a3 = {V : CP 5 → BU(3)c1≡0 | ci(V ) = ai}/ ∼, which is free if only if a2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9 we have a Thomification isomorphism t: π10(BU(3)c1≡0) → π10(Σ−3 tmf(3)), defined relative to an orientation for a generator of π10 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' More precisely, let σ ∈ π10 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) and an take an orientation v0 for σ satisfying condition (∗) from Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8 with respect to the splitting Σ∞ + S10 ≃ S0 ⊕ S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11) t(σ) := ˜ρ ◦ Thv0(σ), 36 MORGAN OPIE Nonzero terms on the E2-page of the Atiyah–Hirzebruch spectral sequence Hp(Σ∞CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' π−qgl1 tmf(3)) =⇒ πp+q MapsS(Σ∞CP 5, gl1 tmf(3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' p = 10 9 8 7 6 5 4 3 2 q = −10 −9 −8 −7 −6 −5 −4 −3 −2 −1 0 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A circle indicates a non-zero three-torsion group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' a square a non- zero torsion-free group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' and a diamond a group Z× (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The stars in bidegree (−10, 10) and (−8, 8) indicate terms along the p + q = 0 line which contribute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' with Thv0 as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Convention 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We write v0 for both the nullhomotopy of σ: Σ∞ + S10 → bgl1S and the associated nullhomotopy of bgl11 ◦ σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Our main goal in this subsection is to prove: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let V be a rank 3 vector bundle on CP 5 with c1 ≡ 0 (mod 3) and c2 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The ρ(σV ) = ρ(V ) + t(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This implies that the invariant ρ separates Chern classes as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' If V1, V2 are rank 3 vector bundles on CP 5 that have the same Chern classes, and such that c1(V1) = c1(V2) ≡ 0 (mod 3), c2(V1) = c2(V2) ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then ρ(V1) = ρ(V2) if and only if V1 and V2 are represented by homotopic maps to BU(3), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' if and only if V1 and V2 are topologically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14 assuming Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since π10(BU(3)c1≡0) acts transitively, there is some element σ ∈ π10(BU(3)c1≡0) such that V1 = σV2 and ρ(V2) = ρ(V1) + t(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Since t is an isomorphism, ρ(V1) = ρ(V2) if and only if σ = 0 if and only if V1 ≃ V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Consider the diagram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='12) Σ∞ + CP 5 bgl1 tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Σ∞ + CP 5 ⊕ S10, σV Σ∞ + Q V ⊕σ A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 37 where Q: CP 5 → CP 5 ∨ S10 is as in Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By Equation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4, Σ∞ + CP 5 ≃ Σ∞ + HP 2 ⊕ Σ2C(α2) ⊕ S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Under this identification, Σ∞ + Q = 1 ⊕ 1 ⊕ ∆: Σ∞ + HP 2 ⊕ Σ2C(α2) ⊕ S10 → Σ∞ + HP 2 ⊕ Σ2C(α2) ⊕ (S10 ⊕ S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let X′ 1 := Σ2C(α2) ⊕ (S10 ⊕ S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We obtain an orientation v′ of V ⊕ σ by summing the j-orientation of (V ⊕ σ|X′ 1)tmf(3) with any nullhomotopy of (V |Σ∞ + HP 2)tmf(3) that lifts the canonical HZ(3)-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' By construction, v′ satisfies (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This induces a nullhomotopy v of (σV )tmf(3) and a nullhomotopy ¯v of V , both of which satisfy condition (∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus we get a commuting diagram of tmf(3)-modules: Σ−3 tmf(3) (CP 5)−σV ⊗ tmf(3) (CP 5 ⊕ S10)−(V ⊕σ) ⊗ tmf(3) Σ∞ + CP 5 ⊗ tmf(3) (Σ∞ + CP 5 ⊕ S10) ⊗ tmf(3), Th(Q) (σV )∗ ˜ρ (V ⊕σ)∗ ˜ρ Q v−1 (v′)−1 where we suppress tensoring with tmf(3) from the horizontal arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Below, the pushout of spaces on the left induces the diagram of Thom spectra on the right: ∗ S10 S0 (S10)−σ CP 5 CP 5 ∨ S10 (CP 5)−V (CP 5 ∨ S10)−V ⊕σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The j-orientation for σS gives an equivalence (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13) (CP5 ∨ S10)−V ⊕σ ≃ (CP 5)−V ⊕ S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The nullhomotopy v′|S10 of σtmf(3) extends the j-orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' So, using the identification (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='13), we have that Th(V ⊕ σ)∗ ˜ρ = Th(V )∗˜ρ ⊕ t(σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thus we get the homotopy commutative Dia- gram (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14) of tmf(3)-modules below: (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='14) (CP 5)−σV ⊗ tmf(3) � (CP 5)−V ⊕ S10� ⊗ tmf(3) Σ−3 tmf(3) Σ∞ + CP 5 ⊗ tmf(3) � Σ∞ + CP 5 ⊕ S10� ⊗ tmf(3) S10 ⊗ tmf(3) � S10 ⊕ S10� ⊗ tmf(3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Th(σV )∗ ˜ρ Th(V )∗ ˜ρ⊕t(σ) Q v−1 ¯v−1⊕1 ρ(V )+t(σ) i ∆⊗tmf(3) i⊕1 38 MORGAN OPIE Comparing the two outer paths from the lower left-hand corner to Σ−3 tmf(3) gives the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Computing ρ on certain sums of line bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ρ be as defined by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For L a line bundle, we define ρ(L) := ρ(L ⊕ C2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Suppose that O(a1), O(a2) and O(a3) are line bundles on CP 5 with ai ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then ρ (O(a1) ⊕ O(a2) ⊕ O(a3)) = ρ(O(a1)) + ρ(O(a2)) + ρ(O(a3)) ∈ Z/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This immediately implies: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let a be an integer divisible by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Then ρ(O(a)⊕3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let V := ⊕3 i=1O(ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Let ˜V be the bundle on (CP 5)×3 given by ˜V := ⊕3 i=1p∗ i O(ai), where pi is the i-th projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' We can factor V : CP 5 → BU(3)c1≡0 as follows V : CP 5 ∆ −→ CP 5×3 ˜V−→ BU(3)c1≡0 and naturally identify Th((CP 5)×3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' − ˜V ) ≃ ⊗i Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −O(ai)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Using tmf(3)-orientations sat- isfying (∗) 7 for each O(ai), we get a diagram of tmf(3)-modules: Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −V ) ⊗i Th(CP 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −O(ai)) Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) Σ−3 tmf(3) Σ∞ + CP 5 (Σ∞ + CP 5)⊗3 S10 (S10)⊗3 S10 S10 ⊕ S10 ⊕ S10 Th(∆) Th( ˜V ) ˜ρ Σ∞ + ∆ ≃tmf(3) ≃tmf(3) Σ∞∆ ρ(V ) 1⊕1⊕1 ⊕iρ(O(ai)) where all terms in the diagram are implicitly tensored with tmf(3) and the maps marked ≃tmf(3) are tmf(3)-Thom isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The diagram is homotopy commutative and comparing the dashed arrows proves the Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ While this section provides some computations of ρ, we do not have a general formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Indeed, it is unclear what a formula for ρ should look like, since ρ cannot be computed from Chern classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Some inspiration can be drawn from [5], where Atiyah and Rees show that the α invariant of a rank 2 bundles on CP 3 can be computed as a holomorphic semi-characteristic [5, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='2], provided we choose an holomorphic representative for the topological class of the bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This leads to the following question: Question 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For V an algebraic vector bundle on CP 5, is there some description of the invariant ρ(V ) in terms of sheaf cohomology of V ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 7See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A CLASSIFICATION OF COMPLEX RANK 3 VECTOR BUNDLES ON CP 5 39 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A 3-torsion tmf(3)-valued invariant for rank 2 bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The homotopy groups of BU(3) through degree 10 are fairly sparse, so a complete analysis of [CP 5, BU(3)] was possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This allowed us to classify rank 3 bundles on CP 5 by first enu- merating such bundles and second defining an invariant to distinguish them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' While unraveling the structure of π∗BU(3), we began to suspect that ρ may also be interesting in the case of rank 2 bundles on CP 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The class ˜ρ: sk26 Th(BU(3)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ3) → Σ−3 tmf(3) factors through sk26 Th(BU(2)c1≡0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' −γ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Moreover: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The map induced map π10BU(2)c1≡0 → π10Σ−3 tmf(3) given by Thomifying with respect to −γ2 followed by ˜ρ is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The invariant V �→ ρ(V ) distinguishes 3-local equivalence classes of rank 2 vector bun- dles on CP 5 with fixed c1, c2 where additionally c1 ≡ 0 (mod 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' For (a), recall Diagram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Note that the unstable generator for π4BU(3) factors through BU(2), so Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9 shows that the image of a generator for π10BU(2)c1≡0 is nonzero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Therefore it suffices to check that π10BU(2) ≃ Z/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' This is classical: since BSU(2) ≃ BS3, the homotopy in the relevant range is given in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' π2 π3 π4 π5 π6 π7 π8 π9 π10 BU(2) Z 0 Z Z/2 Z/2 Z/12 Z/2 Z/2 Z/3 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Homotopy of BU(3) Since CP 5 is even, only even 3-local homotopy gives rise to 3-local invariants (odd 3-local homotopy contributes to constraints on possible Chern classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Therefore a argument as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='9 shows that the π10BU(3)-action as in Construction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='6 is the only source of 3-local invariants beyond Chern classes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' for rank 2 bundles on CP 5 with c1 ≡ 0 (mod 3), these bundles detected by ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' □ Questions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='18 opens up several avenues of inquiry: For which c1, c2 ∈ Z is the action of π10BU(2) nontrivial?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In other words, when is the invariant ρ of rank 2 bundles on CP 5 is occupied?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' What is the 2-local enumeration of rank 2 bundles on CP 5 with fixed Chern data?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Figure 9 shows that there is significant two-local data to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' References [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Ando, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Blumberg, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Gepner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hopkins, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Rezk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Units of ring spectra, orientations and thom spectra via rigid infinite loop space theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Journal of Topology, 7(4):1077–1117, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [2] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Ando, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hopkins, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Rezk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Multiplicative orientations of ko-theory and the spectrum of topo- logical modular forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Available at faculty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='illinois.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='edu/∼mando/papers/koandtmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='pdf, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Antieau and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Elmanto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A primer for unstable motivic homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Surveys on recent developments in algebraic geometry, 95:305–370, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [4] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Atiyah, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Bott, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Shapiro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Clifford modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Topology, 3(1):3–38, 1964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Atiyah and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Rees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Vector bundles on projective 3-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Inventiones Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 35:131–153, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [6] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Bauer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Computation of the homotopy of the spectrum tmf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Topolo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Monogr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 13:11–40, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [7] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Bhattacharya and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Chatham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' On the EO-orientability of vector bundles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Available at arXiv:2003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='03795v3, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Bhattacharya and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Kitchloo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The stable Adams conjecture and higher associative structures on Moore spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 195:375–420, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [9] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Chatham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' An orientation map for height p − 1 real e theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' 40 MORGAN OPIE [10] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Friedlander.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The infinite loop adams conjecture via classification theorems for F-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Cambridge Philos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 87(1):109–150, 1980.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [11] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Friedlander and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Seymour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Two proofs of the stable adams conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 83(6):1300–1302, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [12] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hirzebruch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Topological methods in algebraic geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Springer, 1995.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [13] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Metastable complex vector bundles over complex projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='11800, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Kodaira and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Spencer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Divisor class groups on algebraic varieties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 39:872–877, 1953.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [15] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Kudo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A transgression theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Mem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Fac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Kyusyu Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', A(9):79–81, 1956.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [16] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Mahowald and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Ravenel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Towards a global understanding of the stable homotopy groups of spheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' In S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Gitler, editor, The Leftschets Centennial Conference: Proceedings on Algebraic Topology II, volume 58 of Contemporary Mathematics, pages 109–118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' AMS, 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [17] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Mathew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The homotopy groups of TMF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Available at math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='uchicago.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='edu/∼amathew/tmfhomotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='pdf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [18] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' May and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Ponto.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' More Concise Algebraic Topology: Localization, Completion, and Model Cate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Chicago Lectures in Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Unviersity of Chicago Press, Chicago and London, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [19] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' May and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Quinn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' e∞ ring space and e∞ ring spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lecture Notes in Mathematics, 577, 1977.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [20] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' McCleary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' A User’s Guide to Spectral Sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Cambridge Studies in Advanced Mathematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Cam- bridge University Press, 2 edition, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [21] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Miller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Lectures on algebraic topology ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Available at http://math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='mit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='edu/∼hrm/papers/lectures-905- 906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content='pdf, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [22] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Mimura and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Toda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Homotopy groups of SU(3), SU(4), and Sp(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Kyoto Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 3(2):217– 250, 1963.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [23] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Quillen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' The Adams conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Topology, 10:67–80, 1971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [24] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Sullivan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Genetics of homotopy theory and the Adams conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 100:1–79, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [25] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Thomas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Almost complex structures on complex projective spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=', 193:123– 132, 1974.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' [26] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Zabrodsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' Hopf Spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} +page_content=' North-Holland Publishing Company, Amsterdam, New York, Oxford, 1976.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/cdE3T4oBgHgl3EQfGQmo/content/2301.04313v1.pdf'} diff --git a/fdE0T4oBgHgl3EQfXQBg/content/tmp_files/2301.02289v1.pdf.txt b/fdE0T4oBgHgl3EQfXQBg/content/tmp_files/2301.02289v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..fbea10c1fb7daa4804cbc1c8f3133db77895fc8f --- /dev/null +++ b/fdE0T4oBgHgl3EQfXQBg/content/tmp_files/2301.02289v1.pdf.txt @@ -0,0 +1,2245 @@ +MNRAS 000, 1–15 (2022) +Preprint 9 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Cosmological Fisher forecasts for next-generation spectroscopic surveys +William d’Assignies D.1,2,3★, Cheng Zhao1†, Jiaxi Yu1 and Jean-Paul Kneib1,4 +1 Laboratory of Astrophysics, École Polytechnique Fédérale de Lausanne (EPFL), Observatoire de Sauverny, CH-1290 Versoix, Switzerland. +2 Institut de Física d’Altes Energies (IFAE), The Barcelona Institute of Science and Technology, Campus UAB, 08193 Bellaterra (Barcelona) Spain. +3 Physics institute of the Ecole Normale Supérieure PSL, 24 rue Lhomond, 75005 Paris, France. +4 Aix Marseille Université, CNRS, LAM (Laboratoire d’Astrophysique de Marseille) UMR 7326, F13388, Marseille, France +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +Next-generation spectroscopic surveys such as the MegaMapper, MUltiplexed Survey Telescope (MUST), MaunaKea Spec- +troscopic Explorer (MSE), and Wide Spectroscopic Telescope (WST) are foreseen to increase the number of galaxy/quasar +redshifts by an order of magnitude, with hundred millions of spectra that will be measured at 𝑧 > 2. We perform a Fisher +matrix analysis for these surveys on the baryonic acoustic oscillation (BAO), the redshift-space distortion (RSD) measurement, +the non-Gaussianity amplitude 𝑓NL, and the total neutrino mass 𝑀𝜈. For BAO and RSD parameters, these surveys may achieve +precision at sub-percent level (<0.5 per cent), representing an improvement of factor 10 w.r.t. the latest database. For NG, these +surveys may reach an accuracy of 𝜎( 𝑓NL) ∼ 1. They can also put a tight constraint on 𝑀𝜈 with 𝜎(𝑀𝜈) ∼ 0.02 eV if we do joint +analysis with Planck and even 0.01 eV if combined with other data. In addition, we introduce a general survey model, to derive +the cosmic volume and number density of tracers, given instrumental facilities and survey strategy. Using our Fisher formalism, +we can explore (continuously) a wide range of survey observational parameters, and propose different survey strategies that +optimise the cosmological constraints. Fixing the fibre number and survey duration, we show that the best strategy for 𝑓NL and +𝑀𝜈 measurement is to observe large volumes, despite the noise increase. However, the strategy differs for the apparent magnitude +limit. Finally, we prove that increasing the fibre number improves 𝑀𝜈 measurement but not significantly 𝑓NL. +Key words: techniques: spectroscopic – surveys – neutrinos – early Universe – cosmological parameters – large-scale structure +of Universe +1 INTRODUCTION +Massive high redshift spectroscopic survey aims at exploring baryon +acoustic oscillations (BAO) and the growth of structure through +redshift-space distortions (RSD) with large-scale structures (LSS) in +the Universe, by probing the 3D distribution of galaxies and quasars +in a wide area. LSS also provides one of our best windows on fun- +damental physics, such as properties of the early Universe with the +non-Gaussian primordial fluctuations, or the sum of neutrino mass +(NM). As the product of spectroscopic surveys, the database for +the 3D positions of galaxies and quasars has been growing rapidly. +In the past decade, surveys like the SDSS-III Baryon Oscillation +Spectroscopic Survey (BOSS; Schlegel et al. 2009) and SDSS-IV +extended BOSS (eBOSS; Dawson et al. 2016) have measured mil- +lions of spectra. The ongoing Dark Energy Survey Instrument (DESI; +DESI Collaboration et al. 2016a) is expected to take over 30 million +spectra in 5 years. The next-generation experiments, such as the +MegaMapper (Schlegel et al. 2019), MUltiplexed Survey Telescope1 +(MUST), Widefield Spectroscopic Telescope (WST; Ellis & Dawson +2019), and the MaunaKea Spectroscopic Explorer (MSE; Percival +et al. 2019), are expected to be equipped with a large number of fi- +★ E-mail: wdoumerg@ifae.es +† E-mail: cheng.zhao@epfl.ch +1 https://must.astro.tsinghua.edu.cn +bres (10–20k) thanks to the development of robotic fibre positioners, +and are foreseen to increase the number of observed galaxy/quasar +by another order of magnitude. +These observations will allow us to test the standard ΛCDM +model of cosmology, with parameters constrained at sub-percent- +level precision. The standard ΛCDM model has been able to explain +a large number of observations, from the CMB to low-redshift galax- +ies. However, tensions between measurements have recently become +more and more significant, notably the Hubble constant 𝐻0 (Riess +et al. 2019; Freedman 2021) and the growth parameter 𝜎8 (Macaulay +et al. 2013; Douspis et al. 2019). The origin of these tensions could +come from bias in our measurements, unknown systematics or be the +sign of new physics (Di Valentino et al. 2021; Blanchard et al. 2022). +There are extensions of the standard ΛCDM model, e.g., varying the +dark energy equation of state (Copeland et al. 2006; Tripathi et al. +2017), the primordial non-Gaussianity (NG; Matarrese et al. 2000; +Dalal et al. 2008), the non-zero total neutrino mass (Boyle 2019). +The primordial non-Gaussianity is a test to inflation scenario +(Achúcarro et al. 2022), and the LSS of galaxies/quasars is controlled +by the NG amplitude parameter 𝑓NL through a bias scale dependence +(Maldacena 2003; Desjacques et al. 2009). Therefore measuring the +large-scale clustering of galaxies provides an opportunity to study +the early Universe physics. +Oscillation experiments have shown that at least two families of +neutrinos have non-zero mass (Capozzi et al. 2016), but they can +© 2022 The Authors +arXiv:2301.02289v1 [astro-ph.CO] 5 Jan 2023 + +2 +W. d’Assignies et al. +only constrain the relative mass difference between families. Normal +and inverted hierarchy theories give different predictions of the sum +of neutrino mass respectively: 𝑀𝜈 ∼ 0.06 eV, or 𝑀𝜈 ∼ 0.1 eV (Qian +& Vogel 2015). As massive neutrinos constitute a small fraction of +the energy density of the Universe, a range of cosmological probes +can provide indirect evidence of their mass properties (Lesgourgues +& Pastor 2014). For example, a joint analysis of the Planck cosmic +microwave background (CMB) measurements, and the Baryon Os- +cillation Spectroscopic Survey (BOSS) galaxy clustering data, have +already put an upper limit 𝑀𝜈 < 0.16 eV at 95 per cent confidence +level (Ivanov et al. 2020), and down to 𝑀𝜈 < 0.11 eV with data from +eBOSS (Alam et al. 2021). Thus, the Universe appears to be an ideal +laboratory for the measurement of the neutrino hierarchy. However +it has been recently objected that this measurement depends on the +ΛCDM model (which has started to exhibit weakness), and cannot +categorically exclude a scenario, though a careful study on the de- +pendence of these constraints on the ΛCDM model may be necessary +(Boyle & Komatsu 2018). +The aim of this paper is first to forecast the accuracy of high red- +shift spectroscopic surveys of the next decade, on four cosmological +aspects: BAO scales, the RSD effect, NG amplitude parameters, and +the sum of neutrino mass. We use Fisher information matrix 𝐹𝑖 𝑗 +(Tegmark 1997) for this purpose, assuming its inverse is a typical +covariance matrix of our parameters (which is true in the Gaussian +case, and gives an upper limit in the general one). We will use the lin- +ear theory to evaluate this information matrix. Thus our forecasting +method is not particularly innovative compared to modern Markov +chain Monte Carlo (MCMC; Chudaykin & Ivanov 2019) methods, +and the consideration of non-linearity. Our goal is rather to compare +the different surveys, and to apply this simple formalism in the con- +text of an optimisation of the parameters of a survey. In principle, it +is also possible to perform Fisher forecasts for surveys probing 𝑧 ∼ 1, +with a much higher tracer density. Nonetheless, the main interest of +these surveys is to extract more information from small scales, e.g., +by exploring the power spectrum up to 𝑘max ∼ 0.5ℎ Mpc−1 modes. +The linear model of the power spectrum that we adopt in this study is +not able to describe these scales accurately. Therefore, forecasts for +high-density surveys that focuses on small-scale clustering are left +for a future work. +With the increase of the number of fibre, and as demonstrated by +our forecasts, future surveys will be able to constrain cosmological +parameters such as BAO and RSD at the sub-percent level. Besides, +the measurement of parameters beyond the standard model like 𝑓NL +and 𝑀𝜈, remains a challenge as the error bars are comparable to +the parameter values. That is why, in a second step, we produce +a quantitative optimisation pipeline of the observation strategy of +spectroscopic surveys, for the study of the parameters 𝑓NL and 𝑀𝜈 +as we find their constraints still need improvement despite its large +number of fibres and large cosmic volume. It could be used for the +design of future surveys, but it also provides a point of comparison +between the expected accuracy of some surveys and the technically +optimal one. To do so, we will present a rather general model, of +high redshift spectroscopic survey. We model in a simplified way the +properties of observational targets, the specifications of the telescope, +and the survey strategy. +This paper is organised as follows. In Section 2, we describe in +detail the future high redshift massive spectroscopic surveys. The +methodology used for the science forecasts is outlined in Section +3, as well as our modelling for a general spectroscopic survey. We +present the cosmological parameter constraints and the preferred +improvements for future surveys in Section 4. Finally we summarise +our results in Section 5. +2 SURVEYS +In this section, we describe various spectroscopic galaxy surveys and +cosmological probes considered in our forecasts. These surveys will +observe emission line galaxies (ELGs) that are abundant up to red- +shift ∼ 2 (Madau & Dickinson 2014), Lyman alpha emitter galaxies +(LAEs), Lyman break galaxies (LBGs), and BX galaxies (Steidel +et al. 2004) that can be observed up to redshift 5, or even higher +in theory (Wilson & White 2019). Forecasts presented in this work +do not include constraints on cosmological parameters coming from +cosmic shear, HI intensity mapping and future CMB observations +that will be included in upcoming surveys (Annis et al. 2022) such +as Euclid (Laureijs et al. 2011), DESI (DESI Collaboration et al. +2016a), Puma (Slosar et al. 2019), HETDEX (Adams et al. 2011). +Survey properties considered in our study are listed in Table 1, +and the associated densities are presented in Table 4 in Section 4. +In general, the number density of a given tracer 𝑋 can be (ideally) +modelled by +𝑛(𝑧) = +∫ 𝑚max +𝐸(𝑚)𝜙𝑋 (𝑚, 𝑧)𝑑𝑚, +(1) +where 𝑚 is the apparent magnitude of the tracer, 𝑚max is the max- +imum apparent magnitude of the survey, 𝐸(𝑚) is the efficiency of +observation and 𝜙𝑋 is the tracer luminosity function. If the efficiency +is independent of the magnitude, it reduces to +𝑛(𝑧) = eff · 𝑛𝑋 (𝑚max,𝑧), +(2) +where eff is the constant efficiency of the tracer, and 𝑛𝑋 is the theoret- +ical tracer density (cf Section 3.2). We assume a redshift uncertainty +𝜎𝑧/(1 + 𝑧) = 0.001 for every survey. +2.1 MegaMapper +MegaMapper (Schlegel et al. 2019) is a spectroscopic instrument +that will be located at Las Campanas observatory in the southern +hemisphere. It would target LBG at high redshift 2 < 𝑧 < 5, covering +14,000 square degrees of the sky. Its 6.5-meter telescope, 20,000 +fibres, and a five-year observation period would yield galaxy number +density 𝑛 > 10−4ℎ3Mpc−3 across its redshift range. For the property +of the fiducial LBG sample, we use the values in Table 2 of Ferraro & +Wilson (2019). These values are compatible with the model given by +equation (2), with 𝑛LBG an idealised density distribution introduced +in subsection 3.2 (Eq. (9)), eff = 0.4 for 𝑧 < 4 and eff = 0.9 for +4 < 𝑧 < 5 (see Figure 4 of Sailer et al. 2021), and 𝑚max = 24.5. +2.2 MSE +The MaunaKea Spectroscopic Explorer (MSE; Percival et al. 2019) +will be located in Hawaii in the Northern hemisphere, probing over +10,000 square degrees. It will couple an 11.25-meter mirror with +a 1.5-square-degree field of view (FoV) to 4000 fibres, feeding to +spectrographs that cover 360 to 1300 nm. This design enables the +detection of ELGs at 1.6 < 𝑧 < 2.4, and LBGs at 2.4 < 𝑧 < 4. The +exposure time is 1800s. +The ELG number density shown in Table 4 are taken from Per- +cival et al. (2019). For LBG we will assume a model described +by equation (1). We estimate the efficiency thanks to Figure 2 of +Percival et al. (2019), assuming that 40 per cent of LBG have Equiv- +alent Width values EW< 0, 30 per cent have 0 20, and averaging over EW 2. The effective +efficiency law is then given by 𝐸(𝑚) = −0.18𝑚 + 4.8. Given the +approximate efficiency rate of 0.5, and the required observed den- +sity (𝑛 = 10−4ℎ3/Mpc3), 1400 fibres/deg2 will be allocated to LBG +observations, restricting to a maximum magnitude 𝑚max = 24.2 +(Percival et al. 2019). We introduced our LBG luminosity function +model in subsection 3.2. +2.3 MUST +The MUltiplex Survey Telescope3 (MUST) is a future 6.5-meter +telescope (with a 7 square degree FoV) located in China, in the +Northern hemisphere. Its target can be either LBG+BX (LBGX), or +a combination of LBGX+LAE. Since the exact survey design is still +in flux, our forecast supposes its redshift range to be 2 < 𝑧 < 4, with +a sky coverage between 9,000 and 15,000 deg2, and fibre numbers +to be either 10,000 or 20,000. +2.4 A WST-like NTL survey +The Widefield Spectroscopic Telescope4 (WST; Ellis & Dawson +2019) is a proposed spectroscopic survey in the southern hemisphere +that would couple an 11.4m dish (with a 5 square degree FoV) and +20,000–60,000 fibres, enabling more than a hundred million of fibre +exposures (each ≥ 4,000 seconds long) over its survey period. Its +design would permit observations of LBGs and LAEs up to redshift +5, with number densities 2 to 5 times of those for a MegaMapper-like +survey. Since the exact design of this survey is a work in progress, +we also explore several possible survey parameters. +For the forecast, we consider a similar survey to MegaMapper, +with a sky coverage of 15,000 deg2 for LBG at redshift 2 to 5 +and each tracer can be observed for a period as long as needed +until it reaches the required spectrum quality. We model it with an +efficiency 𝑒ff = 0.9 relative to the theoretical tracer density (cf. +equation (2)), but with different maximum magnitudes – 24.2, 24.5, +and 25 – depending on the final fibre number. This might corresponds +to 20,000–40,000–100,000 fibres for 5 years of observation5. This +forecast somehow represents a cosmological limit on the achievable +parameters accuracy, since we are assuming an efficiency very close +to 1. As we do not really take into account the final properties of +2 We are averaging over the three templates 𝐸 (𝑚) = 0.4𝐸 (𝑚|EW < 0) + +0.3𝐸 (𝑚|0 < EW < 20) + 0.3𝐸 (𝑚|20 < EW) +3 https://must.astro.tsinghua.edu.cn +4 previously named Spectel, https://www.wstelescope.com/ +5 Of course we do not expect surveys to have 100,000 fibres in the next decay. +This forecast rather serves as an upper limit. +WST in our modelling, we will refer to this fiducial survey as NTL +survey (a No-Time-Limit survey) in Section 4. +2.5 A General Survey +We consider also a high redshift (𝑧 > 2) general spectroscopic survey +with a modelling of the survey settings, in order to explore the optimal +strategy that yields the tightest constraints of chosen cosmological +parameters. In a first step, we assume an LBG survey lasting 5 years, +based on a 10-meter telescope equipped with 20,000 fibres. Since +LBGs are abundant mostly at 𝑧 ≳ 2 (Wilson & White 2019), we +consider a redshift window [𝑧min, 𝑧max] that always starts with 𝑧min = +2. The survey volume will be thus described by the fraction of the +survey sky coverage 𝑓sky6 and the redshift span Δ𝑧 = 𝑧max − 𝑧min. In +a second step, we will also vary the ‘observation capacity’ defined +as the product of the survey duration and the fibre number, in to +extend this model to a larger variety of spectroscopic telescopes and +to highlight the improvement of the measurements with the available +technology. We detailed the modelling of such survey in Section 3.5. +3 METHODOLOGY +In this section, we first describe some properties of observed galax- +ies. Then we introduce the commonly used Fisher matrix forecasting +technique. In Section 3.4 we specify the BAO, RSD, non-Gaussianity +and Neutrino mass forecast strategies. We then introduce our mod- +elling of a general survey in 3.5. +3.1 Observed power spectrum +The power spectrum of a dark matter tracer 𝑋 is related to the theo- +retical matter power spectrum 𝑃m(𝑘, 𝑧) with +𝑃𝑋 (𝑘, 𝜇, 𝑧) = (𝑏𝑋 (𝑧) + 𝑓 (𝑧)𝜇2)2𝑃m(𝑘, 𝑧), +(3) +with 𝑏𝑋 being the tracer bias, 𝑓 being the growth rate, and 𝜇 being +the cosine between the line of sight and the 3d mode 𝒌. To take +into account the error in the redshift measurement 𝜎𝑧/(1 + 𝑧) that +propagates to an error in the radial distance via 𝜎𝜒 = 𝜎𝑧𝑐/𝐻(𝑧), we +multiply the power spectrum by a factor exp +� +−𝑘2𝜇2𝜎2𝜒 +� +(Sailer et al. +2021). +We also introduce the cross-power spectrum of two different trac- +ers 𝐴 and 𝐵 following McDonald & Seljak (2009) as +𝑃𝐴𝐵(𝑘, 𝜇, 𝑧) = (𝑏𝐴(𝑧) + 𝑓 (𝑧)𝜇2)(𝑏𝐵(𝑧) + 𝑓 (𝑧)𝜇2)𝑃m(𝑘, 𝑧), +(4) +6 𝑓sky = 1 corresponds to the full sky, 41,253 square degrees. +MNRAS 000, 1–15 (2022) + +4 +W. d’Assignies et al. +Table 2. Fiducial values of cosmological parameters and their Planck Gaus- +sian prior. NG amplitude 𝑓NL is neglected except for the NG forecast. +ℎ +𝜔𝑏 +𝜔𝑐 +𝑛s +𝜏 +ln(𝐴s) +𝑀𝜈 (eV) +Fiducial values +0.677 +0.02247 +0.1192 +0.9675 +0.056 +-13.073 +0.06 +Planck half-width Gaussian prior +0.0054 +0.00015 +0.0012 +0.0042 +0.007 +0.015 +0.5 +where 𝑏𝐴 and 𝑏𝐵 are biases of tracer A and B respectively. We report +in Table 2 fiducial values of six standard cosmological parameters +used in this work, along with the extension model parameter 𝑀𝜈. +Power spectrum will be evaluated using CAMB7 (Howlett et al. +2012) and pyccl8 (Chisari et al. 2019). +3.2 Bias and luminosity function +For ELG, we assume a constant clustering amplitude, based on the +analysis of DESI-selected samples in the DEEP2 data (DESI Col- +laboration et al. 2016a). In that case, the bias can be approximated +as 𝑏ELG = 0.8 × 𝐷(0)/𝐷(𝑧) with 𝐷 the growth function. The factor +0.8 is chosen to be a bit lower than that of DESI (0.84; see DESI +Collaboration et al. 2016b) and eBOSS (1; see Dawson et al. 2016), +as we consider fainter ELGs in this study. +We model LBG and LAE bias following the parametrization of +Wilson & White (2019) using +𝑏LBG/LAE(𝑧, 𝑚) = 𝐴(𝑚)(1 + 𝑧) + 𝐵(𝑚)(1 + 𝑧)2, +(5) +with 𝐴(𝑚) = −0.98(𝑚 −25) +0.11 and 𝐵(𝑚) = 0.12(𝑚 −25) +0.17, +𝑚 being the apparent magnitude. We assume that fainter galaxies +contribute more, since the galaxy abundance grows as the magnitude +increases, and reduce the bias to a one-parameter function 𝑏(𝑧, 𝑚) ≈ +𝑏(𝑧, 𝑚max). For samples with a large magnitude band, we might +separate it into subsamples of different maximal magnitudes, and +adopt a multi-tracer approach. +For the aim of our study, we need to evaluate the LBG density +function. An idealized number density is modelled by +𝑛LBG = +∫ 𝑀𝑐 +−∞ +𝜙(𝑀)𝑑𝑀, +(6) +with 𝜙 the luminosity function (Wilson & White 2019; Sailer et al. +2021), and 𝑀 the absolute magnitude. We use the Schechter model +(Schechter 1976) for the luminosity function: +𝜙(𝑀) = ln 10 +2.5 𝜙★10−0.4(1+𝛼) (𝑀−𝑀★) exp +� +−10−0.4(𝑀−𝑀★)� +(7) +with 𝛼, 𝑀★ and 𝜙★ listed in Table 3 of Wilson & White (2019). The +absolute magnitude cutoff 𝑀𝑐 of galaxies at redshift 𝑧 with apparent +magnitude being 𝑚max is determined as +𝑀𝑐(𝑚max) = 𝑚max − 5 log10 +� 𝐷L(𝑧) +10pc +� ++ 2.5 log10(1 + 𝑧), +(8) +with 𝐷L(𝑧) the luminosity distance. As the observation depends +rather on the apparent magnitude 𝑚 than the absolute magnitude 𝑀, +we rewrite, the density equation, with 𝜙(𝑚, 𝑧) = 𝜙(𝑀(𝑚, 𝑧)), using +𝑑𝑀/𝑑𝑚 = 1, as +𝑛LBG(𝑧) = +∫ 𝑚max +𝜙(𝑚, 𝑧)𝑑𝑚. +(9) +7 https://camb.readthedocs.io/en/latest/ +8 https://ccl.readthedocs.io/en/latest/ +In the rest of the study, we will use this last equation formalism and +refer to 𝑚 as ‘magnitude’ hereafter. +3.3 Fisher Matrix +For a set of cosmological parameters {𝑝𝑖}, the diagonal coefficient +of the inverse of its Fisher matrix F𝑖 𝑗 gives an upper bound on the +variance of each parameter: 𝜎2 +𝑖 ≥ (F )−1 +𝑖𝑖 according to the Cramer- +Rao inequality (with equality for Gaussian likelihood). We follow +the same steps as Zhao et al. (2016) and considered the Fisher matrix +F𝑖 𝑗 = 𝑉sur +4𝜋2 +∫ +1 +−1 +𝑑𝜇 +∫ 𝑘max +𝑘min +𝑘2𝑑𝑘𝐹𝑖 𝑗 (𝑘, 𝜇), +(10) +𝐹𝑖 𝑗 = 1 +2Tr(𝜕𝑝𝑖CC−1𝜕𝑝𝑗 CC−1), +(11) +with 𝑉sur the comoving volume of the survey, and C the data co- +variance matrix. The integration bounds 𝑘min depends on the survey +volume and corresponds to the maximal length, while 𝑘max depends +on the accuracy of the theoretical model on nonlinear scales and on +the shot noise. We take by default +𝑘min = +2𝜋 +𝑉1/3 +sur +[ℎ Mpc−1], 𝑘max = 0.1𝐷(0) +𝐷(𝑧) +[ℎ Mpc−1]. +(12) +Since the neutrino mass, and more generally many new physics prop- +erties, such as the nature of gravity, are significantly encoded at small +scales, we will also consider an ‘optimistic’ integration bound with +𝑘max = 0.3ℎMpc−1. This is motivated by both the reduction of shot +noise in futur data survey, and expected progress in theoretical un- +derstanding and modelling of non-linearities. +3.3.1 One Tracer +When targets are the same type of tracer (MegaMapper LBGs for +example), 𝐶 is a 1 × 1 matrix. We take into account the tracer distri- +bution discreetness by adding a Poissonian shot noise that scales as +the inverse of the number density 1/𝑛 as +𝐶 = 𝑃 + 1/𝑛, +(13) +where 𝑃 is the power spectrum of this tracer. Thus, the Fisher matrix +is +𝐹𝑖 𝑗 = 1 +2 +� +𝑛𝑃 +𝑛𝑃 + 1 +�2 𝜕 ln 𝑃 +𝜕𝑝𝑖 +𝜕 ln 𝑃 +𝜕𝑝 𝑗 +. +(14) +We will report the parameter 𝑛𝑃(𝑘 = 0.14, 𝜇 = 0.6) for different +surveys, an approximate center-of-weight point for BAO measure- +ments.to give a qualitative description of the noise level, following +DESI Collaboration et al. (2016a). +3.3.2 Two tracers +For two tracers (LBG+LAE in MUST for example), 𝐶 is now a 2 × 2 +matrix, and under the same assumption, +C = +� +𝑃𝐴𝐴 + +1 +𝑛𝐴 +𝑃𝐴𝐵 +𝑃𝐴𝐵 +𝑃𝐵𝐵 + +1 +𝑛𝐵 +� +. +(15) +where 𝑃𝐴𝐴 and 𝑃𝐵𝐵 are auto-power spectra of tracer A and B, +𝑃𝐴𝐵 is the cross-power spectrum of tracer A and B. An explicit +expression for the Fisher matrix in the 2 tracers case is given in +Appendix A of Zhao et al. (2016) as combinations of 𝜕 ln 𝑃𝑇 +𝜕𝑝𝑖 +with +𝑇 = 𝐴, 𝐵, 𝐴𝐵. In the case of two independent tracers, C is diagonal +MNRAS 000, 1–15 (2022) + +Next generation spectroscopic survey forecasts +5 +and the two-tracers Fisher matrix is equal to the sum of the two one- +tracer matrix. Similarly, as in the one-tracer case, we will report the +parameter � 𝑛𝑖𝑃(𝑘 = 0.14, 𝜇 = 0.6). +For surveys with extremely massive data sets, whose galaxy appar- +ent magnitudes are spread over a wide band, we separate them into +sub-samples with different magnitude ranges (typically {−∞; 24.5}, +{24.5; 24.8}). Then we adopt a two-tracer forecast approach. Such +a process is mainly motivated by the non-Gaussianity forecast, since +𝛿𝑏 ∝ 𝑓NL𝑏𝐺, cf. section 3.4.3. Indeed the bias decreased with the +magnitude, and a one tracer approach with 𝑏 = 𝑏(𝑚max) assumption +artificially reduce the sensitivity to NG. +3.3.3 Complementary data sets +To combine constrains from two independent surveys9 A and B that +aims to measure the same set of cosmological parameters, one simply +adds their Fisher matrix 𝐹𝑖 𝑗 = 𝐹 𝐴 +𝑖 𝑗 + 𝐹𝐵 +𝑖 𝑗. Furthermore, to introduce +priors from complementary data sets (such as Planck CMB) that has +measured parameters {𝑝𝑖} with accuracies {𝜎𝑖}, one simply add to +the Fisher matrix 𝑃𝑖 𝑗 = 𝛿𝑖 𝑗/𝜎2 +𝑖 (assuming Gaussian uncertainties). +3.3.4 Redshift bins +Forecasts for a survey that covers a large redshift range [𝑧min, 𝑧max], +have to take into account the redshift dependence of parameters. +There are three approaches to do so: +• Split the survey volume into redshift bins {𝑧𝑘} with separation +𝑑𝑧𝑘, and present the forecast for each bin separately, +• Separate the survey volume into redshift bins {𝑧𝑘} with separation +𝑑𝑧𝑘, and sum the all fisher matrices, neglecting cross correlation +between bins: 𝐹𝑖 𝑗 = � +𝑘 𝐹𝑧𝑘 +𝑖 𝑗 , +• Consider one redshift bin, with an effective redshift 𝑧eff ( so with +this approach one neglects redshift dependence of the parameters, +bias and density). +Following Sailer et al. (2021), the effective redshift of of a sub- +sample is calculated with +𝑧eff = +∫ +𝑑𝑧𝐻2(𝑧)𝜒2(𝑧)(𝑑𝜒/𝑑𝑧)3𝑛2(𝑧)𝑧 +∫ +𝑑𝑧𝐻2(𝑧)𝜒2(𝑧)(𝑑𝜒/𝑑𝑧)3𝑛2(𝑧) +. +(16) +None of these approaches is flawless, and we will choose the best one +for different purpose in the following study. Bailoni et al. (2017) has +implemented a multi-bins approach, and show that in some case the +cross-correlation between bins modifies the forecast up to 10-20 per +cent. Nonetheless, Sailer et al. (2021) has shown that for these high- +redshift galaxy surveys, the correction was negligible (< 10 per cent, +cf. Appendix B of their paper). Zhao et al. (2019) have calculated +the optimal redshift weighting scheme for the BOSS survey and a +similar algorithm can be implemented for future surveys, but this is +beyond the scope of this study. +3.4 Cosmological Parameters +3.4.1 BAO +For the BAO forecast, the two parameters are ln(𝐷 𝐴/𝑠) and ln(𝑠𝐻), +with 𝑠 the sound horizon, 𝐷 𝐴 the angular distance, and 𝐻 the Hubble +9 Surveys with non-overlapping redshift ranges, sky coverages or indepen- +dent tracers. +parameter. We assume to have a very good measurement of 𝑠 from +CMB, so that 𝜎(𝐷 𝐴/𝑠) = 𝜎(𝐷 𝐴)/𝑠 and 𝜎(𝑠𝐻) = 𝑠𝜎(𝐻), thus: +𝜎(ln(𝑠𝐻)) = 𝜎(𝐻) +𝐻 +; 𝜎(ln(𝐷 𝐴/𝑠)) = 𝜎(𝐷 𝐴) +𝐷 𝐴 +. +(17) +The distance error on both of the parameters is derived with the Seo +& Eisenstein (2007) approximation of the Fisher matrix, +F𝑖 𝑗 =𝑉sur𝐴2 +0 +∫ 1 +0 +𝑑𝜇 +∫ ∞ +0 +𝑑𝑘 +� +𝑓𝑖(𝜇) 𝑓 𝑗 (𝜇)𝑘2 +× +exp +� +−2 (𝑘Σ𝑠)1.4� +� 𝑃(𝑘) +𝑃(0.2) + +1 +𝑛𝑃(0.2) +�2 exp +� +−𝑘2(1 − 𝜇2)Σ2 +⊥ − 𝑘2𝜇2Σ2 +∥ +� � +, +(18) +where +𝑓𝑖(𝜇) = +� 𝜇2 − 1 +if 𝑖 = 1; +𝜇2 +if 𝑖 = 2. +(19) +Σ∥ and Σ⊥ are the root mean square displacement along and +perpendicular to the line of sight. Σ∥ = Σ0𝐷(𝑧)(1 + 𝑓 (𝑧)) and +Σ⊥ = Σ0𝐷(𝑧) with Σ0 = 10.4𝜎8 ℎ−1Mpc. 10 The Silk-damping ef- +fect is included with the Silk-damping scale Σ𝑠, expressed in ℎ−1Mpc +via +Σ−1 = 1.6 +� +Ω𝑏ℎ2�0.52 � +Ω𝑚ℎ2�0.73 � +1 + +� +10.4Ω𝑚ℎ2�−0.95� +ℎ−1. +(20) +We assume a reduction of the BAO damping scale by a factor 0.5 +w.r.t. the value from Seo & Eisenstein (2007), following section 4.1 +of Font-Ribera et al. (2014). We fixed 𝐴0 = 0.55, the WMAP3 value +given in Seo & Eisenstein (2007).11 +Here for BAO measurements in the two-tracer case, we will as- +sume two independent measurements, and simply sum the two fisher +matrix. +3.4.2 RSD effects +We follow White et al. (2009) for the Redshift Space Distortions fore- +cast. For one tracer, we rewrite the equation of the power spectrum: +𝑃(𝑘, 𝑧) = +� +𝑏(𝑧)𝜎8(𝑧) + 𝑓 (𝑧)𝜎8(𝑧)𝜇2�2 𝑃m(𝑘, 𝑧 = 0) +𝜎8(𝑧 = 0)2 . +(21) +We introduce our parameters: ln[𝑏(𝑧𝑖)𝜎8(𝑧𝑖)] and ln[ 𝑓 (𝑧𝑖)𝜎8(𝑧𝑖)]. +For simplicity, we drop the explicit redshift dependence. In this case, +the derivative of power spectrum w.r.t. parameters are +𝜕 ln 𝑃 +𝜕 ln(𝑏𝜎8) = +2𝑏𝜎8 +𝑏𝜎8 + 𝑓 𝜎8𝜇2 +(22) +𝜕 ln 𝑃 +𝜕 ln( 𝑓 𝜎8) = +2 𝑓 𝜎8𝜇2 +𝑏𝜎8 + 𝑓 𝜎8𝜇2 . +(23) +Everything is similar for the case of two tracers A and B, except that +10 The value in Seo & Eisenstein (2007) for Σ0 is different since we chose to +work with 𝐷(𝑧) instead of 𝐺(𝑧),and we chose a different 𝜎8 value. +11 We have tried varying 𝐴0 between 0.45 and 0.6, the resulting difference +on 𝜎(𝐻) and 𝜎(𝐷𝐴) w.r.t. 𝐴0 = 0.55 was about 10 per cent, which is not +significant for our work. +MNRAS 000, 1–15 (2022) + +6 +W. d’Assignies et al. +we have 3 sets of parameters: ln[𝑏𝐴(𝑧𝑖)𝜎8(𝑧𝑖)], ln[𝑏𝐵(𝑧𝑖)𝜎8(𝑧𝑖)] +and ln[ 𝑓 (𝑧𝑖)𝜎8(𝑧𝑖)], with additional derivatives: +𝜕 ln 𝑃𝐴 +𝜕 ln(𝑏𝐵𝜎8) = +𝜕 ln 𝑃𝐵 +𝜕 ln(𝑏𝐴𝜎8) = 0, +(24) +𝜕 ln 𝑃𝐴𝐵 +𝜕 ln(𝑏𝑥𝜎8) = +𝑏𝑥𝜎8 +𝑏𝑥𝜎8 + 𝑓 𝜎8𝜇2 +(25) +𝜕 ln 𝑃𝐴𝐵 +𝜕 ln( 𝑓 𝜎8) = 𝑓 𝜎8𝜇2 +� +1 +𝑏𝑎𝜎8 + 𝑓 𝜎8𝜇2 + +1 +𝑏𝑏𝜎8 + 𝑓 𝜎8𝜇2 +� +, +(26) +where 𝑥 = 𝐴, 𝐵. +3.4.3 Non Gaussianity +In most NG model (Matarrese et al. 2000; Maldacena 2003; Dalal +et al. 2008; Desjacques et al. 2009), the Bardeen potential is assumed +to contain a quadratic Gaussian field 𝜙 contribution Φ = 𝜙+ 𝑓NL(𝜙2− +⟨𝜙⟩2). The quadratic term induces a non-Gaussian perturbation to the +bias: 𝑏(𝑘, 𝑧) = 𝑏𝐺(𝑧) + Δ𝑏(𝑘, 𝑧) with: +Δ𝑏 = 3 𝑓NL(𝑏𝐺 − 𝑝)𝛿𝑐 +Ω𝑚 +𝑘2𝑇(𝑘)𝐷(𝑧) +� 𝐻0 +𝑐 +�2 +. +(27) +𝑓NL is the NG coupling to evaluate, 𝑇(𝑘) is the transfer function +(with 𝑘2𝑇(𝑘) normalized to 1 at large scales), and 𝑝 is a number +that theoretically depends on the tracer type, and was introduced to +show deviations from the original model of Dalal et al. (2008). We +take 𝑓NL = 0 as a fiducial value. Since 𝑝 is not well characterised +yet, we will assume 𝑝 = 1 for all the different tracers. The lack of +knowledge on 𝑝 value will be discussed in Section 4. We consider +two parameters for the forecast {𝑏𝑔, 𝑓NL}, with +𝜕𝑃 +𝜕 𝑓NL += 2 𝜕Δ𝑏 +𝜕 𝑓NL +(𝑏𝐺 + Δ𝑏 + 𝑓 𝜇2)𝑃m(𝑘). +(28) +For the generalization to two tracers, there is an additional derivative +𝜕𝑃𝐴𝐵 +𝜕 𝑓NL += 𝜕Δ𝑏𝑎 +𝜕 𝑓NL +(𝑏𝑏 + Δ𝑏𝑏 + 𝑓 𝜇2)𝑃m(𝑘) + {𝑏 ↔ 𝑎} +(29) +Karagiannis et al. (2018); Ferraro & Wilson (2019) have sug- +gested that by using bispectrum in addition to the power spectrum, +one might be able to reach 𝜎( 𝑓NL) ∼ 0.1. The modelling of bis- +pectrum observations is relatively complex, and this high-precision +constraint requires indeed more theoretical development from the +modelling side, as well as better understanding of systematics effects +(Yankelevich & Porciani 2019). Our study is far too simplistic to +address these issues, so we leave it for future studies. +3.4.4 Neutrino mass +The evaluation of the sum of the neutrino mass is an active subject, +both in cosmology and particle physics. We adopt a linear-power- +spectrum Fisher approach. It may seem too simple to the current +standard algorithm, which consists of forecasting with an MCMC +and non-linear corrections in the model (e.g., Sailer et al. 2021). +However, our aim in this paper is to study the change of constraints +w.r.t. that of observational parameters in spectroscopic surveys. Fur- +thermore, even the most advanced approaches do not agree with +each other (an issue discussed in Boyle 2019).Indeed, we expect that +the difference between our power spectra and those provided by a +more complex 𝑀𝜈 will be relatively similar for all surveys and our +conclusion should not be affected. +Table 3. The redshift slices and the mean number density at that redshift +range for DESI tracers. +Tracers +Redshift +𝑛 +Range +(10−4 ℎ3 Mpc−3) +LRG +0.65-1.05 +3.0 +ELG +0.75-1.05 +9.8 +1.05-1.35 +4.5 +1.35-1.65 +1.3 +QSO +1.96-2.43 +0.17 +2.43-3.55 +0.063 +The sum of neutrino mass is denoted as +𝑀𝜈 = +∑︁ +𝜈 +𝑚𝜈. +(30) +We consider a cosmology described by the six standards cosmologi- +cal parameters and 𝑀𝜈, +{𝑝𝑖} = {𝑀𝜈, 𝐻0, 𝜔𝑐, 𝜔𝑏, 𝜏, 𝑛𝑠, 𝐴𝑠}. +(31) +To evaluate the derivative of the power spectrum w.r.t. these param- +eters, we use a four-point estimate (we drop the 𝑘, 𝑧, 𝜇 dependency +here): +𝜕𝑃 +𝜕𝜃 |𝜃fid ∼ −𝑃(𝜃 + 2𝛿𝜃) + 8𝑃(𝜃 + 𝛿𝜃) − 8𝑃(𝜃 − 𝛿𝜃) + 𝑃(𝜃 − 2𝛿𝜃) +12𝛿𝜃 +(32) +We use 𝛿𝜃/𝜃 = 0.01 except for 𝛿𝜏/𝜏 = 0.5 and 𝛿𝑀𝜈/𝑀𝜈 = 0.05 for +numerical reasons. Indeed, we want the power spectrum variation (for +every step) to be much larger than this numerical noise induced by +solving Boltzmann equations. Thus for parameters which only have +a small impact on the matter power spectrum (the neutrino mass 𝑀𝜈, +and the optical depth 𝜏), we take a larger parameter step size. We pay +particular attention to the numerical stability and convergence of the +derivatives w.r.t. those steps. We fix the neutrino hierarchy as it is +degenerated in CAMB, for numerical error purposes. +One subtlety not always mentioned is that the power spectrum +used in 𝑀𝜈 study only includes baryons and cold dark matter, and +its associated growth rate 𝑓 . Indeed neutrino perturbations do not +contribute to the formation of galaxies and haloes (Boyle 2019). We +focus on the neutrino mass parameter and marginalised our Fisher +matrix over all the other parameters. +3.4.5 Additional data-sets for NM +We will add a prior, using Planck CMB constraints on our standard +set of parameters: (𝐹Planck)𝑖 𝑗 = 𝛿𝑖 𝑗/𝜎2 +𝑖 (Section 3.3.3 and Table 2). +We will also consider the possibility of combining our forecast with +DESI which has 3 tracers: ELG LRG and QSO. We split DESI ELG +and QSO into several redshift bins and sum the corresponding Fisher +matrix as +𝐹DESI = 𝐹DESI +LRG + 𝐹DESI +ELG + 𝐹DESI +QSO , +(33) +neglecting the correlation between bins. We also neglect cross- +correlation between tracers. Table 3 summarises the redshift binning +of DESI tracers. Since our goal is not to derive the optimal bound, +but rather to make comparisons among different surveys, we do not +include additional information for BOSS or Euclid for example. The +argument is similar to the one for neutrino mass modelling. +MNRAS 000, 1–15 (2022) + +Next generation spectroscopic survey forecasts +7 +3.5 High-Redshift Survey Modelling +In this section we model a general survey (cf. section 2.5), keeping +engineering and observational parameters as variables. We will then +investigate the survey characteristics to get the best constraint on +parameters 𝑓NL and 𝑀𝜈. +3.5.1 Time to observe a single galaxy +The Signal-to-Noise Ratio (SNR) describes how well a source is +measured by an instrument and is given by the CCD equation, +𝑆/𝑁 ∼ +𝐼(𝑚)𝑆tel𝑡 +√︁𝐼(𝑚)𝑆tel𝑡 + 𝜖sky + 𝜖read +, +(34) +with 𝑆tel the telescope surface, 𝑡 the time of observation and 𝐼(𝑚) +the luminous flux from the source. 𝜖sky and 𝜖read are sky noise and +read-out noise which we can neglect in our study since we are dealing +with the deep sky. Thus we have +𝑆/𝑁 ∼ +√︁ +𝐼(𝑚)𝑆tel𝑡. +(35) +During a telescope operation phase, the time for observing an +object is fixed to the exposure time 𝑡exp, independently of its apparent +magnitude. After a first exposure, it is possible that the spectrum is +still too noisy to identify lines for determining the redshift. That +defines the first-exposure magnitude efficiency 𝑝(𝑚|𝑡exp) which is +the probability of getting a redshift-identifiable spectrum during 𝑡exp. +3.5.2 One Exposure +For a single exposure of 𝑡exp = 1800 𝑠, with a telescope of diameter +∼ 10 𝑚, we assume that +𝑝(𝑚|𝑡exp) = 𝐸(𝑚), +(36) +where 𝐸(𝑚) is the observational efficiency introduced in section 2.2 +for the modelling of MSE efficiency (Percival et al. 2019). This law +depends on the exposure time 𝑡exp, the telescope surface 𝑆tel, the +maximum tolerable redshift error |𝑧meas − 𝑧real| = 0.001(1 + 𝑧real), +and spectra simulated with the MSE exposure time calculator given +theredshift finder PandoraEZ (Fumana&Garilli2012).Furthermore, +this law is based on the assumptions of LBG properties that half of +LBG have a detectable Ly𝛼 emission line, the redshift of the other +half can be estimated with their Ly𝛼 and Ly𝛽 absorption features. +Thus, except for the exposure time, and telescope surface, this law +is not specific to MSE survey, and should in principle apply to other +spectroscopic surveys. +3.5.3 Multiple Exposures +After a first exposure, if the spectrum is still too noisy, a target may +get another exposure 𝑡exp. +We assume that if one observes an object with an additional ex- +posure after a first failure, the probability of measuring its redshift +accurately from the stacked spectrum is +𝑝(𝑚|2𝑡exp) = +√ +2𝑝(𝑚|𝑡exp). +(37) +Indeed, in Appendix A, we show that the efficiency of the ELG +detection increases linearly with the SNR for eBOSS ELGs, up to +a very good precision. Thus, we assume a similar trend for high +redshift LBGs, i.e., the efficiency is proportional to the SNR, and +scales with √𝑡 according to Eq (35). If we go even further and decided +to attribute the third exposure of 2 failures, we assume 𝑝(𝑚|3𝑡exp) = +√ +3𝑝(𝑚|𝑡exp) and so on. +Figure 1. Efficiency law for a single exposure 𝐸 (𝑚) (dashed line) and for +multiple exposures 𝑝(𝑚) (solid line), with a minimal efficiency 𝑝min = 0.7 +(reported on the y-axis on the left) as a function of apparent magnitude 𝑚. +Colours represent the fraction of galaxies of magnitude 𝑚 that have been +observed 𝑛 times: 𝜂𝑛 (𝑚), with � +𝑛 𝜂𝑛 (𝑚) = 1. Values of these fractions +are deduced from the y-axis on the right. For e.g. at 𝑚 = 25, 30 per cent of +galaxies are observed once, 50 per cent are observed twice, and 20 per cent +are observed three times. We have also reported two particular magnitude 𝑚1 +and 𝑚2 by vertical pointed lines. +3.5.4 Correcting Efficiency Bias +The one-exposure efficiency of fainter tracers is significantly smaller +than that of bright objects, and one will underestimate the proportion +of high-magnitude tracers. As a consequence, the sample will be +biased to large 𝑚max. That is why we define a minimal observational +efficiency 𝑝min so that ∀𝑚 < 𝑚max, 𝑝(𝑚) ≥ 𝑝min. In practice we +decide to attribute a second exposure after the first failure to a certain +fraction of object: 𝜂2, with 0 ≤ 𝜂2 ≤ 1 − 𝐸(𝑚) (we only assign a +second observation if the first exposure failed). We also define the +fraction of object observed once 𝜂1, with 𝐸(𝑚) ≤ 𝜂1 ≤ 1. Similarly, +if observing all objects that failed during the first exposure twice does +not compensate for the efficiency decreased, a fraction of galaxies 𝜂3 +might need a third observation. This procedure defines an efficiency +law12: 𝑝(𝑚) = max(𝐸(𝑚), 𝑝min). As an example, in Figure 1 we fix +𝑝min = 0.7 and we plot 𝑝(𝑚) (solid line), as well as 𝐸(𝑚) (dashed +line), as functions of apparent magnitude 𝑚. For 𝑚 < 𝑚1 = 22.8, +we have 𝑝(𝑚) = 𝐸(𝑚); 𝑝(𝑚) = 𝑝min for higher magnitude. In the +same figure, we also represent in colour the different fractions of +galaxy observed 𝑛 times 𝜂𝑛,as a function of apparent magnitude 𝑚, +with � +𝑛 𝜂𝑛(𝑚) = 1. The fraction of objects that have been correctly +observed during the first exposure13 is equal to 𝐸(𝑚) by definition. +For 𝑚 < 𝑚1, 𝐸(𝑚) > 𝑝min, and all the object are observed once: +𝜂1(𝑚 < 𝑚1) = 1. For 𝑚1 < 𝑚 < 𝑚2, to compensate that 𝐸(𝑚) < +𝑝min, a fraction 𝜂2 of galaxy are observed twice (the green shade). +Then at 𝑚 = 𝑚2, we have 𝜂1 = 𝐸(𝑚2)(= 0.37), which means +that all the object whose observation failed during the first exposure +are observed twice (so 63 per cent of the sample). Thus for 𝑚 > +24.6, some galaxies might need a third observation (blue shade) to +compensate the gap between 𝐸(𝑚) and 𝑝min . With our model, for +efficiency 𝑝min < 0.85 it is never necessary to observe some object +four times. +12 We neglect the bias induced by the high efficiency of low-magnitude +sources. +13 which is not 𝜂1(𝑚) +MNRAS 000, 1–15 (2022) + +1.0 +1.0 +0.8 +0.8 +Pmin: +law +0.6 +0.6 +Efficiency +0.4 +Distribution +0.4 +Efficiency for one obs E(m) +Efficiency for multiple obs p(m) +0.2 +0.2 +Fraction observed once n(m) +Fraction observed twice nz(m) +Fraction observed 3 times n3(m) +0.0 +0.0 +22.5 +m1 +23.0 +23.5 +24.0 +24.5 m2 +25.0 +Apparent magnitude m8 +W. d’Assignies et al. +The average time dedicated to the observation of an object of +magnitude 𝑚 (whether it is a success or not) is +⟨𝑡(𝑚)⟩ = +∑︁ +𝑛 +𝜂𝑛(𝑚) · 𝑛 · 𝑡exp. +(38) +The analytical expressions of 𝜂𝑛 and ⟨𝑡(𝑚)⟩ as function of 𝑝min, +𝐸(𝑚) and 𝑚 are given in Appendix B. +3.5.5 Survey duration and measured density +The observed volume is described by three numbers: 𝑧min, Δ𝑧=𝑧max− +𝑧min and 𝑓sky. From all available galaxies within this volume, we will +visit a fraction 𝜂til ≲ 1 of them, and the visited tracer number is: +𝑁vis = 𝜂til +∫ 𝑧max +𝑧min +𝑑𝑧 𝑑𝜒 +𝑑𝑧 4𝜋 𝑓sky𝜒2 +∫ 𝑚max +𝑑𝑚𝜙LBG(𝑧, 𝑚). +(39) +This equation is the integration of the number density over the cosmic +volume. The 𝜂til factor takes into account two effects: +• The fibre collision: two close tracers cannot be observed simulta- +neously by two fibres during a single exposure, +• The tilling: the sky is generally not perfectly covered by the suc- +cession of focal planes, if objects are attributed to different exposures +by maximising the survey efficiency, which is typically the case in +reality. +From this last equation, we further define the density of visited tracer +per magnitude, +𝑑𝑁vis(𝑚) = 4𝜋 𝑓sky𝜂til +∫ 𝑧max +𝑧min +𝑑𝑧 𝑑𝜒 +𝑑𝑧 𝜒2𝜙LBG(𝑧, 𝑚)𝑑𝑚. +(40) +Given the average time to observe a tracer with magnitude 𝑚 ⟨𝑡(𝑚)⟩ +(cf. equation (38)), and the fibre number 𝑁fib, the total observational +time of the survey is +𝛼𝑇sur = +∫ 𝑚max +𝑑𝑁vis(𝑚)⟨𝑡(𝑚)⟩/𝑁fib += 𝜂til +4𝜋 𝑓sky +𝑁fib +∫ 𝑧max +𝑧min +𝑑𝑧 𝑑𝜒 +𝑑𝑧 𝜒2 +∫ 𝑚max +𝑑𝑚⟨𝑡(𝑚)⟩𝜙LBG(𝑚, 𝑧), +(41) +where 𝑇sur is the total survey duration (typically 5 years), and 𝛼 +a coefficient to convert it into observational time. For cosmological +observations, we assume we only observe days with a partial moon14, +21 days every 28 days cycle, within practice only 80 per cent of +these nights that can be dedicated to observation (due to weather or +maintenance issues)15, and between 8 and 10 hours of observation +per night, which correspond to 𝛼 = 7 × 106 yr−1s. In equation (41) +we are assuming that every fibre is dedicated to observation, and will +be observed during every exposure. +The density of observed tracers with a good spectroscopic redshift +is +𝑛obs(𝑧) = 𝜂til +∫ 𝑚max +𝑑𝑚𝑝(𝑚)𝜙LBG(𝑚, 𝑧). +(42) +Since 𝑇sur ∝ 1/𝑁fib, the observation for a 5 years survey with 20k +fibre, would be equivalent to that of a 10-year survey with 10k fibres. +Thus we introduce the ‘power’ parameter 𝑁fib 𝑇sur in fibre-year. +14 Days with a full moon can be thus fully dedicated to other astrophysical +observation. +15 For 5 years this corresponds to ≈ 1100 nights. +3.5.6 Optimisation Pipeline +In the first part of our general survey study, we fix the following +parameters: +• 𝑁fib 𝑇sur = 100, 000 in fibre-year; it corresponds to 5 years of +observation with 20,000 fibres for example; +• 𝜂til = 0.96, is the fraction of available tracer in our cosmic volume +that we will observe; +• 𝑡exp = 1800 𝑠, is the exposure time of the instrument; +• 𝑝min = 0.7, is the minimal efficiency rate; +• 𝑧min = 2, is the minimal redshift of our high-redshift surveys; +• 𝛼 = 7 × 106, as explain in section 3.5.5; +• The telescope size to be 10m; +and find the optimal survey volume described by: +• 𝑓sky, the observed fraction of the sky, +• Δ𝑧 = 𝑧max − 𝑧min, the redshift width. +For arbitrary values of these two parameters, the equations (41) and +(42) will constrain: +• 𝑚max the maximal magnitude of observation, +• 𝑛obs(𝑧) the tracer density, +according to the following scheme, +𝑓sky, Δ𝑧 +Eq. (41) +−→ +𝑚max +Eq. (42) +−→ +𝑛obs(𝑧) +F𝑖 𝑗 +−→ 𝜎( 𝑓NL), 𝜎(𝑀𝜈). +(43) +The numerical procedure to get 𝑚max is explained in Appendix C. +In the second step, we fix the optimal survey volume, and free +𝑁fib 𝑇sur and 𝑝min in a similar procedure as previously, mainly for two +reasons. Firstly, it will show the optimal strategy between correctly +measuring most of the objects visited (high 𝑝min), or having a wider +magnitude range and observing fainter objects that are located in the +high redshift region. Secondly, it will quantify the improvement of +the data over the fibre number, and the duration of the survey. We +will vary 𝑁fib 𝑇sur from 60,000 to 220,000 fibre-years. +3.5.7 Observed Density Prediction +Before moving on to the cosmological parameters we show here some +results of our modelling. We plot Figure 2, the measured tracer den- +sity 𝑛obs as a function of the survey cosmic volume, with parameters +described in Section 3.5.6 (first two steps of equation (43)). +We observe hyperbolic trends, as expected since the product +of the two variables scales as the volume at first order (naive +model). For small volumes ( e.g. 𝑓sky = 0.15 and Δ𝑧 = 0.7), the +maximum magnitude of 25 is reached (as illustrated by horizontal +lines), and all tracers are visited by the end of the 5th year, which +leads to saturation (cf Appendix C). We also plot in Figure 3 the +average density as a function of 𝑁fib 𝑇sur, and the minimal efficiency +𝑝min. For a fixed 𝑁fib 𝑇sur, the density of observed tracers 𝑛obs +will be higher with higher minimum efficiency 𝑝min. Indeed if an +observation of a magnitude 𝑚1 object failed in the first exposure, +then its second-exposure-observation is more likely to be successful, +than a first-exposure-observation of another object of magnitude +𝑚2 +≥ 𝑚116. Nonetheless, with lower minimal efficiency, the +16 Indeed, during the second exposure, the initial SNR value will be the one +obtained at the end of the first exposure. +MNRAS 000, 1–15 (2022) + +Next generation spectroscopic survey forecasts +9 +Figure 2. Observed density 𝑛obs as a function of the redshift width Δ𝑧 +and the sky coverage 𝑓sky, for 𝑁fib 𝑇sur = 100, 000 fibre-years, 𝑧min = 2 and +𝑝min = 0.7. We observe hyperbolic trends, with saturation in the small volume +region (bottom left corner), visible with the 20 and 27 ×10−4ℎ3Mpc−3 lines. +Figure 3. Observed density 𝑛obs as a function of 𝑁fib𝑇sur, and minimal +efficiency 𝑝min, for a maximal cosmic volume: 𝑓sky = 0.31, Δ𝑧 = 3,and +𝑧min = 2). The density increases linearly as a functon of 𝑁fib.𝑇sur, and +increases with the efficiency threshold. +maximal magnitude is higher and one is observing more objects +at higher redshift. Thus it is a priori difficult to know which strat- +egy will provide the best constraints on the cosmological parameters. +With this general survey model, one should be able to reproduce +fiducial tracer properties of future surveys with known observational +parameters such as MegaMapper. It is a large volume survey with +𝑓sky ∼ 0.3 and Δ𝑧 = 3, but with a smaller telescope diameter (∼ 6.5 +m) than the one assumed for 𝐸(𝑚) cf. section 3.5.2. Motivated by +the dependence of the CCD equation (35) on +√︁ +𝑆tel, we correct the +efficiency law 𝐸(𝑚) by a factor √︁𝑆Mega/𝑆MSE and require a min- +imum efficiency 𝑝min = 0.5 (Section 2.1). The predicted number +density is 𝑛obs ∼ 2.6 × 10−4 ℎ3 Mpc−3, very close to the expecta- +tion of MegaMapper which is 2.5 × 10−4 ℎ3 Mpc−3. The maximum +magnitude imposed by the model is 24.6, which is in agreement with +MegaMapper’s 24.5 maximal magnitude. It should be noted that this +agreement is remarkable since our modelling is independent of any +MegaMapper settings. +4 RESULTS +In this section, we first present the BAO, RSD, NG, and NM forecasts +for future surveys such as MUST, MegaMapper, and MSE. We then +investigate the NM and NG accuracy given the survey properties with +our model introduced in Section 3.5. We deduce, independently of +any planned survey, the optimal observation strategies, and the limits +when measuring these parameters. +4.1 BAO and RSD Constraints +We summarised in Table 4 the constraints on BAO and RSD param- +eters. We separate MegaMapper and MSE redshift range into several +bins and present the forecast for every bin. We also provide con- +straints in the full redshift range in the last row for each survey. We +derived the constraints for 8 different settings of MUST, and for 3 dif- +ferent settings of an NTL survey, that mimics the observation of the +WST survey (depending on the fibre number). Finally, we consider +a combination of two MegaMapper-like surveys, each located in one +hemisphere. This study shows the power of combined independent +surveys in providing better cosmological constraints. +From this table we can conclude that: +(i) MegaMapper and MUST (20,000 fibres) have relatively similar +accuracy. In theory, they will improve the constraints on BAO and +RSD by a factor of 10 w.r.t. eBOSS constraints (Zhao et al. 2016), and +a factor between 2 and 3 w.r.t. DESI constraints (DESI Collaboration +et al. 2016b). MSE forecasts are not as good as these two because of +its smaller fibre number. +(ii) For MUST, with the same number of fibre, the one- +tracer (LBGX) case gives better constraints than the two tracers +(LBGX+LAE) one. It is mainly because LAEs need a long exposure +time, which decreases the total number of observed LAEs and thus +increases the overall noise, as illustrated by the 𝑛𝑃 parameter values. +(iii) The combination of two independent LBG surveys gives cos- +mological constraints similar to those of an NTL survey with 40,000 +fibres, a factor of +√ +2 smaller than those of a single survey like +MegaMapper. As the two surveys are independent, this corresponds +to the constraints on the BAO and RSD from two independent mea- +surements, combined. +(iv) 100,000 fibres in the NTL survey represent the upper limit for +these high-redshift LBG surveys since it corresponds to the measure- +ment of all galaxies up to 𝑚max = 25. +(v) All the parameters are very well constrained down to the sub-per- +cent level in every survey. This is unrealistic as the real observational +constraints are likely to be dominated by systematics, and not any- +more by the number density of tracers and the cosmic volume. +As the sample size will not bring improvement for cosmological +measurements of future surveys, we do not optimise it in the following +study. +MNRAS 000, 1–15 (2022) + +3.5 +35 +27 +3.0 +[s-d-0] +20 +2.5 +15 +K +2.0 +10 +10 +. +1.5 - +nobs +6 +15 +1.0 - +4 +20 +27 +0.5 +1 +0.10 +0.15 +0.20 +0.25 +0.30 +fsky9 +8 +220 +8 +200 +[s-d4-O] +180 +7 +160 - +6 +140 +5 +120 +nobs +4 +100 +3 +80 +60 +2 +0.50 +0.55 +0.60 +0.65 +0.70 +0.75 +0.80 +0.85 +Minimal success rate pmin10 +W. d’Assignies et al. +Table 4. The predicted 68 per cent Confidence Level (CL) error of the BAO distances and RSD parameters for various survey. We use separate redshift bins for +MegaMapper and MSE, and we also show the forecast using tracers at the whole redshift range in the last row of each survey. We present the forecast for MUST +at 2 < 𝑧 < 4 and for NTL and combined surveys at 2 < 𝑧 < 5. LBGX×LAE denotes a multi-tracer constraint with LBGX and LAE. +BAO and RSD forecast +Fibre +number +Sky area +deg2 +Tracer +Redshift +Number density +10−4 ℎ3 Mpc−3 +𝑛𝑃(0.14, 0.6) +𝜎(𝐷𝐴)/𝐷𝐴 +(%) +𝜎(𝐻)/𝐻 +(%) +𝜎(𝑏𝜎8)/𝑏𝜎8 +(%) +𝜎( 𝑓 𝜎8)/ 𝑓 𝜎8 +(%) +MegaMapper +20k +14k +LBG +2 < 𝑧 < 2.5 +7.9 +1.7 +0.32 +1.0 +0.066 +0.85 +2.5 < 𝑧 < 3 +3.6 +0.68 +0.35 +1.0 +0.073 +1.0 +3 < 𝑧 < 4 +1.1 +0.19 +0.40 +1.0 +0.085 +1.3 +4 < 𝑧 < 5 +0.7 +0.11 +0.45 +0.99 +0.094 +1.7 +2 < 𝑧 < 5 +2.5 +0.5 +0.18 +0.57 +0.039 +0.52 +MSE +4.3k +10k +ELG +1.6 < 𝑧 < 2.4 +1.8 +0.34 +0.79 +2.3 +0.17 +1.4 +LBG +2.4 < 𝑧 < 2.8 +2.3 +0.51 +0.51 +1.5 +0.10 +1.7 +LBG +2.8 < 𝑧 < 3.2 +1.1 +0.22 +0.66 +1.8 +0.14 +2.3 +LBG +3.2 < 𝑧 < 4 +0.43 +0.08 +0.79 +1.9 +0.17 +2.8 +LBG +2.4 < 𝑧 < 4 +1.1 +0.28 +0.28 +0.64 +0.078 +1.2 +MUST (different settings) +20k +15k +LBGX +2 < 𝑧 < 4 +8.9 +2.3 +0.13 +0.41 +0.026 +0.44 +20k +15k +LBGX×LAE +2 < 𝑧 < 4 +4.2–0.84 +1.1 +0.15 +0.49 +0.030–0.14 +0.47 +10k +15k +LBGX +2 < 𝑧 < 4 +5.0 +1.3 +0.15 +0.48 +0.030 +0.52 +10k +15k +LBGX×LAE +2 < 𝑧 < 4 +2.1–0.42 +0.58 +0.19 +0.62 +0.039–0.21 +0.60 +10k +9k +LBG +2 < 𝑧 < 4 +7.1 +2.1 +0.18 +0.56 +0.035 +0.60 +10k +9k +LBGX×LAE +2 < 𝑧 < 4 +3.5–0.71 +0.97 +0.21 +0.68 +0.041–0.21 +0.66 +NTL (WST like survey) +20k +15k +LBG +2 < 𝑧 < 5 +2.5 +0.51 +0.15 +0.48 +0.035 +0.54 +40k +15k +LBG +2 < 𝑧 < 5 +4.9 +0.99 +0.12 +0.38 +0.030 +0.40 +100k +15k +LBG +2 < 𝑧 < 5 +13 +1.9 +0.099 +0.29 +0.027 +0.26 +Combination of two MegaMapper-like surveys +20k +28k +LBG +2 < 𝑧 < 5 +2.5 +0.5 +0.13 +0.40 +0.028 +0.37 +4.2 Non-Gaussianity and Neutrino Mass +4.2.1 Redshift Binning +As mentioned in Section 3.3.4, there are several ways to deal with the +large cosmic volume. In Section 4.1, as their parameters are redshift- +dependent, it is natural to provide forecasts in small redshift bins. +In contrast, non-Gaussianity and neutrino mass are independent of +the redshift. Thus splitting the samples into multiple redshift bins +does not bring more information for their measurements in principle. +However, as the number density of tracers is much higher at low +redshift than that at high redshift, analysing the total redshift range +is not necessarily the best option. Because it may overestimate the +noise that scales as 1/𝑛 at low redshift. +We, therefore, investigate the optimal binning, by exploring two +ways to separate our resdhift interval [𝑧min, 𝑧max] into 𝑁bins: +• define bins with the same comoving volume 𝑉: a ‘fixed volume +approach’; +• define bins with the same number of targets 𝑁gal. In this case, the +low-redshift bins have volumes smaller than those at higher-redshift +bins. +A model including a continuous dependence of the redshift and +density is more promising, but we leave this more complex option +for a future study. We calculate 𝜎( 𝑓NL) and 𝜎(𝑀𝜈) for 𝑁bins = +1, 2, ..., 10, and report their relation in Figure 4. We fix the tracer +density and cosmic volume for these forecasts to be the fiducial +MegaMapper ones.We check that the optimal binning scheme is +independent of these parameters. +For non-Gaussianity, the fixed volume approach provides the best +constraints with 2 or 3 bins. Indeed 𝑓NL describes a large-scale +phenomenon, and reducing bin sizes increases the low integration +limit value 𝑘min, which leads to a rapid increase in variances for a +large number of bins. The drop in 𝑓NL accuracy from one bin to two +bins is due to the reduction in noise at the low redshift bin which +compensates for its small volume17. +For neutrino mass measurement, contrary to 𝑓NL, there is no clear +pattern, and therefore no number of bin has to be favoured. Indeed, +except for one bin, the results fluctuate by less than 5 per cent from +the mean value, which is not large in view of our method. This is a +confirmation that this forecast does not depend on large scales, but +on small ones as theoretically expected. +In the following studies, we adopt a volume-fixed approach with 3 +bins for both forecasts. +4.2.2 Small scale choice +We do the forecast with two different 𝑘max values: 0.1 and 0.3 +ℎMpc−1. The 𝑘max reached in the future survey analysis may be +between these two, or even larger. However, for 𝑘max = 0.3 ℎMpc−1, +our linear approach is already insufficient, and a proper future anal- +ysis would have to take into account non-linear correction as what +Sailer et al. (2021); Boyle & Schmidt (2021) have done. This will +mainly affect the neutrino forecast since non-Gaussianity is a large- +scale phenomenon, whereas the power spectrum is more affected by +the massive neutrino contribution at large 𝑘. +17 This difference is smaller in the case of the fixed number of tracer. Because +in that case, the volume of the low-redshift bin is small, and we have a small +bin with low noise, and a large bin with high noise at high redshift. +MNRAS 000, 1–15 (2022) + +Next generation spectroscopic survey forecasts +11 +Figure 4. NM (the left panel) and NG (the right panel) forecast w.r.t. the bin number, for two binning schemes (𝑉 -based in blue dots and 𝑁gal-based in pink +dots). For NM we also include the average value for the fixed volume approach (solid lines) and delimit the ±5 per cent region around it by the two dashed lines. +For NG, the approach with fixed volume is clearly the optimal one with 2 or 3 bins, whereas, for NM, there is not the best choice. +4.2.3 Forecasts +In Table 5 we present forecasts for the same surveys as Table 4, but +for the neutrino mass and non-Gaussianity. For the neutrino forecast, +we provide constraints with only a prior from Planck CMB, and those +with a Planck CMB prior and measurements from DESI, as described +in Section 3.4.5. +For non-Gaussianity, we may reach with MegaMapper-like sur- +vey 𝜎( 𝑓NL) ∼ 1.2, which is a factor of 4 better than the DESI +forecast (DESI Collaboration et al. 2016b). In particular, the con- +straints depend mainly on the volume of the survey. For example, +with the number of fibres reduced by a factor 2 (which propagates +to the tracer density), constraints of MUST vary from 1.6 to 1.7 +for 𝑘max = 0.1ℎMpc−1; whereas reducing the observation window +from 15,000 to 9,000 deg2 leads to a weaker constraint from 1.7 to +2.1 for 10,000 fibres (despite a higher tracer density for the reduced +window). 𝑘max values have little impact on 𝑓NL, which is within +expectation since it describes a large scale deviation from classi- +cal Gaussian prediction. A combination of two surveys (coverage of +28,000 square degrees) results in a smaller 𝜎( 𝑓NL), than the NTL +with 100,000 fibres, which shows that the measurement of this pa- +rameter is limited by the volume and not the tracer density. That is +also why, MSE constrains are much weaker than the ones of the other +surveys. +As already mentioned in Section 3.4.3, we assume a value 𝑝 = 1, +but indeed different values are theoretically possible. The NG ampli- +tude is degenerated with 𝑝, as it scales as 𝑓NL𝑝, as well as the vari- +ance We illustrate this via the 𝜎( 𝑓NL) forecast for a MegaMapper-like +survey as a function of the 𝑝 values in Figure 5. In particular, for rel- +atively reasonable values like 𝑝 = 1.6, the variance on NG amplitude +is already significantly degraded (∼ 30 per cent w.r.t. 𝑝 = 1) . Thus, +in order to have a reliable measurement of the 𝑓NL parameter (value +and uncertainty), one needs robust theoretical priors, provided by +simulations for example. One can find a complete discussion about +this issue in Barreira (2022). +The neutrino forecast depends strongly on the value of the small +scale limit 𝑘max (by a factor of two in most cases). With a CMB +prior, 𝜎(𝑀𝜈) values are relatively similar for most surveys of the +order of 0.04eV for 𝑘max = 0.1, and 0.025eV for 𝑘max = 0.3. With +our approach, we find an improvement of about 50 per cent w.r.t. +CMB+DESI forecast only. Combining both, it is possible to reach +0.015 eV. However forecasting this parameter with our naive approach +aims to show global trends, and values should not be taken in the +strict sense for the reasons mentioned in section 3.4.4, but rather as +indicators. Unlike the NG, the NM measurement of the combination +of two surveys is worse than the NTL 100,000 fibres. Thus this +parameter is more impacted by noise. +The neutrino forecast varies slightly with the characteristics of the +survey. Indeed, the limiting factor for the accuracy of this parameter +is the weak prior of some parameters from the standard model, rather +than the survey settings. As discussed in Boyle (2019); Boyle & +Schmidt (2021), it is due to our limited knowledge of 𝐴𝑠, which is +linked to 𝜏 because of a strong CMB degeneracy between these two +parameters 18. Table 2 of Liu et al. (2016) summarises forecasts of +the expected future knowledge on ln(𝐴𝑠) and 𝜏 with future 21cm +surveys. We illustrate this issue in Figure 6, by plotting 𝜎(𝑀𝜈) for +a MegaMapper-like survey, as a function of the prior on ln 𝐴𝑠, with +values credible according to Table 2 of Liu et al. (2016). For example, +a difference in the 𝜎(ln 𝐴𝑠) prior from 0.014 to 0.005 leads to an +improvement from 0.024eV to 0.018eV, in the optimistic scenario. +Thus improving the knowledge on 𝐴𝑠 is the best way to detect the +neutrino mass hierarchy as large as a 5𝜎 confidence level. For other +cosmological parameters, we find that the dependencies are much +weaker, and we simply take the CMB prior values. +4.3 Optimised NG surveys +We now study the optimized survey characteristics to minimise +𝜎( 𝑓NL), with the survey model described in Section 3.5. The fixed- +volume redshift bin number is given by +𝑁bin = +��� +��� +1 +if Δ𝑧 < 1 +2 +if 1 ≤ Δ𝑧 < 2 +3 +if 2 ≤ Δ𝑧. +(44) +We start this analysis by varying the cosmic volume in Figure 7, +18 Indeed 𝐴𝑠 is not directly constrained but 𝜏 and 𝐴𝑠 exp(−2𝜏) are. +MNRAS 000, 1–15 (2022) + +2.00 +0.051 +binswithfixedv +average valuefor fixed y +bins with fixed Ngal +average+-5% +1.95 +binswithfixedv +0.049 +bins with fixed Ngal +1.90 +0.047 +[ev +fNL) +1.85 +6 +6 +0.045 +1.80 +0.043 +1.75 +1.70 +0.041 +1 +3 +5 +7 +9 +1 +3 +5 +7 +9 +Number of bins +Numberofbins12 +W. d’Assignies et al. +Table 5. The same Table as as Table 4 but for 𝑓NL and 𝑀𝜈. The forecast is based on a high mode integration limit 𝑘max = 0.1 − 0.3 ℎMpc−1, which corresponds +to a pessimistic and an optimistic scenario (that is why we provide two values for each parameter). The left column for 𝑀𝜈 is the forecast with a CMB prior, +whereas the right column is the result of a combination of DESI and a CMB prior. +Non Gaussianity and neutrino mass forecast for 𝑘max = 0.1–0.3 ℎMpc−1 +Fibre +number +Sky area +deg2 +Tracer +Redshift +Number density +10−4 ℎ3 Mpc−3 +𝜎( 𝑓NL) +𝜎(𝑀𝜈) in 10−2eV +Planck +𝜎(𝑀𝜈) in 10−2eV +Planck and DESI +MegaMapper +20k +14k +LBG +2 < 𝑧 < 5 +2.5 +1.3–1.2 +4.2–2.4 +3.3–1.6 +MSE +4.3k +10k +ELG +1.6 < 𝑧 < 2.4 +1.8 +8.9–8.1 +5.0–4.5 +4.1–3.0 +LBG +2.4 < 𝑧 < 4 +1.1 +2.8–2.5 +5.1–4.3 +3.7–2.1 +MUST (different settings) +20k +15k +LBGX +2 < 𝑧 < 4 +8.9 +1.6–1.4 +4.2–2.4 +3.1–1.6 +20k +15k +LBGX×LAE +2 < 𝑧 < 4 +4.2–0.84 +1.7–1.5 +4.4–2.9 +3.2–1.8 +10k +15k +LBGX +2 < 𝑧 < 4 +5.0 +1.7–1.5 +4.3–2.8 +3.2–1.7 +10k +15k +LBGX×LAE +2 < 𝑧 < 4 +2.1–0.42 +1.9–1.8 +4.5–3.4 +3.3–2.0 +10k +9k +LBG +2 < 𝑧 < 4 +7.1 +2.1–1.8 +4.5–2.8 +3.2–1.8 +10k +9k +LBGX×LAE +2 < 𝑧 < 4 +3.5–0.71 +2.2–2.0 +4.6–3.4 +3.3–2.0 +NTL (WST like survey) +20k +15k +LBG +2 < 𝑧 < 5 +2.5 +1.4–1.4 +4.1–2.1 +3.3–1.5 +40k +15k +LBG +2 < 𝑧 < 5 +4.9 +1.1–1.0 +3.6–1.7 +3.0–1.3 +100k +15k +LBG +2 < 𝑧 < 5 +13 +1.0–0.90 +3.2–1.3 +2.7–1.0 +Combination of two MegaMapper-like surveys +20k +28k +LBG +2 < 𝑧 < 5 +2.5 +0.92–0.85 +3.7–1.8 +3.0–1.3 +Figure 5. The predicted constraints on non-Gaussianity for a high-redshift +LBG survey w.r.t. the fiducial p value. +for a survey with 100,000 fibre-years, and a minimum efficiency of +0.7. The improvement of the precision goes hand in hand with the +increase of the volume. Thus for the study of 𝑓NL, one should always +favour a larger survey volume. It is not beyond expectation since +𝑓NL is a parameter related to the large scale. In addition to that, it +depended little on tracer density as we show in Table 5. This remains +true when varying the survey strategy parameters, such as the survey +duration and the minimum efficiency. +Then we set a maximum volume among all the surveys and vary +the fibre-year parameter 𝑁fib𝑇sur, as well as the minimum efficiency +𝑝min. The forecast is reported in Figure 8. The NM accuracy depends +weakly on the efficiency threshold but still deprives the high ones. +Thus it is optimal to maximise the tracer average density and to +restrict the maximum magnitude of observation, as it reduces the +noise. However, high-redshift bins are populated with fainter galaxies +Figure 6. The predicted neutrino mass constraints for a MegaMapper-like +survey w.r.t. the ln 𝐴𝑠 prior. For the other cosmological parameters, prior +values are taken from Planck CMB measurements. +and limiting the maximal magnitude also limits the tracer density at +high redshift. This mutual restriction may explain why the tendency +is weak. There is also a second contribution: we consider that the +effective bias is the bias of tracers with the maximum magnitude +𝑏(𝑧, 𝑚) ≈ 𝑏LBG(𝑧, 𝑚max) (Section 3.2). Since bias decreases with +magnitude, we underestimate the small-magnitude tracer bias. Since +the NG effect scales as Δ𝑏 = 𝑓NL(𝑏−𝑝), it also reduces the amplitude +of the NG. +Furthermore, multiplying the fibre-year parameter by four only +increases the accuracy by 50 per cent. The main limitation neither +comes from 𝑁fib 𝑇sur nor 𝑘max, but the cosmic volume of the survey +𝑉sur. However, since with a longer survey time, one is able to go to +a higher magnitude, it underestimates the variance of NG amplitude +because the bias decreases as describe previously. +MNRAS 000, 1–15 (2022) + +kmax = 0.1h/MpcC +kmax = 0.3h/Mpc +3.0 +2.5- +o(fnL) +2.0 +1.5 - +1.0 +-1.0 +0.5 +1.0 +1.5 +2.0 +2.5 +0.0 +3.0 +—0.5 +dMega+Planck, kmax = 0.1h/Mpc +Mega+Planck, kmax = 0.3h/Mpc +4.0 +3.5 +3.0 +2.5 +2.0 +0.006 +0.008 +0.010 +0.012 +0.014 +Prior o(lnAs)Next generation spectroscopic survey forecasts +13 +Figure 7. Non-Gaussianity (left panel) and massive neutrino (right panel) general survey forecasts as function of the cosmic volume describes by the fraction +of the sky 𝑓sky and redshift width Δ𝑧, for 100,000 fibre-year survey, 𝑧min = 2 and 𝑝min = 0.7. +Figure 8. Non-Gaussianity (left panel) and massive neutrino (right panel) general survey forecasts as function of the efficiency threshold 𝑝min and 𝑁fib.𝑇sur, for +maximal cosmic volume: 𝑓sky = 0.31, Δ𝑧 = 3, and 𝑧min = 2. +4.4 Optimised NM surveys +We adopt the same binning choice as for the NG study in Section 4.3. +We first show the results for different survey volumes in Figure 7. The +variance of 𝑀𝜈 is less correlated with volume than for NG, especially +for 𝑓sky between 0.2 and 0.3, and Δ𝑧 between 2.5 and 3.5. The optimal +region seems to be around Δ𝑧 = 3 and 𝑓sky = 0.3, but the variation +of 𝜎(𝑀𝜈) in the nearby if these parameters is comparable with the +accuracy of our naive model (∼ 10 per cent). Thus these results show +a relatively good region, rather than a clear-cut result, with a trade-off +between the volume and noise. Furthermore, the density of LBGs at +high redshift is low, and including high redshift galaxies does not +help constrain NM given the large shot noise, beyond Δ𝑧 = 3. +In accordance with the previously observed optimal region, we set +the volume at 𝑓sky = 0.31 and Δ𝑧 = 3. In Figure 8 we present the +forecast varying the fibre-year parameter and the minimum efficiency. +The NM results are quite different from those of NG. Indeed, it is +clear that the NM constraint improves with an increasing fibre-year. +The error is divided by 2 when the fibre-year parameter is multiplied +by 4, which means that the shot noise level plays an important role. +Moreover, the best is to work with a minimum efficiency, in order to +observe objects with a low luminosity, located at high redshift. We +have verified that for slightly different cosmic volumes close to the +maximal one, these results are unchanged. +5 CONCLUSION +As successors of DESI, MegaMapper, MSE, and MUST will become +the largest spectroscopic surveys in the world in the next decades. +MNRAS 000, 1–15 (2022) + +3.5 +12.00 +5.00 +4.00 +3.0 +6.00 +3.0 +2 +2.5 +3.00 +2.5 +2.20 +-2 +2.50 + in[10′ +2.00 +2.00 +2.50 +2.20 +o(Mv) +1.5 - +1.50 +1.5 - +00 +2.10 +1.0 +1.25 +1.0 +2.05 +4.00 +0.5 +-1.10 +0.5 +2.00 +0.10 +0.15 +0.20 +0.25 +0.30 +0.10 +0.15 +0.20 +0.25 +0.30 +fsky +fsky1.80 +3.60 +200 +05 +200 +1.65 +3.10 + fibre.year] +175 +1.50 +175 +150 +150 +1.30 +-1.90 +2.10 +in[10 +in[103 +(fnL)) +125 +125 +[103 +1.20 +1.20 +100 +100 +1.90 +α(Mv)1 +1.10 +2.10 +75 +1.30 +75 +1.75 +2.50 +1.05 +50 +50 +1.65 +1.50 +3.10 +25 +1.00 +25. +1.50 +0.50 +0.55 +0.600.650.700.75 +0.800.85 +0.50 +0.600.650.700.75 +0.80 +0.85 +Minimal efficiency pmin +Minimal efficiency Pmin14 +W. d’Assignies et al. +They will map the Universe in the redshift range 2 < 𝑧 < 5 using +multiple tracers and thereby improve our knowledge of cosmological +distances and the structure growth (through BAO and RSD), as well +as constrain the initial conditions of the Universe and measure the +sum of the mass of neutrinos. It is also possible that these surveys +make observations for 1 (N+) +Zn2VN3 + Zn (evaporation) (17). AIMD simulations conducted in this work show that a Zn2VN3 +structure containing an extra (unevaporated) Zn atom can be formed at 180 K within ~1.3 ps +(Movie 2). Therefore, by controlling the Zn evaporation rate and the ionized nitrogen rate it can +be possible to utilize chemical vapor deposition for 2D Zn2VN3 synthesis. During the +manufacturing, the thickness of sputter-deposited thin films is increased with sputtering power (for +constant deposition time). For example, the thickness of the Mo-incorporated Cu2ZnSnS4 absorber +layer increased with Mo co-sputtering power, as shown in the cross-section scanning transmission +electron microscope study (25). More recently, the ternary metal-zinc-nitride thin films of about +235 nm thickness were demonstrated experimentally and implemented in a photodetector device + +(26). Proper choice of a substrate is an important step for production of 2D Zn2VN3. For instance, +bulk Zn2VN3 deposited on glass and sapphire substrates appears to be phase-pure, while the +epitaxial stabilization on Al2O3 (0001) increases its crystallinity and texture (25). + + +Figure 1. (a) Schematic representation of Zn2VN3 transformation from bulk to 2D, and the 2D Zn2VN3 unit cell +combined with ELF. The Zn, V, and N atoms are colored in gray, red and violet, respectively. (b) Phonon dispersion +curves of 2D Zn2VN3. + +Figure 2a shows the band structure of 2D Zn2VN3 obtained using the Heyd–Scuseria– +Ernzerhof (HSE06) exchange-correlation functional. For the comparison, the band structure of 2D +Zn2VN3 obtained via the Perdew-Burke-Ernzerhof (PBE) functional under the generalized +gradient approximation is plotted in Figure S3a. Based on the HSE06 calculations, it is found that +2D Zn2VN3 is a semiconductor with an indirect band gap of 2.75 eV and a direct band gap of 2.85 +eV. Both gaps are slightly higher than in bulk Zn2VN3 (17). Similarly to its bulk counterpart, 2D +Zn2VN3 is a p-type semiconductor with the Fermi level located slightly above the valence band +edge. For the indirect gap, the conduction band minimum (CBM) is located in the vicinity of the +Γ point, while the valence band maximum (VBM) is located at the S point. In the case of the direct +gap, both CBM and VBM are located at the S point. The partial density of states (PDOS) of 2D +Zn2VN3 (Figure S3b) demonstrates that d states of V atoms give the main contribution to the CBM, +while the VBM is formed by p states of N atoms. From the band structure and PDOS plots the +CBM localization is noticed. Such localization of states around the CBM is commonly observed +in materials due to cation disorder (27, 28). Localization on the d states of V atoms has been found +in bulk Zn2VN3 (17). It is also supposed that the origin of the CBM localization in 2D Zn2VN3, as +reflected in the PDOS plot (Figure S3), is due to the cation disordering. +The optical response of a material at a given frequency can be determined via its +frequency-dependent complex dielectric function that can be defined as a function of incident +photon energy having real and imaginary parts (29). This function describes the process of light +propagation through the material. Light absorption of a material at a given frequency ω can be + +(a) +(b) +25 +Frequency (THz) +20 +a = b = 5.63 A +15 +(001) +10 +5 +0 +X +S +Y +YT +zrsimply realized by the positive value of the real part of the dielectric function. Therefore, the real +part refers to the ability of a material to store the electric energy. The imaginary part of the +dielectric function refers to the ohmic resistance of the material. For instance, the imaginary part +of the dielectric function of a pure dielectric has a zero value. The real and imaginary parts of the +complex dielectric function for 2D Zn2VN3 are presented in Figure 2b. The real part of the +dielectric function for the incident electromagnetic field normal to the 2D Zn2VN3 surface is +positive in the considered region of the electromagnetic spectra from 0 to 16 eV. This suggests +that photons propagate through the monolayer. The static dielectric function, which represents the +dielectric response of the material to a static electric field, is found to be 1.28. The real dielectric +constant has a considerable peak at 4.9 eV with the maximum value of 1.55, and it exhibits a few +peaks at higher energies. The imaginary part of the dielectric function shows that 2D Zn2VN3 +absorbs light in the visible and ultraviolet regions. Considering the electric and optical properties +of the predicted 2D Zn2VN3, it can be regarded as a transparent material for the visible lights and +a useful shielding material in ultraviolet region. +Figure 2c shows the diagram of the 2D Zn2VN3 work function in comparison with other +2D materials and bulk metals possessing the highest known work function values. The work +function of 2D Zn2VN3 is high, 5.27 eV, comparable to that of borophene (30). The high work +function of 2D Zn2VN3 can be attributed to the nature of its atomic states around the Fermi level +consisting of not only the out-of-plane pz states but also the in-plane p hybridized states (Figure +S4). Thus, the ionization of 2D Zn2VN3 requires significant energy and is comparable to that of, +for example, borophene. + + +(a) +c) +4. +Pt +2D B +2 +2.85 eV +2D Zn,VN +Energy (eV) +Ni +2D PC +0 +2D BCP +2D P +2 +2D C +Ti +-4 +3.6 4.0 4.4 4.8 5.2 5.6 6.0 +Work function value (eV) +X +s +Y +YT +ZI +(b) +2.1 +1.2 + xx direction + xx direction +1.8 + zz direction +(0) +. - zz direction +0.9 +(0) +1.5 +Imaginary +1.2 +0.6. +eal +0.9 +R +0.3. +0.6. +0.3. +0.0. +0 +2 +4 +6 +810 12 14 16 +0 +2 +4. +68 10 12 14 16 +Energy (eV) +Energy (eV)Figure 2. (a) Band structure of 2D Zn2VN3 calculated by the HSE approach. (b) Real and +imaginary parts of dielectric function versus energy for 2D Zn2VN3. (c) Comparison of the work +function of 2D Zn2VN3 with those of common 2D materials and bulk metals. + +Mazdziarz formulated the mechanical stability of 2D materials acquired in rectangular +lattices as follows (31): +1 +2 (𝐶11+ 𝐶22 + √4𝐶12 +2 − (𝐶11 − 𝐶22)2 ) > 0 +1 +2 (𝐶11+ 𝐶22 − √4𝐶12 +2 − (𝐶11 − 𝐶22)2 ) > 0 (2) + 𝐶66 > 0 +The above presented criteria are met in the case of 2D Zn2VN3 confirming its mechanical stability. +The calculated elastic constants Cij for 2D Zn2VN3 are collected in Table S1. +Mechanical properties of 2D Zn2VN3 such as Young’s modulus, Poisson’s ratio and shear +modulus are also considered. Young’s modulus of 2D Zn2VN3 in the x and y directions is +calculated as (32,33): +𝐸[𝑥]= +𝐶11𝐶22−𝐶12 +2 +𝐶11 +, and 𝐸[𝑦]= +𝐶11𝐶22−𝐶12 +2 +𝐶22 + (3) +Shear modulus of 2D Zn2VN3 is calculated as (30,31): +G = C66 (4) +Poisson’s ratio of 2D Zn2VN3 in the x and y directions is calculated as (30,31): +𝑣[𝑥]= +𝐶12 +𝐶11, and 𝑣[𝑦]= +𝐶12 +𝐶22 (5) +Figure 3 presents the spatial dependence of Young’s modulus, shear modulus and +Poisson’s ratio for 2D Zn2VN3. These parameters are almost direction independent. The values of +Young’s modulus of 2D Zn2VN3 in the x and y directions are found to be ~99.7 N/m. Shear +modulus of 2D Zn2VN3 is found to be 37.6 N/m. Poisson’s ratio of 2D Zn2VN3 in both x and y +directions is found to be 0.375. For comparison, 2D Zn2VN3 has lower stiffness and higher +elasticity relative to graphene (34). + + +Figure 3. Spatial dependencies of (a) Young’s modulus (N/m), (b) shear modulus (N/m), and (c) +Poisson’s ratio for 2D Zn2VN3. + +(a)Young'smodulusinxy-plane +(b)Shearmodulusinxy-plane +()Poisson'sratioinxy-plane +100 +40 +0.4 +50 +20 +0.2 +0 +0 +0.0 +-50 +-20 +-0.2 +-100 +40 +-0.4 +-100 +-50 +0 +50 +100 +-40 +-20 +0 +20 +40 +-0.4 +-0.2 +0.0 +0.2 +0.42D Zn2VN3 can be switched from a direct band gap semiconductor to an indirect band gap +semiconductor and its band gap size can be increased/decreased up to ~50% via strain engineering. +Although the PBE functional underestimates the band gap compared to the HSE functional, PBE +(Figure S3a) and HSE (Figure 2a) display similar trends in the band alignment in the band structure +plots. Thus, the effect of strain engineering on the band structure of 2D Zn2VN3 can be studied +using the PBE approach. Figure 4a shows changes in the band gap of 2D Zn2VN3 under strain +applied along the surface plane. The applied strain is in the range from -10% (compressive strain) +to 10% (tensile strain) that can be realized experimentally (35-40). The band gap of 2D Zn2VN3 +decreases rapidly from 1.57 eV to 0.76 eV when the compressive strain increases from 0 to 10%. +As the VBM shifts from the S point to the Γ point (Figure S5), an indirect-direct band gap transition +is observed in 2D Zn2VN3 under the compressive strain higher than 6%. A rapid decrease of the +2D Zn2VN3 band gap from 1.57 eV to 1.07 eV is observed with the increase of the tensile strain +from 0% to 8%. The applied tensile strain in the range from 8% to 10% leads to a slight increase +of the band gap from 1.07 eV to 1.19 eV (indirect band gap) and 1.22 eV (direct band gap) (Figure +S5). +As 2D materials always contain surface defects that may affect their structure stability and +performance (41-45), it is necessary to investigate point defects in 2D Zn2VN3. Several common +point defects are found to be stable in 2D Zn2VN3: a single vacancy of a N atom (SVN); a single +vacancy of a Zn atom (SVZn); a single vacancy of a V atom (SVV); in-plane and out-of-plane +double vacancies of one V atom and one N atom (DVV-N); and in-plane and out-of-plane double +vacancies of one Zn atom and one N atom (DVZn-N). SV defects can be created by removing one +Zn, V or N atom from the 2D Zn2VN3 surface. The DVV-N and DVZn-N defects in the 2D Zn2VN3 +surface are formed when V and N or Zn and N atoms are removed from the surface (in-plane DV) +or from the plane perpendicular to the surface (out-of-plane DV) of 2D Zn2VN3. The stability of +point defects in 2D Zn2VN3 is evaluated based on their formation energy, Eform. Eform of the +considered stable defects in 2D Zn2VN3 are collected in Table 1. According to Table 1, the SVZn +and SVN defects have the lowest Eform of 4.27 eV and 5.27 eV, respectively. The out-of-plane +DVZn-N, in-plane DVZn-N, SVV, in-plane DVV-N, and out-of-plane DVV-N defects have comparably +high formation energies of 7.83 eV, 8.54 eV, 10.25 eV, 10.96 eV, and 11.92 eV, respectively. Eform +of SVs in 2D Zn2VN3 is comparable to that in graphene (~7.50 eV) (46) and MoS2 (2.10-6.20 eV) +(47), while the formation of the DVs in 2D Zn2VN3 is less favorable than in graphene (~8.0 eV) +(46) and MoS2 (~4.0 eV) (47). +Table 1. Eform (eV) of point defects in 2D Zn2VN3. +SVN +SVZn +SVV +in-plane +DVV-N +out-of-plane +DVV-N +in-plane +DVZn-N +out-of-plane +DVZn-N +5.27 +4.27 +10.25 +10.96 +11.92 +8.54 +7.83 +Proper identification and classification of defects is a key capability of atomically resolved +scanning tunneling microscopy (STM) (48,49). To facilitate experimental needs it is possible to +utilize DFT simulated STM images for the differentiation of point defects in 2D Zn2VN3. The +atomic structures and corresponding STM images of the pristine and defect-containing 2D Zn2VN3 +are presented in Figures S6a-d and S7a-d. The STM images of 2D Zn2VN3 containing the +considered defects correlate with the corresponding atomic structures, and the defects are easily +identified. The STM image in Figure S6b (right panel) shows that the SVN defect in 2D Zn2VN3 +can be identified by the triangle formed with two bright spots and one dark spot, arising from two +Zn atoms and one V atom, with one N atom missing inside the triangle. Similarly, according to + +Figures S6c and d (right panels), the SVV and SVZn defects in 2D Zn2VN3 appear as triangles +formed of three small dark spots characterizing three N atoms, with V or Zn atoms missing inside +the triangles. The in-plane DVV-N defect in 2D Zn2VN3 is presented in Figure S7a. Here, two semi- +hexagons, formed of two bright spots (Zn atoms) and two dark spots (N atoms) each, have two +missing atoms (one V atom and one N atom) at their border. The out-of-plane DVV-N defect in 2D +Zn2VN3 (Figure S7b) is characterized by missing V and Zn atoms in-between five big bright spots +(Zn atoms) and one dark spots (N atom) slightly shifted from their positions compared to the +perfect case. The in-plane DVZn-N defect in 2D Zn2VN3 is seen in Figure S7c as two missing atoms +(Zn and N) inside a square formed of three dark spots (two N atoms and one V atom). The out-of- +plane DVZn-N defect in 2D Zn2VN3 (Figure S7d) may be identified as missing atoms within a +triangle formed of two bright spots and one dark spot due to two Zn atoms and one V atoms. This +pattern is similar to the STM image the SVN defect. To deeper understand the changes in the +electronic structure of 2D Zn2VN3 induced by the defects, the density of states resolved in space, +known as local density of states (LDOS), calculated for peripheral atoms in the defect core and for +atoms far from the defect core, are shown in Figures S8 and S9. It is found that the defects induce +significant changes in the electronic structure of 2D Zn2VN3. Such changes facilitate defect +identification via photoemission spectroscopy techniques. +Figure 4b presents the temperature-depended surface density of point defects in 2D +Zn2VN3. The SVN and SVZn defects possess significantly higher surface densities compared to the +other defects found to be stable in 2D Zn2VN3. The surface density of the SVV, out-of-plane DVZn- +N, in-plane DVZn-N, in-plane DVV-N, and out-of-plane DVV-N defects in 2D Zn2VN3 is slightly lower +than that in graphene (50) and MoS2 (51), making their formation less energetically favorable. On +the other hand, the surface density of the SVN and SVZn defects in 2D Zn2VN3 is comparable to +that of the SV defects in graphene (50) and MoS2 (51). +Structural degradation of 2D materials can also be caused by their interaction with the +humid environment, particularly, with H2O and O2 molecules contained in the air. Therefore, the +interaction of the H2O and O2 molecules with the 2D Zn2VN3 surface is further evaluated. As it +has been shown previously, for most of 2D materials, oxidation is the most dangerous process that +can lead to degradation of their surface (52-54), while H2O-saturated surfaces can exhibit higher +stability as such saturation prevents the oxidation (55,56). It is found that the adsorption energy, +Eads, of O2 on the 2D Zn2VN3 surface is as high as -0.14 eV, while Eads of H2O (-0.49 eV) is ~3.5 +lower (Figure S10). Hence, the adsorption of H2O on 2D Zn2VN3 is more favorable compared to +O2. It should be noted, that according to the LDOS plots (Figure S11) remarkable changes in H2O +and O2 molecular states upon their interaction with the 2D Zn2VN3 surface are observed. +Therefore, the kinetic analysis of H2O and O2 splitting on the 2D Zn2VN3 surface is further +conducted. Figure 4c presents the result from the CI-NEB calculation. The energy barrier, Eb, for +the H2O and O2 molecule dissociation on 2D Zn2VN3 is found to be 2.82 eV and 2.04 eV, +respectively. Considering the obtained high values of Eb for the dissociation of H2O and O2 on the +2D Zn2VN3 surface, which are comparable to those of the H2O and O2 molecule dissociation on +InSe (57), and based of the Eads analysis, it can be concluded that 2D Zn2VN3 possesses high +structural integrity under environmental conditions. A discussion on the potential application of +2D Zn2VN3 to water splitting is presented in Supporting Information, and the calculated VBM and +CBM positions of 2D Zn2VN3 with the redox potential of H2O and oxidation levels are shown in +Figure S12. + + +Figure 4. (a) Band gap of 2D Zn2VN3 as a function of strain. (b) Surface density of point defects +in 2D Zn2VN3 as a function of temperature. (c) Activation barriers for H2O and O2 molecule +splitting on 2D Zn2VN3. +In summary, a new material, a 2D analog of the recently predicted and synthesized ternary +nitride semiconductor Zn2VN3 (17), is predicted computationally. The fabrication of 2D Zn2VN3 +is highly possible due to is relatively low exfoliation energy of 105 meV/Å2. It is also proposed +that the chemical vapor deposition approach can be utilized for the synthesis of 2D Zn2VN3 +similarly to the bulk Zn2VN3 (17) and Cu2ZnSnS4 thin film (25) manufacturing. The environmental +stability of 2D Zn2VN3 should be high due to resistivity of its surface to oxygen and defect +formation. 2D Zn2VN3 has an indirect band gap of 2.75 eV, which can be tuned up to 50% under +applied stains. 2D Zn2VN3 possesses a high work function of 5.27 eV, absorbs visible and +ultraviolet light, and exhibits moderate mechanical properties. All this makes 2D Zn2VN3 a good +candidate for application in opto-electronic and straintronic devices. +Computational Methods +The spin-polarized first-principles calculations were performed within the framework of +density functional theory as implemented in the plane-wave the Vienna Ab initio Simulation +Package (VASP) (58). The Perdew-Burke-Ernzerhof functional (PBE) (59) under the generalized +gradient approximation and the HSE06 hybrid exchange-correlation functional (60) were used. +Dispersive interactions were included using the van der Waals corrected functional (61). The +geometry optimization was stopped once the atomic forces and total energy values were smaller +than 10−4 eV/Å and 10−8 eV, respectively. The plane-wave cut-off energy was set to 520 eV. The +periodic boundary conditions were applied for the two in-plane transverse directions. A vacuum +space of 25 Å was introduced to the direction perpendicular to the surface plane. +The electron localization function (ELF) was calculated to obtain the distribution of +electrons in 2D Zn2VN3. The Phonopy code associated with VASP was used for the simulation of +the phonon spectrum (62). The 3×3×1 supercell of 2D Zn2VN3 was used to perform the +calculations based on finite displacement methods with the atomic displacement distance of 0.01 +Å. The dielectric function of 2D Zn2VN3 was calculated based on the TD-HSE06 approach (63). +Ab initio molecular dynamics simulations controlled by the Nose–Hoover thermostat were +performed for 5 ps at the temperature of 300 K and with a time step of 1.0 fs (64). + +(a) +(b) +Bandgap size (eV) +(c) +0.8 1.0 1.2 1.4 1.6 +1E-20 +3 +Q'H . +1E-40" +-10 +:02 +indirect +1E-60- +-8 +8 +2 +1F-80 +9- +density +(eV) +1L-100 +-4 +direct +1E-120 +Energy +-2 +1E-140 +Areal +Strain +0 +1E-160 +SVn +0 +2 +.SVzn +1E-180 +indirect +.SVy +4 +1E-200 +6 +1E-220 +DV +8 +0 +200 +400 +600800 +Reaction path +10 +direct +: indirect +Temperature (K)The stress-strain relation (65) was used to calculate the components of the stiffness matrix +from which the Young’s modulus, shear modulus, and Poisson’s ratio were obtained and +directional dependencies of these quantities were defined using the ELATE software for analysis +of elastic tensors (66). +To consider point defects in 2D Zn2VN3 a supercell composed of 3×3×1 unit cells (36 Zn, +18 V and 54 N atoms) was created to avoid non-physical interactions between periodic images +while keeping affordable computational demand. Under such conditions, the concentration of MV +defects was 0.93% and the concentration of DV defects was 1.85%. The Tersoff-Hamann approach +was implemented to simulate Scanning Tunneling Microscope (STM) images of pure and defect- +containing 2D Zn2VN3 (67). +The formation energy Eform of point defects in 2D Zn2VN3 was calculated as +Eform = Edefect − Epure + NZn·EZn + NV·EV+ NN·EN, (6) +where Edefect and Epure are the total energies of pure and defect-containing 2D Zn2VN3, EZn, EV and +EN are the energies of single Zn, V and N atoms, and NZn, NV·and NN· correspond to the number +of the removed Zn, V and N atoms. +The surface density of defects Nd in 2D Zn2VN3, at a finite temperature, was calculated +according to the Arrhenius equation: +𝑁𝑑 = 𝑁𝑝ure 𝑒−𝐸𝑓orm/(𝑘𝐵𝑇), (7) +where Npure is the surface density of atoms in pure 2D Zn2VN3, kB is the Boltzmann constant, and +T is temperature. Note that only defects presented at the surface were shown. +The climbing image–nudged elastic band (CI-NEB) method was used to obtain the +reaction pathway of the H2O and O2 molecules on the 2D Zn2VN3 surface (68). + +ASSOCIATED CONTENT + +Notes +The authors declare no competing financial interest. + +ACKNOWLEDGMENTS +S.A.Sh. is thankful for the funding provided by the Russian Science Foundation (grant No. 21-71- +10129). E.A.K. acknowledges the support of Grant NSh-4320.2022.1.2 of the President of the +Russian Federation for state support of young Russian scientists - candidates of sciences and +doctors of sciences. А.А.K. is grateful for financial support to the Ministry of Science and Higher +Education of the Russian Federation within the framework of the state task of the USATU (No. +075-03-2022-318/1) of the youth research laboratory «Metals and Alloys under Extreme Impacts». +Authors acknowledge Peter the Great Saint Petersburg Polytechnic University Supercomputer +Center “Polytechnic” and Joint Supercomputer Center of the Russian Academy of Sciences for +computational resources. A.A.K. grateful Dr. Siarhei Zhuk for useful discussions on the synthesis + +and application of Zn–V–N compounds. O.V.P. acknowledges support of the US National Science +Foundation (grant CHE-1900510). + +REFERENCES +(1) Campi, D.; Kumari, S.; Marzari, N. Prediction of phonon-mediated superconductivity with +high critical temperature in the two-dimensional topological semimetal W2N3. Nano Lett. +2021, 21, 3435−3442. +(2) Yang, Y.; Fang, W.H.; Benderskii, A.; Long, R.; Prezhdo, O. V. Strain controls charge +carrier lifetimes in monolayer WSe2: Ab initio time domain analysis. J. Phys. Chem. Lett. +2019, 10(24), 7732−7739. +(3) Manzeli, S.; Ovchinnikov, D.; Pasquier, D.; Yazyev, O. V.; Kis, A. 2D transition metal +dichalcogenides. Nat. Rev. Mater. 2017, 2, 17033. +(4) Gibertini, M.; Koperski, M.; Morpurgo, A. F.; Novoselov, K. S. Magnetic 2D materials +and heterostructures. Nat. Nanotechnol. 2019, 14, 408−419. +(5) Jiang, C.; Li, Q.; Huang, J.; Bi, S.; Ji, R.; Guo, Q. Single-layer MoS2 mechanical resonant +piezo-sensors with high mass sensitivity. ACS Appl. Mater. Interfaces 2020, 12(37), +41991−41998. +(6) Šiškins, M.; Lee, M.; Alijani, F.; van Blankenstein, M. R.; Davidovikj, D.; van der Zant, +H. S. J.; Steeneken, P. G. Highly anisotropic mechanical and optical properties of 2D +layered As2S3 membranes. ACS Nano 2019, 13(9), 10845–10851. +(7) Kou, L.; Ma, Y.; Sun, Z.; Heine, T.; Chen, C. Two-dimensional topological insulators: +Progress and prospects. J. Phys. Chem. Lett. 2017, 8, 1905−1919. +(8) Friedrich, R.; Ghorbani-Asl, M.; Curtarolo, S.; Krasheninnikov, A. V. Data-driven quest +for two-dimensional non-van der Waals materials. Nano Lett. 2022, 22(3), 989−997. +(9) Novoselov, K. S.; Jiang, D.; Schedin, F.; Booth, T. J.; Khotkevich, V. V.; Morozov, S. V.; +Geim, A. K. Two-dimensional atomic crystals. Proc. Natl. Acad. Sci. U. S. A. 2005, 102, +10451−10453. +(10) +Nicolosi, V.; Chhowalla, M.; Kanatzidis, M. G.; Strano, M. S.; Coleman, J. N. +Liquid exfoliation of layered materials. Science 2013, 340, 1226419. +(11) +Lemme, M. C., Akinwande, D., Huyghebaert, C.; Stampfer, C. 2D materials for +future heterogeneous electronics. Nat Commun. 2022, 13, 1392. +(12) +Carvalho, A.; Trevisanutto, P. E.; Taioli, S.; Castro Neto, A. H. Computational +methods for 2D materials modelling. Rep. Prog. Phys. 2021, 84, 106501. +(13) +Mellaerts, S.; Meng, R.; Menghini, M.; Afanasiev, V.; Seo, J. W.; Houssa, M.; +Locquet, J. P. Two dimensional V2O3 and its experimental feasibility as robust room- +temperature magnetic Chern insulator. npj 2D Mater. Appl. 2021, 5, 65. +(14) +Rybkovskiy, D. V.; Kvashnin, A.G.; Kvashnina, Y. A.; Oganov, A. R. Structure, +stability, and mechanical properties of boron-rich Mo–B phases: A computational study. +J. Phys. Chem. Lett. 2020, 11(7), 2393−2401. +(15) +Kvashnin, A.G.; Tantardini, C.; Zakaryan, H. A.; Kvashnina, Y. A.; Oganov, A. R. +Computational search for new W–Mo–B compounds. Chem. Mater. 2020, 32(16), +7028−7035. +(16) +Tikhomirova, K. A.; Tantardini, C.; Sukhanova, E. V.; Popov, Z. I.; Evlashin, S. +A.; Tarkhov, M. A.; Zhdanov, V. L.; Dudin, A. A.; Oganov, A. R.; Kvashnin, D. G.; +Kvashnin, A. G. Exotic two-dimensional structure: The first case of hexagonal NaCl. J. +Phys. Chem. Lett. 2020, 11(10), 3821−3827. + +(17) +Zhuk, S.; Kistanov, A. A.; Boehme, S. C.; Ott, N.; La Mattina, F.; Stiefel, M.; +Kovalenko, M. V.; Siol, S. Synthesis and characterization of the ternary nitride +semiconductor Zn2VN3: Theoretical prediction, combinatorial screening, and epitaxial +stabilization. Chem. Mater. 2021, 33(23), 9306–9316. +(18) +Fu, F.; Li, J.; Yang, C. J. T.; Liang, H.; Faes, A. Jeangros, Q.; Ballif, C.; Hou, Y. +Monolithic perovskite-silicon tandem solar cells: From the Lab to Fab? Adv. Mater. 2022, +34, 2106540. +(19) +Kistanov, A. A.; Shcherbinin, S. A.; Botella, R.; Davletshin, A.; Cao, W. Family of +two-dimensional transition metal dichlorides: Fundamental properties, structural defects, +and environmental stability. J. Phys. Chem. Lett. 2022, 13(9), 2165−2172. +(20) +Sun, W.; Li, Y.; Wang, B.; Jiang, X.; Katsnelson, M. I.; Korzhavyi, P.; Erikssona, +O.; Di Marco, I. A new 2D monolayer BiXene, M2C (M = Mo, Tc, Os). Nanoscale 2016, +8, 15753−15762. +(21) +Jung, J. H.; Park, C.-H.; Ihm, J. A. Rigorous method of calculating exfoliation +energies from first principles. Nano Lett. 2018, 18, 2759−2765. +(22) +Mohanty, B.; Wei, Y.; Ghorbani-Asl, M.; Krasheninnikov, A. V.; Rajput, P.; Jena, +B. K. Revealing the defect-dominated oxygen evolution activity of hematene. J. Mater. +Chem. A 2020, 8, 6709−6716. +(23) +Mounet, N.; Gibertini, M.; Schwaller, P.; Campi, D.; Merkys, A.; Marrazzo, A.; +Sohier, T.; Castelli, I. E.; Cepellotti, A.; Pizzi, G.; Marzari, N. Two-dimensional materials +from high-throughput computational exfoliation of experimentally known compounds. +Nat. Nanotechnol. 2018, 13, 246−252. +(24) +Choudhary, K.; Kalish, I.; Beams, R.; Tavazza, F. Highthroughput identification +and characterization of two-dimensional materials using density functional theory. Sci. +Rep. 2017, 7, 5179. +(25) +Zhuk, S.; Wong, T. K. S.; Hadke, S. S.; Lie, S.; Guchhait, A.; Gao, Y. Wong, H. +L.; Cheng, S.; Wang, X.; Dalapati, G. K. Molybdenum incorporated Cu1.69ZnSnS4 kesterite +photovoltaic devices with bilayer microstructure and tunable optical-electronic properties. +Sol. Energy 2019, 194, 777−787. +(26) +George, J.; Vikraman, H. K.; Reji, R. P.; Ghanashyam, K. M.; Jayaraman, S. V.; +Sivalingam, Y.; Narasimha. K. M. S. R. Novel ternary nitride thin film-based self-powered, +broad spectral responsive photodetector with a high detectivity for weak light. Adv. Mater. +Technol. 2023, 8, 2200645. +(27) +Cordell, J. J.; Tucker, G. J.; Tamboli, A. Bandgap analysis and carrier localization +in cation-disordered ZnGeN2. APL Mater. 2022, 10, 011112. +(28) +Kistanov, A. A.; Cao, W.; Huttula, M.; Khadiullin, S. Kh.; Korznikova, E. A.; +Smirnov, A.; Wang, X.; Zhuk, S. Impact of various dopant elements on the electronic +structure of Cu2ZnSnS4 (CZTS) thin films: a DFT study. CrystEngComm, 2020, 22, +5786−5791. +(29) +Naseria, M.; Hoat, D.M. Prediction of a new 2D B2CO monolayer from density +functional theory. Comput. Mater. Sci. 2021, 186, 109975. +(30) +Kistanov, A. A.; Cai, Y.; Zhou, K.; Srikanth, N.; Dmitriev, S. V.; Zhang, Y. W. +Exploring the charge localization and band gap opening of borophene: A first-principles +study. Nanoscale, 2018, 10, 1403−1410. +(31) +Maździarz, M. Comment on ‘The Computational 2D Materials Database: high- +throughput modeling and discovery of atomically thin crystals’. 2D Mater. 2019, 6, +048001. + +(32) +Andrew, R. C.; Mapasha, R. E.; Ukpong, A. M.; Chetty, N. Mechanical properties +of graphene and boronitrene. Phys. Rev. B 2012, 85, 125428. +(33) +Zhao, T.; Zhang, Sh.; Guo, Y.; Wang, Q. TiC2: a new two-dimensional sheet +beyond MXenes. Nanoscale 2016, 8, 233−242. +(34) +Cadelano, E.; Palla, P. L.; Giordano, S.; Colombo, L. Elastic properties of +hydrogenated graphene. Phys. Rev. B 2010, 82, 235414. +(35) +Blundo, E.; Cappelluti, E.; Felici, M.; Pettinari, G.; Polimeni, A. Strain-tuning of +the electronic, optical, and vibrational properties of two-dimensional crystals featured. +Appl. Phys. Rev. 2021, 8, 021318. +(36) +Ryu, Y. K.; Carrascoso, F.; López-Nebreda, R.; Agraït, N.; Frisenda, R.; +Castellanos-Gomez, A.; Microheater actuators as a versatile platform for strain engineering +in 2D materials. Nano Lett. 2020, 20(7), 5339–5345. +(37) +Shcherbinin, S. A.; Zhou, K.; Dmitriev, S. V.; Korznikova, E. A.; Davletshin, A. +R.; Kistanov, A. A. Two-dimensional black phosphorus carbide: Rippling and formation +of nanotubes. J. Phys. Chem. C 2020, 124(18), 10235–10243. +(38) +Kripalani, D. R.; Sun, P. P.; Lin, P.; Xue, M.; Zhou, K. Strain-driven superplasticity +of ultrathin tin (II) oxide films and the modulation of their electronic properties: A first- +principles study. Phys. Rev. B 2019, 100(21), 214112. +(39) +Wang, Y.; Lei, J.; Bai, L.; Zhou, K.; Liu, Z. Effects of tensile strain rate and grain +size on the mechanical properties of nanocrystalline T-carbon. Comput. Mater. Sci. 2019, +170, 109188. +(40) +Chernozatonskii, L. A.; Artyukh, A. A.; Kvashnin, A. G.; Kvashnin D. G. +Mechanical engineering effect in electronic and optical properties of graphene +nanomeshes. ACS Appl. Mater. Interfaces 2020, 12(49), 55189–55194. +(41) +Perez, C. M.; Ghosh, D.; Prezhdo, O.; Nie, W.; Tretiak, S.; Neukirch, A. Point +defects in two-dimensional ruddlesden–popper perovskites explored with ab initio +calculations. J. Phys. Chem. Lett. 2022, 13, 5213–5219. +(42) +Wu, Y.; Chu, W.; Vasenko, A. S.; Prezhdo O. V. Common defects accelerate charge +carrier recombination in CsSnI3 without creating mid-gap states. J. Phys. Chem. Lett. 2021, +12 (36), 8699–8705. +(43) +Kretschmer, S.; Ghaderzadeh, S.; Facsko, S.; Krasheninnikov, A. V. Threshold ion +energies for creating defects in 2D materials from first-principles calculations: Chemical +interactions are important. J. Phys. Chem. Lett. 2022, 13(2), 514–519. +(44) +Komsa, H. P.; Krasheninnikov, A. V. Physics and theory of defects in 2D materials: +the role of reduced dimensionality, Materials Today, Defects in Two-Dimensional +Materials, Elsevier, 2022, Pages 7-41. +(45) +Xia, J.; Gu, H.; Liang, C.; Cai, Y.; Xing. G. Manipulation of band alignment in two- +dimensional vertical WSe2/BA2PbI4 ruddlesden–popper perovskite heterojunctions via +defect engineering. J. Phys. Chem. Lett. 2022, 13, 4579–4588. +(46) +Krasheninnikov, A. V.; Lehtinen, P. O.; Foster, A. S.; Nieminen, R. M. Bending +the rules: Contrasting vacancy energetics and migration in graphite and carbon nanotubes. +Chem. Phys. Lett. 2006, 418, 132–136. +(47) +Hong, J.; Hu, Z.; Probert, M.; Li, K.; Lv, D.; Yang, X.; Gu, L.; Mao, N.; Feng, Q.; +Xie, L.; Zhang, J.; Wu, D.; Zhang, Z.; Jin, C.; Ji, W.; Zhang, X.; Yuan, J.; Zhang, Z. +Exploring atomic defects in molybdenum disulphide monolayers. Nat Commun. 2015, 6, +6293. + +(48) +Dan, J.; Zhao, X.; Ning, S.; Lu, J.; Loh, K. P.; He, Q.; Loh, N. D.; Pennycook, S. J. +Learning motifs and their hierarchies in atomic resolution microscopy. Sci. Adv. 2022, 8, +15. +(49) +Zhussupbekov, K.; Ansari, L.; McManus, J. B.; Zhussupbekova, A.; Shvets, I. V.; +Duesberg, G. S.; Hurley, P. K.; Gity, F.; Coileáin, C. Ó; McEvoy, N. Imaging and +identification of point defects in PtTe2. npj 2D Mater Appl 2021, 5, 14. +(50) +Hu, W.; Yang, J. Defects in phosphorene. J. Phys. Chem. C 2015, 119, +20474−20480. +(51) +Jeong, H. Y.; Lee, S. Y.; Ly, T. H.; Han, G. H.; Kim, H.; Nam, H.; Jiong, Z.; Shin, +B. G.; Yun, S. J.; Kim, J.; Kim, U. J; Hwang, S.; Lee, Y. H. Visualizing point defects in +transition-metal dichalcogenides using optical microscopy. ACS Nano 2016, 10, 770−777. +(52) +Lloret, V.; Rivero-Crespo, M. A.; Vidal-Moya, J. A.; Wild, S.; Domenech-Carbo, +A.; Heller, B. S. J.; Shin, S.; Steinruck, H. P.; Maier, F.; Hauke, F.; Varela, M.; Hirsch, A.; +Leyva-Perez, A.; Abellan, G. Few layer 2D pnictogens catalyze the alkylation of soft +nucleophiles with esters. Nat. Commun. 2019, 10, 509. +(53) +Gibaja, C.; Rodriguez-San-Miguel, D.; Ares, P.; Glmez-Herrero, J.; Varela, M.; +Gillen, R.; Maultzsch, J.; Hauke, F.; Hirsch, A.; Abellan, G.; Zamora, F. Few-layer +antimonene by liquid-phase exfoliation. Angew. Chem. Int. Ed. 2016, 55, 14345–14349. +(54) +Kistanov, A. A.; Khadiullin, S. Kh.; Zhou, K.; Dmitriev, S. V.; Korznikova E. A. +Environmental stability of bismuthene: Oxidation mechanism and structural stability of 2D +pnictogens. J. Mater. Chem. C 2019, 7, 9195–9202. +(55) +Botella, R.; Chiter, F.; Costa, D.; Nakashima, S.; Lefevre, G. Influence of +hydration/dehydration on adsorbed molecules: Case of phthalate on goethite. Colloids Surf. +A Physicochem. Eng. 2021, 625, 126872. +(56) +Gardeh, M. Gh.; Kistanov, A. A.; Nguyen, H.; Manzano, H.; Cao, W.; Kinnunen, +P. Exploring mechanisms of hydration and carbonation of MgO and Mg(OH)2 in reactive +magnesium oxide-based cements. J. Phys. Chem. C 2022, 126(14), 6196–6206. +(57) +Ma, D.; Li, T.; Yuan, D.; He, C.; Lu, Z.; Lu, Z.; Yang, Z.; Wang, Y. The role of the +intrinsic Se and In vacancies in the interaction of O2 and H2O molecules with the InSe +monolayer. Appl. Surf. Sci. 2018, 434, 215–227. +(58) +Kresse, G.; Furthmuller, J. Efficient iterative schemes for ab initio total-energy +calculations using a plane-wave basis set. Phys. Rev. B 1996, 54, 11169. +(59) +Perdew, J. P.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made +simple. Phys. Rev. Lett. 1996, 77, 3865−3868. +(60) +Heyd, J.; Scuseria, G. E.; Ernzerhof, M. Hybrid functionals based on a screened +coulomb potential. J. Chem. Phys. 2003, 118, 8207. +(61) +Becke, A. D. Density-functional exchange-energy approximation with correct +asymptotic behavior. Phys. Rev. A: At., Mol., Opt. Phys. 1988, 38, 3098. +(62) +Togo, A.; Tanaka, I. First-principles phonon calculations in materials science. Scr. +Mater. 2015, 108, 1−5. +(63) +Ketolainen, T.; Macháčová, N.; Karlický, F. Optical gaps and excitonic properties +of 2D materials by hybrid time-dependent density functional theory: Evidences for +monolayers and prospects for van der Waals heterostructures. J. Chem. Theory Comput. +2020, 16, 5876–5883. +(64) +Nosé, S. A unified formulation of the constant temperature molecular dynamics +methods. J. Chem. Phys. 1984, 81(1), 511–519. + +(65) +Le Page, Y.; Saxe, P. Symmetry-general least-squares extraction of elastic data for +strained materials from ab initio calculations of stress. Phys. Rev. B: Condens. Matter +Mater. Phys. 2001, 63, 174103. +(66) +Gaillac, R.; Pullumbi, P.; Coudert, F.-C. ELATE: An open-source online +application for analysis and visualization of elastic tensors. J. Phys.: Condens. Matter. +2016, 28, 275201. +(67) +Tersoff, J., Hamann, D. R. Theory of the scanning tunneling microscope. Phys. Rev. +B 1985, 31, 805. +(68) +Henkelman, G.; Uberuaga, B. P.; Jónsson, H. A climbing image nudged elastic band +method for finding saddle points and minimum energy paths. J. Chem. Phys. 2000, +113(22), 9901–9904. + + + + + + + + + + + + + + + + + + + + + + + + + +Supporting Information +Section 1. Conventional cell of 2D Zn2VN3 + +Figure S1. (a) Conventional cell of 2D Zn2VN3. (b) Energy fluctuation of the 2D Zn2VN3 system +from AIMD calculations at 300 K. + + +Section 2. Binding energy of 2D Zn2VN3 + +Figure S2. Binding energy required for exfoliation of (a) 2D Zn2VN3 and (b) graphene. + + +a) +(b) +-676 +Energy (eV) +a=9.70 A +-678 +680 +-682 +0 +1 +2 +3 +4 +5 +Time (ps) +b=5.74 A(a) +(b) +eV +3 +12 +0Section 3. GGA band sructure and PDOS of 2D Zn2VN3 + + +Figure S3. (a) Band structure and (b) PDOS of 2D Zn2VN3 obtained via the Perdew-Burke- +Ernzerhof (PBE) exchange-correlation functional. + + + + +Figure S4. PDOS of 2D Zn2VN3. + + + + +(a) +(b) +4 +Total DOS +d-states Zn +p-states N +d-states V +2 +Energy (eV) +1.74 eV +-2 +.4 +X +S +Y +VT +Z +0 +10 +20 +PDOS (eV/states)PDOS (eV/states) +N (px) +N (py) +20 +N (pz) +N (s) +10 +4 +0 +Energy (eV) +Figure S5. Band structure of 2D Zn2VN3 under compressive (upper row) and tensile (lower row) +strain obtained via the PBE approach. + + + + + +Section 4. The calculated elastic constants Cij for 2D Zn2VN3 + +Table S1. The calculated elastic constants Cij for 2D Zn2VN3. +C11, N/m +117 +C22, N/m +117 +C12, N/m +44.5 +C44, N/m +37.6 + + + +-10% +-8% +-6% +-4% +-2% +0% +Energy (eV) +.74ev +1.29 eV +1.57ev +0 +X +s +V +YTZ +11 +X +s +YTZ +X +s +Y +r +X +s +Z +X +10% +8% +6% +4% +2% +0% +Energy (eV) +.74 el +.22 ev +1.19 eV +.42 ev +0 +2 +X +s +YT +Z +T +X +s +7 +X +S +x +s +X +s +Z +X +Y +ZSection 5. Atomic stricture, STM images and LDOSs of defect-containing 2D Zn2VN3 + + +Figure S6. Atomic stricture (left) and STM image (right) of (a) pure, (b) SVN, (c) SVV, and (d) +SVZn defect-containing 2D Zn2VN3. + + +(a) +(b) +(c) +(d) +Figure S7. Atomic stricture (left) and STM image (right) of (a) in-plane DVV_N, (b) out-of-plane +DVV_N, (c) in-plane DVZn_N, and (d) out-of-plane DVZn_N defect-containing 2D Zn2VN3. + + + + + + +(a) +(b) +(c) +(d)To deeper understand the changes in the electronic structure of 2D Zn2VN3 induced by the defects, +the density of states resolved in space, known as local density of states (LDOS), is calculated for +peripheral atoms in the defect core and for atoms far from the defect core, as shown in Figures S8 +and S9. The states introduced by the SVN defect are mainly contributed by the V atom (Figure +S8a). The changes in the CBM and the VBM depicted in the LDOS plot for the defect (Figure +S8b) arise mainly from the N1 and N2 atoms and N3, respectively. The defect-induced states in the +vicinity of the VBM in the LDOS plot of the SVZn system (Figure S8c) mainly originate from the +N1, N2 and N3 atoms. In the case of the in-plane DVV_N defect in 2D Zn2VN3 (Figure S9a), mainly +the N1 and N2 atoms surrounding the defect have partially occupied/unoccupied states contributing +into the valence/conduction bands of 2D Zn2VN3. The N1 atom (Figure S9b) is responsible for the +in-gap states appearing in 2D Zn2VN3 containing the out-of-plane DVV_N defect. In the case of the +in-plane DVZn_N in 2D Zn2VN3, in-gap states appear mainly due to the V atom located in the +vicinity of the defect core (Figure S9c). In-gap states in 2D Zn2VN3, appearing due to the out-of- +plane DVZn_N, are formed by the V, Zn2, and N2 atoms, as shown in Figure S9d. + +Figure S8. Atomic stricture (left) and LDOS (right) of (a) SVN, (b) SVV, and (c) SVZn defect- +containing 2D Zn2VN3. + + +20 +(a) +LDOS(states/eV) +0 +Total Dos +Zn. +Zn2 +-20 +-2 +0 +2 +Energy (eV) +(b) +20 +Total DOS +N, +N. +LDOS(states/eV) +N. +0 +-20 +-2 +-1 +0 +1 +2 +Energy (eV) +(c) +20 +Total DOS +N1 +N +N2 +LDOS(states/eV) +N. +0 +N +20 +-2 +0 +2 +Energy (eV) +Figure S9. Atomic stricture (left) and LDOS (right) of (a) in-plane DVV_N, (b) out-of-plane +DVV_N, (c) in-plane DVZn_N, and (d) out-of-plane DVZn_N defect-containing 2D Zn2VN3. + + +20 +(a) +LDOS(states/eV) +Zn2 +N +Total DOS +Zn. +Znz +N, +N. +-20. +-2 +.1 +0 +2 +Energy (eV) +20 +(b) +LDOS(states/eV) +N +Total DOS +Zn +N +V2 +Zn +N +-20 +-2 +0 +2 +Energy (eV) +20 +LDOS(states/eV) +0 +Total DOS +21 +N. +"N +V +Zn +-20. +-2 +-1 +0 +2 +Energy (eV) +20 +Total DOS +(d) +N. +LDOS(states/eV) +Zn2 +N3 +0 +-20. +-2 +0 +2 +Energy (eV)Section 6. H2O and O2 on 2D Zn2VN3 +According to Figure S10a, the most favorable adsorption position of H2O on 2D Zn2VN3 +corresponds to the position of the molecule above the Zn-V bond on the side of the hexagon at the +distance of 2.01 Å above the surface of 2D Zn2VN3, and the lowest Eads of H2O on 2D Zn2VN3 is +-0.49 eV. Figure S10b shows the most favorable adsorption position of O2 on 2D Zn2VN3. In that +case, O2 is in the ring of the hexagon at the distance of 2.73 Å, and the lowest Eads of O2 on 2D +Zn2VN3 is -0.14 eV. + +Figure S10. Atomic configurations of (a) H2O and (b) O2 on 2D Zn2VN3. + + + + + + + +Eads= -0.49 eV eV +Eads= -0.39 eV +Eads= -0.40 eV +b +Eads=-0.14 eV +Eads=-0.13 eV +Eads=-0.13 eV +The three highest occupied molecular orbitals (HOMO) of the H2O molecule, named according to +the irreducible representation of the point group of H2O, are 1b1 (HOMO), 3a1 (HOMO-1), and +1b2 (HOMO-2). According to the LDOS plot (Figure S10a) the 3a1 orbital is most broadened due +to its favored orbital mixing with the N atoms, confirming the interaction ability of 2D Zn2VN3 +with H2O. The LDOS plot for the O2 molecule on 2D Zn2VN3 (Figure S10b) reflects additional +O2-induced states within the band gap. The half-filled 2π HOMO state aligns within the valence +band maximum and allows the electrons to be excited to the O2 molecule, thereby creating holes +in 2D Zn2VN3. The 2π* LUMO state is located above the Fermi level at ~0.90 eV. The presence +of the O2-induced states within the band gap of 2D Zn2VN3 and the non-trivial +adsorption/oxidation ability of O2 to 2D Zn2VN3 can alter its optical and electronic properties. + +Figure S11. LDOS of (a) H2O and (b) O2 on 2D Zn2VN3. + + + +(a) +100 +Total +1b +H,0 +Zn +LDOS (states/eV) +N +-100 +3 +0 +2 +Energy (eV) +(b) +100 +Total +0 +T +Zn +LDOS (states/eV) +N +2元 +-100 +-3 +-2 +0 +2 +3 +Energy (eV)Section 7. Water splitting application of 2D Zn2VN3 +Figure S12 shows the calculated VBM and CBM positions of 2D Zn2VN3 with the redox potential +of H2O and oxidation levels. It is found that the VBM of 2D Zn2VN3 is below the oxidation +potential of O2/H2O, while the CBM is also lower than the reduction potential of H2/ H2O, which +indicates that 2D Zn2VN3 is suitable only for oxygen production. + + +Figure S12. The calculated VBM and CBM positions of 2D Zn2VN3 with respect to the water +redox potentials. + + + +-1.5- +-2.0 +CBM +-2.5 +-3.0 +-3.5 +vac +-4.0 +E +- +-4.5 + -H2/H20 +E +VBM +-5.0 +-5.5 +- 02/H20 +-6.0 \ No newline at end of file diff --git a/i9FKT4oBgHgl3EQfwC46/content/tmp_files/load_file.txt b/i9FKT4oBgHgl3EQfwC46/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f73c12b08bd40d5da89675463578a917938b0dc --- /dev/null +++ b/i9FKT4oBgHgl3EQfwC46/content/tmp_files/load_file.txt @@ -0,0 +1,1535 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf,len=1534 +page_content='Prediction and Characterization of Two-Dimensional Zn2VN3 Andrey A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kistanov,1,* Stepan A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Shcherbinin,2,3 Elena A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Korznikova1, Oleg V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prezhdo4 1 The Laboratory of Metals and Alloys Under Extreme Impacts, Ufa University of Science and Technology, 450076 Ufa, Russia 2 Peter the Great Saint Petersburg Polytechnical University, 195251 Saint Petersburg, Russia 3 Institute for Problems in Mechanical Engineering RAS, 199178Saint Petersburg, Russia 4 Department of Chemistry, University of Southern California, Los Angeles, California 90089, United States Corresponding author: andrei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='kistanov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='ufa@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='com Keywords: DFT, prediction, electronic properties, strain, defects Abstract A two-dimensional (2D) monolayer of a novel ternary nitride Zn2VN3 is computationally designed, and its dynamical and thermal stability is demonstrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A synthesis strategy is proposed based on experimental works on production of ternary nitride thin films, calculations of formation and exfoliation energies, and ab initio molecular dynamics simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A comprehensive characterization of 2D Zn2VN3, including investigation of its opto-electronic and mechanical properties, is conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It is shown that 2D Zn2VN3 is a semiconductor with an indirect band gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='75 eV and a high work function of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Its light absorption covers visible and ultraviolet regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The band gap of 2D Zn2VN3 is found to be well tunable by applied strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' At the same time 2D Zn2VN3 possesses high stability against mechanical loads, point defects, and environmental impacts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Considering the unique properties found for 2D Zn2VN3, it can be used for application in opto-electronic and straintronic nanodevices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' TOC Two-dimensional (2D) materials remain an active field of research in science and engineering over the last 15 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' During that time countless 2D structures with unique superconducting (1), opto-electronic (2,3), magnetic (4), mechanical (5,6), and topological (7) properties have been found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For the most part, the research effort has been directed to the exploration of 2D materials which have bulk counterparts representing anisotropic crystals with layers folded together by van der Waals (vdW) forces (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Weak vdW interaction between the layers of such 2D materials supports a natural structural separation of the 2D subcells in the crystals via mechanical (9) or liquid-phase (10) exfoliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Although the deposition and growth technologies are generally available for of 2D materials, control of defects and contamination is not yet compliant with the specifications defined for production (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' High processing temperatures are typically required for high quality 2D materials, complicating their production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Despite these challenges, existing and newly developed 2D materials carry the promise of successful integration into technology and commercial devices during the next decade (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Various methods to predict and fabricate novel bulk and 2D materials, including experimental, ab initio, and machine learning approaches, are currently in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' To facilitate experimental realization of unexplored structures their proper characterization is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It can be successfully realized in computational simulations that have become an effective tool in the prediction and characterization of unknown compounds (12,13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A combination of density functional theory (DFT) calculations and machine learning algorisms allows one to discover exotic low-dimensional structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A broad range of computational techniques has been utilized to describe the crystal structure of potentially superhard boron-rich MoBx phases (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Later, using the evolutionary algorithm, five new superhard ternary compounds in the W−Mo−B system have been predicted at different temperatures, and the composition−temperature phase diagrams have been plotted (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A thin film of NaCl on the (110) diamond surface has been crystallized in the experiment based on a theoretical guidance provided by the ab initio calculations combined with an evolutionary algorithm (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Data-driven high-throughput investigations have led to identification of 8 binary and 20 ternary non-vdW potentially synthesizable candidates with the hematite and ilmenite type structures (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Recently, computationally guided high-throughput synthesis has been used to explore the Zn–V–N phase space, resulting in the synthesis of a novel ternary nitride Zn2VN3 film (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wide band gap wurtzite Zn2VN3 thin films exhibit p-type conductivity, charge carrier concentration of ~1017 cm-3, and promising Hall mobility of about 80 cm2/V·s (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Furthermore, charge carrier concentration in Zn2VN3 is controlled by Zn/V ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The favorable set of optical and electronic properties makes Zn2VN3 thin films an interesting candidate for hole-selective contacts and hole transport layer applications in solar cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tunability of carrier concentration in Zn2VN3 may also be used to fabricate solar cells with back surface field to facilitate charge transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Demonstration of epitaxial stabilization of sputter-deposited Zn2VN3 thin films at low synthesis temperatures (<200°C) and chemical stability of the nitride material may be suitable for application in tandem perovskite-Si solar cells, in which a diffusion barrier is desired to protect the bottom cell (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' However, the synthesized thin films are characterized by cation disorder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Potentially this can be avoided in 2D Zn2VN3, which may open up additional functionalities for this novel semiconductor material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In this work, following the recent prediction and synthesis of wurtzite Zn2VN3 thin films, the existence of 2D Zn2VN3 is investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A comprehensive study of its dynamic and thermal stability as well as the grows mechanism is reported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A thorough characterization of 2D Zn2VN3 is conducted, including identification of its opto-electronic and mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In addition, structural defects in 2D Zn2VN3 and its environmental stability are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The reported study both predicts a new 2D material and offers its characterization and possible applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A monolayer of 2D Zn2VN3 is obtained from the recently predicted and synthesized bulk Zn2VN3 (17) by cutting it along the (001) direction, as shown in Figure 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The top and side views of the 2D Zn2VN3 unit cell are also presented in Figure 1a, and the conventional cell is shown in Figure S1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The unit cell of 2D Zn2VN3 consisting of 24 atoms (8 Zn atoms, 12 N atoms and 4 V atoms) stabilizes in a 2D orthorhombic lattice with the space group 36 Cmc21 and the lattice parameters are a = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='72 and b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='63 Å (see cif file in SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The electronic localization function (ELF) reflects the degree of charge localization in the real space, where 0 represents a free electronic state while 1 represents a perfect localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated ELF for 2D Zn2VN3 with the isosurface value of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 (Figure 1a) reflects electron density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The electron localization basin is spherical and completely migrates to the Zn atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' All basins surround the respective cores, suggesting an ionic bond in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The existence of strong ionic bonds in 2D Zn2VN3 suggests its high stability against formation of most point defects (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated phonon dispersion spectra of 2D Zn2VN3 along the high symmetry path of the Brillouin zone (Figure 1b) shows its kinetic stability, as the transverse, longitudinal and out-of-plane z-direction acoustic modes have real frequencies and display normal linear dispersion around the Г point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Thermal stability of 2D Zn2VN3 is confirmed via AIMD simulations showing that the structure remains stable after 5 ps at 300 K (Figure S1b and Movie 1) The majority of 2D materials are exfoliated from powders/thin films or designed via special methods such as chemical vapor deposition (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Feasibility of exfoliation of 2D Zn2VN3 needs to be evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated binding energy that needs to be overcome to achieve exfoliation of a 2D Zn2VN3 monolayer from bulk Zn2VN3 is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='83 eV, which is ~20 times higher than that of graphene (Figure S2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The exfoliation energy ΔEexf for 2D Zn2VN3, that is, the binding energy between layers in the bulk, is computed as the energy difference between the relaxed 2D and bulk systems (21), Δ𝐸exf = 𝐸2𝐷−𝐸𝑏𝑢𝑙𝑘 𝐴 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (1) Here, E2D and Ebulk correspond to the total energies of the optimized 2D and bulk Zn2VN3, respectively, and A is the in-plane surface area according to the optimized bulk Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ΔEexf of 2D Zn2VN3 is found to be 105 meV/Å2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' This value is ~5 times higher than the ΔEexf value of graphene (∼20 meV/Å2) (21,22), while it is below the 130-200 meV/ Å2 limit proposed for “potentially exfoliable” systems (22,24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Therefore, exfoliation of the 2D Zn2VN3 monolayer should be possible at certain conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The recently reported approach for deposition of bulk Zn2VN3 suggests that it can be formed from evaporation of Zn3N2 and VN at a temperature of 390- 490 K and in the presence of ionized nitrogen, roughly following the reaction Zn3N2 + VN -> (N+) Zn2VN3 + Zn (evaporation) (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' AIMD simulations conducted in this work show that a Zn2VN3 structure containing an extra (unevaporated) Zn atom can be formed at 180 K within ~1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='3 ps (Movie 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Therefore, by controlling the Zn evaporation rate and the ionized nitrogen rate it can be possible to utilize chemical vapor deposition for 2D Zn2VN3 synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' During the manufacturing, the thickness of sputter-deposited thin films is increased with sputtering power (for constant deposition time).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For example, the thickness of the Mo-incorporated Cu2ZnSnS4 absorber layer increased with Mo co-sputtering power, as shown in the cross-section scanning transmission electron microscope study (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' More recently, the ternary metal-zinc-nitride thin films of about 235 nm thickness were demonstrated experimentally and implemented in a photodetector device (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Proper choice of a substrate is an important step for production of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For instance, bulk Zn2VN3 deposited on glass and sapphire substrates appears to be phase-pure, while the epitaxial stabilization on Al2O3 (0001) increases its crystallinity and texture (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) Schematic representation of Zn2VN3 transformation from bulk to 2D, and the 2D Zn2VN3 unit cell combined with ELF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The Zn, V, and N atoms are colored in gray, red and violet, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (b) Phonon dispersion curves of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 2a shows the band structure of 2D Zn2VN3 obtained using the Heyd–Scuseria– Ernzerhof (HSE06) exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For the comparison, the band structure of 2D Zn2VN3 obtained via the Perdew-Burke-Ernzerhof (PBE) functional under the generalized gradient approximation is plotted in Figure S3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Based on the HSE06 calculations, it is found that 2D Zn2VN3 is a semiconductor with an indirect band gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='75 eV and a direct band gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='85 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Both gaps are slightly higher than in bulk Zn2VN3 (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Similarly to its bulk counterpart, 2D Zn2VN3 is a p-type semiconductor with the Fermi level located slightly above the valence band edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For the indirect gap, the conduction band minimum (CBM) is located in the vicinity of the Γ point, while the valence band maximum (VBM) is located at the S point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In the case of the direct gap, both CBM and VBM are located at the S point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The partial density of states (PDOS) of 2D Zn2VN3 (Figure S3b) demonstrates that d states of V atoms give the main contribution to the CBM, while the VBM is formed by p states of N atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' From the band structure and PDOS plots the CBM localization is noticed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Such localization of states around the CBM is commonly observed in materials due to cation disorder (27, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Localization on the d states of V atoms has been found in bulk Zn2VN3 (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It is also supposed that the origin of the CBM localization in 2D Zn2VN3, as reflected in the PDOS plot (Figure S3), is due to the cation disordering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The optical response of a material at a given frequency can be determined via its frequency-dependent complex dielectric function that can be defined as a function of incident photon energy having real and imaginary parts (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' This function describes the process of light propagation through the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Light absorption of a material at a given frequency ω can be (a) (b) 25 Frequency (THz) 20 a = b = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='63 A 15 (001) 10 5 0 X S Y YT zrsimply realized by the positive value of the real part of the dielectric function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Therefore, the real part refers to the ability of a material to store the electric energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The imaginary part of the dielectric function refers to the ohmic resistance of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For instance, the imaginary part of the dielectric function of a pure dielectric has a zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The real and imaginary parts of the complex dielectric function for 2D Zn2VN3 are presented in Figure 2b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The real part of the dielectric function for the incident electromagnetic field normal to the 2D Zn2VN3 surface is positive in the considered region of the electromagnetic spectra from 0 to 16 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' This suggests that photons propagate through the monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The static dielectric function, which represents the dielectric response of the material to a static electric field, is found to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The real dielectric constant has a considerable peak at 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='9 eV with the maximum value of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='55, and it exhibits a few peaks at higher energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The imaginary part of the dielectric function shows that 2D Zn2VN3 absorbs light in the visible and ultraviolet regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Considering the electric and optical properties of the predicted 2D Zn2VN3, it can be regarded as a transparent material for the visible lights and a useful shielding material in ultraviolet region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 2c shows the diagram of the 2D Zn2VN3 work function in comparison with other 2D materials and bulk metals possessing the highest known work function values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The work function of 2D Zn2VN3 is high, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 eV, comparable to that of borophene (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The high work function of 2D Zn2VN3 can be attributed to the nature of its atomic states around the Fermi level consisting of not only the out-of-plane pz states but also the in-plane p hybridized states (Figure S4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Thus, the ionization of 2D Zn2VN3 requires significant energy and is comparable to that of, for example, borophene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) c) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Pt 2D B 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='85 eV 2D Zn,VN Energy (eV) Ni 2D PC 0 2D BCP 2D P 2 2D C Ti 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 Work function value (eV) X s Y YT ZI (b) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 xx direction xx direction 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='8 zz direction (0) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' - zz direction 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='9 (0) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 Imaginary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' eal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='9 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 0 2 4 6 810 12 14 16 0 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 68 10 12 14 16 Energy (eV) Energy (eV)Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) Band structure of 2D Zn2VN3 calculated by the HSE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (b) Real and imaginary parts of dielectric function versus energy for 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (c) Comparison of the work function of 2D Zn2VN3 with those of common 2D materials and bulk metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mazdziarz formulated the mechanical stability of 2D materials acquired in rectangular lattices as follows (31): 1 2 (𝐶11+ 𝐶22 + √4𝐶12 2 − (𝐶11 − 𝐶22)2 ) > 0 1 2 (𝐶11+ 𝐶22 − √4𝐶12 2 − (𝐶11 − 𝐶22)2 ) > 0 (2) 𝐶66 > 0 The above presented criteria are met in the case of 2D Zn2VN3 confirming its mechanical stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated elastic constants Cij for 2D Zn2VN3 are collected in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mechanical properties of 2D Zn2VN3 such as Young’s modulus, Poisson’s ratio and shear modulus are also considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Young’s modulus of 2D Zn2VN3 in the x and y directions is calculated as (32,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='33): 𝐸[𝑥]= 𝐶11𝐶22−𝐶12 2 𝐶11 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' and 𝐸[𝑦]= 𝐶11𝐶22−𝐶12 2 𝐶22 (3) Shear modulus of 2D Zn2VN3 is calculated as (30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='31): G = C66 (4) Poisson’s ratio of 2D Zn2VN3 in the x and y directions is calculated as (30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='31): 𝑣[𝑥]= 𝐶12 𝐶11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' and 𝑣[𝑦]= 𝐶12 𝐶22 (5) Figure 3 presents the spatial dependence of Young’s modulus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' shear modulus and Poisson’s ratio for 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' These parameters are almost direction independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The values of Young’s modulus of 2D Zn2VN3 in the x and y directions are found to be ~99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='7 N/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Shear modulus of 2D Zn2VN3 is found to be 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='6 N/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Poisson’s ratio of 2D Zn2VN3 in both x and y directions is found to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='375.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' For comparison, 2D Zn2VN3 has lower stiffness and higher elasticity relative to graphene (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Spatial dependencies of (a) Young’s modulus (N/m), (b) shear modulus (N/m), and (c) Poisson’s ratio for 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=" (a)Young'smodulusinxy-plane (b)Shearmodulusinxy-plane ()Poisson'sratioinxy-plane 100 40 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='4 50 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 0 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 50 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 100 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='4 100 50 0 50 100 40 20 0 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='42D Zn2VN3 can be switched from a direct band gap semiconductor to an indirect band gap semiconductor and its band gap size can be increased/decreased up to ~50% via strain engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Although the PBE functional underestimates the band gap compared to the HSE functional, PBE (Figure S3a) and HSE (Figure 2a) display similar trends in the band alignment in the band structure plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Thus, the effect of strain engineering on the band structure of 2D Zn2VN3 can be studied using the PBE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 4a shows changes in the band gap of 2D Zn2VN3 under strain applied along the surface plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The applied strain is in the range from -10% (compressive strain) to 10% (tensile strain) that can be realized experimentally (35-40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The band gap of 2D Zn2VN3 decreases rapidly from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='57 eV to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='76 eV when the compressive strain increases from 0 to 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' As the VBM shifts from the S point to the Γ point (Figure S5), an indirect-direct band gap transition is observed in 2D Zn2VN3 under the compressive strain higher than 6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A rapid decrease of the 2D Zn2VN3 band gap from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='57 eV to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='07 eV is observed with the increase of the tensile strain from 0% to 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The applied tensile strain in the range from 8% to 10% leads to a slight increase of the band gap from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='07 eV to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='19 eV (indirect band gap) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='22 eV (direct band gap) (Figure S5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' As 2D materials always contain surface defects that may affect their structure stability and performance (41-45), it is necessary to investigate point defects in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Several common point defects are found to be stable in 2D Zn2VN3: a single vacancy of a N atom (SVN);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' a single vacancy of a Zn atom (SVZn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' a single vacancy of a V atom (SVV);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' in-plane and out-of-plane double vacancies of one V atom and one N atom (DVV-N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' and in-plane and out-of-plane double vacancies of one Zn atom and one N atom (DVZn-N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' SV defects can be created by removing one Zn, V or N atom from the 2D Zn2VN3 surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The DVV-N and DVZn-N defects in the 2D Zn2VN3 surface are formed when V and N or Zn and N atoms are removed from the surface (in-plane DV) or from the plane perpendicular to the surface (out-of-plane DV) of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The stability of point defects in 2D Zn2VN3 is evaluated based on their formation energy, Eform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Eform of the considered stable defects in 2D Zn2VN3 are collected in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' According to Table 1, the SVZn and SVN defects have the lowest Eform of 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 eV and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The out-of-plane DVZn-N, in-plane DVZn-N, SVV, in-plane DVV-N, and out-of-plane DVV-N defects have comparably high formation energies of 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='83 eV, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='54 eV, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='25 eV, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='96 eV, and 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='92 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Eform of SVs in 2D Zn2VN3 is comparable to that in graphene (~7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='50 eV) (46) and MoS2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='10-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='20 eV) (47), while the formation of the DVs in 2D Zn2VN3 is less favorable than in graphene (~8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 eV) (46) and MoS2 (~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 eV) (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Eform (eV) of point defects in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' SVN SVZn SVV in-plane DVV-N out-of-plane DVV-N in-plane DVZn-N out-of-plane DVZn-N 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='25 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='96 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='92 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='54 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='83 Proper identification and classification of defects is a key capability of atomically resolved scanning tunneling microscopy (STM) (48,49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' To facilitate experimental needs it is possible to utilize DFT simulated STM images for the differentiation of point defects in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The atomic structures and corresponding STM images of the pristine and defect-containing 2D Zn2VN3 are presented in Figures S6a-d and S7a-d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The STM images of 2D Zn2VN3 containing the considered defects correlate with the corresponding atomic structures, and the defects are easily identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The STM image in Figure S6b (right panel) shows that the SVN defect in 2D Zn2VN3 can be identified by the triangle formed with two bright spots and one dark spot, arising from two Zn atoms and one V atom, with one N atom missing inside the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Similarly, according to Figures S6c and d (right panels), the SVV and SVZn defects in 2D Zn2VN3 appear as triangles formed of three small dark spots characterizing three N atoms, with V or Zn atoms missing inside the triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The in-plane DVV-N defect in 2D Zn2VN3 is presented in Figure S7a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Here, two semi- hexagons, formed of two bright spots (Zn atoms) and two dark spots (N atoms) each, have two missing atoms (one V atom and one N atom) at their border.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The out-of-plane DVV-N defect in 2D Zn2VN3 (Figure S7b) is characterized by missing V and Zn atoms in-between five big bright spots (Zn atoms) and one dark spots (N atom) slightly shifted from their positions compared to the perfect case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The in-plane DVZn-N defect in 2D Zn2VN3 is seen in Figure S7c as two missing atoms (Zn and N) inside a square formed of three dark spots (two N atoms and one V atom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The out-of- plane DVZn-N defect in 2D Zn2VN3 (Figure S7d) may be identified as missing atoms within a triangle formed of two bright spots and one dark spot due to two Zn atoms and one V atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' This pattern is similar to the STM image the SVN defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' To deeper understand the changes in the electronic structure of 2D Zn2VN3 induced by the defects, the density of states resolved in space, known as local density of states (LDOS), calculated for peripheral atoms in the defect core and for atoms far from the defect core, are shown in Figures S8 and S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It is found that the defects induce significant changes in the electronic structure of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Such changes facilitate defect identification via photoemission spectroscopy techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 4b presents the temperature-depended surface density of point defects in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The SVN and SVZn defects possess significantly higher surface densities compared to the other defects found to be stable in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The surface density of the SVV, out-of-plane DVZn- N, in-plane DVZn-N, in-plane DVV-N, and out-of-plane DVV-N defects in 2D Zn2VN3 is slightly lower than that in graphene (50) and MoS2 (51), making their formation less energetically favorable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' On the other hand, the surface density of the SVN and SVZn defects in 2D Zn2VN3 is comparable to that of the SV defects in graphene (50) and MoS2 (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Structural degradation of 2D materials can also be caused by their interaction with the humid environment, particularly, with H2O and O2 molecules contained in the air.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Therefore, the interaction of the H2O and O2 molecules with the 2D Zn2VN3 surface is further evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' As it has been shown previously, for most of 2D materials, oxidation is the most dangerous process that can lead to degradation of their surface (52-54), while H2O-saturated surfaces can exhibit higher stability as such saturation prevents the oxidation (55,56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It is found that the adsorption energy, Eads, of O2 on the 2D Zn2VN3 surface is as high as -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='14 eV, while Eads of H2O (-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='49 eV) is ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 lower (Figure S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hence, the adsorption of H2O on 2D Zn2VN3 is more favorable compared to O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It should be noted, that according to the LDOS plots (Figure S11) remarkable changes in H2O and O2 molecular states upon their interaction with the 2D Zn2VN3 surface are observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Therefore, the kinetic analysis of H2O and O2 splitting on the 2D Zn2VN3 surface is further conducted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 4c presents the result from the CI-NEB calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The energy barrier, Eb, for the H2O and O2 molecule dissociation on 2D Zn2VN3 is found to be 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='82 eV and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='04 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Considering the obtained high values of Eb for the dissociation of H2O and O2 on the 2D Zn2VN3 surface, which are comparable to those of the H2O and O2 molecule dissociation on InSe (57), and based of the Eads analysis, it can be concluded that 2D Zn2VN3 possesses high structural integrity under environmental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A discussion on the potential application of 2D Zn2VN3 to water splitting is presented in Supporting Information, and the calculated VBM and CBM positions of 2D Zn2VN3 with the redox potential of H2O and oxidation levels are shown in Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) Band gap of 2D Zn2VN3 as a function of strain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (b) Surface density of point defects in 2D Zn2VN3 as a function of temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (c) Activation barriers for H2O and O2 molecule splitting on 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In summary, a new material, a 2D analog of the recently predicted and synthesized ternary nitride semiconductor Zn2VN3 (17), is predicted computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The fabrication of 2D Zn2VN3 is highly possible due to is relatively low exfoliation energy of 105 meV/Å2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It is also proposed that the chemical vapor deposition approach can be utilized for the synthesis of 2D Zn2VN3 similarly to the bulk Zn2VN3 (17) and Cu2ZnSnS4 thin film (25) manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The environmental stability of 2D Zn2VN3 should be high due to resistivity of its surface to oxygen and defect formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2D Zn2VN3 has an indirect band gap of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='75 eV, which can be tuned up to 50% under applied stains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2D Zn2VN3 possesses a high work function of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='27 eV, absorbs visible and ultraviolet light, and exhibits moderate mechanical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' All this makes 2D Zn2VN3 a good candidate for application in opto-electronic and straintronic devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Computational Methods The spin-polarized first-principles calculations were performed within the framework of density functional theory as implemented in the plane-wave the Vienna Ab initio Simulation Package (VASP) (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The Perdew-Burke-Ernzerhof functional (PBE) (59) under the generalized gradient approximation and the HSE06 hybrid exchange-correlation functional (60) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Dispersive interactions were included using the van der Waals corrected functional (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The geometry optimization was stopped once the atomic forces and total energy values were smaller than 10−4 eV/Å and 10−8 eV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The plane-wave cut-off energy was set to 520 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The periodic boundary conditions were applied for the two in-plane transverse directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A vacuum space of 25 Å was introduced to the direction perpendicular to the surface plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The electron localization function (ELF) was calculated to obtain the distribution of electrons in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The Phonopy code associated with VASP was used for the simulation of the phonon spectrum (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The 3×3×1 supercell of 2D Zn2VN3 was used to perform the calculations based on finite displacement methods with the atomic displacement distance of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='01 Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The dielectric function of 2D Zn2VN3 was calculated based on the TD-HSE06 approach (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ab initio molecular dynamics simulations controlled by the Nose–Hoover thermostat were performed for 5 ps at the temperature of 300 K and with a time step of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 fs (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) (b) Bandgap size (eV) (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content="6 1E-20 3 Q'H ." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 1E-40" 10 :02 indirect 1E-60- 8 8 2 1F-80 9- density (eV) 1L-100 4 direct 1E-120 Energy 2 1E-140 Areal Strain 0 1E-160 SVn 0 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='SVzn 1E-180 indirect .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='SVy 4 1E-200 6 1E-220 DV 8 0 200 400 600800 Reaction path 10 direct : indirect Temperature (K)The stress-strain relation (65) was used to calculate the components of the stiffness matrix from which the Young’s modulus, shear modulus, and Poisson’s ratio were obtained and directional dependencies of these quantities were defined using the ELATE software for analysis of elastic tensors (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' To consider point defects in 2D Zn2VN3 a supercell composed of 3×3×1 unit cells (36 Zn, 18 V and 54 N atoms) was created to avoid non-physical interactions between periodic images while keeping affordable computational demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Under such conditions, the concentration of MV defects was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='93% and the concentration of DV defects was 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='85%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The Tersoff-Hamann approach was implemented to simulate Scanning Tunneling Microscope (STM) images of pure and defect- containing 2D Zn2VN3 (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The formation energy Eform of point defects in 2D Zn2VN3 was calculated as Eform = Edefect − Epure + NZn·EZn + NV·EV+ NN·EN, (6) where Edefect and Epure are the total energies of pure and defect-containing 2D Zn2VN3, EZn, EV and EN are the energies of single Zn, V and N atoms, and NZn, NV·and NN· correspond to the number of the removed Zn, V and N atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The surface density of defects Nd in 2D Zn2VN3, at a finite temperature, was calculated according to the Arrhenius equation: 𝑁𝑑 = 𝑁𝑝ure 𝑒−𝐸𝑓orm/(𝑘𝐵𝑇), (7) where Npure is the surface density of atoms in pure 2D Zn2VN3, kB is the Boltzmann constant, and T is temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Note that only defects presented at the surface were shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The climbing image–nudged elastic band (CI-NEB) method was used to obtain the reaction pathway of the H2O and O2 molecules on the 2D Zn2VN3 surface (68).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ASSOCIATED CONTENT Notes The authors declare no competing financial interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ACKNOWLEDGMENTS S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' is thankful for the funding provided by the Russian Science Foundation (grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 21-71- 10129).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' acknowledges the support of Grant NSh-4320.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='2 of the President of the Russian Federation for state support of young Russian scientists - candidates of sciences and doctors of sciences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='А.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' is grateful for financial support to the Ministry of Science and Higher Education of the Russian Federation within the framework of the state task of the USATU (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 075-03-2022-318/1) of the youth research laboratory «Metals and Alloys under Extreme Impacts».' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Authors acknowledge Peter the Great Saint Petersburg Polytechnic University Supercomputer Center “Polytechnic” and Joint Supercomputer Center of the Russian Academy of Sciences for computational resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' grateful Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Siarhei Zhuk for useful discussions on the synthesis and application of Zn–V–N compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' acknowledges support of the US National Science Foundation (grant CHE-1900510).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' REFERENCES (1) Campi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kumari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Marzari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prediction of phonon-mediated superconductivity with high critical temperature in the two-dimensional topological semimetal W2N3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 21, 3435−3442.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (2) Yang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Fang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Benderskii, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Long, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prezhdo, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Strain controls charge carrier lifetimes in monolayer WSe2: Ab initio time domain analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2019, 10(24), 7732−7739.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (3) Manzeli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ovchinnikov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Pasquier, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yazyev, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kis, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2D transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2017, 2, 17033.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (4) Gibertini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Koperski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Morpurgo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Novoselov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Magnetic 2D materials and heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2019, 14, 408−419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (5) Jiang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Li, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Huang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Bi, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Guo, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Single-layer MoS2 mechanical resonant piezo-sensors with high mass sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Interfaces 2020, 12(37), 41991−41998.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (6) Šiškins, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Alijani, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' van Blankenstein, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Davidovikj, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' van der Zant, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Steeneken, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Highly anisotropic mechanical and optical properties of 2D layered As2S3 membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ACS Nano 2019, 13(9), 10845–10851.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (7) Kou, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ma, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sun, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Heine, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chen, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Two-dimensional topological insulators: Progress and prospects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2017, 8, 1905−1919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (8) Friedrich, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ghorbani-Asl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Curtarolo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Krasheninnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Data-driven quest for two-dimensional non-van der Waals materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 22(3), 989−997.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (9) Novoselov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jiang, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Schedin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Booth, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Khotkevich, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Morozov, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Geim, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Two-dimensional atomic crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Natl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2005, 102, 10451−10453.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (10) Nicolosi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chhowalla, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kanatzidis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Strano, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Coleman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Liquid exfoliation of layered materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Science 2013, 340, 1226419.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (11) Lemme, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=', Akinwande, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=', Huyghebaert, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Stampfer, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2D materials for future heterogeneous electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nat Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 13, 1392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (12) Carvalho, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Trevisanutto, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Taioli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Castro Neto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Computational methods for 2D materials modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 84, 106501.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (13) Mellaerts, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Meng, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Menghini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Afanasiev, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Seo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Houssa, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Locquet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Two dimensional V2O3 and its experimental feasibility as robust room- temperature magnetic Chern insulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' npj 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 5, 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (14) Rybkovskiy, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnina, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Oganov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Structure, stability, and mechanical properties of boron-rich Mo–B phases: A computational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2020, 11(7), 2393−2401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (15) Kvashnin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tantardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zakaryan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnina, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Oganov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Computational search for new W–Mo–B compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2020, 32(16), 7028−7035.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (16) Tikhomirova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tantardini, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sukhanova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Popov, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Evlashin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tarkhov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhdanov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Dudin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Oganov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Exotic two-dimensional structure: The first case of hexagonal NaCl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2020, 11(10), 3821−3827.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (17) Zhuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Boehme, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ott, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' La Mattina, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Stiefel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kovalenko, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Siol, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Synthesis and characterization of the ternary nitride semiconductor Zn2VN3: Theoretical prediction, combinatorial screening, and epitaxial stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 33(23), 9306–9316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (18) Fu, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Liang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Faes, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jeangros, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ballif, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hou, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Monolithic perovskite-silicon tandem solar cells: From the Lab to Fab?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 34, 2106540.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (19) Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Shcherbinin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Botella, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Davletshin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Family of two-dimensional transition metal dichlorides: Fundamental properties, structural defects, and environmental stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 13(9), 2165−2172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (20) Sun, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wang, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jiang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Katsnelson, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Korzhavyi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Erikssona, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Di Marco, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A new 2D monolayer BiXene, M2C (M = Mo, Tc, Os).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nanoscale 2016, 8, 15753−15762.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (21) Jung, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Park, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ihm, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rigorous method of calculating exfoliation energies from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2018, 18, 2759−2765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (22) Mohanty, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wei, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ghorbani-Asl, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Krasheninnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rajput, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jena, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Revealing the defect-dominated oxygen evolution activity of hematene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A 2020, 8, 6709−6716.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (23) Mounet, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gibertini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Schwaller, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Campi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Merkys, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Marrazzo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sohier, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Castelli, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cepellotti, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Pizzi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Marzari, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Two-dimensional materials from high-throughput computational exfoliation of experimentally known compounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nanotechnol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2018, 13, 246−252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (24) Choudhary, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kalish, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Beams, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tavazza, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Highthroughput identification and characterization of two-dimensional materials using density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2017, 7, 5179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (25) Zhuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hadke, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lie, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Guchhait, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cheng, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Dalapati, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Molybdenum incorporated Cu1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='69ZnSnS4 kesterite photovoltaic devices with bilayer microstructure and tunable optical-electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Energy 2019, 194, 777−787.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (26) George, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Vikraman, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Reji, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ghanashyam, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jayaraman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sivalingam, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Narasimha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Novel ternary nitride thin film-based self-powered, broad spectral responsive photodetector with a high detectivity for weak light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2023, 8, 2200645.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (27) Cordell, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tucker, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tamboli, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Bandgap analysis and carrier localization in cation-disordered ZnGeN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' APL Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 10, 011112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (28) Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Huttula, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Khadiullin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Korznikova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Smirnov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhuk, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Impact of various dopant elements on the electronic structure of Cu2ZnSnS4 (CZTS) thin films: a DFT study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' CrystEngComm, 2020, 22, 5786−5791.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (29) Naseria, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hoat, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prediction of a new 2D B2CO monolayer from density functional theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 186, 109975.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (30) Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Srikanth, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Dmitriev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Exploring the charge localization and band gap opening of borophene: A first-principles study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nanoscale, 2018, 10, 1403−1410.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (31) Maździarz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Comment on ‘The Computational 2D Materials Database: high- throughput modeling and discovery of atomically thin crystals’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2D Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2019, 6, 048001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (32) Andrew, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mapasha, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ukpong, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chetty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mechanical properties of graphene and boronitrene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B 2012, 85, 125428.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (33) Zhao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhang, Sh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' TiC2: a new two-dimensional sheet beyond MXenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nanoscale 2016, 8, 233−242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (34) Cadelano, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Palla, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Giordano, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Colombo, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Elastic properties of hydrogenated graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B 2010, 82, 235414.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (35) Blundo, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cappelluti, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Felici, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Pettinari, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Polimeni, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Strain-tuning of the electronic, optical, and vibrational properties of two-dimensional crystals featured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 8, 021318.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (36) Ryu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Carrascoso, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' López-Nebreda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Agraït, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Frisenda, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Castellanos-Gomez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Microheater actuators as a versatile platform for strain engineering in 2D materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2020, 20(7), 5339–5345.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (37) Shcherbinin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Dmitriev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Korznikova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Davletshin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Two-dimensional black phosphorus carbide: Rippling and formation of nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C 2020, 124(18), 10235–10243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (38) Kripalani, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sun, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lin, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Xue, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Strain-driven superplasticity of ultrathin tin (II) oxide films and the modulation of their electronic properties: A first- principles study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B 2019, 100(21), 214112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (39) Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Bai, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Liu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Effects of tensile strain rate and grain size on the mechanical properties of nanocrystalline T-carbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2019, 170, 109188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (40) Chernozatonskii, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Artyukh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kvashnin D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mechanical engineering effect in electronic and optical properties of graphene nanomeshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Interfaces 2020, 12(49), 55189–55194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (41) Perez, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ghosh, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prezhdo, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nie, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tretiak, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Neukirch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Point defects in two-dimensional ruddlesden–popper perovskites explored with ab initio calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 13, 5213–5219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (42) Wu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Vasenko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Prezhdo O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Common defects accelerate charge carrier recombination in CsSnI3 without creating mid-gap states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 12 (36), 8699–8705.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (43) Kretschmer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ghaderzadeh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Facsko, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Krasheninnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Threshold ion energies for creating defects in 2D materials from first-principles calculations: Chemical interactions are important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 13(2), 514–519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (44) Komsa, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Krasheninnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Physics and theory of defects in 2D materials: the role of reduced dimensionality, Materials Today, Defects in Two-Dimensional Materials, Elsevier, 2022, Pages 7-41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (45) Xia, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Liang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cai, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Xing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Manipulation of band alignment in two- dimensional vertical WSe2/BA2PbI4 ruddlesden–popper perovskite heterojunctions via defect engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 13, 4579–4588.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (46) Krasheninnikov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lehtinen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Foster, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nieminen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Bending the rules: Contrasting vacancy energetics and migration in graphite and carbon nanotubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2006, 418, 132–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (47) Hong, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Probert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lv, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Feng, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Xie, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ji, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Exploring atomic defects in molybdenum disulphide monolayers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nat Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2015, 6, 6293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (48) Dan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhao, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ning, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Loh, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Loh, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Pennycook, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Learning motifs and their hierarchies in atomic resolution microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2022, 8, 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (49) Zhussupbekov, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ansari, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' McManus, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhussupbekova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Shvets, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Duesberg, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hurley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gity, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Coileáin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ó;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' McEvoy, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Imaging and identification of point defects in PtTe2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' npj 2D Mater Appl 2021, 5, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (50) Hu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Defects in phosphorene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C 2015, 119, 20474−20480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (51) Jeong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ly, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Han, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nam, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jiong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Shin, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yun, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kim, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kim, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hwang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lee, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Visualizing point defects in transition-metal dichalcogenides using optical microscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ACS Nano 2016, 10, 770−777.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (52) Lloret, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rivero-Crespo, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Vidal-Moya, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wild, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Domenech-Carbo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Heller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Shin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Steinruck, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Maier, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hauke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Varela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hirsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Leyva-Perez, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Abellan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Few layer 2D pnictogens catalyze the alkylation of soft nucleophiles with esters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2019, 10, 509.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (53) Gibaja, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rodriguez-San-Miguel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ares, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Glmez-Herrero, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Varela, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gillen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Maultzsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hauke, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hirsch, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Abellan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zamora, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Few-layer antimonene by liquid-phase exfoliation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2016, 55, 14345–14349.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (54) Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Khadiullin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zhou, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Dmitriev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Korznikova E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Environmental stability of bismuthene: Oxidation mechanism and structural stability of 2D pnictogens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C 2019, 7, 9195–9202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (55) Botella, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chiter, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Costa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nakashima, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lefevre, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Influence of hydration/dehydration on adsorbed molecules: Case of phthalate on goethite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Colloids Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A Physicochem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2021, 625, 126872.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (56) Gardeh, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Gh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kistanov, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Nguyen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Manzano, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Cao, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Kinnunen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Exploring mechanisms of hydration and carbonation of MgO and Mg(OH)2 in reactive magnesium oxide-based cements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C 2022, 126(14), 6196–6206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (57) Ma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Li, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yuan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' He, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lu, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Yang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The role of the intrinsic Se and In vacancies in the interaction of O2 and H2O molecules with the InSe monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Surf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2018, 434, 215–227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (58) Kresse, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Furthmuller, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B 1996, 54, 11169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (59) Perdew, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Burke, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Generalized gradient approximation made simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 1996, 77, 3865−3868.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (60) Heyd, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Scuseria, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Ernzerhof, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Hybrid functionals based on a screened coulomb potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2003, 118, 8207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (61) Becke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Density-functional exchange-energy approximation with correct asymptotic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A: At.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=', Mol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=', Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 1988, 38, 3098.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (62) Togo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Tanaka, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' First-principles phonon calculations in materials science.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Scr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2015, 108, 1−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (63) Ketolainen, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Macháčová, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Karlický, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Optical gaps and excitonic properties of 2D materials by hybrid time-dependent density functional theory: Evidences for monolayers and prospects for van der Waals heterostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Theory Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2020, 16, 5876–5883.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (64) Nosé, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A unified formulation of the constant temperature molecular dynamics methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 1984, 81(1), 511–519.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (65) Le Page, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Saxe, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Symmetry-general least-squares extraction of elastic data for strained materials from ab initio calculations of stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B: Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Matter Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2001, 63, 174103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (66) Gaillac, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Pullumbi, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Coudert, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' ELATE: An open-source online application for analysis and visualization of elastic tensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' : Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2016, 28, 275201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (67) Tersoff, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=', Hamann, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Theory of the scanning tunneling microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' B 1985, 31, 805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (68) Henkelman, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Uberuaga, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Jónsson, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' A climbing image nudged elastic band method for finding saddle points and minimum energy paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2000, 113(22), 9901–9904.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Supporting Information Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Conventional cell of 2D Zn2VN3 Figure S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) Conventional cell of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (b) Energy fluctuation of the 2D Zn2VN3 system from AIMD calculations at 300 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Binding energy of 2D Zn2VN3 Figure S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Binding energy required for exfoliation of (a) 2D Zn2VN3 and (b) graphene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' a) (b) 676 Energy (eV) a=9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='70 A 678 680 682 0 1 2 3 4 5 Time (ps) b=5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='74 A(a) (b) eV 3 12 0Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' GGA band sructure and PDOS of 2D Zn2VN3 Figure S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) Band structure and (b) PDOS of 2D Zn2VN3 obtained via the Perdew-Burke- Ernzerhof (PBE) exchange-correlation functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' PDOS of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) (b) 4 Total DOS d-states Zn p-states N d-states V 2 Energy (eV) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='74 eV 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='4 X S Y VT Z 0 10 20 PDOS (eV/states)PDOS (eV/states) N (px) N (py) 20 N (pz) N (s) 10 4 0 Energy (eV) Figure S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Band structure of 2D Zn2VN3 under compressive (upper row) and tensile (lower row) strain obtained via the PBE approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated elastic constants Cij for 2D Zn2VN3 Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated elastic constants Cij for 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' C11, N/m 117 C22, N/m 117 C12, N/m 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 C44, N/m 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='6 10% 8% 6% 4% 2% 0% Energy (eV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='74ev 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='29 eV 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='57ev 0 X s V YTZ 11 X s YTZ X s Y r X s Z X 10% 8% 6% 4% 2% 0% Energy (eV) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='74 el .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='22 ev 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='19 eV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='42 ev 0 2 X s YT Z T X s 7 X S x s X s Z X Y ZSection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Atomic stricture, STM images and LDOSs of defect-containing 2D Zn2VN3 Figure S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Atomic stricture (left) and STM image (right) of (a) pure, (b) SVN, (c) SVV, and (d) SVZn defect-containing 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) (b) (c) (d) Figure S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Atomic stricture (left) and STM image (right) of (a) in-plane DVV_N, (b) out-of-plane DVV_N, (c) in-plane DVZn_N, and (d) out-of-plane DVZn_N defect-containing 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) (b) (c) (d)To deeper understand the changes in the electronic structure of 2D Zn2VN3 induced by the defects, the density of states resolved in space, known as local density of states (LDOS), is calculated for peripheral atoms in the defect core and for atoms far from the defect core, as shown in Figures S8 and S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The states introduced by the SVN defect are mainly contributed by the V atom (Figure S8a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The changes in the CBM and the VBM depicted in the LDOS plot for the defect (Figure S8b) arise mainly from the N1 and N2 atoms and N3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The defect-induced states in the vicinity of the VBM in the LDOS plot of the SVZn system (Figure S8c) mainly originate from the N1, N2 and N3 atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In the case of the in-plane DVV_N defect in 2D Zn2VN3 (Figure S9a), mainly the N1 and N2 atoms surrounding the defect have partially occupied/unoccupied states contributing into the valence/conduction bands of 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The N1 atom (Figure S9b) is responsible for the in-gap states appearing in 2D Zn2VN3 containing the out-of-plane DVV_N defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In the case of the in-plane DVZn_N in 2D Zn2VN3, in-gap states appear mainly due to the V atom located in the vicinity of the defect core (Figure S9c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In-gap states in 2D Zn2VN3, appearing due to the out-of- plane DVZn_N, are formed by the V, Zn2, and N2 atoms, as shown in Figure S9d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure S8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Atomic stricture (left) and LDOS (right) of (a) SVN, (b) SVV, and (c) SVZn defect- containing 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 20 (a) LDOS(states/eV) 0 Total Dos Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Zn2 20 2 0 2 Energy (eV) (b) 20 Total DOS N, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' LDOS(states/eV) N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 0 20 2 1 0 1 2 Energy (eV) (c) 20 Total DOS N1 N N2 LDOS(states/eV) N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 0 N 20 2 0 2 Energy (eV) Figure S9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Atomic stricture (left) and LDOS (right) of (a) in-plane DVV_N, (b) out-of-plane DVV_N, (c) in-plane DVZn_N, and (d) out-of-plane DVZn_N defect-containing 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 20 (a) LDOS(states/eV) Zn2 N Total DOS Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Znz N, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='1 0 2 Energy (eV) 20 (b) LDOS(states/eV) N Total DOS Zn N V2 Zn N 20 2 0 2 Energy (eV) 20 LDOS(states/eV) 0 Total DOS 21 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' "N V Zn 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2 1 0 2 Energy (eV) 20 Total DOS (d) N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' LDOS(states/eV) Zn2 N3 0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 2 0 2 Energy (eV)Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' H2O and O2 on 2D Zn2VN3 According to Figure S10a, the most favorable adsorption position of H2O on 2D Zn2VN3 corresponds to the position of the molecule above the Zn-V bond on the side of the hexagon at the distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='01 Å above the surface of 2D Zn2VN3, and the lowest Eads of H2O on 2D Zn2VN3 is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='49 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure S10b shows the most favorable adsorption position of O2 on 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' In that case, O2 is in the ring of the hexagon at the distance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='73 Å, and the lowest Eads of O2 on 2D Zn2VN3 is -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='14 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure S10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Atomic configurations of (a) H2O and (b) O2 on 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Eads= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='49 eV eV Eads= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='39 eV Eads= -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='40 eV b Eads=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='14 eV Eads=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='13 eV Eads=-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='13 eV The three highest occupied molecular orbitals (HOMO) of the H2O molecule, named according to the irreducible representation of the point group of H2O, are 1b1 (HOMO), 3a1 (HOMO-1), and 1b2 (HOMO-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' According to the LDOS plot (Figure S10a) the 3a1 orbital is most broadened due to its favored orbital mixing with the N atoms, confirming the interaction ability of 2D Zn2VN3 with H2O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The LDOS plot for the O2 molecule on 2D Zn2VN3 (Figure S10b) reflects additional O2-induced states within the band gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The half-filled 2π HOMO state aligns within the valence band maximum and allows the electrons to be excited to the O2 molecule, thereby creating holes in 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The 2π* LUMO state is located above the Fermi level at ~0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='90 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The presence of the O2-induced states within the band gap of 2D Zn2VN3 and the non-trivial adsorption/oxidation ability of O2 to 2D Zn2VN3 can alter its optical and electronic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure S11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' LDOS of (a) H2O and (b) O2 on 2D Zn2VN3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' (a) 100 Total 1b H,0 Zn LDOS (states/eV) N 100 3 0 2 Energy (eV) (b) 100 Total 0 T Zn LDOS (states/eV) N 2元 100 3 2 0 2 3 Energy (eV)Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Water splitting application of 2D Zn2VN3 Figure S12 shows the calculated VBM and CBM positions of 2D Zn2VN3 with the redox potential of H2O and oxidation levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' It is found that the VBM of 2D Zn2VN3 is below the oxidation potential of O2/H2O, while the CBM is also lower than the reduction potential of H2/ H2O, which indicates that 2D Zn2VN3 is suitable only for oxygen production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' Figure S12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' The calculated VBM and CBM positions of 2D Zn2VN3 with respect to the water redox potentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5- 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 CBM 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 vac 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 E 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 H2/H20 E VBM 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='5 02/H20 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} +page_content='0' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/i9FKT4oBgHgl3EQfwC46/content/2301.11897v1.pdf'} diff --git a/itAzT4oBgHgl3EQfbPyr/vector_store/index.faiss b/itAzT4oBgHgl3EQfbPyr/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..ca9ba7267eceabfdb345a37329f4710a10de9e6e --- /dev/null +++ b/itAzT4oBgHgl3EQfbPyr/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a3717268df9dd6d4f9948361c7ec576b42433b81a4f4a43535b6307d4bcacd72 +size 4587565 diff --git a/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf b/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..0feabfc08f3c471a9c4244b296a526294ad7c116 --- /dev/null +++ b/j9E1T4oBgHgl3EQf0QVw/content/2301.03453v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8d32598e22bd4cacaf590ed198ed7d21fc299fc03b91928f43aed73259881179 +size 1857502 diff --git a/j9E1T4oBgHgl3EQf0QVw/vector_store/index.faiss b/j9E1T4oBgHgl3EQf0QVw/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..966c30ea6cdd8ad30049f1fe986afa93fb54e42b --- /dev/null +++ b/j9E1T4oBgHgl3EQf0QVw/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:6982e1bb38e193e5c4df2052e41224a03afa6c42f0731cd04b59e67e54af1d2c +size 2752557 diff --git a/j9E1T4oBgHgl3EQf0QVw/vector_store/index.pkl b/j9E1T4oBgHgl3EQf0QVw/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b73ddbfbe8009a3cc7e336fc25d7049d17268d7b --- /dev/null +++ b/j9E1T4oBgHgl3EQf0QVw/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:cacafd33bb72e618ce70c3b2a3551e9884fedfacd4df7377dc613392e8a0179a +size 107161 diff --git a/jdE4T4oBgHgl3EQfsg0n/content/2301.05216v1.pdf b/jdE4T4oBgHgl3EQfsg0n/content/2301.05216v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..64654a91eee1d63b0cd6e6e0a61c9ca9c1f4aa24 --- /dev/null +++ b/jdE4T4oBgHgl3EQfsg0n/content/2301.05216v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:73422d82bd157d363a20a5e2c78fd34f42f06a069a36398a0afc6dc0ce04b552 +size 163273 diff --git a/jdE4T4oBgHgl3EQfsg0n/vector_store/index.pkl b/jdE4T4oBgHgl3EQfsg0n/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..b60adbe544d34127564d1fb877f0b18e702630a4 --- /dev/null +++ b/jdE4T4oBgHgl3EQfsg0n/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:00f361695f805fd6e742446c15fbb920ac99e2f0563ede97b7a27610709601f8 +size 99718 diff --git a/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf b/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5050473d06929ca2b7befd42bf81546f9c02a761 --- /dev/null +++ b/jdFPT4oBgHgl3EQfFjQr/content/2301.13000v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:502b4697c52d80f9842b3c449924515315addd8120edeacd8211ed521028d652 +size 4604664 diff --git a/kb_37/content/kb_37.pdf b/kb_37/content/kb_37.pdf new file mode 100644 index 0000000000000000000000000000000000000000..b556237f3e01ca645f00f82f53bf75e77ac87a7c --- /dev/null +++ b/kb_37/content/kb_37.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0b76bfa0836e8a70c446428adddb288e8bd135ecf2831803234e50ad866ae980 +size 739819 diff --git a/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf b/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4eedf36e4b06b9aac4be33f6cb71971b213bfec3 --- /dev/null +++ b/kdE3T4oBgHgl3EQfJgnA/content/2301.04345v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b05e9612c3da569f84b9fe9e33d38ed3284067d3b238e0cbb23a539ae5018c01 +size 1086826 diff --git a/kdE3T4oBgHgl3EQfJgnA/vector_store/index.faiss b/kdE3T4oBgHgl3EQfJgnA/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..763429c68f1f8bb0e63d033bed39dc37e5a1bf20 --- /dev/null +++ b/kdE3T4oBgHgl3EQfJgnA/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:60cff8f942b98c2dcb8165ec73ca94bbf40b57c14bf0d0d32e1a93b2cb7b81ce +size 2621485 diff --git a/kdE3T4oBgHgl3EQfJgnA/vector_store/index.pkl b/kdE3T4oBgHgl3EQfJgnA/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..20b4d53ba6583a81b17c419c29890e43a637f0ed --- /dev/null +++ b/kdE3T4oBgHgl3EQfJgnA/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8681bd3aa5543078f8938e128ea3d3dfc9f796532595006249df1071de4eeaf2 +size 85483 diff --git a/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf b/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..a0c62350f36eee3f295600b03a590198f4333158 --- /dev/null +++ b/l9FPT4oBgHgl3EQf3jX-/content/2301.13191v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:b42c70ab4ebe6fc4de00197b79990dc8b297134df1a421ca9f21e1968646ea25 +size 890406 diff --git a/l9FPT4oBgHgl3EQf3jX-/vector_store/index.pkl b/l9FPT4oBgHgl3EQf3jX-/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..679ffdda3fc3daad21cf3e66cafcb2237f8873ca --- /dev/null +++ b/l9FPT4oBgHgl3EQf3jX-/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f905ab8b753b77611748b5568dcae32c6a4de84f6cd0c7f21b99bd8b769ec1fe +size 189494 diff --git a/lNFPT4oBgHgl3EQf2zXD/content/tmp_files/2301.13188v1.pdf.txt b/lNFPT4oBgHgl3EQf2zXD/content/tmp_files/2301.13188v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..54d14a2f19c18d597baa44a1ba78df99f44bacc9 --- /dev/null +++ b/lNFPT4oBgHgl3EQf2zXD/content/tmp_files/2301.13188v1.pdf.txt @@ -0,0 +1,2577 @@ +Extracting Training Data from Diffusion Models +Nicholas Carlini∗1 +Jamie Hayes∗2 +Milad Nasr∗1 +Matthew Jagielski+1 +Vikash Sehwag+4 +Florian Tram`er+3 +Borja Balle†2 +Daphne Ippolito†1 +Eric Wallace†5 +1Google +2DeepMind +3ETHZ +4Princeton +5UC Berkeley +∗Equal contribution ++Equal contribution +†Equal contribution +Abstract +Image diffusion models such as DALL-E 2, Imagen, and +Stable Diffusion have attracted significant attention due +to their ability to generate high-quality synthetic images. +In this work, we show that diffusion models memorize +individual images from their training data and emit them +at generation time. With a generate-and-filter pipeline, +we extract over a thousand training examples from state- +of-the-art models, ranging from photographs of individ- +ual people to trademarked company logos. We also train +hundreds of diffusion models in various settings to an- +alyze how different modeling and data decisions affect +privacy. Overall, our results show that diffusion models +are much less private than prior generative models such +as GANs, and that mitigating these vulnerabilities may +require new advances in privacy-preserving training. +1 +Introduction +Denoising diffusion models are an emerging class of +generative neural networks that produce images from +a training distribution via an iterative denoising pro- +cess [64, 66, 33]. Compared to prior approaches such +as GANs [30] or VAEs [46], diffusion models produce +higher-quality samples [18] and are easier to scale [56] +and control [51]. Consequently, they have rapidly be- +come the de-facto method for generating high-resolution +images, and large-scale models such as DALL-E 2 [56] +have attracted significant public interest. +The appeal of generative diffusion models is rooted +in their ability to synthesize novel images that are os- +tensibly unlike anything in the training set. Indeed, past +large-scale training efforts “do not find overfitting to be +an issue”, [60] and researchers in privacy-sensitive do- +mains have even suggested that diffusion models could +“protect[] the privacy [...] of real images” [37] by gen- +erating synthetic examples [13, 14, 59, 2, 53]. This line +of work relies on the assumption that diffusion models +do not memorize and regenerate their training data. If +they did, it would violate all privacy guarantees and raise +numerous questions regarding model generalization and +“digital forgery” [65]. +Training Set +Generated Image +Caption: Living in the light +with Ann Graham Lotz +Prompt: +Ann Graham Lotz +Figure 1: Diffusion models memorize individual train- +ing examples and generate them at test time. Left: an +image from Stable Diffusion’s training set (licensed CC +BY-SA 3.0, see [49]). Right: a Stable Diffusion gen- +eration when prompted with “Ann Graham Lotz”. The +reconstruction is nearly identical (ℓ2 distance = 0.031). +In this work, we demonstrate that state-of-the-art dif- +fusion models do memorize and regenerate individual +training examples. To begin, we propose and implement +new definitions for “memorization” in image models. We +then devise a two-stage data extraction attack that gener- +ates images using standard approaches, and flags those +that exceed certain membership inference scoring crite- +ria. Applying this method to Stable Diffusion [58] and +Imagen [60], we extract over a hundred near-identical +replicas of training images that range from personally +identifiable photos to trademarked logos (e.g., Figure 1). +To better understand how and why memorization oc- +curs, we train hundreds of diffusion models on CIFAR- +10 to analyze the impact of model accuracy, hyperparam- +eters, augmentation, and deduplication on privacy. Dif- +fusion models are the least private form of image mod- +els that we evaluate—for example, they leak more than +twice as much training data as GANs. Unfortunately, we +also find that existing privacy-enhancing techniques do +not provide an acceptable privacy-utility tradeoff. Over- +all, our paper highlights the tension between increasingly +powerful generative models and data privacy, and raises +questions on how diffusion models work and how they +should be responsibly deployed. +1 +arXiv:2301.13188v1 [cs.CR] 30 Jan 2023 + +2 +Background +Diffusion models. Generative image models have a long +history (see [29, Chapter 20]). Generative Adversarial +Networks (GANs) [30] were the breakthrough that first +enabled the generation of high-fidelity images at scale [6, +44]. But over the last two years, diffusion models [64] +have largely displaced GANs: they achieve state-of-the- +art results on academic benchmarks [18] and form the +basis of all recently popularized image generators such as +Stable Diffusion [58], DALL-E 2 [57, 56], Runway [58], +Midjourney [67] and Imagen [60]. +Denoising Diffusion Probabilistic Models [33]1 are +conceptually simple: they are nothing more than im- +age denoisers. During training, given a clean image x, +we sample a time-step t ∈ [0,T] and a Gaussian noise +vector ε ∼ N (0,I), to produce a noised image x′ ← +√atx+√1−atε, for some decaying parameter at ∈ [0,1] +where a0 = 1 and aT = 0. A diffusion model fθ removes +the noise ε to recover the original image x by predicting +the noise that was added by stochastically minimizing the +objective 1 +N ∑i Et,ε L (xi,t,ε; fθ), where +L (xi,t,ε; fθ) = ∥ε − fθ(√atxi + +� +1−atε,t)∥2 +2 . (1) +Despite being trained with this simple denoising ob- +jective, diffusion models can generate high-quality im- +ages by first sampling a random vector zT ∼ N (0,I) and +then applying the diffusion model fθ to remove the noise +from this random “image”. To make the denoising pro- +cess easier, we do not remove all of the noise at once— +we instead iteratively apply the model to slowly remove +noise. Formally, the final image z0 is obtained from zT by +iterating the rule zt−1 = fθ(zt,t)+σtN (0,I) for a noise +schedule σt (dependent on at) with σ1 = 0. This process +relies on the fact that the model fθ was trained to denoise +images with varying degrees of noise. Overall, running +this iterative generation process (which we will denote +by Gen) with large-scale diffusion models produces re- +sults that resemble natural images. +Some diffusion models are further conditioned to gen- +erate a particular type of image. Class-conditional dif- +fusion models take as input a class-label (e.g., “dog” or +“cat”) alongside the noised image to produce a particu- +lar class of image. Text-conditioned models take this one +step further and take as input the text embedding of some +prompt (e.g., “a photograph of a horse on the moon”) us- +ing a pre-trained language encoder (e.g., CLIP [54]). +1Our description of diffusion models below omits a number of sig- +nificant details. However, these details are orthogonal to the results of +our attacks and we omit them for simplicity. +Training data privacy attacks. +Neural networks of- +ten leak details of their training datasets. Membership +inference attacks [62, 80, 8] answer the question “was +this example in the training set?” and present a mild +privacy breach. Neural networks are also vulnerable to +more powerful attacks such as inversion attacks [27, 81] +that extract representative examples from a target class, +attribute inference attacks [28] that reconstruct subsets +of attributes of training examples, and extraction attacks +[10, 11, 5] that completely recover training examples. In +this paper, we focus on each of these three attacks when +applied to diffusion models. +Concurrent work explores the privacy of diffusion +models. +Wu et al. +[78] and Hu et al. +[34] perform +membership inference attacks on diffusion models; our +results use more sophisticated attack methods and study +stronger privacy risks such as data extraction. Somepalli +et al. [65] show several cases where (non-adversarially) +sampling from a diffusion model can produce memorized +training examples. However, they focus mainly on com- +paring the semantic similarity of generated images to the +training set, i.e., “style copying”. In contrast, we focus +on worst-case privacy under a much more restrictive no- +tion of memorization, and perform our attacks on a wider +range of models. +3 +Motivation and Threat Model +There are two distinct motivations for understanding how +diffusion models memorize and regenerate training data. +Understanding privacy risks. +Diffusion models that +regenerate data scraped from the Internet can pose sim- +ilar privacy and copyright risks as language models [11, +7, 31]. For example, memorizing and regenerating copy- +righted text [11] and source code [35] has been pointed +to as indicators of potential copyright infringement [76]. +Similarly, copying images from professional artists has +been called “digital forgery” [65] and has spurred debate +in the art community. +Future diffusion models might also be trained on more +sensitive private data. Indeed, GANs have already been +applied to medical imagery [73, 20, 45], which under- +lines the importance of understanding the risks of gener- +ative models before we apply them to private domains. +Worse, a growing literature suggests that diffusion +models could create synthetic training data to “protect +the privacy and usage rights of real images” [37], and +production tools already claim to use diffusion models to +protect data privacy [71, 17, 12]. Our work shows diffu- +sion models may be unfit for this purpose. +2 + +Understanding generalization. +Beyond data privacy, +understanding how and why diffusion models memorize +training data may help us understand their generalization +capabilities. For instance, a common question for large- +scale generative models is whether their impressive re- +sults arise from truly novel generations, or are instead +the result of direct copying and remixing of their train- +ing data. By studying memorization, we can provide a +concrete empirical characterization of the rates at which +generative models perform such data copying. +In their diffusion model, Saharia et al. “do not find +over-fitting to be an issue, and believe further training +might improve overall performance“ [60], and yet we +will show that this model memorizes individual exam- +ples. It may thus be necessary to broaden our definitions +of overfitting to include memorization and related pri- +vacy metrics. Our results also suggest that Feldman’s +theory that memorization is necessary for generalization +in classifiers [24] may extend to generative models, rais- +ing the question of whether the improved performance +of diffusion models compared to prior approaches is pre- +cisely because diffusion models memorize more. +3.1 +Threat Model +Our threat model considers an adversary A that interacts +with a diffusion model Gen (backed by a neural network +fθ) to extract images from the model’s training set D. +Image-generation systems. +Unconditional diffusion +models are trained on a dataset D = {x1,x2,...,xn}. +When queried, the system outputs a generated image +xgen ← Gen(r) using a fresh random noise r as input. +Conditional models are trained on annotated images +(e.g., labeled or captioned) D = {(x1,c1),...,(xn,cn)} +and when queried with a prompt p, the system outputs +xgen ← Gen(p;r) using the prompt p and noise r. +Adversary capabilities. We consider two adversaries: +• A black-box adversary can query Gen to generate +images. If Gen is a conditional generator, the adver- +sary can provide arbitrary prompts p. The adversary +cannot control the system’s internal randomness r. +• A white-box adversary gets full access to the system +Gen and its internal diffusion model fθ. They can +control the model’s randomness and can thus use +the model to denoise arbitrary input images. +In both cases, we assume that an adversary who attacks a +conditional image generator knows the captions for some +images in the training set—thus allowing us to study the +worst-case privacy risk in diffusion models. +Adversary goals. We consider three broad types of ad- +versarial goals, from strongest to weakest attacks: +1. Data extraction: The adversary aims to recover an +image from the training set x ∈ D. The attack is +successful if the adversary extracts an image ˆx that +is almost identical (see Section 4.1) to some x ∈ D. +2. Data reconstruction: +The adversary has partial +knowledge of a training image x ∈ D (e.g., a sub- +set of the image) and aims to recover the full image. +This is an image-analog of an attribute inference at- +tack [80], which aims to recover unknown features +from partial knowledge of an input. +3. Membership inference: Given an image x, the ad- +versary aims to infer whether x is in the training set. +3.2 +Ethics and Broader Impact +Training data extraction attacks can present a threat to +user privacy. We take numerous steps to mitigate any +possible harms from our paper. First, we study mod- +els that are trained on publicly-available images (e.g., +LAION and CIFAR-10) and therefore do not expose any +data that was not already available online. +Nevertheless, data that is available online may not +have been intended to be available online. LAION, for +example, contains unintentionally released medical im- +ages of several patients [23]. +We also therefore en- +sure that all images shown in our paper are of pub- +lic figures (e.g., politicians, musicians, actors, or au- +thors) who knowingly chose to place their images on- +line. As a result, inserting these images in our paper +is unlikely to cause any unintended privacy violation. +For example, Figure 1 comes from Ann Graham Lotz’s +Wikipedia profile picture and is licensed under Creative +Commons, which allows us to “redistribute the material +in any medium” and “remix, transform, and build upon +the material for any purpose, even commercially”. +Third, we shared an advance copy of this paper with +the authors of each of the large-scale diffusion models +that we study. +This gave the authors and their corre- +sponding organizations the ability to consider possible +safeguards and software changes ahead of time. +In total, we believe that publishing our paper and pub- +licly disclosing these privacy vulnerabilities is both eth- +ical and responsible. Indeed, at the moment, no one ap- +pears to be immediately harmed by the (lack of) privacy +of diffusion models; our goal with this work is thus to +make sure to preempt these harms and encourage respon- +sible training of diffusion models in the future. +3 + +4 +Extracting Training Data from State-of- +the-art Diffusion Models +We begin our paper by extracting training images from +large, pre-trained, high-resolution diffusion models. +4.1 +Defining Image Memorization +Most existing literature on training data extraction fo- +cuses on text language models, where a sequence is said +to be “extracted” and “memorized” if an adversary can +prompt the model to recover a verbatim sequence from +the training set [11, 41]. Because we work with high- +resolution images, verbatim definitions of memorization +are not suitable. Instead, we define a notion of approxi- +mate memorization based on image similarity metrics. +Definition 1 ((ℓ,δ)-Diffusion Extraction) [adapted +from [11]]. We say that an example x is extractable from +a diffusion model fθ if there exists an efficient algorithm +A (that does not receive x as input) such that ˆx = A ( fθ) +has the property that ℓ(x, ˆx) ≤ δ. +Here, ℓ is a distance function and δ is a threshold that +determines whether we count two images as being iden- +tical. In this paper, unless otherwise noted we follow +Balle et al. [5] and use the Euclidean 2-norm distance +ℓ2(a,b) = +� +∑i(ai −bi)2/d where d is the dimension of +the inputs to normalize ℓ ∈ [0,1]. Given this definition of +extractability, we can now define memorization. +Definition 2 ((k,ℓ,δ)-Eidetic Memorization) [adapted +from [11]]. We say that an example x is (k,ℓ,δ)-Eidetic +memorized 2 by a diffusion model if x is extractable from +the diffusion model, and there are at most k training +examples ˆx ∈ X where ℓ(x, ˆx) ≤ δ. +Again, ℓ is a distance function and δ is its correspond- +ing threshold. The constant k quantifies the number of +near-duplicates of x in the dataset. If k is a small frac- +tion of the data, then memorization is likely problematic. +When k is a larger fraction of data, memorization might +be expected—but it could still be problematic, e.g., if the +duplicated data is copyrighted. +2This paper covers a very restricted definition of “memorization”: +whether diffusion models can be induced to generate near-copies of +some training examples when prompted with appropriate instructions. +We will describe an approach that can generate images that are close +approximations of some training images (especially images that are fre- +quently represented in the training dataset through duplication or other +means). There is active discussion within the technical and legal com- +munities about whether the presence of this type of “memorization” +suggests that generative neural networks “contain” their training data. +Figure 2: We do not count the generated image of Obama +(at left) as memorized because it has a high ℓ2 distance +to every training image. The four nearest training images +are shown at right, each has a distance above 0.3. +Restrictions of our definition. Our definition of extrac- +tion is intentionally conservative as compared to what +privacy concerns one might ultimately have. +For ex- +ample, if we prompt Stable Diffusion to generate “A +Photograph of Barack Obama,” it produces an entirely +recognizable photograph of Barack Obama but not an +near-identical reconstruction of any particular training +image. +Figure 2 compares the generated image (left) +to the 4 nearest training images under the Euclidean 2- +norm (right). Under our memorization definition, this +image would not count as memorized. Nevertheless, the +model’s ability to generate (new) recognizable pictures +of certain individuals could still cause privacy harms. +4.2 +Extracting Data from Stable Diffusion +We now extract training data from Stable Diffusion: the +largest and most popular open-source diffusion model +[58]. +This model is an 890 million parameter text- +conditioned diffusion model trained on 160 million im- +ages. +We generate from the model using the default +PLMS sampling scheme at a resolution of 512×512 pix- +els. As the model is trained on publicly-available images, +we can easily verify our attack’s success and also mit- +igate potential harms from exposing the extracted data. +We begin with a black-box attack. +Identifying duplicates in the training data. To reduce +the computational load of our attack, as is done in [65], +we bias our search towards duplicated training examples +because these are orders of magnitude more likely to be +memorized than non-duplicated examples [47, 41]. +If we search for images that are bit-for-bit identically +duplicated in the training dataset, we would significantly +undercount the true rate of duplication. Instead, we ac- +count for near-duplication. Ideally, we would search for +any training examples that are nearly duplicated with a +4 + +Original: +Generated: +Figure 3: Examples of the images that we extract from Stable Diffusion v1.4 using random sampling and our mem- +bership inference procedure. The top row shows the original images and the bottom row shows our extracted images. +pixel-level ℓ2 distance below some threshold. But this +is computationally intractable, as it would require an all- +pairs comparison of 160 million images in Stable Dif- +fusion’s training set, each of which is a 512 × 512 × 3 +dimensional vector. Instead, we first embed each image +to a 512 dimensional vector using CLIP [54], and then +perform the all-pairs comparison between images in this +lower-dimensional space (increasing efficiency by over +1500×). We count two examples as near-duplicates if +their CLIP embeddings have a high cosine similarity. For +each of these near-duplicated images, we use the corre- +sponding captions as the input to our extraction attack. +4.2.1 +Extraction Methodology +Our extraction approach adapts the methodology from +prior work [11] to images and consists of two steps: +1. Generate many examples using the diffusion model +in the standard sampling manner and with the +known prompts from the prior section. +2. Perform membership inference to separate the +model’s novel generations from those generations +which are memorized training examples. +Generating many images. +The first step is trivial but +computationally expensive: we query the Gen function +in a black-box manner using the selected prompts as in- +put. To reduce the computational overhead of our experi- +ments, we use the timestep-resampled generation imple- +mentation that is available in the Stable Diffusion code- +base [58]. This process generates images in a more ag- +gressive fashion by removing larger amounts of noise at +each time step and results in slightly lower visual fidelity +at a significant (∼ 10×) performance increase. We gener- +ate 500 candidate images for each text prompt to increase +the likelihood that we find memorization. +Performing membership inference. +The second step +requires flagging generations that appear to be memo- +rized training images. +Since we assume a black-box +threat model in this section, we do not have access to +the loss and cannot exploit techniques from state-of-the- +art membership inference attacks [11]. We instead de- +sign a new membership inference attack strategy based +on the intuition that for diffusion models, with high prob- +ability Gen(p;r1) ̸= Gen(p;r2) for two different random +initial seeds r1,r2. On the other hand, if Gen(p;r1) ≈d +Gen(p;r2) under some distance measure d, it is likely +that these generated samples are memorized examples. +The 500 images that we generate for each prompt have +different (but unknown) random seeds. We can therefore +construct a graph over the 500 generations by connect- +ing an edge between generation i and j if xi ≈d xj. If +the largest clique in this graph is at least size 10 (i.e., +≥ 10 of the 500 generations are near-identical), we pre- +dict that this clique is a memorized image. Empirically, +clique-finding is more effective than searching for pairs +of images x1 ≈d x2 as it has fewer false positives. +To compute the distance measure d among the images +in the clique, we use a modified Euclidean ℓ2 distance. +In particular, we found that many generations were often +spuriously similar according to ℓ2 distance (e.g., they all +had gray background). We therefore instead divide each +image into 16 non-overlapping 128×128 tiles and mea- +sure the maximum of the ℓ2 distance between any pair of +image tiles between the two images. +4.2.2 +Extraction Results +In order to evaluate the effectiveness of our attack, we +select the 350,000 most-duplicated examples from the +training dataset and generate 500 candidate images for +each of these prompts (totaling 175 million generated im- +ages). We first sort all of these generated images by or- +dering them by the mean distance between images in the +clique to identify generations that we predict are likely to +be memorized training data. We then take each of these +generated images and annotate each as either “extracted” +or “not extracted” by comparing it to the training images +under Definition 1. We find 94 images are (ℓ2,0.15)- +extracted. To ensure that these images not only match +5 + +200 +20 +40 +60 +80 +100 +Memorized Examples Extracted +0.6 +0.7 +0.8 +0.9 +1.0 +Attack Precision +Manual Inspection +(ℓ2, 0.15)-Extraction +Figure 4: Our attack reliably separates novel genera- +tions from memorized training examples, under two def- +initions of memorization—either (ℓ2,0.15)-extraction or +manual human inspection of generated images. +some arbitrary definition, we also manually annotate the +top-1000 generated images as either memorized or not +memorized by visual analysis, and find that a further 13 +(for a total of 109 images) are near-copies of training +examples even if they do not fit our 2-norm definition. +Figure 3 shows a subset of the extracted images that are +reproduced with near pixel-perfect accuracy; all images +have an ℓ2 difference under 0.05. (As a point of refer- +ence, re-encoding a PNG as a JPEG with quality level 50 +results in an ℓ2 difference of 0.02 on average.) +Given our ordered set of annotated images, we can +also compute a curve evaluating the number of extracted +images to the attack’s false positive rate. +Our attack +is exceptionally precise: out of 175 million generated +images, we can identify 50 memorized images with 0 +false positives, and all our memorized images can be ex- +tracted with a precision above 50%. Figure 4 contains the +precision-recall curve for both memorization definitions. +Measuring (k,ℓ,δ)-eidetic memorization. +In Defini- +tion 2 we introduced an adaptation of Eidetic memo- +rization [11] tailored to the domain of generative im- +age models. As mentioned earlier, we compute similar- +ity between pairs of images with a direct ℓ2 pixel-space +similarity. This analysis is computationally expensive3 +as it requires comparing each of our memorized images +against each of the 160 million training examples. We +set δ = 0.1 as this threshold is sufficient to identify al- +3In practice it is even more challenging: for non-square images, +Stable Diffusion takes a random square crop, and so to check if the +generated image x matches a non-square training image y we must try +all possible alignments between x on top of the image y. +10 +30 +100 +300 +1000 +3000 +Number of duplicates +0 +10 +20 +30 +Frequency +Figure 5: Our attack extracts images from Stable Diffu- +sion most often when they have been duplicated at least +k = 100 times; although this should be taken as an upper +bound because our methodology explicitly searches for +memorization of duplicated images. +most all small image corruptions (e.g., JPEG compres- +sion, small brightness/contrast adjustments) but has very +few false positives. +Figure 5 shows the results of this analysis. While we +identify little Eidetic memorization for k < 100, this is +expected due to the fact we choose prompts of highly- +duplicated images. Note that at this level of duplication, +the duplicated examples still make up just one in a mil- +lion training examples. These results show that duplica- +tion is a major factor behind training data extraction. +Qualitative analysis. +The majority of the images that +we extract (58%) are photographs with a recognizable +person as the primary subject; the remainder are mostly +either products for sale (17%), logos/posters (14%), or +other art or graphics. We caution that if a future diffusion +model were trained on sensitive (e.g., medical) data, then +the kinds of data that we extract would likely be drawn +from this sensitive data distribution. +Despite the fact that these images are publicly acces- +sible on the Internet, not all of them are permissively li- +censed. We find that a significant number of these im- +ages fall under an explicit non-permissive copyright no- +tice (35%). Many other images (61%) have no explicit +copyright notice but may fall under a general copyright +protection for the website that hosts them (e.g., images +of products on a sales website). Several of the images +that we extracted are licensed CC BY-SA, which requires +“[to] give appropriate credit, provide a link to the li- +cense, and indicate if changes were made.” Stable Dif- +fusion thus memorizes numerous copyrighted and non- +6 + +permissive-licensed images, which the model may repro- +duce without the accompanying license. +4.3 +Extracting Data from Imagen +While Stable Diffusion is the best publicly-available +diffusion model, +there are non-public models that +achieve stronger performance using larger models and +datasets [56, 60]. Prior work has found that larger mod- +els are more likely to memorize training data [11, 9] and +we thus study Imagen [60], a 2 billion parameter text- +to-image diffusion model. While individual details dif- +fer between Imagen’s and Stable Diffusion’s implemen- +tation and training scheme, these details are independent +of our extraction results. +We follow the same procedure as earlier but focus +on the top-1000 most duplicated prompts for computa- +tional reasons. We then generate 500 images for each +of these prompts, and compute the ℓ2 similarity between +each generated image and the corresponding training +image. +By repeating the same membership inference +steps as above—searching for cliques under patched ℓ2 +distance–we identify 23 of these 1,000 images as mem- +orized training examples.4 This is significantly higher +than the rate of memorization in Stable Diffusion, and +clearly demonstrates that memorization across diffusion +models is highly dependent on training settings such as +the model size, training time, and dataset size. +4.4 +Extracting Outlier Examples +The attacks presented above succeed, but only at extract- +ing images that are highly duplicated. +This “high k” +memorization may be problematic, but as we mentioned +previously, the most compelling practical attack would +be to demonstrate memorization in the “low k” regime. +We now set out to achieve this goal. In order to find +non-duplicated examples likely to be memorized, we +take advantage of the fact that while on average models +often respect the privacy of the majority of the dataset, +there often exists a small set of “outlier” examples whose +privacy is more significantly exposed [24]. And so in- +stead of searching for memorization across all images, +we are more likely to succeed if we focus our effort on +these outlier examples. +But how should we find which images are poten- +tially outliers? Prior work was able to train hundreds +of models on subsets of the training dataset and then +4Unfortunately, because the Imagen training dataset is not public, +we are unable to provide visual examples of successful reconstructions. +use an influence-function-style approach to identify ex- +amples that have a significant impact on the final model +weights [25]. Unfortunately, given the cost of training +even a single large diffusion model is in the millions-of- +dollars, this approach will not be feasible here. +Therefore we take a simpler approach. We first com- +pute the CLIP embedding of each training example, and +then compute the “outlierness” of each example as the +average distance (in CLIP embedding space) to its 1,000 +nearest neighbors in the training dataset. +Results. +Surprisingly, we find that attacking out-of- +distribution images is much more effective for Imagen +than it is for Stable Diffusion. On Imagen, we attempted +extraction of the 500 images with the highest out-of- +distribution score. Imagen memorized and regurgitated +3 of these images (which were unique in the training +dataset). In contrast, we failed to identify any memo- +rization when applying the same methodology to Stable +Diffusion—even after attempting to extract the 10,000 +most-outlier samples. +Thus, Imagen appears less pri- +vate than Stable Diffusion both on duplicated and non- +duplicated images. We believe this is due to the fact that +Imagen uses a model with a much higher capacity com- +pared to Stable diffusion, which allows for more memo- +rization [9]. Moreover, Imagen is trained for more iter- +ations and on a smaller dataset, which can also result in +higher memorization. +5 +Investigating Memorization +The above experiments are visually striking and clearly +indicate that memorization is pervasive in large diffusion +models—and that data extraction is feasible. But these +experiments do not explain why and how these models +memorize training data. In this section we train smaller +diffusion models and perform controlled experiments in +order to more clearly understand memorization. +Experimental setup. +For the remainder of this sec- +tion, we focus on diffusion models trained on CIFAR-10. +We use state-of-the-art training code 5 to train 16 diffu- +sion models, each on a randomly-partitioned half of the +CIFAR-10 training dataset. We run three types of pri- +vacy attacks: membership inference attacks, attribute in- +5We either directly use OpenAI’s Improved Diffusion repos- +itory (https://github.com/openai/improved-diffusion) in +Section 5.1, or our own re-implementation in all following sections. +Models trained with our re-implementation achieve almost identical +FID to the open-sourced models. We use half the dataset as is stan- +dard in privacy analyses [8]. +7 + +Figure 6: Direct 2-norm measurement fails to identify +memorized CIFAR-10 examples. Each of the above im- +ages have a ℓ2 distance of less than 0.05, yet only one +(the car) is actually a memorized training example. +ference attacks, and data reconstruction attacks. For the +membership inference attacks, we train class-conditional +models that reach an FID below 3.5 (see Figure 11), plac- +ing them in the top-30 generative models on CIFAR-10 +[16]. +For reconstruction attacks (Section 5.1) and at- +tribute inference attacks with inpainting (Section 5.3), +we train unconditional models with an FID below 4. +5.1 +Untargeted Extraction +Before devling deeper into understanding memorization, +we begin by validating that memorization does still occur +in our smaller models. Because these models are not text +conditioned, we focus on untargeted extraction. Specif- +ically, given our 16 diffusion models trained on CIFAR- +10, we unconditionally generate 216 images from each +model for a total of 220 candidate images. Because we +will later develop high-precision membership inference +attacks, in this section we directly search for memorized +training examples among all our million generated exam- +ples. Thus this is not an attack per se, but rather verifying +the capability of these models to memorize. +Identifying matches. In the prior section, we performed +targeted attacks and could therefore check for successful +memorization by simply computing the ℓ2 distance be- +tween the target image and the generated image. Here, +as we perform an all-pairs comparison, we find that us- +ing an uncalibrated ℓ2 threshold fails to accurately iden- +tify memorized training examples. For example, if we set +a highly-restrictive threshold of 0.05, then nearly all “ex- +tracted” images are of entirely blue skies or green land- +scapes (see Figure 6). We explored several other met- +rics (including perceptual distances like SSIM or CLIP +embedding distance) but found that none could reliably +identify memorized training images for CIFAR-10. +We instead define an image as extracted if the ℓ2 dis- +tance to its nearest neighbor in the training set is abnor- +mally low compared to all other training images. Fig- +ure 7 illustrates this by computing the ℓ2 distance be- +tween two different generated images and every image +in the CIFAR-10 training dataset. The left figure shows a +failed extraction attempt; despite the fact that the nearest +0.00 +0.25 +0.50 +0.75 +1.00 + L2 distance between generated and training images +100 +101 +102 +103 +Frequency +0.00 +0.25 +0.50 +0.75 +1.00 +Figure 7: Per-image ℓ2 thresholds are necessary to sep- +arate memorized images from novel generations on a +CIFAR-10 model. Each plot shows the distribution of ℓ2 +distances from a generated image to all training images +(along with the image and the nearest training image). +Left shows a typical distribution for a non-memorized +image. Right shows a memorized image distribution; +while the most similar training image has high absolute +ℓ2 distance, it is abnormally low for this distribution. The +dashed black line shows our adaptive ℓ2 threshold. +training image has an ℓ2 distance of just 0.06, this dis- +tance is on par with the distance to many other training +images (i.e., all images that contain a blue sky). In con- +trast, the right plot shows a successful extraction attack. +Here, even though the ℓ2 distance to the nearest train- +ing image is higher than for the prior failed attack (0.07), +this value is unusually small compared to other training +images which almost all are at a distance above 0.2. +We thus slightly modify our attack to use the distance +ℓ(ˆx,x;Sˆx) = +ℓ2(ˆx,x) +α ·Ey∈Sˆx[ℓ2(ˆx,y)]. +where Sˆx is the set containing the n closest elements from +the training dataset to the example ˆx. This distance is +small if the extracted image x is much closer to the train- +ing image ˆx compared to the n closest neighbors of ˆx in +the training set. We run our attack with α = 0.5 and +n = 50. Our attack was not sensitive to these choices. +Results. +Using the above methodology we iden- +tify 1,280 unique extracted images from the CIFAR-10 +dataset (2.5% of the entire dataset).6 In Figure 8 we show +a selection of training examples that we extract and full +results are shown in Figure 17 in the Appendix. +6Some CIFAR-10 training images are generated multiple times. In +these cases, we only count the first generation as a successful attack. +Further, because the CIFAR-10 training dataset contains many dupli- +cate images, we do not count two generations of two different (but du- +plicated) images in the training dataset. +8 + +Figure 8: Selected training examples that we extract from a diffusion model trained on CIFAR-10 by sampling from +the model 1 million times. Top row: generated output from a diffusion model. Bottom row: nearest (ℓ2) example +from the training dataset. Figure 17 in the Appendix contains all 1,280 unique extracted images. +5.2 +Membership Inference Attacks +We now evaluate membership inference with more tra- +ditional attack techniques that use white-box access, as +opposed to Section 4.2.1 that assumed black-box access. +We will show that all examples have significant privacy +leakage under membership inference attacks, compared +to the small fraction that are sensitive to data extraction. +We consider two membership inference attacks on our +class-conditional CIFAR-10-trained diffusion models.7 +The loss threshold attack. Yeom et al. [80] introduce +the simplest membership inference attack: because mod- +els are trained to minimize their loss on the training set, +we should expect that training examples have lower loss +than non-training examples. The loss threshold attack +thus computes the loss l = L (x; f) and reports “mem- +ber” if l < τ for some chosen threshold τ and otherwise +“non-member’. The value of τ can be selected to max- +imize a desired metric (e.g., true positive rate at some +fixed false positive rate or the overall attack accuracy). +The Likelihood Ratio Attack (LiRA). +Carlini et al. +[8] introduce the state-of-the-art approach to performing +membership inference attacks. LiRA first trains a col- +lection of shadow models, each model on random sub- +sets of the training dataset. LiRA then computes the loss +L (x; fi) for the example x under each of these shadow +models fi. These losses are split into two sets: the losses +IN = {lini} for the example x under the shadow models +{ fi} that did see the example x during training, and the +losses OUT = {louti} for the example x under the shadow +models { f j} that did not see the example x during train- +ing. LiRA finishes the initialization process by fitting +Gaussians NIN to the IN set and NOUT to OUT set of +losses. Finally, to predict membership inference for a +new model f ∗, we compute l∗ = L (x, f ∗) and then mea- +sure whether Pr[l∗|NIN] > Pr[l∗|NOUT]. +Choosing a loss function. Both membership inference +attacks use a loss function L . In the case of classifica- +tion models, Carlini et al. [8] find that choosing a loss +7Appendix C.4 replicates these results for unconditional models. +function is one of the most important components of the +attack. We find that this effect is even more pronounced +for diffusion models. In particular, unlike classifiers that +have a single loss function (e.g., cross entropy) used to +train the model, diffusion models are trained to minimize +the reconstruction loss when a random quantity of Gaus- +sian noise ε has been added to an image. This means that +“the loss” of an image is not well defined—instead, we +can only ask for the loss L (x,t,ε) of an image x for a +certain timestep t with a corresponding amount of noise +ε (cf. Equation (1)). +We must thus compute the optimal timestep t at which +we should measure the loss. +To do so, we train 16 +shadow models each on a random 50% of the CIFAR- +10 training dataset. We then compute the loss for every +model, for every example in the training dataset, and ev- +ery timestep t ∈ [1,T] (T = 1,000 in the models we use). +Figure 9 plots the timestep used to compute the loss +against the attack success rate, measured as the true pos- +itive rate (TPR), i.e., the number of examples which truly +are members over the total number of members, at a fixed +false positive rate (FPR) of 1%, i.e., the fraction of exam- +ples which are incorrectly identified as members. Eval- +uating L at t ∈ [50,300] leads to the most successful +attacks. We conjecture that this a “Goldilock’s zone” for +membership inference: if t is too small, and so the noisy +image is similar to the original, then predicting the added +noise is easy regardless if the input was in the training +set; if t is too large, and so the noisy image is similar to +Gaussian noise, then the task is too difficult. Our remain- +ing experiments will evaluate L (·,t,·) at t = 100, where +we observed a TPR of 71% at an FPR of 1%. +5.2.1 +Baseline Attack Results +We now evaluate membership inference using our speci- +fied loss function. We follow recent advice [8] and evalu- +ate the efficacy of membership inference attacks by com- +paring their true positive rate to the false positive rate +on a log-log scale. In Figure 10, we plot the member- +ship inference ROC curve for the loss threshold attack +and LiRA. An out-of-the-box implementation of LiRA +9 + +1 +200 +400 +600 +800 +1000 +Diffusion timestep +0.0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +TPR@FPR=1% +Figure 9: We run membership inference using LiRA +and compute the diffusion model loss at different noise +timesteps on CIFAR-10. +Evaluating L (·,t,·) at t ∈ +[50,300] produces the best results. +achieves a true positive rate of over 70% at a false posi- +tive rate of just 1%. As a point of reference, state-of-the- +art classifiers are much more private, e.g., with a < 20% +TPR at 1% FPR [8]. This shows that diffusion models +are significantly less private than classifiers trained on +the same data. (In part this may be because diffusion +models are often trained far longer than classifiers.) +Qualitative analysis. +In Figure 20, we visualize the +least- and most-private images as determined by their +easiness to detect via LiRA. We find that the easiest- +to-attack examples are all extremely out-of-distribution +visually from the CIFAR-10 dataset. These images are +even more visually out-of-distribution compared to the +outliers identified by Feldman et al. [24] who produce a +similar set of images but for image classifiers. In con- +trast, the images that are hardest to attack are all dupli- +cated images. It is challenging to detect the presence or +absence of each of these images in the training dataset +because there is another identical image in the training +dataset that may have been present or absent—therefore +making the membership inference question ill-defined. +5.2.2 +Augmentations Improve Attacks +Membership inference attacks can also be improved by +reducing the variance in the loss signal [8, 79]. +We +study two ways to achieve this for diffusion models. +First, because our loss function has randomness (re- +call that to compute the reconstruction loss we mea- +sure the quantity L (x,t,ε) for a random noise sam- +ple ε ∼ N (0,I)), we can compute a better estimate of +the true loss by averaging over different noise samples: +L (x,t) = Eε∼N (0,I)[L (x,t,ε)]. +10 +3 +10 +2 +10 +1 +100 +False positive rate +10 +3 +10 +2 +10 +1 +100 +True positive rate +Strong LiRA. AUC: 0.997 +LiRA. AUC: 0.982 +Threshold. AUC: 0.613 +Figure 10: Membership inference ROC curve for a diffu- +sion model trained on CIFAR-10 using the loss threshold +attack, baseline LiRA, and “Strong LiRA” with repeated +queries and augmentation (§5.2.2). +By varying the number of point samples taken to es- +timate this expectation we can potentially increase the +attack success rate. And second, because our diffusion +models train on augmented versions of training images +(e.g., by flipping images horizontally), it makes sense +to compute the loss averaged over all possible augmen- +tations. Prior work has found that both of these attack +strategies are effective at increasing the efficacy of mem- +bership inference attacks for classifiers [8, 39], and we +find they are effective here as well. +Improved attack results. +Figure 10 shows the effect +of combining both these strategies. Together they are re- +markably successful, and at a false positive rate of 0.1% +they increase the true positive rate by over a factor of +six from 7% to 44%. Figure 19 in the Appendix breaks +down the impact of each component: in Figure 19a we +increase the number of Monte Carlo samples from 1 (the +base LiRA attack) to 20, and in Figure 19b we augment +samples with a horizontal flip. +5.2.3 +Memorization Versus Utility +We train our diffusion models to reach state-of-the-art +levels of performance. +Prior work on language mod- +els has found that better models are often easier to at- +tack than less accurate models—intuitively, because they +extract more information from the same training dataset +[9]. Here we perform a similar experiment. +Attack results vs. FID. +To evaluate our generative +models, we use the standard Fr´echet Inception Distance +(FID) [32], where lower scores indicate higher qual- +ity. +Our previous CIFAR-10 results used models that +10 + +4 +6 +8 +FID +0.2 +0.4 +0.6 +0.8 +1.0 +TPR@FPR=1% +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Update step +1e6 +Figure 11: Better diffusion models are more vulnerable +to membership inference attacks; evaluating with TPR +at an FPR of 1%. As the FID decreases (corresponding +to a quality increase) the membership inference attack +success rate grows from 7% to nearly 100%. +achieved the best FID (on average 3.5) based on early +stopping. Here we evaluate models over the course of +training in Figure 11. We compute the attack success +rate as a function of FID, and we find that as the quality +of the diffusion model increases so too does the privacy +leakage. These results are concerning because they sug- +gest that stronger diffusion models of the future may be +even less private. +5.3 +Inpainting Attacks +Having performed untargeted extraction on CIFAR-10 +models, we now construct a targeted version of our at- +tack. +As mentioned earlier, performing a targeted at- +tack is complicated by the fact that these models do not +support textual prompting. +We instead provide guid- +ance by performing a form of attribute inference attack +[38, 80, 81] that we call an “inpainting attack”. Given +an image, we first mask out a portion of this image; our +attack objective is to recover the masked region. We then +run this attack on both training and testing images, and +compare the attack efficacy on each. Specifically, for +an image x, we mask some fraction of pixels to create +a masked image xm, and then use the trained model to re- +construct the image as xrec. The exact algorithm we use +for inpainting is given in Lugmayr et al. [48]. +Because diffusion model inpainting is stochastic (it de- +pends on the random sample ε ∼ N (0,I)), we create a +set of inpainted images Xrec = {x1 +rec,x2 +rec,...,xn +rec}, where +we set n = 5,000. For each xrec ∈ Xrec, we compute the +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +2 distance when x isn't in training +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +2 distance when x is in training +Cat example +Bird example +100 other samples +Figure 12: Evaluating inpainting attacks on 100 CIFAR- +10 examples, measuring the ℓ2 distance between images +and their inpainted reconstructions when we mask out +the left half of the image for 100 randomly selected im- +ages. We also plot the ℓ2 distances for the bird and cat +examples shown in Figure 13. When an adversary has +partial knowledge of an image, inpainting attacks work +far better than typical data extraction. +diffusion model’s loss on this sample (at timestep 100) +divided by a shadow model’s loss that was not trained +on the sample. We then use this score to identify the +highest-scoring reconstructions xrec ∈ Xrec. +Results. +Our specific attack masks out the left half of +an image and applies the diffusion model on the right +half of the image to inpaint the rest. We repeat this pro- +cess 5000 times and take the top-10 scoring reconstruc- +tions using a membership inference attack. We repeat +this attack for 100 images using diffusion models that +are trained with and without the images. Figure 12 com- +pares the average distance between the sample and the +ten highest scoring inpainted samples. This allows us +to show our inpainting attacks have succeed: the recon- +struction loss is substantially better in terms of ℓ2 dis- +tance when the image is in the training set than when not. +Figure 13 also shows qualitative examples of this attack. +The highest-scoring reconstruction looks visually similar +to the target image when the target is in training and does +not resemble the target when it is not in training. Over- +all, these results show that an adversary who has partial +knowledge of an image can substantially improve their +extraction results. We conduct a more thorough analysis +of inpainting attacks in Appendix D. +11 + +Target: x +Masked: xm +Reconstruction when x +is in training. +Reconstruction when x +is not in training. +Target: x +Masked: xm +Reconstruction when x +is in training. +Reconstruction when x +is not in training. +Figure 13: +Inpainting-based reconstruction attack on +CIFAR-10. Given an image from CIFAR-10 (first col- +umn), we randomly mask half of the image (second col- +umn), and then inpaint the image for a model which con- +tained this image in the training set (third column) versus +inpainting the image for a model which did not contain +this image in the training set (fourth column). +6 +Comparing Diffusion Models to GANs +Are diffusion models more or less private than compet- +ing generative modeling approaches? In this section we +take a first look at this question by comparing diffu- +sion models to Generative Adversarial Networks (GANs) +[30, 61, 55], an approach that has held the state-of-the-art +results for image generation for nearly a decade. +Unlike diffusion models that are explicitly trained to +memorize and reconstruct their training datasets, GANs +are not. Instead, GANs consist of two competing neu- +ral networks: a generator and a discriminator. Similar to +diffusion models, the generator receives random noise as +input, but unlike a diffusion model, it must convert this +noise to a valid image in a single forward pass. To train +a GAN, the discriminator is trained to predict if an im- +age comes from the generator or not, and the generator +is trained to fool the discriminator. As a result, GANs +differ from diffusion models in that their generators are +only trained using indirect information about the train- +ing data (i.e., using gradients from the discriminator) be- +cause they never receive training data as input, whereas +diffusion models are explicitly trained to reconstruct the +training set. +Membership inference attacks. +We first propose a +privacy attack methodology for GANs.8 We initially fo- +cus on membership inference attacks, where following +Balle et al. [5], we assume access to both the discrimi- +nator and generator. We perform membership inference +using the loss threshold [80] and LiRA [8] attacks, where +8While existing privacy attacks exist for GANs, they were proposed +before the latest advancements in privacy attack techniques, requiring +us to develop our own methods which out-perform prior work. +Architecture +Images Extracted +FID +GANs +StyleGAN-ADA [43] +150 +2.9 +DiffBigGAN [82] +57 +4.6 +E2GAN [69] +95 +11.3 +NDA [63] +70 +12.6 +WGAN-ALP [68] +49 +13.0 +DDPMs +OpenAI-DDPM [52] +301 +2.9 +DDPM [33] +232 +3.2 +Table 1: The number of training images that we extract +from different off-the-shelf pretrained generative mod- +els out of 1 million unconditional generations. We show +GAN models sorted by FID (lower is better) on the top +and diffusion models on the bottom. Overall, we find +that diffusion models memorize more than GAN models. +Moreover, better generative models (lower FID) tend to +memorize more data. +we use the discriminator’s loss as the metric. To per- +form LiRA, we follow a similar methodology as Sec- +tion 5 and train 256 individual GAN models each on a +random 50% split of the CIFAR-10 training dataset but +otherwise leave training hyperparameters unchanged. +We study three GAN architectures, all implemented +using the StudioGAN framework [42]: BigGAN [6], +MHGAN [74], and StyleGAN [44]. Figure 14 shows the +membership inference results. Overall, diffusion models +have higher membership inference leakage, e.g., diffu- +sion models had 50% TPR at a FPR of 0.1% as compared +to < 30% TPR for GANs. This suggests that diffusion +models are less private than GANs for membership in- +ference attacks under default training settings, even when +the GAN attack is strengthened due to having access to +the discriminator (which would be unlikely in practice, +as only the generator is necessary to create new images). +Data extraction results. +We next turn our attention +away from measuring worst-case privacy risk and focus +our attention on more practical black-box extraction at- +tacks. +We follow the same procedure as Section 5.1, +where we generate 220 images from each model architec- +ture and identify those that are near-copies of the training +data using the same similarity function as before. Again +we only consider non-duplicated CIFAR-10 training im- +ages in our counting. For this experiment, instead of us- +ing models we train ourselves (something that was neces- +sary to run LiRA), we study five off-the-shelf pre-trained +GANs: WGAN-ALP [68], E2GAN [69], NDA [63], +DiffBigGAN [82], and StyleGAN-ADA [43]. We also +evaluate two off-the-shelf DDPM diffusion model re- +leased by Ho et al. [33] and Nichol et al. [52]. Note that +all of these pre-trained models are trained by the origi- +12 + +10−5 +10−4 +10−3 +10−2 +10−1 +100 +False Positive Rate +10−5 +10−4 +10−3 +10−2 +10−1 +100 +True Positive Rate +LiRA +auc=0.891, TPR@FPR=0.001: 0.109 +Global threshold +auc=0.878, TPR@FPR=0.001: 0.021 +(a) StyleGAN FID avg = 3.7 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +False Positive Rate +10−5 +10−4 +10−3 +10−2 +10−1 +100 +True Positive Rate +LiRA +auc=0.971, TPR@FPR=0.001: 0.258 +Global threshold +auc=0.511, TPR@FPR=0.001: 0.001 +(b) MHGAN FID avg = 7.9 +10−5 +10−4 +10−3 +10−2 +10−1 +100 +False Positive Rate +10−5 +10−4 +10−3 +10−2 +10−1 +100 +True Positive Rate +LiRA +auc=0.989, TPR@FPR=0.001: 0.418 +Global threshold +auc=0.967, TPR@FPR=0.001: 0.003 +(c) BigGAN FID avg = 7.7 +Figure 14: Membership inference results on GAN models using the loss threshold and LiRA attacks on the discrimi- +nator. Overall, GANs are significantly more private than diffusion models under default training configurations. +(a) StyleGAN +(b) MHGAN +(c) BigGAN +Figure 15: Selected training examples we extract from three GANs trained on CIFAR-10 for different architectures. +Top row: generated output from a diffusion model. Bottom row: nearest (ℓ2) example from the training dataset. +Figure 25 in the Appendix contains all unique extracted images. +nal authors to maximize utility on the entire CIFAR-10 +dataset rather than a random 50% split as in our prior +models trained for MIA. +Table 1 shows the number of extracted images for each +model and their corresponding FID. Overall, we find +that diffusion models memorize more data than GANs, +even when the GANs reach similar performance, e.g., the +best DDPM model memorizes 2× more than StyleGAN- +ADA but reaches the same FID. Moreover, generative +models (both GANs and diffusion models) tend to mem- +orize more data as their quality (FID) improves, e.g., +StyleGAN-ADA memorizes 3× more images than the +weakest GANs. +Using the GANs we trained ourselves, we show ex- +amples of the near-copy generations in Figure 15 for the +three GANs that we trained ourselves, and Figure 24 in +the Appendix shows every sample that we extract for +those models. +The Appendix also contains near-copy +generations from the five off-the-shelf GANs. Overall, +these results further reinforce the conclusion that diffu- +sion models are less private than GAN models. +We also surprisingly find that diffusion models and +GANs memorize many of the same images. In particular, +despite the fact that our diffusion model memorizes 1280 +images and a StyleGAN model we train on half of the +dataset memorizes 361 images, we find that 244 unique +images are memorized in common. If images were mem- +orized uniformly at random, we should expect on average +10 images would be memorized by both, giving excep- +tionally strong evidence that some images (p < 10−261) +are inherently less private than others. Understanding +why this phenomenon occurs is a fruitful direction for +future work. +13 + +7 +Defenses and Recommendations +Given the degree to which diffusion models memorize +and regenerate training examples, in this section we ex- +plore various defenses and practical strategies that may +help to reduce and audit model memorization. +7.1 +Deduplicating Training Data +In Section 4.2, we showed that many examples that are +easy to extract are duplicated many times (e.g., > 100) +in the training data. Similar results have been shown for +language models for text [11, 40] and data deduplica- +tion has been shown to be an effective mitigation against +memorization for those models [47, 41]. In the image +domain, simple deduplication is common, where images +with identical URLs and captions are removed, but most +datasets do not compute other inter-image similarity met- +rics such as ℓ2 distance or CLIP similarity. We thus en- +courage practitioners to deduplicate future datasets using +these more advanced notions of duplication. +Unfortunately, deduplication is not a perfect solution. +To better understand the effectiveness of data deduplica- +tion, we deduplicate CIFAR-10 and re-train a diffusion +model on this modified dataset. We compute image sim- +ilarity using the imagededup tool and deduplicate any +images that have a similarity above > 0.85. +This re- +moves 5,275 examples from the 50,000 total examples +in CIFAR-10. We repeat the same generation procedure +as Section 5.1, where we generate 220 images from the +model and count how many examples are regenerated +from the training set. The model trained on the dedu- +plicated data regenerates 986 examples, as compared to +1280 for the original model. +While not a substantial +drop, these results show that deduplication can mitigate +memorization. Moreover, we also expect that deduplica- +tion will be much more effective for models trained on +larger-scale datasets (e.g., Stable Diffusion), as we ob- +served a much stronger correlation between data extrac- +tion and duplication rates for those models. +7.2 +Differentially-Private Training +The gold standard technique to defend against privacy +attacks is by training with differential privacy (DP) guar- +antees [21, 22]. Diffusion models can be trained with +differentially-private stochastic gradient descent (DP- +SGD) [1], where the model’s gradients are clipped and +noised to prevent the model from leaking substantial in- +formation about the presence of any individual image in +the dataset. Applying DP-SGD induces a trade-off be- +tween privacy and utility, and recent work shows that +1 +2 +3 +4 +8 +16 +32 +64 +Duplicate Count +2 +4 +6 +8 +10 +Maximum Exposure +Random +Figure 16: Canary exposure (a measure of non-privacy) +as a function of duplicate count. Inserting a canary twice +is sufficient to reach maximum exposure. +DP-SGD can be applied to small-scale diffusion models +without substantial performance degradation [19]. +Unfortunately, we applied DP-SGD to our diffusion +model codebase and found that it caused the training on +CIFAR-10 to consistently diverge, even at high values for +ε (the privacy budget, around 50). In fact, even applying +a non-trivial gradient clipping or noising on their own +(both are required in DP-SGD) caused the training to fail. +We leave a further investigation of these failures to future +work, and we believe that new advances in DP-SGD and +privacy-preserving training techniques may be required +to train diffusion models in privacy-sensitive settings. +7.3 +Auditing with Canaries +In addition to implementing defenses, it is important +for practitioners to empirically audit their models to de- +termine how vulnerable they are in practice [36]. Our +attacks above represent one method to evaluate model +privacy. Nevertheless, our attacks are expensive, e.g., +our membership inference results require training many +shadow models, and thus lighter weight alternatives may +be desired. +One such alternative is to insert canary examples into +the training set, a common approach to evaluate mem- +orization in language models [10]. Here, one creates a +large “pool” of canaries, e.g., by randomly generating +noise images, and inserts a subset of the canaries into +the training set. After training, one computes the expo- +sure of the canaries, which roughly measures how many +bits were learned about the inserted canaries as compared +to the larger pool of not inserted canaries. This loss- +based metric only requires training one model and can +also be designed in a worst-case way (e.g., adversarial +worst-case images could be used). +To evaluate exposure for diffusion models, we gen- +14 + +erate canaries consisting of uniformly generated noise. +We then duplicate the canaries in the training set at dif- +ferent rates and measure the maximum exposure. Fig- +ure 16 shows the results. Here, the maximum exposure +is 10, and some canaries reach this exposure after being +inserted only twice. The exposure is not strictly increas- +ing with duplicate count, which may be a result of some +canaries being “harder” than others, and, ultimately, ran- +dom canaries we generate may not be the most effective +canaries to use to test memorization for diffusion models. +8 +Related Work +Memorization in language models. +Numerous past +works study memorization in generative models across +different domains, architectures, and threat models. One +area of recent interest is memorization in language mod- +els for text, where past work shows that adversaries can +extract training samples using two-step attack techniques +that resemble our approach [11, 47, 41, 40]. Our work +differs from these past results because we focus on the +image domain and also use more semantic notions of +data regeneration (e.g., using CLIP scores) as opposed +to focusing on exact verbatim repetition (although recent +language modeling work has begun to explore approxi- +mate memorization as well [35]). +Memorization in image generation. +Aside from lan- +guage modeling, past work also analyzes memorization +in image generation, mainly from the perspective of gen- +eralization in GANs (i.e., the novelty of model gener- +ations). For instance, numerous metrics exist to mea- +sure similarity with the training data [32, 3], the extent +of mode collapse [61, 15], and the impact of individual +training samples [4, 75]. Moreover, other work provides +insights into when and why GANs may replicate train- +ing examples [50, 26], as well as how to mitigate such +effects [50]. Our work extends these lines of inquiry to +conditional diffusion models, where we measure novelty +by computing how frequently models regenerate training +instances when provided with textual prompts. +Recent and concurrent work also studies privacy in im- +age generation for both GANs [70] and diffusion mod- +els [65, 78, 34]. Tinsley et al. [70] show that StyleGAN +can generate individuals’ faces, and Somepalli et al. [65] +show that Stable Diffusion can output semantically sim- +ilar images to its training set. Compared to these works, +we identify privacy vulnerabilities in a wider range of +systems (e.g., Imagen and CIFAR models) and threat +models (e.g., membership inference attacks). +9 +Discussion and Conclusion +State-of-the-art diffusion models memorize and regen- +erate individual training images, allowing adversaries +to launch training data extraction attacks. By training +our own models we find that increasing utility can de- +grade privacy, and simple defenses such as deduplication +are insufficient to completely address the memorization +challenge. We see that state-of-the-art diffusion models +memorize 2× more than comparable GANs, and more +useful diffusion models memorize more than weaker dif- +fusion models. This suggests that the vulnerability of +generative image models may grow over time. Going +forward, our work raises questions around the memoriza- +tion and generalization capabilities of diffusion models. +Questions of generalization. +Do large-scale models +work by generating novel output, or do they just copy +and interpolate between individual training examples? +If our extraction attacks had failed, it may have refuted +the hypothesis that models copy and interpolate training +data; but because our attacks succeed, this question re- +mains open. Given that different models memorize vary- +ing amounts of data, we hope future work will explore +how diffusion models copy from their training datasets. +Our work also highlights the difficulty in defining +memorization. +While we have found extensive mem- +orization with a simple ℓ2-based measurement, a more +comprehensive analysis will be necessary to accurately +capture more nuanced definitions of memorization that +allow for more human-aligned notions of data copying. +Practical consequences. +We raise four practical con- +sequences for those who train and deploy diffusion mod- +els. +First, while not a perfect defense, we recom- +mend deduplicating training datasets and minimizing +over-training. Second, we suggest using our attack—or +other auditing techniques—to estimate the privacy risk of +trained models. Third, once practical privacy-preserving +techniques become possible, we recommend their use +whenever possible. Finally, we hope our work will tem- +per the heuristic privacy expectations that have come to +be associated with diffusion model outputs: synthetic +data does not give privacy for free [13, 14, 59, 2, 53]. +On the whole, our work contributes to a growing body +of literature that raises questions regarding the legal, eth- +ical, and privacy issues that arise from training on web- +scraped public data [7, 65, 72, 77]. Researchers and prac- +titioners should be wary of training on uncurated public +data without first taking steps to understand the underly- +ing ethics and privacy implications. +15 + +NC +MN +JH +MJ +FT +VS +BB +DI +EW +Conceived Project +X +X +X +X +Formalized Memorization Definition +X +X +X +X +X +X +Experimented with Stable Diffusion +X +X +Experimented with Imagen +X +Experimented with CIFAR-10 Diffusion +X +X +Experimented with GANs +X +X +X +Experimented with Defenses +X +X +X +Prepared Figures +X +X +X +X +X +X +X +Analyzed Data +X +X +X +X +X +X +Wrote Paper +X +X +X +X +X +X +X +X +X +Managed the Project +X +Table 2: Contributions of each author in the paper. +Contributions +• Nicholas, Jamie, Vikash, and Eric each indepen- +dently proposed the problem statement of extracting +training data from diffusion models. +• Nicholas, Eric, and Florian performed preliminary +experiments to identify cases of data extraction in +diffusion models. +• Milad performed most of the experiments on Stable +Diffusion and Imagen, and Nicholas counted dupli- +cates in the LAION training dataset; each wrote the +corresponding sections of the paper. +• Jamie performed the membership inference attacks +and inpainting attacks on CIFAR-10 diffusion mod- +els, and Nicholas performed the diffusion extraction +experiments; each wrote the corresponding sections +of the paper. +• Matthew ran experiments for canary memorization +and wrote the corresponding section of the paper. +• Florian and Vikash performed preliminary experi- +ments on memorization in GANs, and Milad and +Vikash ran the experiments included in the paper. +• Milad ran the membership inference experiments on +GANs. +• Vikash ran extraction experiments on pretrained +GANs. +• Daphne and Florian improved figure clarity and pre- +sentation. +• Daphne, Borja, and Eric edited the paper and con- +tributed to paper framing. +• Nicholas organized the project and wrote the initial +paper draft. +Acknowledgements and Conflicts of Interest +The authors are grateful to Tom Goldstein, Olivia +Wiles, Katherine Lee, Austin Tarango, Ian Wilbur, Jeff +Dean, Andreas Terzis, Robin Rombach, and Andreas +Blattmann for comments on early drafts of this paper. +Nicholas, Milad, Matthew, and Daphne are employed +at Google, and Jamie and Borja are employed at Deep- +Mind, companies that both train large machine learning +models (including diffusion models) on both public and +private datasets. +Eric Wallace is supported by the Apple Scholars in +AI/ML Fellowship. +References +[1] Mart´ın Abadi, Andy Chu, Ian Goodfellow, H Bren- +dan McMahan, Ilya Mironov, Kunal Talwar, and +Li Zhang. Deep learning with differential privacy. +In ACM CCS, 2016. +[2] Hazrat Ali, Shafaq Murad, and Zubair Shah. Spot +the fake lungs: Generating synthetic medical im- +ages using neural diffusion models. arXiv preprint +arXiv:2211.00902, 2022. +[3] Sanjeev Arora, Andrej Risteski, and Yi Zhang. Do +GANs learn the distribution? Some theory and em- +pirics. +In International Conference on Learning +Representations, 2018. +[4] Yogesh Balaji, Hamed Hassani, Rama Chellappa, +and Soheil Feizi. Entropic GANs meet VAEs: A +statistical approach to compute sample likelihoods +in GANs. In International Conference on Machine +Learning, 2019. +16 + +[5] Borja Balle, Giovanni Cherubin, and Jamie Hayes. +Reconstructing training data with informed adver- +saries. In IEEE Symposium on Security and Pri- +vacy, 2022. +[6] Andrew Brock, Jeff Donahue, and Karen Si- +monyan. Large scale GAN training for high fidelity +natural image synthesis. In International Confer- +ence on Learning Representations, 2019. +[7] Hannah Brown, +Katherine Lee, +Fatemehsadat +Mireshghallah, Reza Shokri, and Florian Tram`er. +What does it mean for a language model to pre- +serve privacy? +In ACM Conference on Fairness, +Accountability, and Transparency, 2022. +[8] Nicholas Carlini, Steve Chien, Milad Nasr, Shuang +Song, Andreas Terzis, and Florian Tramer. Mem- +bership inference attacks from first principles. In +IEEE Symposium on Security and Privacy. IEEE, +2022. +[9] Nicholas +Carlini, +Daphne +Ippolito, +Matthew +Jagielski, Katherine Lee, Florian Tramer, and +Chiyuan +Zhang. +Quantifying +memorization +across neural language models. +arXiv preprint +arXiv:2202.07646, 2022. +[10] Nicholas Carlini, Chang Liu, +´Ulfar Erlingsson, +Jernej Kos, and Dawn Song. +The secret sharer: +Evaluating and testing unintended memorization in +neural networks. In USENIX Security Symposium, +2019. +[11] Nicholas Carlini, Florian Tramer, Eric Wallace, +Matthew Jagielski, Ariel Herbert-Voss, Katherine +Lee, Adam Roberts, Tom Brown, Dawn Song, Ul- +far Erlingsson, et al. Extracting training data from +large language models. In USENIX Security Sym- +posium, 2021. +[12] Andrew Carr. +Gretel.ai: +Diffusion models for +document synthesis. https://gretel.ai/blog/ +diffusion-models-for-document-synthesis, +2022. +[13] Pierre Chambon, Christian Bluethgen, Jean-Benoit +Delbrouck, Rogier Van der Sluijs, Małgorzata +Połacin, Juan Manuel Zambrano Chaves, Tan- +ishq Mathew Abraham, Shivanshu Purohit, Cur- +tis P. Langlotz, and Akshay Chaudhari. RoentGen: +Vision-language foundation model for chest X- +ray generation. arXiv preprint arXiv:2211.12737, +2022. +[14] Pierre Chambon, Christian Bluethgen, Curtis P. +Langlotz, +and Akshay Chaudhari. +Adapting +pretrained vision-language foundational models +to medical imaging domains. +arXiv preprint +arXiv:2210.04133, 2022. +[15] Tong Che, Yanran Li, Athul Paul Jacob, Yoshua +Bengio, and Wenjie Li. Mode regularized gener- +ative adversarial networks. In International Con- +ference on Learning Representations, 2016. +[16] Papers With Code. https://paperswithcode. +com/sota/image-generation-on-cifar-10, +2023. +[17] Elise Devaux. +List of synthetic data vendors— +2022. +https://elise-deux.medium.com/ +new-list-of-synthetic-data-vendors-2022-f06dbe91784, +2022. +[18] Prafulla Dhariwal and Alexander Nichol. Diffusion +models beat GANs on image synthesis. Advances +in Neural Information Processing Systems, 2021. +[19] Tim Dockhorn, Tianshi Cao, Arash Vahdat, and +Karsten Kreis. Differentially private diffusion mod- +els. arXiv preprint arXiv:2210.09929, 2022. +[20] August DuMont Sch¨utte, J¨urgen Hetzel, Sergios +Gatidis, Tobias Hepp, Benedikt Dietz, Stefan +Bauer, and Patrick Schwab. Overcoming barriers +to data sharing with medical image generation: a +comprehensive evaluation. NPJ Digital Medicine, +2021. +[21] C Dwork, F McSherry, K Nissim, and A Smith. +Calibrating noise to sensitivity in private data anal- +ysis. In TCC, 2006. +[22] Cynthia Dwork. Differential privacy: A survey of +results. In TAMC, 2008. +[23] Benj +Edwards. +Artist +finds +private +med- +ical +record +photos +in +popular +AI +train- +ing +data +set. +https://arstechnica. +com/information-technology/2022/09/ +artist-finds-private-medical-record-photos-in-popular-ai-training-data-set, +2022. +[24] Vitaly Feldman. +Does learning require memo- +rization? +A short tale about a long tail. +In +ACM SIGACT Symposium on Theory of Comput- +ing, 2020. +17 + +[25] Vitaly Feldman and Chiyuan Zhang. What neural +networks memorize and why: Discovering the long +tail via influence estimation. Advances in Neural +Information Processing Systems, 2020. +[26] Qianli Feng, Chenqi Guo, Fabian Benitez-Quiroz, +and Aleix M Martinez. When do GANs replicate? +on the choice of dataset size. In IEEE/CVF Inter- +national Conference on Computer Vision, 2021. +[27] Matt Fredrikson, Somesh Jha, and Thomas Risten- +part. +Model inversion attacks that exploit confi- +dence information and basic countermeasures. In +ACM Conference on Computer and Communica- +tions Security (CCS), 2015. +[28] Matthew Fredrikson, Eric Lantz, Somesh Jha, Si- +mon Lin, David Page, and Thomas Ristenpart. +Privacy in pharmacogenetics: An end-to-end case +study of personalized warfarin dosing. In USENIX +Security Symposium, 2014. +[29] Ian Goodfellow, +Yoshua Bengio, +and Aaron +Courville. Deep learning. MIT press, 2016. +[30] Ian +Goodfellow, +Jean +Pouget-Abadie, +Mehdi +Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, +Aaron Courville, and Yoshua Bengio. Generative +adversarial networks. Advances in Neural Informa- +tion Processing Systems, 2014. +[31] Peter +Henderson, +Koustuv +Sinha, +Nicolas +Angelard-Gontier, Nan Rosemary Ke, Genevieve +Fried, Ryan Lowe, and Joelle Pineau. +Ethical +challenges in data-driven dialogue systems. +In +AAAI/ACM Conference on AI, Ethics, and Society, +2018. +[32] Martin Heusel, Hubert Ramsauer, Thomas Un- +terthiner, Bernhard Nessler, and Sepp Hochreiter. +GANs trained by a two time-scale update rule con- +verge to a local nash equilibrium. Advances in Neu- +ral Information Processing Systems, 2017. +[33] Jonathan Ho, Ajay Jain, and Pieter Abbeel. De- +noising diffusion probabilistic models. Advances +in Neural Information Processing Systems, 2020. +[34] Hailong Hu and Jun Pang. +Membership in- +ference of diffusion models. +arXiv preprint +arXiv:2301.09956, 2023. +[35] Daphne Ippolito, Florian Tram`er, Milad Nasr, +Chiyuan Zhang, Matthew Jagielski, Katherine Lee, +Christopher A Choquette-Choo, and Nicholas Car- +lini. +Preventing verbatim memorization in lan- +guage models gives a false sense of privacy. arXiv +preprint arXiv:2210.17546, 2022. +[36] Matthew Jagielski, Jonathan Ullman, and Alina +Oprea. +Auditing differentially private machine +learning: How private is private SGD? Advances +in Neural Information Processing Systems, 2020. +[37] Ali Jahanian, Xavier Puig, Yonglong Tian, and +Phillip Isola. Generative models as a data source for +multiview representation learning. +International +Conference on Learning Representations, 2021. +[38] Bargav Jayaraman and David Evans. Are attribute +inference attacks just imputation? +ACM Confer- +ence on Computer and Communications Security +(CCS), 2022. +[39] Bargav Jayaraman, Lingxiao Wang, Katherine +Knipmeyer, Quanquan Gu, and David Evans. Re- +visiting membership inference under realistic as- +sumptions. +Proceedings on Privacy Enhancing +Technologies, 2020. +[40] Nikhil Kandpal, Haikang Deng, Adam Roberts, +Eric Wallace, and Colin Raffel. +Large language +models struggle to learn long-tail knowledge. arXiv +preprint arXiv:2211.08411, 2022. +[41] Nikhil Kandpal, Eric Wallace, and Colin Raffel. +Deduplicating training data mitigates privacy risks +in language models. International Conference on +Machine Learning, 2022. +[42] MinGuk Kang, Joonghyuk Shin, and Jaesik Park. +StudioGAN: A Taxonomy and Benchmark of +GANs for Image Synthesis. +arXiv preprint +arXiv:2206.09479, 2022. +[43] Tero Karras, Miika Aittala, Janne Hellsten, Samuli +Laine, Jaakko Lehtinen, and Timo Aila. Training +generative adversarial networks with limited data. +In Advances in Neural Information Processing Sys- +tems, 2020. +[44] Tero Karras, Samuli Laine, and Timo Aila. A style- +based generator architecture for generative adver- +sarial networks. In IEEE/CVF conference on com- +puter vision and pattern recognition, 2019. +[45] Salome Kazeminia, Christoph Baur, Arjan Kuijper, +Bram van Ginneken, Nassir Navab, Shadi Albar- +qouni, and Anirban Mukhopadhyay. +GANs for +18 + +medical image analysis. Artificial Intelligence in +Medicine, 2020. +[46] Diederik P Kingma and Max Welling. +Auto- +encoding variational bayes. In International Con- +ference on Learning Representations, 2014. +[47] Katherine Lee, Daphne Ippolito, Andrew Nystrom, +Chiyuan Zhang, Douglas Eck, Chris Callison- +Burch, and Nicholas Carlini. Deduplicating train- +ing data makes language models better. In Associ- +ation for Computational Linguistics, 2022. +[48] Andreas Lugmayr, +Martin Danelljan, +Andres +Romero, +Fisher Yu, +Radu Timofte, +and Luc +Van Gool. RePaint: Inpainting using denoising dif- +fusion probabilistic models. In IEEE/CVF Confer- +ence on Computer Vision and Pattern Recognition, +2022. +[49] AnGeL +Ministries. +File:Anne +Gra- +ham +Lotz +(October +2008). +https: +//commons.wikimedia.org/wiki/File: +Anne_Graham_Lotz_(October_2008).jpg. +Accessed on December 2022. +[50] Vaishnavh Nagarajan, Colin Raffel, and Ian J +Goodfellow. Theoretical insights into memoriza- +tion in GANs. In Neural Information Processing +Systems Workshop, 2018. +[51] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, +Pranav Shyam, Pamela Mishkin, Bob McGrew, +Ilya Sutskever, and Mark Chen. +GLIDE: To- +wards photorealistic image generation and editing +with text-guided diffusion models. arXiv preprint +arXiv:2112.10741, 2021. +[52] Alexander Quinn Nichol and Prafulla Dhariwal. +Improved denoising diffusion probabilistic models. +In International Conference on Machine Learning, +2021. +[53] Walter H. L. Pinaya, Petru-Daniel Tudosiu, Jes- +sica Dafflon, Pedro F da Costa, Virginia Fer- +nandez, Parashkev Nachev, Sebastien Ourselin, +and M. Jorge Cardoso. +Brain imaging genera- +tion with latent diffusion models. arXiv preprint +arXi:2209.07162, 2022. +[54] Alec Radford, Jong Wook Kim, Chris Hallacy, +Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, +Girish Sastry, Amanda Askell, Pamela Mishkin, +Jack Clark, et al. Learning transferable visual mod- +els from natural language supervision. In Interna- +tional Conference on Machine Learning, 2021. +[55] Alec Radford, Luke Metz, and Soumith Chintala. +Unsupervised representation learning with deep +convolutional generative adversarial networks. In +International Conference on Learning Representa- +tions, 2016. +[56] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, +Casey Chu, and Mark Chen. +Hierarchical text- +conditional image generation with CLIP latents. +arXiv preprint arXiv:2204.06125, 2022. +[57] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, +Scott Gray, Chelsea Voss, Alec Radford, Mark +Chen, and Ilya Sutskever. Zero-shot text-to-image +generation. +In International Conference on Ma- +chine Learning, 2021. +[58] Robin Rombach, Andreas Blattmann, Dominik +Lorenz, Patrick Esser, and Bj¨orn Ommer. +High- +resolution image synthesis with latent diffusion +models. In IEEE/CVF Conference on Computer Vi- +sion and Pattern Recognition, 2022. +[59] Pouria +Rouzrokh, +Bardia +Khosravi, +Shahriar +Faghani, Mana Moassefi, Sanaz Vahdati, and +Bradley J. Erickson. Multitask brain tumor inpaint- +ing with diffusion models: A methodological re- +port. arXiv preprint arXiv:2210.12113, 2022. +[60] Chitwan Saharia, William Chan, Saurabh Sax- +ena, Lala Li, Jay Whang, Emily Denton, Seyed +Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, +S Sara Mahdavi, Rapha Gontijo Lopes, et al. +Photorealistic text-to-image diffusion models with +deep language understanding. +arXiv preprint +arXiv:2205.11487, 2022. +[61] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, +Vicki Cheung, Alec Radford, and Xi Chen. +Im- +proved techniques for training GANs. Advances in +Neural Information Processing Systems, 2016. +[62] Reza Shokri, Marco Stronati, Congzheng Song, +and Vitaly Shmatikov. Membership inference at- +tacks against machine learning models. In IEEE +Symposium on Security and Privacy, 2017. +[63] Abhishek Sinha, Kumar Ayush, Jiaming Song, Bu- +rak Uzkent, Hongxia Jin, and Stefano Ermon. Neg- +ative data augmentation. In International Confer- +ence on Learning Representations, 2021. +[64] Jascha Sohl-Dickstein, Eric Weiss, Niru Mah- +eswaranathan, and Surya Ganguli. +Deep unsu- +pervised learning using nonequilibrium thermody- +19 + +namics. In International Conference on Machine +Learning, 2015. +[65] Gowthami Somepalli, Vasu Singla, Micah Gold- +blum, Jonas Geiping, and Tom Goldstein. +Dif- +fusion art or digital forgery? +Investigating data +replication in diffusion models. +arXiv preprint +arXiv:2212.03860, 2022. +[66] Yang Song and Stefano Ermon. Generative mod- +eling by estimating gradients of the data distribu- +tion. Advances in Neural Information Processing +Systems, 2019. +[67] Midjourney Team. https://www.midjourney. +com/, 2022. +[68] D´avid Terj´ek. Adversarial lipschitz regularization. +In International Conference on Learning Represen- +tations, 2019. +[69] Yuan Tian, Qin Wang, Zhiwu Huang, Wen Li, +Dengxin Dai, Minghao Yang, Jun Wang, and Olga +Fink. +Off-policy reinforcement learning for effi- +cient and effective gan architecture search. In Eu- +ropean Conference on Computer Vision, 2020. +[70] Patrick Tinsley, Adam Czajka, and Patrick Flynn. +This face does not exist... but it might be +yours! Identity leakage in generative models. In +IEEE/CVF Winter Conference on Applications of +Computer Vision, 2021. +[71] Rob Toews. +Synthetic data is about to trans- +form +artificial +intelligence. +https://www. +forbes.com/sites/robtoews/2022/06/12/ +synthetic-data-is-about-to-transform-artificial-intelligence/, +2022. +[72] Florian Tram`er, Gautam Kamath, and Nicholas +Carlini. +Considerations for differentially private +learning with large-scale public pretraining. arXiv +preprint arXiv:2212.06470, 2022. +[73] Allan Tucker, Zhenchen Wang, Ylenia Rotalinti, +and Puja Myles. Generating high-fidelity synthetic +patient data for assessing machine learning health- +care software. NPJ Digital Medicine, 2020. +[74] Ryan Turner, Jane Hung, Eric Frank, Yunus +Saatchi, and Jason Yosinski. Metropolis-hastings +generative adversarial networks. In International +Conference on Machine Learning, 2019. +[75] Gerrit van den Burg and Chris Williams. On mem- +orization in probabilistic deep generative models. +Advances in Neural Information Processing Sys- +tems, 2021. +[76] James +Vincent. +The +lawsuit +that +could +rewrite the rules of AI copyright. +https: +//www.theverge.com/2022/11/8/23446821/ +microsoft-openai-github-copilot-class-action-lawsuit-ai-copyright-violation-training-data, +2022. +[77] Eric Wallace, Florian Tram`er, Matthew Jagielski, +and Ariel Herbert-Voss. Does GPT-2 know your +phone number? BAIR Blog, 2020. +[78] Yixin Wu, Ning Yu, Zheng Li, Michael Backes, and +Yang Zhang. Membership inference attacks against +text-to-image generation models. +arXiv preprint +arXiv:2210.00968, 2022. +[79] Jiayuan Ye, Aadyaa Maddi, Sasi Kumar Mu- +rakonda, Vincent Bindschaedler, and Reza Shokri. +Enhanced membership inference attacks against +machine learning models. In ACM SIGSAC Con- +ference on Computer and Communications Security +(CCS), 2021. +[80] Samuel Yeom, Irene Giacomelli, Matt Fredrikson, +and Somesh Jha. Privacy risk in machine learning: +Analyzing the connection to overfitting. In IEEE +Computer Security Foundations Symposium (CSF), +2018. +[81] Yuheng Zhang, Ruoxi Jia, Hengzhi Pei, Wenxiao +Wang, Bo Li, and Dawn Song. +The secret re- +vealer: Generative model-inversion attacks against +deep neural networks. In IEEE/CVF conference on +computer vision and pattern recognition, 2020. +[82] Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, +and Song Han. +Differentiable augmentation for +data-efficient GAN training. Advances in Neural +Information Processing Systems, 2020. +20 + +A +Collected Details for Figures +Table 3: Catalog of figures containing qualitative examples. +Figure # +Model +Dataset +Who trained it? +Sampling strategy +Figure 1 +Stable Diffusion +LAION +Stability AI +PLMS +Figure 2 +Stable Diffusion +LAION +Stability AI +PLMS +Figure 3 +Stable Diffusion +LAION +Stability AI +PLMS +Figure 6 +Uncond Diffusion +CIFAR-10 +Ours +DDIM +Figure 7 +Uncond Diffusion +CIFAR-10 +Ours +DDIM +Figure 8 +Uncond Diffusion +CIFAR-10 +Ours +DDIM +Figure 12 +Uncond Diffusion +CIFAR-10 +Ours +Inpainting +Figure 13 +Uncond Diffusion +CIFAR-10 +Ours +Inpainting +Figure 15 +StyleGAN, MHGAN, BigGAN +CIFAR-10 +Ours +GAN default +Figure 17 +Uncond Diffusion +CIFAR-10 +Ours +DDIM +Figure 20 +Uncond Diffusion +CIFAR-10 +Ours +DDIM +Figure 22 +Uncond Diffusion +CIFAR-10 +Ours +Inpainting +Figure 23 +Uncond Diffusion +CIFAR-10 +Ours +Inpainting +Figure 24 +Several different GANs +CIFAR-10 +Original paper authors +GAN default +21 + +B +All CIFAR-10 Memorized Images +Figure 17: All 1280 images we extract from diffusion models trained on CIFAR-10, after 1 million generations from +16 diffusion models. +22 + +C +Additional Attacks on CIFAR-10 +Here, we expand on our investigation of memorization of training data on CIFAR-10. +C.1 +Membership Inference at Different Training Steps +1000 +2000 +3000 +4000 +Each train example processed X times +0.2 +0.4 +0.6 +0.8 +1.0 +TPR@FPR=1% +(a) How membership attack success +changes as a training example is +processed repeatedly throughout +training. +0.2 +0.4 +0.6 +0.8 +1.0 +Training data seen +1e8 +0.2 +0.4 +0.6 +0.8 +1.0 +TPR@FPR=1% +(b) How membership attack success +changes as more data is processed +throughout training. +10 +3 +10 +2 +10 +1 +100 +False positive rate +10 +3 +10 +2 +10 +1 +100 +True positive rate +Data seen: 5M +AUC: 0.742 +TPR@FPR=1%: 0.050 +Data seen: 102M +AUC: 0.997 +TPR@FPR=1%: 0.989 +(c) ROC curve for the membership +attack for different training steps. +Figure 18: Membership inference attacks as a function of the amount of training data processed on +CIFAR-10. +In Section 5.2.3, we implicitly investigated membership attack success as a function of the number update steps +when training a diffusion model. We explicitly model this relationship in Figure 18. First, in Figure 18a we plot +membership attack success as a function of the number of times that an example was processed over training. If +an example is processed more than 2000 times during training, invariably membership attacks are perfect against that +example. Second, in Figure 18b, we plot membership attack success as a function of the total amount of data processed +during training. Unsurprisingly, membership attack success increases as more training data is processed. This is +highlighted in Figure 18c, where we plot the membership attack ROC curve. At 5M training examples processed, +at a FPR of 1% the TPR is 5%, and increases to 99% after 102M examples are processed. Note that this number of +processed training inputs is commonly used in diffusion model training. For example, the OpenAI CIFAR-10 diffusion +model 9 is trained for 500,000 steps at a batch size of 128, meaning 64M training examples are processed. Even at this +number of processed training examples, our membership attack has a TPR > 95% at a FPR of 1%. +9https://github.com/openai/improved-diffusion +23 + +C.2 +Membership Inference with Different Augmentation Strategies +10 +3 +10 +2 +10 +1 +100 +False positive rate +10 +3 +10 +2 +10 +1 +100 +True positive rate +n: 1 AUC: 0.982 +TPR@FPR=0.1%: 0.071 +n: 2 AUC: 0.991 +TPR@FPR=0.1%: 0.128 +n: 5 AUC: 0.995 +TPR@FPR=0.1%: 0.210 +n: 10 AUC: 0.996 +TPR@FPR=0.1%: 0.260 +n: 20 AUC: 0.997 +TPR@FPR=0.1%: 0.294 +(a) +10 +3 +10 +2 +10 +1 +100 +False positive rate +10 +3 +10 +2 +10 +1 +100 +True positive rate +w/o Aug +n: 20 +AUC: 0.997 +TPR@FPR=0.1%: 0.294 +w/ Aug +n: 20 +AUC: 0.997 +TPR@FPR=0.1%: 0.437 +(b) +Figure 19: We can improve membership inference attack success rates on CIFAR-10 by reducing noise. In (a), +membership inference attacks are improved by averaging the loss over multiple noise samples in the diffusion process. +In (b), attacks are improved by querying on augmented versions of the candidate image. +24 + +C.3 +Membership Inference Inliers and Outliers +Figure 20: When performing our membership inference attack, the hardest-to-attack examples (left) are all duplicates +in the CIFAR-10 training set, and the easiest-to-attack examples (right) are visually outliers from CIFAR-10 images. +25 + +C.4 +Membership Inference on Conditional and Unconditional Models +Diffusion models can be conditioned on labels (or prompts for text-to-image models). We compare the difference +in membership inference on a CIFAR-10 diffusion model trained unconditionally with a model conditionally trained +on CIFAR-10 labels. The conditional and unconditional models reach approximately the same FID after training; +between 3.5-4.2 FID. We plot the membership attack ROC curve in Figure 21 and note that the conditional model +is marginally more vulnerable. However, it is difficult to tell if this is a fundamental difference between conditional +and unconditional models, or because the conditional model contains more parameters than unconditional model (the +conditional models contains an extra embedding layer for the one-hot label input). +10 +3 +10 +2 +10 +1 +100 +False positive rate +10 +3 +10 +2 +10 +1 +100 +True positive rate +Unconditional. +AUC: 0.970 +TPR@FPR=1%: 0.549 +Conditional. +AUC: 0.982 +TPR@FPR=1%: 0.714 +Figure 21: Membership attack against a conditional and +unconditional diffusion model on CIFAR-10. +26 + +D +More Inpainting Attacks on CIFAR-10 +Here, we take a deeper dive into the inpainting attacks introduced in Section 5.3. As previously explained, for a target +x, we create Xrec where |Xrec| = 5000. In Figure 22a, for every xrec ∈ Xrec, we plot the normalized ℓ2 distance between +the reconstruction and target, against the loss (at diffusion timestep 100) of xrec. We also plot in Figure 22d, the eight +examples from Xrec that have the smallest loss on the main model. There is a small positive correlation between loss +and ℓ2 distance; although some appear to be similar to x, there are notable differences. +In Figure 22b we compare the loss of each reconstruction on the main model against the support model we will use +to form the contrastive loss. We make this correlation more pronounced by dividing the main loss by the support loss +in Figure 22c. This has the effect of increasing the correlation between the (now contrastive) loss and ℓ2 distance. This +has the effect of filtering out examples that are seen as likely under both models, and can be seen by inspecting the +eight examples from Xrec that have have the smallest +main model loss +support model loss in Figure 22e. These examples look more visually +similar to x in comparison to examples in Figure 22d. +Figure 22 inspected the attack success when x was in the training set. We show in Figure 23 that the attack fails +when x was not included in training; using a contrastive loss doesn’t signficantly increase the Pearson correlation +coefficient. This means our attack is indeed exploiting the fact that the model can only inpaint correctly because of +memorisation and not due to generalisation. +27 + +(a) Loss (using the main model at diffusion +timestep 100) on all 5,000 inpainted +examples Xrec. +(b) Comparison of loss on main and +support models (at diffusion timestep 100) +on all 5,000 inpainted examples. +(c) Contrastive loss ( main model loss +support model loss) on +all 5,000 inpainted examples Xrec. +(d) 8 inpainted examples with the smallest loss. Leftmost +is the original example, second to left is the masked +example and the rest are inpainted examples. +(e) 8 inpainted examples with the smallest +main model loss +support model loss. +Leftmost is the original example, second to left is the +masked example and the rest are inpainted examples. +Figure 22: Example of an inpainting attack (against a model we refer to as the main model) on an image +of a bird from CIFAR-10 when that image is included in training, and we mask out 60% of the central +pixels. In (a) we plot the L2 distance between 5,000 inpainted reconstructions and the original +(non-masked out) image and compare this to the loss with respect to the (main) model. In (b), we +compare the loss of these reconstructions on the (main) model with a support model for which we know +the image wasn’t contained in the training set. In (c), we compare L2 distances between reconstructions +with a contrastive loss which is given as the loss of the image with respect to the main model divided by +the loss of the image with respect to the support model, and find there is stronger relationship between +smaller L2 distances and smaller losses compared to (a). Figure (d) gives examples of reconstructions +with small loss and Figure (e) gives examples of reconstructions with small contrastive loss. +28 + +Pearson camelstion ccefficient: Q.41 +0.D6 +0.04 +(of +0.03 +0.10 +0.15 +0.20 +0.25 +0.30 + distance to target0.D-6 + (of support model) +0.D5 +0.03, +0.02 +0.02 +0.03 +0.04 +0.05 +0.06 +Lauss (of miain mkdel)Pearson camelstion ccefficient: 0.63 +11 +Contrastive +60 +0.B +0.7 +0.10 +0.15 +0.20 +0.25 +0.30 + distance to target(a) Loss (using the main model at diffusion +timestep 100) on all 5,000 inpainted +examples Xrec. +(b) Comparison of loss on main and +support models (at diffusion timestep 100) +on all 5,000 inpainted examples. +(c) Contrastive loss ( main model loss +support model loss) on +all 5,000 inpainted examples Xrec. +(d) 8 inpainted examples with the smallest loss. Leftmost +is the original example, second to left is the masked +example and the rest are inpainted examples. +(e) 8 inpainted examples with the smallest +main model loss +support model loss. +Leftmost is the original example, second to left is the +masked example and the rest are inpainted examples. +Figure 23: Example of an inpainting attack (against a model we refer to as the main model) on an image of a +bird from CIFAR-10 when that image is not included in training, and we mask out 60% of the central pixels. In +(a) we plot the L2 distance between 5,000 inpainted reconstructions and the original (non-masked out) image +and compare this to the loss with respect to the (main) model. In (b), we compare the loss of these +reconstructions on the (main) model with a support model for which we know the image wasn’t contained in +the training set. In (c), we compare L2 distances between reconstructions with a contrastive loss which is given +as the loss of the image with respect to the main model divided by the loss of the image with respect to the +support model, and find there is stronger relationship between smaller L2 distances and smaller losses +compared to (a). Figure (d) gives examples of reconstructions with small loss and Figure (e) gives examples of +reconstructions with small contrastive loss. +E +GAN Training Setup +We used on StudioGAN10 codebase for training GAN in this work. For the StyleGAN and MHGAN architectures, we +followed the default hyper-parameters provided in the StudioGAN repository. However, for the BigGAN architecture, +we increased the number of training steps to 200,000, which is different from the original hyper-parameters, to increase +image fidelity. We trained a total of 256 models for each GAN architecture, with each model being trained on a +randomly selected half of the CIFAR-10 dataset. We selected the iteration that achieved the highest FID score on the +test set for each model. +F +Additional GAN Extraction Results +Figure 24 and Figure 25 contain additional examples extracted from GANs trained on CIFAR-10. +10https://github.com/POSTECH-CVLab/PyTorch-StudioGAN +29 + +Pearson camelstion ccefficient: o.15 +0.08 - +0.07 +main +0.0G +0.05 +0.03 +0.15 +0.200.25 +0.30 +0.35 + distance to targetLass (of support model) +500 +500 +t +0.02 +0.02 +0.03 0.D40.050.D6 0.D7 +0.b8 +Lass (of miain mkdel)Pearson canelstion ccefficient: 0.18 +13 +11 +1D +0.9 +0.15 +0.20 +0.25 +0.30 +0.35 + distance to target(a) StyleGAN +(b) MHGAN +(c) BigGAN +Figure 24: Training examples extracted from a CIFAR-10 GAN for different architectures across 107 generations. +30 + +A +. +一 +十 +1(a) WGAN +(b) E2GAN +(c) NDA +(d) DiffAugment-BigGAN +(e) StyleGAN-ADA +(f) DDPM +Figure 25: Training examples extracted from different publicly available pretrained GANs and diffusion (DDPM) +models. We use normalized ℓ2 distance in pixel space to find memorized training samples. In each pair of images, left +and right image corresponds to real and it closely synthetic image. For StyleGAN-ADA and DDPM model we display +120 pairs with smallest normalized ℓ2 distance. For others we display all memorized training images. 1M generations +31 + +B \ No newline at end of file diff --git a/lNFPT4oBgHgl3EQf2zXD/content/tmp_files/load_file.txt b/lNFPT4oBgHgl3EQf2zXD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c1b2a1aaae73e76c9e686e316b567caf14f48e82 --- /dev/null +++ b/lNFPT4oBgHgl3EQf2zXD/content/tmp_files/load_file.txt @@ -0,0 +1,1231 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf,len=1230 +page_content='Extracting Training Data from Diffusion Models Nicholas Carlini∗1 Jamie Hayes∗2 Milad Nasr∗1 Matthew Jagielski+1 Vikash Sehwag+4 Florian Tram`er+3 Borja Balle†2 Daphne Ippolito†1 Eric Wallace†5 1Google 2DeepMind 3ETHZ 4Princeton 5UC Berkeley ∗Equal contribution +Equal contribution †Equal contribution Abstract Image diffusion models such as DALL-E 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Imagen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' and Stable Diffusion have attracted significant attention due to their ability to generate high-quality synthetic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In this work, we show that diffusion models memorize individual images from their training data and emit them at generation time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' With a generate-and-filter pipeline, we extract over a thousand training examples from state- of-the-art models, ranging from photographs of individ- ual people to trademarked company logos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We also train hundreds of diffusion models in various settings to an- alyze how different modeling and data decisions affect privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, our results show that diffusion models are much less private than prior generative models such as GANs, and that mitigating these vulnerabilities may require new advances in privacy-preserving training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 1 Introduction Denoising diffusion models are an emerging class of generative neural networks that produce images from a training distribution via an iterative denoising pro- cess [64, 66, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Compared to prior approaches such as GANs [30] or VAEs [46], diffusion models produce higher-quality samples [18] and are easier to scale [56] and control [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Consequently, they have rapidly be- come the de-facto method for generating high-resolution images, and large-scale models such as DALL-E 2 [56] have attracted significant public interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The appeal of generative diffusion models is rooted in their ability to synthesize novel images that are os- tensibly unlike anything in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Indeed, past large-scale training efforts “do not find overfitting to be an issue”, [60] and researchers in privacy-sensitive do- mains have even suggested that diffusion models could “protect[] the privacy [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='] of real images” [37] by gen- erating synthetic examples [13, 14, 59, 2, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This line of work relies on the assumption that diffusion models do not memorize and regenerate their training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If they did, it would violate all privacy guarantees and raise numerous questions regarding model generalization and “digital forgery” [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Training Set Generated Image Caption: Living in the light with Ann Graham Lotz Prompt: Ann Graham Lotz Figure 1: Diffusion models memorize individual train- ing examples and generate them at test time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Left: an image from Stable Diffusion’s training set (licensed CC BY-SA 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0, see [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Right: a Stable Diffusion gen- eration when prompted with “Ann Graham Lotz”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The reconstruction is nearly identical (ℓ2 distance = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='031).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In this work, we demonstrate that state-of-the-art dif- fusion models do memorize and regenerate individual training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To begin, we propose and implement new definitions for “memorization” in image models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then devise a two-stage data extraction attack that gener- ates images using standard approaches, and flags those that exceed certain membership inference scoring crite- ria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Applying this method to Stable Diffusion [58] and Imagen [60], we extract over a hundred near-identical replicas of training images that range from personally identifiable photos to trademarked logos (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To better understand how and why memorization oc- curs, we train hundreds of diffusion models on CIFAR- 10 to analyze the impact of model accuracy, hyperparam- eters, augmentation, and deduplication on privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Dif- fusion models are the least private form of image mod- els that we evaluate—for example, they leak more than twice as much training data as GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unfortunately, we also find that existing privacy-enhancing techniques do not provide an acceptable privacy-utility tradeoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Over- all, our paper highlights the tension between increasingly powerful generative models and data privacy, and raises questions on how diffusion models work and how they should be responsibly deployed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='13188v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='CR] 30 Jan 2023 2 Background Diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generative image models have a long history (see [29, Chapter 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generative Adversarial Networks (GANs) [30] were the breakthrough that first enabled the generation of high-fidelity images at scale [6, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' But over the last two years, diffusion models [64] have largely displaced GANs: they achieve state-of-the- art results on academic benchmarks [18] and form the basis of all recently popularized image generators such as Stable Diffusion [58], DALL-E 2 [57, 56], Runway [58], Midjourney [67] and Imagen [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Denoising Diffusion Probabilistic Models [33]1 are conceptually simple: they are nothing more than im- age denoisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' During training, given a clean image x, we sample a time-step t ∈ [0,T] and a Gaussian noise vector ε ∼ N (0,I), to produce a noised image x′ ← √atx+√1−atε, for some decaying parameter at ∈ [0,1] where a0 = 1 and aT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' A diffusion model fθ removes the noise ε to recover the original image x by predicting the noise that was added by stochastically minimizing the objective 1 N ∑i Et,ε L (xi,t,ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' fθ), where L (xi,t,ε;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' fθ) = ∥ε − fθ(√atxi + � 1−atε,t)∥2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (1) Despite being trained with this simple denoising ob- jective, diffusion models can generate high-quality im- ages by first sampling a random vector zT ∼ N (0,I) and then applying the diffusion model fθ to remove the noise from this random “image”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To make the denoising pro- cess easier, we do not remove all of the noise at once— we instead iteratively apply the model to slowly remove noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Formally, the final image z0 is obtained from zT by iterating the rule zt−1 = fθ(zt,t)+σtN (0,I) for a noise schedule σt (dependent on at) with σ1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This process relies on the fact that the model fθ was trained to denoise images with varying degrees of noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, running this iterative generation process (which we will denote by Gen) with large-scale diffusion models produces re- sults that resemble natural images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Some diffusion models are further conditioned to gen- erate a particular type of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Class-conditional dif- fusion models take as input a class-label (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', “dog” or “cat”) alongside the noised image to produce a particu- lar class of image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Text-conditioned models take this one step further and take as input the text embedding of some prompt (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', “a photograph of a horse on the moon”) us- ing a pre-trained language encoder (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', CLIP [54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 1Our description of diffusion models below omits a number of sig- nificant details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' However, these details are orthogonal to the results of our attacks and we omit them for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Training data privacy attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Neural networks of- ten leak details of their training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Membership inference attacks [62, 80, 8] answer the question “was this example in the training set?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' and present a mild privacy breach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Neural networks are also vulnerable to more powerful attacks such as inversion attacks [27, 81] that extract representative examples from a target class, attribute inference attacks [28] that reconstruct subsets of attributes of training examples, and extraction attacks [10, 11, 5] that completely recover training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In this paper, we focus on each of these three attacks when applied to diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Concurrent work explores the privacy of diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [78] and Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [34] perform membership inference attacks on diffusion models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' our results use more sophisticated attack methods and study stronger privacy risks such as data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Somepalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [65] show several cases where (non-adversarially) sampling from a diffusion model can produce memorized training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' However, they focus mainly on com- paring the semantic similarity of generated images to the training set, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', “style copying”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In contrast, we focus on worst-case privacy under a much more restrictive no- tion of memorization, and perform our attacks on a wider range of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 3 Motivation and Threat Model There are two distinct motivations for understanding how diffusion models memorize and regenerate training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Understanding privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Diffusion models that regenerate data scraped from the Internet can pose sim- ilar privacy and copyright risks as language models [11, 7, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For example, memorizing and regenerating copy- righted text [11] and source code [35] has been pointed to as indicators of potential copyright infringement [76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Similarly, copying images from professional artists has been called “digital forgery” [65] and has spurred debate in the art community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Future diffusion models might also be trained on more sensitive private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Indeed, GANs have already been applied to medical imagery [73, 20, 45], which under- lines the importance of understanding the risks of gener- ative models before we apply them to private domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Worse, a growing literature suggests that diffusion models could create synthetic training data to “protect the privacy and usage rights of real images” [37], and production tools already claim to use diffusion models to protect data privacy [71, 17, 12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our work shows diffu- sion models may be unfit for this purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 2 Understanding generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Beyond data privacy, understanding how and why diffusion models memorize training data may help us understand their generalization capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For instance, a common question for large- scale generative models is whether their impressive re- sults arise from truly novel generations, or are instead the result of direct copying and remixing of their train- ing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' By studying memorization, we can provide a concrete empirical characterization of the rates at which generative models perform such data copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In their diffusion model, Saharia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' “do not find over-fitting to be an issue, and believe further training might improve overall performance“ [60], and yet we will show that this model memorizes individual exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' It may thus be necessary to broaden our definitions of overfitting to include memorization and related pri- vacy metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our results also suggest that Feldman’s theory that memorization is necessary for generalization in classifiers [24] may extend to generative models, rais- ing the question of whether the improved performance of diffusion models compared to prior approaches is pre- cisely because diffusion models memorize more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Threat Model Our threat model considers an adversary A that interacts with a diffusion model Gen (backed by a neural network fθ) to extract images from the model’s training set D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Image-generation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unconditional diffusion models are trained on a dataset D = {x1,x2,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=',xn}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' When queried, the system outputs a generated image xgen ← Gen(r) using a fresh random noise r as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Conditional models are trained on annotated images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', labeled or captioned) D = {(x1,c1),.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=',(xn,cn)} and when queried with a prompt p, the system outputs xgen ← Gen(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='r) using the prompt p and noise r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Adversary capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We consider two adversaries: A black-box adversary can query Gen to generate images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If Gen is a conditional generator, the adver- sary can provide arbitrary prompts p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The adversary cannot control the system’s internal randomness r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' A white-box adversary gets full access to the system Gen and its internal diffusion model fθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' They can control the model’s randomness and can thus use the model to denoise arbitrary input images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In both cases, we assume that an adversary who attacks a conditional image generator knows the captions for some images in the training set—thus allowing us to study the worst-case privacy risk in diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Adversary goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We consider three broad types of ad- versarial goals, from strongest to weakest attacks: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Data extraction: The adversary aims to recover an image from the training set x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The attack is successful if the adversary extracts an image ˆx that is almost identical (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1) to some x ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Data reconstruction: The adversary has partial knowledge of a training image x ∈ D (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', a sub- set of the image) and aims to recover the full image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This is an image-analog of an attribute inference at- tack [80], which aims to recover unknown features from partial knowledge of an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Membership inference: Given an image x, the ad- versary aims to infer whether x is in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Ethics and Broader Impact Training data extraction attacks can present a threat to user privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We take numerous steps to mitigate any possible harms from our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' First, we study mod- els that are trained on publicly-available images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', LAION and CIFAR-10) and therefore do not expose any data that was not already available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Nevertheless, data that is available online may not have been intended to be available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' LAION, for example, contains unintentionally released medical im- ages of several patients [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We also therefore en- sure that all images shown in our paper are of pub- lic figures (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', politicians, musicians, actors, or au- thors) who knowingly chose to place their images on- line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As a result, inserting these images in our paper is unlikely to cause any unintended privacy violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For example, Figure 1 comes from Ann Graham Lotz’s Wikipedia profile picture and is licensed under Creative Commons, which allows us to “redistribute the material in any medium” and “remix, transform, and build upon the material for any purpose, even commercially”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Third, we shared an advance copy of this paper with the authors of each of the large-scale diffusion models that we study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This gave the authors and their corre- sponding organizations the ability to consider possible safeguards and software changes ahead of time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In total, we believe that publishing our paper and pub- licly disclosing these privacy vulnerabilities is both eth- ical and responsible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Indeed, at the moment, no one ap- pears to be immediately harmed by the (lack of) privacy of diffusion models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' our goal with this work is thus to make sure to preempt these harms and encourage respon- sible training of diffusion models in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 3 4 Extracting Training Data from State-of- the-art Diffusion Models We begin our paper by extracting training images from large, pre-trained, high-resolution diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Defining Image Memorization Most existing literature on training data extraction fo- cuses on text language models, where a sequence is said to be “extracted” and “memorized” if an adversary can prompt the model to recover a verbatim sequence from the training set [11, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Because we work with high- resolution images, verbatim definitions of memorization are not suitable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Instead, we define a notion of approxi- mate memorization based on image similarity metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Definition 1 ((ℓ,δ)-Diffusion Extraction) [adapted from [11]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We say that an example x is extractable from a diffusion model fθ if there exists an efficient algorithm A (that does not receive x as input) such that ˆx = A ( fθ) has the property that ℓ(x, ˆx) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here, ℓ is a distance function and δ is a threshold that determines whether we count two images as being iden- tical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In this paper, unless otherwise noted we follow Balle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [5] and use the Euclidean 2-norm distance ℓ2(a,b) = � ∑i(ai −bi)2/d where d is the dimension of the inputs to normalize ℓ ∈ [0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Given this definition of extractability, we can now define memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Definition 2 ((k,ℓ,δ)-Eidetic Memorization) [adapted from [11]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We say that an example x is (k,ℓ,δ)-Eidetic memorized 2 by a diffusion model if x is extractable from the diffusion model, and there are at most k training examples ˆx ∈ X where ℓ(x, ˆx) ≤ δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Again, ℓ is a distance function and δ is its correspond- ing threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The constant k quantifies the number of near-duplicates of x in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If k is a small frac- tion of the data, then memorization is likely problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' When k is a larger fraction of data, memorization might be expected—but it could still be problematic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', if the duplicated data is copyrighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 2This paper covers a very restricted definition of “memorization”: whether diffusion models can be induced to generate near-copies of some training examples when prompted with appropriate instructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We will describe an approach that can generate images that are close approximations of some training images (especially images that are fre- quently represented in the training dataset through duplication or other means).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' There is active discussion within the technical and legal com- munities about whether the presence of this type of “memorization” suggests that generative neural networks “contain” their training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 2: We do not count the generated image of Obama (at left) as memorized because it has a high ℓ2 distance to every training image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The four nearest training images are shown at right, each has a distance above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Restrictions of our definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our definition of extrac- tion is intentionally conservative as compared to what privacy concerns one might ultimately have.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For ex- ample, if we prompt Stable Diffusion to generate “A Photograph of Barack Obama,” it produces an entirely recognizable photograph of Barack Obama but not an near-identical reconstruction of any particular training image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 2 compares the generated image (left) to the 4 nearest training images under the Euclidean 2- norm (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Under our memorization definition, this image would not count as memorized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Nevertheless, the model’s ability to generate (new) recognizable pictures of certain individuals could still cause privacy harms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Extracting Data from Stable Diffusion We now extract training data from Stable Diffusion: the largest and most popular open-source diffusion model [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This model is an 890 million parameter text- conditioned diffusion model trained on 160 million im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We generate from the model using the default PLMS sampling scheme at a resolution of 512×512 pix- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As the model is trained on publicly-available images, we can easily verify our attack’s success and also mit- igate potential harms from exposing the extracted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We begin with a black-box attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Identifying duplicates in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To reduce the computational load of our attack, as is done in [65], we bias our search towards duplicated training examples because these are orders of magnitude more likely to be memorized than non-duplicated examples [47, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If we search for images that are bit-for-bit identically duplicated in the training dataset, we would significantly undercount the true rate of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Instead, we ac- count for near-duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Ideally, we would search for any training examples that are nearly duplicated with a 4 Original: Generated: Figure 3: Examples of the images that we extract from Stable Diffusion v1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 using random sampling and our mem- bership inference procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The top row shows the original images and the bottom row shows our extracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' pixel-level ℓ2 distance below some threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' But this is computationally intractable, as it would require an all- pairs comparison of 160 million images in Stable Dif- fusion’s training set, each of which is a 512 × 512 × 3 dimensional vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Instead, we first embed each image to a 512 dimensional vector using CLIP [54], and then perform the all-pairs comparison between images in this lower-dimensional space (increasing efficiency by over 1500×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We count two examples as near-duplicates if their CLIP embeddings have a high cosine similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For each of these near-duplicated images, we use the corre- sponding captions as the input to our extraction attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Extraction Methodology Our extraction approach adapts the methodology from prior work [11] to images and consists of two steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generate many examples using the diffusion model in the standard sampling manner and with the known prompts from the prior section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Perform membership inference to separate the model’s novel generations from those generations which are memorized training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generating many images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The first step is trivial but computationally expensive: we query the Gen function in a black-box manner using the selected prompts as in- put.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To reduce the computational overhead of our experi- ments, we use the timestep-resampled generation imple- mentation that is available in the Stable Diffusion code- base [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This process generates images in a more ag- gressive fashion by removing larger amounts of noise at each time step and results in slightly lower visual fidelity at a significant (∼ 10×) performance increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We gener- ate 500 candidate images for each text prompt to increase the likelihood that we find memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Performing membership inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The second step requires flagging generations that appear to be memo- rized training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Since we assume a black-box threat model in this section, we do not have access to the loss and cannot exploit techniques from state-of-the- art membership inference attacks [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We instead de- sign a new membership inference attack strategy based on the intuition that for diffusion models, with high prob- ability Gen(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='r1) ̸= Gen(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='r2) for two different random initial seeds r1,r2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' On the other hand, if Gen(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='r1) ≈d Gen(p;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='r2) under some distance measure d, it is likely that these generated samples are memorized examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The 500 images that we generate for each prompt have different (but unknown) random seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We can therefore construct a graph over the 500 generations by connect- ing an edge between generation i and j if xi ≈d xj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If the largest clique in this graph is at least size 10 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', ≥ 10 of the 500 generations are near-identical), we pre- dict that this clique is a memorized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Empirically, clique-finding is more effective than searching for pairs of images x1 ≈d x2 as it has fewer false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To compute the distance measure d among the images in the clique, we use a modified Euclidean ℓ2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In particular, we found that many generations were often spuriously similar according to ℓ2 distance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', they all had gray background).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We therefore instead divide each image into 16 non-overlapping 128×128 tiles and mea- sure the maximum of the ℓ2 distance between any pair of image tiles between the two images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Extraction Results In order to evaluate the effectiveness of our attack, we select the 350,000 most-duplicated examples from the training dataset and generate 500 candidate images for each of these prompts (totaling 175 million generated im- ages).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We first sort all of these generated images by or- dering them by the mean distance between images in the clique to identify generations that we predict are likely to be memorized training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then take each of these generated images and annotate each as either “extracted” or “not extracted” by comparing it to the training images under Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We find 94 images are (ℓ2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15)- extracted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To ensure that these images not only match 5 200 20 40 60 80 100 Memorized Examples Extracted 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 Attack Precision Manual Inspection (ℓ2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15)-Extraction Figure 4: Our attack reliably separates novel genera- tions from memorized training examples, under two def- initions of memorization—either (ℓ2,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15)-extraction or manual human inspection of generated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' some arbitrary definition, we also manually annotate the top-1000 generated images as either memorized or not memorized by visual analysis, and find that a further 13 (for a total of 109 images) are near-copies of training examples even if they do not fit our 2-norm definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 3 shows a subset of the extracted images that are reproduced with near pixel-perfect accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' all images have an ℓ2 difference under 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (As a point of refer- ence, re-encoding a PNG as a JPEG with quality level 50 results in an ℓ2 difference of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='02 on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=') Given our ordered set of annotated images, we can also compute a curve evaluating the number of extracted images to the attack’s false positive rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our attack is exceptionally precise: out of 175 million generated images, we can identify 50 memorized images with 0 false positives, and all our memorized images can be ex- tracted with a precision above 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 4 contains the precision-recall curve for both memorization definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Measuring (k,ℓ,δ)-eidetic memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Defini- tion 2 we introduced an adaptation of Eidetic memo- rization [11] tailored to the domain of generative im- age models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As mentioned earlier, we compute similar- ity between pairs of images with a direct ℓ2 pixel-space similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This analysis is computationally expensive3 as it requires comparing each of our memorized images against each of the 160 million training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We set δ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 as this threshold is sufficient to identify al- 3In practice it is even more challenging: for non-square images, Stable Diffusion takes a random square crop, and so to check if the generated image x matches a non-square training image y we must try all possible alignments between x on top of the image y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 10 30 100 300 1000 3000 Number of duplicates 0 10 20 30 Frequency Figure 5: Our attack extracts images from Stable Diffu- sion most often when they have been duplicated at least k = 100 times;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' although this should be taken as an upper bound because our methodology explicitly searches for memorization of duplicated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' most all small image corruptions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', JPEG compres- sion, small brightness/contrast adjustments) but has very few false positives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 5 shows the results of this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' While we identify little Eidetic memorization for k < 100, this is expected due to the fact we choose prompts of highly- duplicated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Note that at this level of duplication, the duplicated examples still make up just one in a mil- lion training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' These results show that duplica- tion is a major factor behind training data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The majority of the images that we extract (58%) are photographs with a recognizable person as the primary subject;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' the remainder are mostly either products for sale (17%), logos/posters (14%), or other art or graphics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We caution that if a future diffusion model were trained on sensitive (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', medical) data, then the kinds of data that we extract would likely be drawn from this sensitive data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Despite the fact that these images are publicly acces- sible on the Internet, not all of them are permissively li- censed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We find that a significant number of these im- ages fall under an explicit non-permissive copyright no- tice (35%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Many other images (61%) have no explicit copyright notice but may fall under a general copyright protection for the website that hosts them (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', images of products on a sales website).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Several of the images that we extracted are licensed CC BY-SA, which requires “[to] give appropriate credit, provide a link to the li- cense, and indicate if changes were made.” Stable Dif- fusion thus memorizes numerous copyrighted and non- 6 permissive-licensed images, which the model may repro- duce without the accompanying license.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 Extracting Data from Imagen While Stable Diffusion is the best publicly-available diffusion model, there are non-public models that achieve stronger performance using larger models and datasets [56, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Prior work has found that larger mod- els are more likely to memorize training data [11, 9] and we thus study Imagen [60], a 2 billion parameter text- to-image diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' While individual details dif- fer between Imagen’s and Stable Diffusion’s implemen- tation and training scheme, these details are independent of our extraction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We follow the same procedure as earlier but focus on the top-1000 most duplicated prompts for computa- tional reasons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then generate 500 images for each of these prompts, and compute the ℓ2 similarity between each generated image and the corresponding training image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' By repeating the same membership inference steps as above—searching for cliques under patched ℓ2 distance–we identify 23 of these 1,000 images as mem- orized training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 This is significantly higher than the rate of memorization in Stable Diffusion, and clearly demonstrates that memorization across diffusion models is highly dependent on training settings such as the model size, training time, and dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 Extracting Outlier Examples The attacks presented above succeed, but only at extract- ing images that are highly duplicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This “high k” memorization may be problematic, but as we mentioned previously, the most compelling practical attack would be to demonstrate memorization in the “low k” regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We now set out to achieve this goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In order to find non-duplicated examples likely to be memorized, we take advantage of the fact that while on average models often respect the privacy of the majority of the dataset, there often exists a small set of “outlier” examples whose privacy is more significantly exposed [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' And so in- stead of searching for memorization across all images, we are more likely to succeed if we focus our effort on these outlier examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' But how should we find which images are poten- tially outliers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Prior work was able to train hundreds of models on subsets of the training dataset and then 4Unfortunately, because the Imagen training dataset is not public, we are unable to provide visual examples of successful reconstructions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' use an influence-function-style approach to identify ex- amples that have a significant impact on the final model weights [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unfortunately, given the cost of training even a single large diffusion model is in the millions-of- dollars, this approach will not be feasible here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Therefore we take a simpler approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We first com- pute the CLIP embedding of each training example, and then compute the “outlierness” of each example as the average distance (in CLIP embedding space) to its 1,000 nearest neighbors in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Surprisingly, we find that attacking out-of- distribution images is much more effective for Imagen than it is for Stable Diffusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' On Imagen, we attempted extraction of the 500 images with the highest out-of- distribution score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Imagen memorized and regurgitated 3 of these images (which were unique in the training dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In contrast, we failed to identify any memo- rization when applying the same methodology to Stable Diffusion—even after attempting to extract the 10,000 most-outlier samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Thus, Imagen appears less pri- vate than Stable Diffusion both on duplicated and non- duplicated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We believe this is due to the fact that Imagen uses a model with a much higher capacity com- pared to Stable diffusion, which allows for more memo- rization [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Moreover, Imagen is trained for more iter- ations and on a smaller dataset, which can also result in higher memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5 Investigating Memorization The above experiments are visually striking and clearly indicate that memorization is pervasive in large diffusion models—and that data extraction is feasible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' But these experiments do not explain why and how these models memorize training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In this section we train smaller diffusion models and perform controlled experiments in order to more clearly understand memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For the remainder of this sec- tion, we focus on diffusion models trained on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We use state-of-the-art training code 5 to train 16 diffu- sion models, each on a randomly-partitioned half of the CIFAR-10 training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We run three types of pri- vacy attacks: membership inference attacks, attribute in- 5We either directly use OpenAI’s Improved Diffusion repos- itory (https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='com/openai/improved-diffusion) in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1, or our own re-implementation in all following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Models trained with our re-implementation achieve almost identical FID to the open-sourced models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We use half the dataset as is stan- dard in privacy analyses [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 7 Figure 6: Direct 2-norm measurement fails to identify memorized CIFAR-10 examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Each of the above im- ages have a ℓ2 distance of less than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05, yet only one (the car) is actually a memorized training example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' ference attacks, and data reconstruction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For the membership inference attacks, we train class-conditional models that reach an FID below 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5 (see Figure 11), plac- ing them in the top-30 generative models on CIFAR-10 [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For reconstruction attacks (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1) and at- tribute inference attacks with inpainting (Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3), we train unconditional models with an FID below 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Untargeted Extraction Before devling deeper into understanding memorization, we begin by validating that memorization does still occur in our smaller models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Because these models are not text conditioned, we focus on untargeted extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Specif- ically, given our 16 diffusion models trained on CIFAR- 10, we unconditionally generate 216 images from each model for a total of 220 candidate images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Because we will later develop high-precision membership inference attacks, in this section we directly search for memorized training examples among all our million generated exam- ples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Thus this is not an attack per se, but rather verifying the capability of these models to memorize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Identifying matches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In the prior section, we performed targeted attacks and could therefore check for successful memorization by simply computing the ℓ2 distance be- tween the target image and the generated image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here, as we perform an all-pairs comparison, we find that us- ing an uncalibrated ℓ2 threshold fails to accurately iden- tify memorized training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For example, if we set a highly-restrictive threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05, then nearly all “ex- tracted” images are of entirely blue skies or green land- scapes (see Figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We explored several other met- rics (including perceptual distances like SSIM or CLIP embedding distance) but found that none could reliably identify memorized training images for CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We instead define an image as extracted if the ℓ2 dis- tance to its nearest neighbor in the training set is abnor- mally low compared to all other training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Fig- ure 7 illustrates this by computing the ℓ2 distance be- tween two different generated images and every image in the CIFAR-10 training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The left figure shows a failed extraction attempt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' despite the fact that the nearest 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='00 L2 distance between generated and training images 100 101 102 103 Frequency 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='00 Figure 7: Per-image ℓ2 thresholds are necessary to sep- arate memorized images from novel generations on a CIFAR-10 model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Each plot shows the distribution of ℓ2 distances from a generated image to all training images (along with the image and the nearest training image).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Left shows a typical distribution for a non-memorized image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Right shows a memorized image distribution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' while the most similar training image has high absolute ℓ2 distance, it is abnormally low for this distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The dashed black line shows our adaptive ℓ2 threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' training image has an ℓ2 distance of just 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='06, this dis- tance is on par with the distance to many other training images (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', all images that contain a blue sky).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In con- trast, the right plot shows a successful extraction attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here, even though the ℓ2 distance to the nearest train- ing image is higher than for the prior failed attack (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='07), this value is unusually small compared to other training images which almost all are at a distance above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We thus slightly modify our attack to use the distance ℓ(ˆx,x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='Sˆx) = ℓ2(ˆx,x) α ·Ey∈Sˆx[ℓ2(ˆx,y)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' where Sˆx is the set containing the n closest elements from the training dataset to the example ˆx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This distance is small if the extracted image x is much closer to the train- ing image ˆx compared to the n closest neighbors of ˆx in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We run our attack with α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5 and n = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our attack was not sensitive to these choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Using the above methodology we iden- tify 1,280 unique extracted images from the CIFAR-10 dataset (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5% of the entire dataset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 In Figure 8 we show a selection of training examples that we extract and full results are shown in Figure 17 in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 6Some CIFAR-10 training images are generated multiple times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In these cases, we only count the first generation as a successful attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Further, because the CIFAR-10 training dataset contains many dupli- cate images, we do not count two generations of two different (but du- plicated) images in the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 8 Figure 8: Selected training examples that we extract from a diffusion model trained on CIFAR-10 by sampling from the model 1 million times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Top row: generated output from a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Bottom row: nearest (ℓ2) example from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 17 in the Appendix contains all 1,280 unique extracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Membership Inference Attacks We now evaluate membership inference with more tra- ditional attack techniques that use white-box access, as opposed to Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 that assumed black-box access.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We will show that all examples have significant privacy leakage under membership inference attacks, compared to the small fraction that are sensitive to data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We consider two membership inference attacks on our class-conditional CIFAR-10-trained diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='7 The loss threshold attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Yeom et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [80] introduce the simplest membership inference attack: because mod- els are trained to minimize their loss on the training set, we should expect that training examples have lower loss than non-training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The loss threshold attack thus computes the loss l = L (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' f) and reports “mem- ber” if l < τ for some chosen threshold τ and otherwise “non-member’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The value of τ can be selected to max- imize a desired metric (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', true positive rate at some fixed false positive rate or the overall attack accuracy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The Likelihood Ratio Attack (LiRA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Carlini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [8] introduce the state-of-the-art approach to performing membership inference attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' LiRA first trains a col- lection of shadow models, each model on random sub- sets of the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' LiRA then computes the loss L (x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' fi) for the example x under each of these shadow models fi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' These losses are split into two sets: the losses IN = {lini} for the example x under the shadow models { fi} that did see the example x during training, and the losses OUT = {louti} for the example x under the shadow models { f j} that did not see the example x during train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' LiRA finishes the initialization process by fitting Gaussians NIN to the IN set and NOUT to OUT set of losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Finally, to predict membership inference for a new model f ∗, we compute l∗ = L (x, f ∗) and then mea- sure whether Pr[l∗|NIN] > Pr[l∗|NOUT].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Choosing a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Both membership inference attacks use a loss function L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In the case of classifica- tion models, Carlini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [8] find that choosing a loss 7Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 replicates these results for unconditional models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' function is one of the most important components of the attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We find that this effect is even more pronounced for diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In particular, unlike classifiers that have a single loss function (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', cross entropy) used to train the model, diffusion models are trained to minimize the reconstruction loss when a random quantity of Gaus- sian noise ε has been added to an image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This means that “the loss” of an image is not well defined—instead, we can only ask for the loss L (x,t,ε) of an image x for a certain timestep t with a corresponding amount of noise ε (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Equation (1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We must thus compute the optimal timestep t at which we should measure the loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To do so, we train 16 shadow models each on a random 50% of the CIFAR- 10 training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then compute the loss for every model, for every example in the training dataset, and ev- ery timestep t ∈ [1,T] (T = 1,000 in the models we use).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 9 plots the timestep used to compute the loss against the attack success rate, measured as the true pos- itive rate (TPR), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', the number of examples which truly are members over the total number of members, at a fixed false positive rate (FPR) of 1%, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', the fraction of exam- ples which are incorrectly identified as members.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Eval- uating L at t ∈ [50,300] leads to the most successful attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We conjecture that this a “Goldilock’s zone” for membership inference: if t is too small, and so the noisy image is similar to the original, then predicting the added noise is easy regardless if the input was in the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' if t is too large, and so the noisy image is similar to Gaussian noise, then the task is too difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our remain- ing experiments will evaluate L (·,t,·) at t = 100, where we observed a TPR of 71% at an FPR of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Baseline Attack Results We now evaluate membership inference using our speci- fied loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We follow recent advice [8] and evalu- ate the efficacy of membership inference attacks by com- paring their true positive rate to the false positive rate on a log-log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Figure 10, we plot the member- ship inference ROC curve for the loss threshold attack and LiRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' An out-of-the-box implementation of LiRA 9 1 200 400 600 800 1000 Diffusion timestep 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='7 TPR@FPR=1% Figure 9: We run membership inference using LiRA and compute the diffusion model loss at different noise timesteps on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Evaluating L (·,t,·) at t ∈ [50,300] produces the best results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' achieves a true positive rate of over 70% at a false posi- tive rate of just 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As a point of reference, state-of-the- art classifiers are much more private, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', with a < 20% TPR at 1% FPR [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This shows that diffusion models are significantly less private than classifiers trained on the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (In part this may be because diffusion models are often trained far longer than classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=') Qualitative analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Figure 20, we visualize the least- and most-private images as determined by their easiness to detect via LiRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We find that the easiest- to-attack examples are all extremely out-of-distribution visually from the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' These images are even more visually out-of-distribution compared to the outliers identified by Feldman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [24] who produce a similar set of images but for image classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In con- trast, the images that are hardest to attack are all dupli- cated images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' It is challenging to detect the presence or absence of each of these images in the training dataset because there is another identical image in the training dataset that may have been present or absent—therefore making the membership inference question ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Augmentations Improve Attacks Membership inference attacks can also be improved by reducing the variance in the loss signal [8, 79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We study two ways to achieve this for diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' First, because our loss function has randomness (re- call that to compute the reconstruction loss we mea- sure the quantity L (x,t,ε) for a random noise sam- ple ε ∼ N (0,I)), we can compute a better estimate of the true loss by averaging over different noise samples: L (x,t) = Eε∼N (0,I)[L (x,t,ε)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 10 3 10 2 10 1 100 False positive rate 10 3 10 2 10 1 100 True positive rate Strong LiRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='997 LiRA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='982 Threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='613 Figure 10: Membership inference ROC curve for a diffu- sion model trained on CIFAR-10 using the loss threshold attack, baseline LiRA, and “Strong LiRA” with repeated queries and augmentation (§5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' By varying the number of point samples taken to es- timate this expectation we can potentially increase the attack success rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' And second, because our diffusion models train on augmented versions of training images (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', by flipping images horizontally), it makes sense to compute the loss averaged over all possible augmen- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Prior work has found that both of these attack strategies are effective at increasing the efficacy of mem- bership inference attacks for classifiers [8, 39], and we find they are effective here as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Improved attack results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 10 shows the effect of combining both these strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Together they are re- markably successful, and at a false positive rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1% they increase the true positive rate by over a factor of six from 7% to 44%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 19 in the Appendix breaks down the impact of each component: in Figure 19a we increase the number of Monte Carlo samples from 1 (the base LiRA attack) to 20, and in Figure 19b we augment samples with a horizontal flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 Memorization Versus Utility We train our diffusion models to reach state-of-the-art levels of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Prior work on language mod- els has found that better models are often easier to at- tack than less accurate models—intuitively, because they extract more information from the same training dataset [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here we perform a similar experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Attack results vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' FID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To evaluate our generative models, we use the standard Fr´echet Inception Distance (FID) [32], where lower scores indicate higher qual- ity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our previous CIFAR-10 results used models that 10 4 6 8 FID 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 TPR@FPR=1% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 Update step 1e6 Figure 11: Better diffusion models are more vulnerable to membership inference attacks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' evaluating with TPR at an FPR of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As the FID decreases (corresponding to a quality increase) the membership inference attack success rate grows from 7% to nearly 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' achieved the best FID (on average 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5) based on early stopping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here we evaluate models over the course of training in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We compute the attack success rate as a function of FID, and we find that as the quality of the diffusion model increases so too does the privacy leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' These results are concerning because they sug- gest that stronger diffusion models of the future may be even less private.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 Inpainting Attacks Having performed untargeted extraction on CIFAR-10 models, we now construct a targeted version of our at- tack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As mentioned earlier, performing a targeted at- tack is complicated by the fact that these models do not support textual prompting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We instead provide guid- ance by performing a form of attribute inference attack [38, 80, 81] that we call an “inpainting attack”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Given an image, we first mask out a portion of this image;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' our attack objective is to recover the masked region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then run this attack on both training and testing images, and compare the attack efficacy on each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Specifically, for an image x, we mask some fraction of pixels to create a masked image xm, and then use the trained model to re- construct the image as xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The exact algorithm we use for inpainting is given in Lugmayr et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Because diffusion model inpainting is stochastic (it de- pends on the random sample ε ∼ N (0,I)), we create a set of inpainted images Xrec = {x1 rec,x2 rec,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=',xn rec}, where we set n = 5,000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For each xrec ∈ Xrec, we compute the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content="30 2 distance when x isn't in training 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='30 2 distance when x is in training Cat example Bird example 100 other samples Figure 12: Evaluating inpainting attacks on 100 CIFAR- 10 examples, measuring the ℓ2 distance between images and their inpainted reconstructions when we mask out the left half of the image for 100 randomly selected im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We also plot the ℓ2 distances for the bird and cat examples shown in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' When an adversary has partial knowledge of an image, inpainting attacks work far better than typical data extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' diffusion model’s loss on this sample (at timestep 100) divided by a shadow model’s loss that was not trained on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then use this score to identify the highest-scoring reconstructions xrec ∈ Xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our specific attack masks out the left half of an image and applies the diffusion model on the right half of the image to inpaint the rest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We repeat this pro- cess 5000 times and take the top-10 scoring reconstruc- tions using a membership inference attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We repeat this attack for 100 images using diffusion models that are trained with and without the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 12 com- pares the average distance between the sample and the ten highest scoring inpainted samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This allows us to show our inpainting attacks have succeed: the recon- struction loss is substantially better in terms of ℓ2 dis- tance when the image is in the training set than when not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 13 also shows qualitative examples of this attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The highest-scoring reconstruction looks visually similar to the target image when the target is in training and does not resemble the target when it is not in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Over- all, these results show that an adversary who has partial knowledge of an image can substantially improve their extraction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We conduct a more thorough analysis of inpainting attacks in Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 11 Target: x Masked: xm Reconstruction when x is in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Reconstruction when x is not in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Target: x Masked: xm Reconstruction when x is in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Reconstruction when x is not in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 13: Inpainting-based reconstruction attack on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Given an image from CIFAR-10 (first col- umn), we randomly mask half of the image (second col- umn), and then inpaint the image for a model which con- tained this image in the training set (third column) versus inpainting the image for a model which did not contain this image in the training set (fourth column).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 6 Comparing Diffusion Models to GANs Are diffusion models more or less private than compet- ing generative modeling approaches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In this section we take a first look at this question by comparing diffu- sion models to Generative Adversarial Networks (GANs) [30, 61, 55], an approach that has held the state-of-the-art results for image generation for nearly a decade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unlike diffusion models that are explicitly trained to memorize and reconstruct their training datasets, GANs are not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Instead, GANs consist of two competing neu- ral networks: a generator and a discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Similar to diffusion models, the generator receives random noise as input, but unlike a diffusion model, it must convert this noise to a valid image in a single forward pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To train a GAN, the discriminator is trained to predict if an im- age comes from the generator or not, and the generator is trained to fool the discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As a result, GANs differ from diffusion models in that their generators are only trained using indirect information about the train- ing data (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', using gradients from the discriminator) be- cause they never receive training data as input, whereas diffusion models are explicitly trained to reconstruct the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Membership inference attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We first propose a privacy attack methodology for GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='8 We initially fo- cus on membership inference attacks, where following Balle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [5], we assume access to both the discrimi- nator and generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We perform membership inference using the loss threshold [80] and LiRA [8] attacks, where 8While existing privacy attacks exist for GANs, they were proposed before the latest advancements in privacy attack techniques, requiring us to develop our own methods which out-perform prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Architecture Images Extracted FID GANs StyleGAN-ADA [43] 150 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='9 DiffBigGAN [82] 57 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 E2GAN [69] 95 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 NDA [63] 70 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 WGAN-ALP [68] 49 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 DDPMs OpenAI-DDPM [52] 301 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='9 DDPM [33] 232 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Table 1: The number of training images that we extract from different off-the-shelf pretrained generative mod- els out of 1 million unconditional generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We show GAN models sorted by FID (lower is better) on the top and diffusion models on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, we find that diffusion models memorize more than GAN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Moreover, better generative models (lower FID) tend to memorize more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' we use the discriminator’s loss as the metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To per- form LiRA, we follow a similar methodology as Sec- tion 5 and train 256 individual GAN models each on a random 50% split of the CIFAR-10 training dataset but otherwise leave training hyperparameters unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We study three GAN architectures, all implemented using the StudioGAN framework [42]: BigGAN [6], MHGAN [74], and StyleGAN [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 14 shows the membership inference results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, diffusion models have higher membership inference leakage, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', diffu- sion models had 50% TPR at a FPR of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1% as compared to < 30% TPR for GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This suggests that diffusion models are less private than GANs for membership in- ference attacks under default training settings, even when the GAN attack is strengthened due to having access to the discriminator (which would be unlikely in practice, as only the generator is necessary to create new images).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Data extraction results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We next turn our attention away from measuring worst-case privacy risk and focus our attention on more practical black-box extraction at- tacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We follow the same procedure as Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1, where we generate 220 images from each model architec- ture and identify those that are near-copies of the training data using the same similarity function as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Again we only consider non-duplicated CIFAR-10 training im- ages in our counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For this experiment, instead of us- ing models we train ourselves (something that was neces- sary to run LiRA), we study five off-the-shelf pre-trained GANs: WGAN-ALP [68], E2GAN [69], NDA [63], DiffBigGAN [82], and StyleGAN-ADA [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We also evaluate two off-the-shelf DDPM diffusion model re- leased by Ho et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [33] and Nichol et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Note that all of these pre-trained models are trained by the origi- 12 10−5 10−4 10−3 10−2 10−1 100 False Positive Rate 10−5 10−4 10−3 10−2 10−1 100 True Positive Rate LiRA auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='891, TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='109 Global threshold auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='878, TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='021 (a) StyleGAN FID avg = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='7 10−5 10−4 10−3 10−2 10−1 100 False Positive Rate 10−5 10−4 10−3 10−2 10−1 100 True Positive Rate LiRA auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='971, TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='258 Global threshold auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='511, TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001 (b) MHGAN FID avg = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='9 10−5 10−4 10−3 10−2 10−1 100 False Positive Rate 10−5 10−4 10−3 10−2 10−1 100 True Positive Rate LiRA auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='989, TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='418 Global threshold auc=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='967, TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='001: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='003 (c) BigGAN FID avg = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='7 Figure 14: Membership inference results on GAN models using the loss threshold and LiRA attacks on the discrimi- nator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, GANs are significantly more private than diffusion models under default training configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (a) StyleGAN (b) MHGAN (c) BigGAN Figure 15: Selected training examples we extract from three GANs trained on CIFAR-10 for different architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Top row: generated output from a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Bottom row: nearest (ℓ2) example from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 25 in the Appendix contains all unique extracted images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' nal authors to maximize utility on the entire CIFAR-10 dataset rather than a random 50% split as in our prior models trained for MIA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Table 1 shows the number of extracted images for each model and their corresponding FID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, we find that diffusion models memorize more data than GANs, even when the GANs reach similar performance, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', the best DDPM model memorizes 2× more than StyleGAN- ADA but reaches the same FID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Moreover, generative models (both GANs and diffusion models) tend to mem- orize more data as their quality (FID) improves, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', StyleGAN-ADA memorizes 3× more images than the weakest GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Using the GANs we trained ourselves, we show ex- amples of the near-copy generations in Figure 15 for the three GANs that we trained ourselves, and Figure 24 in the Appendix shows every sample that we extract for those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The Appendix also contains near-copy generations from the five off-the-shelf GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overall, these results further reinforce the conclusion that diffu- sion models are less private than GAN models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We also surprisingly find that diffusion models and GANs memorize many of the same images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In particular, despite the fact that our diffusion model memorizes 1280 images and a StyleGAN model we train on half of the dataset memorizes 361 images, we find that 244 unique images are memorized in common.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If images were mem- orized uniformly at random, we should expect on average 10 images would be memorized by both, giving excep- tionally strong evidence that some images (p < 10−261) are inherently less private than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Understanding why this phenomenon occurs is a fruitful direction for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 13 7 Defenses and Recommendations Given the degree to which diffusion models memorize and regenerate training examples, in this section we ex- plore various defenses and practical strategies that may help to reduce and audit model memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Deduplicating Training Data In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2, we showed that many examples that are easy to extract are duplicated many times (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', > 100) in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Similar results have been shown for language models for text [11, 40] and data deduplica- tion has been shown to be an effective mitigation against memorization for those models [47, 41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In the image domain, simple deduplication is common, where images with identical URLs and captions are removed, but most datasets do not compute other inter-image similarity met- rics such as ℓ2 distance or CLIP similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We thus en- courage practitioners to deduplicate future datasets using these more advanced notions of duplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unfortunately, deduplication is not a perfect solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To better understand the effectiveness of data deduplica- tion, we deduplicate CIFAR-10 and re-train a diffusion model on this modified dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We compute image sim- ilarity using the imagededup tool and deduplicate any images that have a similarity above > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This re- moves 5,275 examples from the 50,000 total examples in CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We repeat the same generation procedure as Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1, where we generate 220 images from the model and count how many examples are regenerated from the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The model trained on the dedu- plicated data regenerates 986 examples, as compared to 1280 for the original model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' While not a substantial drop, these results show that deduplication can mitigate memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Moreover, we also expect that deduplica- tion will be much more effective for models trained on larger-scale datasets (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', Stable Diffusion), as we ob- served a much stronger correlation between data extrac- tion and duplication rates for those models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Differentially-Private Training The gold standard technique to defend against privacy attacks is by training with differential privacy (DP) guar- antees [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Diffusion models can be trained with differentially-private stochastic gradient descent (DP- SGD) [1], where the model’s gradients are clipped and noised to prevent the model from leaking substantial in- formation about the presence of any individual image in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Applying DP-SGD induces a trade-off be- tween privacy and utility, and recent work shows that 1 2 3 4 8 16 32 64 Duplicate Count 2 4 6 8 10 Maximum Exposure Random Figure 16: Canary exposure (a measure of non-privacy) as a function of duplicate count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Inserting a canary twice is sufficient to reach maximum exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' DP-SGD can be applied to small-scale diffusion models without substantial performance degradation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unfortunately, we applied DP-SGD to our diffusion model codebase and found that it caused the training on CIFAR-10 to consistently diverge, even at high values for ε (the privacy budget, around 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In fact, even applying a non-trivial gradient clipping or noising on their own (both are required in DP-SGD) caused the training to fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We leave a further investigation of these failures to future work, and we believe that new advances in DP-SGD and privacy-preserving training techniques may be required to train diffusion models in privacy-sensitive settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 Auditing with Canaries In addition to implementing defenses, it is important for practitioners to empirically audit their models to de- termine how vulnerable they are in practice [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our attacks above represent one method to evaluate model privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Nevertheless, our attacks are expensive, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', our membership inference results require training many shadow models, and thus lighter weight alternatives may be desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' One such alternative is to insert canary examples into the training set, a common approach to evaluate mem- orization in language models [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here, one creates a large “pool” of canaries, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', by randomly generating noise images, and inserts a subset of the canaries into the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' After training, one computes the expo- sure of the canaries, which roughly measures how many bits were learned about the inserted canaries as compared to the larger pool of not inserted canaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This loss- based metric only requires training one model and can also be designed in a worst-case way (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', adversarial worst-case images could be used).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' To evaluate exposure for diffusion models, we gen- 14 erate canaries consisting of uniformly generated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We then duplicate the canaries in the training set at dif- ferent rates and measure the maximum exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Fig- ure 16 shows the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Here, the maximum exposure is 10, and some canaries reach this exposure after being inserted only twice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The exposure is not strictly increas- ing with duplicate count, which may be a result of some canaries being “harder” than others, and, ultimately, ran- dom canaries we generate may not be the most effective canaries to use to test memorization for diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 8 Related Work Memorization in language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Numerous past works study memorization in generative models across different domains, architectures, and threat models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' One area of recent interest is memorization in language mod- els for text, where past work shows that adversaries can extract training samples using two-step attack techniques that resemble our approach [11, 47, 41, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our work differs from these past results because we focus on the image domain and also use more semantic notions of data regeneration (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', using CLIP scores) as opposed to focusing on exact verbatim repetition (although recent language modeling work has begun to explore approxi- mate memorization as well [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Memorization in image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Aside from lan- guage modeling, past work also analyzes memorization in image generation, mainly from the perspective of gen- eralization in GANs (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', the novelty of model gener- ations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For instance, numerous metrics exist to mea- sure similarity with the training data [32, 3], the extent of mode collapse [61, 15], and the impact of individual training samples [4, 75].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Moreover, other work provides insights into when and why GANs may replicate train- ing examples [50, 26], as well as how to mitigate such effects [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our work extends these lines of inquiry to conditional diffusion models, where we measure novelty by computing how frequently models regenerate training instances when provided with textual prompts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Recent and concurrent work also studies privacy in im- age generation for both GANs [70] and diffusion mod- els [65, 78, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Tinsley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [70] show that StyleGAN can generate individuals’ faces, and Somepalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [65] show that Stable Diffusion can output semantically sim- ilar images to its training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Compared to these works, we identify privacy vulnerabilities in a wider range of systems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', Imagen and CIFAR models) and threat models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=', membership inference attacks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 9 Discussion and Conclusion State-of-the-art diffusion models memorize and regen- erate individual training images, allowing adversaries to launch training data extraction attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' By training our own models we find that increasing utility can de- grade privacy, and simple defenses such as deduplication are insufficient to completely address the memorization challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We see that state-of-the-art diffusion models memorize 2× more than comparable GANs, and more useful diffusion models memorize more than weaker dif- fusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This suggests that the vulnerability of generative image models may grow over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Going forward, our work raises questions around the memoriza- tion and generalization capabilities of diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Questions of generalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Do large-scale models work by generating novel output, or do they just copy and interpolate between individual training examples?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If our extraction attacks had failed, it may have refuted the hypothesis that models copy and interpolate training data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' but because our attacks succeed, this question re- mains open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Given that different models memorize vary- ing amounts of data, we hope future work will explore how diffusion models copy from their training datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Our work also highlights the difficulty in defining memorization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' While we have found extensive mem- orization with a simple ℓ2-based measurement, a more comprehensive analysis will be necessary to accurately capture more nuanced definitions of memorization that allow for more human-aligned notions of data copying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Practical consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We raise four practical con- sequences for those who train and deploy diffusion mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' First, while not a perfect defense, we recom- mend deduplicating training datasets and minimizing over-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Second, we suggest using our attack—or other auditing techniques—to estimate the privacy risk of trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Third, once practical privacy-preserving techniques become possible, we recommend their use whenever possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Finally, we hope our work will tem- per the heuristic privacy expectations that have come to be associated with diffusion model outputs: synthetic data does not give privacy for free [13, 14, 59, 2, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' On the whole, our work contributes to a growing body of literature that raises questions regarding the legal, eth- ical, and privacy issues that arise from training on web- scraped public data [7, 65, 72, 77].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Researchers and prac- titioners should be wary of training on uncurated public data without first taking steps to understand the underly- ing ethics and privacy implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 15 NC MN JH MJ FT VS BB DI EW Conceived Project X X X X Formalized Memorization Definition X X X X X X Experimented with Stable Diffusion X X Experimented with Imagen X Experimented with CIFAR-10 Diffusion X X Experimented with GANs X X X Experimented with Defenses X X X Prepared Figures X X X X X X X Analyzed Data X X X X X X Wrote Paper X X X X X X X X X Managed the Project X Table 2: Contributions of each author in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Contributions Nicholas, Jamie, Vikash, and Eric each indepen- dently proposed the problem statement of extracting training data from diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Nicholas, Eric, and Florian performed preliminary experiments to identify cases of data extraction in diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Milad performed most of the experiments on Stable Diffusion and Imagen, and Nicholas counted dupli- cates in the LAION training dataset;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' each wrote the corresponding sections of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Jamie performed the membership inference attacks and inpainting attacks on CIFAR-10 diffusion mod- els, and Nicholas performed the diffusion extraction experiments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' each wrote the corresponding sections of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Matthew ran experiments for canary memorization and wrote the corresponding section of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Florian and Vikash performed preliminary experi- ments on memorization in GANs, and Milad and Vikash ran the experiments included in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Milad ran the membership inference experiments on GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Vikash ran extraction experiments on pretrained GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Daphne and Florian improved figure clarity and pre- sentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Daphne, Borja, and Eric edited the paper and con- tributed to paper framing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Nicholas organized the project and wrote the initial paper draft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Acknowledgements and Conflicts of Interest The authors are grateful to Tom Goldstein, Olivia Wiles, Katherine Lee, Austin Tarango, Ian Wilbur, Jeff Dean, Andreas Terzis, Robin Rombach, and Andreas Blattmann for comments on early drafts of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Nicholas, Milad, Matthew, and Daphne are employed at Google, and Jamie and Borja are employed at Deep- Mind, companies that both train large machine learning models (including diffusion models) on both public and private datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Eric Wallace is supported by the Apple Scholars in AI/ML Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' References [1] Mart´ın Abadi, Andy Chu, Ian Goodfellow, H Bren- dan McMahan, Ilya Mironov, Kunal Talwar, and Li Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Deep learning with differential privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In ACM CCS, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [2] Hazrat Ali, Shafaq Murad, and Zubair Shah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Spot the fake lungs: Generating synthetic medical im- ages using neural diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='00902, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [3] Sanjeev Arora, Andrej Risteski, and Yi Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Do GANs learn the distribution?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Some theory and em- pirics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [4] Yogesh Balaji, Hamed Hassani, Rama Chellappa, and Soheil Feizi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Entropic GANs meet VAEs: A statistical approach to compute sample likelihoods in GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Machine Learning, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 16 [5] Borja Balle, Giovanni Cherubin, and Jamie Hayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Reconstructing training data with informed adver- saries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE Symposium on Security and Pri- vacy, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [6] Andrew Brock, Jeff Donahue, and Karen Si- monyan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Large scale GAN training for high fidelity natural image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Confer- ence on Learning Representations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [7] Hannah Brown, Katherine Lee, Fatemehsadat Mireshghallah, Reza Shokri, and Florian Tram`er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' What does it mean for a language model to pre- serve privacy?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In ACM Conference on Fairness, Accountability, and Transparency, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [8] Nicholas Carlini, Steve Chien, Milad Nasr, Shuang Song, Andreas Terzis, and Florian Tramer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Mem- bership inference attacks from first principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE Symposium on Security and Privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' IEEE, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [9] Nicholas Carlini, Daphne Ippolito, Matthew Jagielski, Katherine Lee, Florian Tramer, and Chiyuan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Quantifying memorization across neural language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='07646, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [10] Nicholas Carlini, Chang Liu, ´Ulfar Erlingsson, Jernej Kos, and Dawn Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The secret sharer: Evaluating and testing unintended memorization in neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In USENIX Security Symposium, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [11] Nicholas Carlini, Florian Tramer, Eric Wallace, Matthew Jagielski, Ariel Herbert-Voss, Katherine Lee, Adam Roberts, Tom Brown, Dawn Song, Ul- far Erlingsson, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Extracting training data from large language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In USENIX Security Sym- posium, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [12] Andrew Carr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Gretel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='ai: Diffusion models for document synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https://gretel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='ai/blog/ diffusion-models-for-document-synthesis, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [13] Pierre Chambon, Christian Bluethgen, Jean-Benoit Delbrouck, Rogier Van der Sluijs, Małgorzata Połacin, Juan Manuel Zambrano Chaves, Tan- ishq Mathew Abraham, Shivanshu Purohit, Cur- tis P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Langlotz, and Akshay Chaudhari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' RoentGen: Vision-language foundation model for chest X- ray generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='12737, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [14] Pierre Chambon, Christian Bluethgen, Curtis P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Langlotz, and Akshay Chaudhari.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Adapting pretrained vision-language foundational models to medical imaging domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='04133, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [15] Tong Che, Yanran Li, Athul Paul Jacob, Yoshua Bengio, and Wenjie Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Mode regularized gener- ative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Con- ference on Learning Representations, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [16] Papers With Code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https://paperswithcode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' com/sota/image-generation-on-cifar-10, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [17] Elise Devaux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' List of synthetic data vendors— 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https://elise-deux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='com/ new-list-of-synthetic-data-vendors-2022-f06dbe91784, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [18] Prafulla Dhariwal and Alexander Nichol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Diffusion models beat GANs on image synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [19] Tim Dockhorn, Tianshi Cao, Arash Vahdat, and Karsten Kreis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Differentially private diffusion mod- els.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='09929, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [20] August DuMont Sch¨utte, J¨urgen Hetzel, Sergios Gatidis, Tobias Hepp, Benedikt Dietz, Stefan Bauer, and Patrick Schwab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Overcoming barriers to data sharing with medical image generation: a comprehensive evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' NPJ Digital Medicine, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [21] C Dwork, F McSherry, K Nissim, and A Smith.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Calibrating noise to sensitivity in private data anal- ysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In TCC, 2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [22] Cynthia Dwork.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Differential privacy: A survey of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In TAMC, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [23] Benj Edwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Artist finds private med- ical record photos in popular AI train- ing data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https://arstechnica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' com/information-technology/2022/09/ artist-finds-private-medical-record-photos-in-popular-ai-training-data-set, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [24] Vitaly Feldman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Does learning require memo- rization?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' A short tale about a long tail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In ACM SIGACT Symposium on Theory of Comput- ing, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 17 [25] Vitaly Feldman and Chiyuan Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' What neural networks memorize and why: Discovering the long tail via influence estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [26] Qianli Feng, Chenqi Guo, Fabian Benitez-Quiroz, and Aleix M Martinez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' When do GANs replicate?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' on the choice of dataset size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE/CVF Inter- national Conference on Computer Vision, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [27] Matt Fredrikson, Somesh Jha, and Thomas Risten- part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Model inversion attacks that exploit confi- dence information and basic countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In ACM Conference on Computer and Communica- tions Security (CCS), 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [28] Matthew Fredrikson, Eric Lantz, Somesh Jha, Si- mon Lin, David Page, and Thomas Ristenpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Privacy in pharmacogenetics: An end-to-end case study of personalized warfarin dosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In USENIX Security Symposium, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [29] Ian Goodfellow, Yoshua Bengio, and Aaron Courville.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Deep learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' MIT press, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [30] Ian Goodfellow, Jean Pouget-Abadie, Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Informa- tion Processing Systems, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [31] Peter Henderson, Koustuv Sinha, Nicolas Angelard-Gontier, Nan Rosemary Ke, Genevieve Fried, Ryan Lowe, and Joelle Pineau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Ethical challenges in data-driven dialogue systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In AAAI/ACM Conference on AI, Ethics, and Society, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [32] Martin Heusel, Hubert Ramsauer, Thomas Un- terthiner, Bernhard Nessler, and Sepp Hochreiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' GANs trained by a two time-scale update rule con- verge to a local nash equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neu- ral Information Processing Systems, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [33] Jonathan Ho, Ajay Jain, and Pieter Abbeel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' De- noising diffusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [34] Hailong Hu and Jun Pang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Membership in- ference of diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='09956, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [35] Daphne Ippolito, Florian Tram`er, Milad Nasr, Chiyuan Zhang, Matthew Jagielski, Katherine Lee, Christopher A Choquette-Choo, and Nicholas Car- lini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Preventing verbatim memorization in lan- guage models gives a false sense of privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='17546, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [36] Matthew Jagielski, Jonathan Ullman, and Alina Oprea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Auditing differentially private machine learning: How private is private SGD?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [37] Ali Jahanian, Xavier Puig, Yonglong Tian, and Phillip Isola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generative models as a data source for multiview representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' International Conference on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [38] Bargav Jayaraman and David Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Are attribute inference attacks just imputation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' ACM Confer- ence on Computer and Communications Security (CCS), 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [39] Bargav Jayaraman, Lingxiao Wang, Katherine Knipmeyer, Quanquan Gu, and David Evans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Re- visiting membership inference under realistic as- sumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Proceedings on Privacy Enhancing Technologies, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [40] Nikhil Kandpal, Haikang Deng, Adam Roberts, Eric Wallace, and Colin Raffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Large language models struggle to learn long-tail knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='08411, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [41] Nikhil Kandpal, Eric Wallace, and Colin Raffel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Deduplicating training data mitigates privacy risks in language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' International Conference on Machine Learning, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [42] MinGuk Kang, Joonghyuk Shin, and Jaesik Park.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' StudioGAN: A Taxonomy and Benchmark of GANs for Image Synthesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='09479, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [43] Tero Karras, Miika Aittala, Janne Hellsten, Samuli Laine, Jaakko Lehtinen, and Timo Aila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Training generative adversarial networks with limited data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Advances in Neural Information Processing Sys- tems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [44] Tero Karras, Samuli Laine, and Timo Aila.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' A style- based generator architecture for generative adver- sarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE/CVF conference on com- puter vision and pattern recognition, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [45] Salome Kazeminia, Christoph Baur, Arjan Kuijper, Bram van Ginneken, Nassir Navab, Shadi Albar- qouni, and Anirban Mukhopadhyay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' GANs for 18 medical image analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Artificial Intelligence in Medicine, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [46] Diederik P Kingma and Max Welling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Auto- encoding variational bayes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Con- ference on Learning Representations, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [47] Katherine Lee, Daphne Ippolito, Andrew Nystrom, Chiyuan Zhang, Douglas Eck, Chris Callison- Burch, and Nicholas Carlini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Deduplicating train- ing data makes language models better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Associ- ation for Computational Linguistics, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [48] Andreas Lugmayr, Martin Danelljan, Andres Romero, Fisher Yu, Radu Timofte, and Luc Van Gool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' RePaint: Inpainting using denoising dif- fusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE/CVF Confer- ence on Computer Vision and Pattern Recognition, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [49] AnGeL Ministries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' File:Anne Gra- ham Lotz (October 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https: //commons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='wikimedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='org/wiki/File: Anne_Graham_Lotz_(October_2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='jpg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Accessed on December 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [50] Vaishnavh Nagarajan, Colin Raffel, and Ian J Goodfellow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Theoretical insights into memoriza- tion in GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Neural Information Processing Systems Workshop, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [51] Alex Nichol, Prafulla Dhariwal, Aditya Ramesh, Pranav Shyam, Pamela Mishkin, Bob McGrew, Ilya Sutskever, and Mark Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' GLIDE: To- wards photorealistic image generation and editing with text-guided diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='10741, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [52] Alexander Quinn Nichol and Prafulla Dhariwal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Improved denoising diffusion probabilistic models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Machine Learning, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [53] Walter H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Pinaya, Petru-Daniel Tudosiu, Jes- sica Dafflon, Pedro F da Costa, Virginia Fer- nandez, Parashkev Nachev, Sebastien Ourselin, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Jorge Cardoso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Brain imaging genera- tion with latent diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXi:2209.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='07162, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [54] Alec Radford, Jong Wook Kim, Chris Hallacy, Aditya Ramesh, Gabriel Goh, Sandhini Agarwal, Girish Sastry, Amanda Askell, Pamela Mishkin, Jack Clark, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Learning transferable visual mod- els from natural language supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Interna- tional Conference on Machine Learning, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [55] Alec Radford, Luke Metz, and Soumith Chintala.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unsupervised representation learning with deep convolutional generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Learning Representa- tions, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [56] Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, and Mark Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Hierarchical text- conditional image generation with CLIP latents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2204.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='06125, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [57] Aditya Ramesh, Mikhail Pavlov, Gabriel Goh, Scott Gray, Chelsea Voss, Alec Radford, Mark Chen, and Ilya Sutskever.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Zero-shot text-to-image generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Ma- chine Learning, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [58] Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, and Bj¨orn Ommer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' High- resolution image synthesis with latent diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE/CVF Conference on Computer Vi- sion and Pattern Recognition, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [59] Pouria Rouzrokh, Bardia Khosravi, Shahriar Faghani, Mana Moassefi, Sanaz Vahdati, and Bradley J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Erickson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Multitask brain tumor inpaint- ing with diffusion models: A methodological re- port.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='12113, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [60] Chitwan Saharia, William Chan, Saurabh Sax- ena, Lala Li, Jay Whang, Emily Denton, Seyed Kamyar Seyed Ghasemipour, Burcu Karagol Ayan, S Sara Mahdavi, Rapha Gontijo Lopes, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Photorealistic text-to-image diffusion models with deep language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='11487, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [61] Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, and Xi Chen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Im- proved techniques for training GANs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [62] Reza Shokri, Marco Stronati, Congzheng Song, and Vitaly Shmatikov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Membership inference at- tacks against machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE Symposium on Security and Privacy, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [63] Abhishek Sinha, Kumar Ayush, Jiaming Song, Bu- rak Uzkent, Hongxia Jin, and Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Neg- ative data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Confer- ence on Learning Representations, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [64] Jascha Sohl-Dickstein, Eric Weiss, Niru Mah- eswaranathan, and Surya Ganguli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Deep unsu- pervised learning using nonequilibrium thermody- 19 namics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Machine Learning, 2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [65] Gowthami Somepalli, Vasu Singla, Micah Gold- blum, Jonas Geiping, and Tom Goldstein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Dif- fusion art or digital forgery?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Investigating data replication in diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='03860, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [66] Yang Song and Stefano Ermon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generative mod- eling by estimating gradients of the data distribu- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [67] Midjourney Team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='midjourney.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' com/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [68] D´avid Terj´ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Adversarial lipschitz regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Learning Represen- tations, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [69] Yuan Tian, Qin Wang, Zhiwu Huang, Wen Li, Dengxin Dai, Minghao Yang, Jun Wang, and Olga Fink.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Off-policy reinforcement learning for effi- cient and effective gan architecture search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Eu- ropean Conference on Computer Vision, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [70] Patrick Tinsley, Adam Czajka, and Patrick Flynn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This face does not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' but it might be yours!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Identity leakage in generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE/CVF Winter Conference on Applications of Computer Vision, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [71] Rob Toews.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Synthetic data is about to trans- form artificial intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' forbes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='com/sites/robtoews/2022/06/12/ synthetic-data-is-about-to-transform-artificial-intelligence/, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [72] Florian Tram`er, Gautam Kamath, and Nicholas Carlini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Considerations for differentially private learning with large-scale public pretraining.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='06470, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [73] Allan Tucker, Zhenchen Wang, Ylenia Rotalinti, and Puja Myles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Generating high-fidelity synthetic patient data for assessing machine learning health- care software.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' NPJ Digital Medicine, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [74] Ryan Turner, Jane Hung, Eric Frank, Yunus Saatchi, and Jason Yosinski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Metropolis-hastings generative adversarial networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In International Conference on Machine Learning, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [75] Gerrit van den Burg and Chris Williams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' On mem- orization in probabilistic deep generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Sys- tems, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [76] James Vincent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The lawsuit that could rewrite the rules of AI copyright.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' https: //www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='theverge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='com/2022/11/8/23446821/ microsoft-openai-github-copilot-class-action-lawsuit-ai-copyright-violation-training-data, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [77] Eric Wallace, Florian Tram`er, Matthew Jagielski, and Ariel Herbert-Voss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Does GPT-2 know your phone number?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' BAIR Blog, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [78] Yixin Wu, Ning Yu, Zheng Li, Michael Backes, and Yang Zhang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Membership inference attacks against text-to-image generation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' arXiv preprint arXiv:2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='00968, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [79] Jiayuan Ye, Aadyaa Maddi, Sasi Kumar Mu- rakonda, Vincent Bindschaedler, and Reza Shokri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Enhanced membership inference attacks against machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In ACM SIGSAC Con- ference on Computer and Communications Security (CCS), 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [80] Samuel Yeom, Irene Giacomelli, Matt Fredrikson, and Somesh Jha.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Privacy risk in machine learning: Analyzing the connection to overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE Computer Security Foundations Symposium (CSF), 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [81] Yuheng Zhang, Ruoxi Jia, Hengzhi Pei, Wenxiao Wang, Bo Li, and Dawn Song.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The secret re- vealer: Generative model-inversion attacks against deep neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In IEEE/CVF conference on computer vision and pattern recognition, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' [82] Shengyu Zhao, Zhijian Liu, Ji Lin, Jun-Yan Zhu, and Song Han.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Differentiable augmentation for data-efficient GAN training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Advances in Neural Information Processing Systems, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 20 A Collected Details for Figures Table 3: Catalog of figures containing qualitative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure # Model Dataset Who trained it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Sampling strategy Figure 1 Stable Diffusion LAION Stability AI PLMS Figure 2 Stable Diffusion LAION Stability AI PLMS Figure 3 Stable Diffusion LAION Stability AI PLMS Figure 6 Uncond Diffusion CIFAR-10 Ours DDIM Figure 7 Uncond Diffusion CIFAR-10 Ours DDIM Figure 8 Uncond Diffusion CIFAR-10 Ours DDIM Figure 12 Uncond Diffusion CIFAR-10 Ours Inpainting Figure 13 Uncond Diffusion CIFAR-10 Ours Inpainting Figure 15 StyleGAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' MHGAN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' BigGAN CIFAR-10 Ours GAN default Figure 17 Uncond Diffusion CIFAR-10 Ours DDIM Figure 20 Uncond Diffusion CIFAR-10 Ours DDIM Figure 22 Uncond Diffusion CIFAR-10 Ours Inpainting Figure 23 Uncond Diffusion CIFAR-10 Ours Inpainting Figure 24 Several different GANs CIFAR-10 Original paper authors GAN default 21 B All CIFAR-10 Memorized Images Figure 17: All 1280 images we extract from diffusion models trained on CIFAR-10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' after 1 million generations from 16 diffusion models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 22 C Additional Attacks on CIFAR-10 Here, we expand on our investigation of memorization of training data on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1 Membership Inference at Different Training Steps 1000 2000 3000 4000 Each train example processed X times 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 TPR@FPR=1% (a) How membership attack success changes as a training example is processed repeatedly throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 Training data seen 1e8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0 TPR@FPR=1% (b) How membership attack success changes as more data is processed throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 10 3 10 2 10 1 100 False positive rate 10 3 10 2 10 1 100 True positive rate Data seen: 5M AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='742 TPR@FPR=1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='050 Data seen: 102M AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='997 TPR@FPR=1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='989 (c) ROC curve for the membership attack for different training steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 18: Membership inference attacks as a function of the amount of training data processed on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3, we implicitly investigated membership attack success as a function of the number update steps when training a diffusion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We explicitly model this relationship in Figure 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' First, in Figure 18a we plot membership attack success as a function of the number of times that an example was processed over training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' If an example is processed more than 2000 times during training, invariably membership attacks are perfect against that example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Second, in Figure 18b, we plot membership attack success as a function of the total amount of data processed during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Unsurprisingly, membership attack success increases as more training data is processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This is highlighted in Figure 18c, where we plot the membership attack ROC curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' At 5M training examples processed, at a FPR of 1% the TPR is 5%, and increases to 99% after 102M examples are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Note that this number of processed training inputs is commonly used in diffusion model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For example, the OpenAI CIFAR-10 diffusion model 9 is trained for 500,000 steps at a batch size of 128, meaning 64M training examples are processed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Even at this number of processed training examples, our membership attack has a TPR > 95% at a FPR of 1%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 9https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='com/openai/improved-diffusion 23 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 Membership Inference with Different Augmentation Strategies 10 3 10 2 10 1 100 False positive rate 10 3 10 2 10 1 100 True positive rate n: 1 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='982 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='071 n: 2 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='991 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='128 n: 5 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='995 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='210 n: 10 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='996 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='260 n: 20 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='997 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='294 (a) 10 3 10 2 10 1 100 False positive rate 10 3 10 2 10 1 100 True positive rate w/o Aug n: 20 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='997 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='294 w/ Aug n: 20 AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='997 TPR@FPR=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='437 (b) Figure 19: We can improve membership inference attack success rates on CIFAR-10 by reducing noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (a), membership inference attacks are improved by averaging the loss over multiple noise samples in the diffusion process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (b), attacks are improved by querying on augmented versions of the candidate image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3 Membership Inference Inliers and Outliers Figure 20: When performing our membership inference attack, the hardest-to-attack examples (left) are all duplicates in the CIFAR-10 training set, and the easiest-to-attack examples (right) are visually outliers from CIFAR-10 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 25 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='4 Membership Inference on Conditional and Unconditional Models Diffusion models can be conditioned on labels (or prompts for text-to-image models).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We compare the difference in membership inference on a CIFAR-10 diffusion model trained unconditionally with a model conditionally trained on CIFAR-10 labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' The conditional and unconditional models reach approximately the same FID after training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' between 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='5-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='2 FID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We plot the membership attack ROC curve in Figure 21 and note that the conditional model is marginally more vulnerable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' However, it is difficult to tell if this is a fundamental difference between conditional and unconditional models, or because the conditional model contains more parameters than unconditional model (the conditional models contains an extra embedding layer for the one-hot label input).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 10 3 10 2 10 1 100 False positive rate 10 3 10 2 10 1 100 True positive rate Unconditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='970 TPR@FPR=1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='549 Conditional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' AUC: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='982 TPR@FPR=1%: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='714 Figure 21: Membership attack against a conditional and unconditional diffusion model on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 26 D More Inpainting Attacks on CIFAR-10 Here, we take a deeper dive into the inpainting attacks introduced in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' As previously explained, for a target x, we create Xrec where |Xrec| = 5000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Figure 22a, for every xrec ∈ Xrec, we plot the normalized ℓ2 distance between the reconstruction and target, against the loss (at diffusion timestep 100) of xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We also plot in Figure 22d, the eight examples from Xrec that have the smallest loss on the main model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' There is a small positive correlation between loss and ℓ2 distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' although some appear to be similar to x, there are notable differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In Figure 22b we compare the loss of each reconstruction on the main model against the support model we will use to form the contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We make this correlation more pronounced by dividing the main loss by the support loss in Figure 22c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This has the effect of increasing the correlation between the (now contrastive) loss and ℓ2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This has the effect of filtering out examples that are seen as likely under both models, and can be seen by inspecting the eight examples from Xrec that have have the smallest main model loss support model loss in Figure 22e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' These examples look more visually similar to x in comparison to examples in Figure 22d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 22 inspected the attack success when x was in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We show in Figure 23 that the attack fails when x was not included in training;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' using a contrastive loss doesn’t signficantly increase the Pearson correlation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' This means our attack is indeed exploiting the fact that the model can only inpaint correctly because of memorisation and not due to generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 27 (a) Loss (using the main model at diffusion timestep 100) on all 5,000 inpainted examples Xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (b) Comparison of loss on main and support models (at diffusion timestep 100) on all 5,000 inpainted examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (c) Contrastive loss ( main model loss support model loss) on all 5,000 inpainted examples Xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (d) 8 inpainted examples with the smallest loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Leftmost is the original example, second to left is the masked example and the rest are inpainted examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (e) 8 inpainted examples with the smallest main model loss support model loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Leftmost is the original example, second to left is the masked example and the rest are inpainted examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 22: Example of an inpainting attack (against a model we refer to as the main model) on an image of a bird from CIFAR-10 when that image is included in training, and we mask out 60% of the central pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (a) we plot the L2 distance between 5,000 inpainted reconstructions and the original (non-masked out) image and compare this to the loss with respect to the (main) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (b), we compare the loss of these reconstructions on the (main) model with a support model for which we know the image wasn’t contained in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (c), we compare L2 distances between reconstructions with a contrastive loss which is given as the loss of the image with respect to the main model divided by the loss of the image with respect to the support model, and find there is stronger relationship between smaller L2 distances and smaller losses compared to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure (d) gives examples of reconstructions with small loss and Figure (e) gives examples of reconstructions with small contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 28 Pearson camelstion ccefficient: Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='D6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='04 (of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='30 distance to target0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='D-6 (of support model) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='D5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='06 Lauss (of miain mkdel)Pearson camelstion ccefficient: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='63 11 Contrastive 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='B 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='30 distance to target(a) Loss (using the main model at diffusion timestep 100) on all 5,000 inpainted examples Xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (b) Comparison of loss on main and support models (at diffusion timestep 100) on all 5,000 inpainted examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (c) Contrastive loss ( main model loss support model loss) on all 5,000 inpainted examples Xrec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (d) 8 inpainted examples with the smallest loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Leftmost is the original example, second to left is the masked example and the rest are inpainted examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' (e) 8 inpainted examples with the smallest main model loss support model loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Leftmost is the original example, second to left is the masked example and the rest are inpainted examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure 23: Example of an inpainting attack (against a model we refer to as the main model) on an image of a bird from CIFAR-10 when that image is not included in training, and we mask out 60% of the central pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (a) we plot the L2 distance between 5,000 inpainted reconstructions and the original (non-masked out) image and compare this to the loss with respect to the (main) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (b), we compare the loss of these reconstructions on the (main) model with a support model for which we know the image wasn’t contained in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In (c), we compare L2 distances between reconstructions with a contrastive loss which is given as the loss of the image with respect to the main model divided by the loss of the image with respect to the support model, and find there is stronger relationship between smaller L2 distances and smaller losses compared to (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' Figure (d) gives examples of reconstructions with small loss and Figure (e) gives examples of reconstructions with small contrastive loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' E GAN Training Setup We used on StudioGAN10 codebase for training GAN in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For the StyleGAN and MHGAN architectures, we followed the default hyper-parameters provided in the StudioGAN repository.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' However, for the BigGAN architecture, we increased the number of training steps to 200,000, which is different from the original hyper-parameters, to increase image fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We trained a total of 256 models for each GAN architecture, with each model being trained on a randomly selected half of the CIFAR-10 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We selected the iteration that achieved the highest FID score on the test set for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' F Additional GAN Extraction Results Figure 24 and Figure 25 contain additional examples extracted from GANs trained on CIFAR-10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 10https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='com/POSTECH-CVLab/PyTorch-StudioGAN 29 Pearson camelstion ccefficient: o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='08 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='07 main 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='0G 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='35 distance to targetLass (of support model) 500 500 t 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='D40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='050.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='D6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='D7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='b8 Lass (of miain mkdel)Pearson canelstion ccefficient: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='18 13 11 1D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content='35 distance to target(a) StyleGAN (b) MHGAN (c) BigGAN Figure 24: Training examples extracted from a CIFAR-10 GAN for different architectures across 107 generations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 30 A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 一 十 1(a) WGAN (b) E2GAN (c) NDA (d) DiffAugment-BigGAN (e) StyleGAN-ADA (f) DDPM Figure 25: Training examples extracted from different publicly available pretrained GANs and diffusion (DDPM) models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' We use normalized ℓ2 distance in pixel space to find memorized training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' In each pair of images, left and right image corresponds to real and it closely synthetic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For StyleGAN-ADA and DDPM model we display 120 pairs with smallest normalized ℓ2 distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' For others we display all memorized training images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} +page_content=' 1M generations 31 B' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/lNFPT4oBgHgl3EQf2zXD/content/2301.13188v1.pdf'} diff --git a/m9E1T4oBgHgl3EQf1QUs/content/tmp_files/2301.03465v1.pdf.txt b/m9E1T4oBgHgl3EQf1QUs/content/tmp_files/2301.03465v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eb4cefb216b192ce71e563982a240a23264e9cc6 --- /dev/null +++ b/m9E1T4oBgHgl3EQf1QUs/content/tmp_files/2301.03465v1.pdf.txt @@ -0,0 +1,1650 @@ +1 +Deep Learning for Short-Latency Epileptic +Seizure Detection with Probabilistic Classification +Yankun Xu, Jie Yang, Wenjie Ming, Shuang Wang, and Mohamad Sawan, Fellow, IEEE +Abstract—In this manuscript, we propose a novel deep learning +(DL)-based framework intended for obtaining short latency in +real-time electroencephalogram-based epileptic seizure detection +using multiscale 3D convolutional neural networks. We pioneer +converting seizure detection task from traditional binary classifi- +cation of samples from ictal and interictal periods to probabilistic +classification of samples from interictal, ictal, and crossing peri- +ods. We introduce a crossing period from seizure-oriented EEG +recording and propose a labelling rule using soft-label for samples +from the crossing period to build a probabilistic classification +task. A novel multiscale short-time Fourier transform feature +extraction method and 3D convolution neural network architec- +ture are proposed to accurately capture predictive probabilities +of samples. Furthermore, we also propose rectified weighting +strategy to enhance predictive probabilities, and accumulative +decision-making rule to achieve short detection latency. We +implement leave-one-seizure-out cross validation on two prevalent +datasets – CHB-MIT scalp EEG dataset and SWEC-ETHZ +intracranial EEG dataset. Eventually, the proposed algorithm +achieved 94 out of 99 seizures detected during the crossing period, +averaged 14.84% rectified predictive ictal probability (RPIP) +errors of crossing samples, 2.3 s detection latency, 0.32/h false +detection rate on CHB-MIT dataset, meanwhile 84 out of 89 +detected seizures, 16.17% RPIP errors, 4.7 s detection latency, +and 0.75/h FDR are achieved on SWEC-ETHZ dataset. The +obtained detection latencies are at least 50% faster than state- +of-the-art results reported in previous studies. +Index +Terms—Epilepsy, +seizure +detection, +EEG, +brain- +computer interface, detection latency, probabilistic classification, +deep learning +I. INTRODUCTION +E +PILEPTIC patients suffering from unprovoked recurrent +seizures occupy approximately 1% population worldwide +[1], [2]. Seizure is originated from abnormal discharged neu- +ron inside small brain region, then discharging current is +spread to other regions. Two main seizure types, named focal +and generalized seizure, depend on seizures begin in one area +then spread to other areas or seizures begin throughout brain +cortex simultaneously. Long-term drug therapy is one of the +major treatment intended for epileptic patients, however, one- +third of patients face drug-resistant epilepsy [3], [4]. +This work was supported in part by the Westlake University, in part by +the Zhejiang Key R&D Program under Grant 2021C03002, and in part by +the Zhejiang Leading Innovative and Entrepreneur Team Introduction under +Grant 2020R01005. (Corresponding authors: Jie Yang; Mohamad Sawan.) +Yankun Xu, Jie Yang and Mohamad Sawan are with the Center of +Excellence in Biomedical Research on Advanced Integrated-on-chips Neu- +rotechnologies (CenBRAIN Neurotech), School of Engineering, Westlake +University, Hangzhou, Zhejiang, China, (e-mail: yangjie@westlake.edu.cn; +sawan@westlake.edu.cn). +Wenjie Ming and Shuang Wang are with the Epilepsy Center, Department +of Neurology, Second Affiliated Hospital, School of Medicine, Zhejiang +University, Hangzhou, China. +EEG Onset +Clinical Onset +Many +seconds +Samples +… +… +… +1 +0 +Times (s） +0 +Times (s） +EEG +Recordings +: True Label +: Binary Detection +Labelling +Decision +1 +Prob. +Prob. +: ∑ ������������������������ (Sum of Prob.) +: ������������������������ (Predictive Ictal Prob.) +Decision Threshold +Interictal +Crossing +Ictal +Latency +������������������������ +������������������������ +������������������������ : Expected Detection Latency +������������������������ : Latency in Binary Way +Alarm +Fig. 1. Schematic figure for illustrating EEG recordings, segmented samples, +traditional seizure detection challenges, and expected decision-making system. +Recently, brain-computer interface (BCI) has been broadly +applied to healthcare and neuroscience domain, such as re- +habilitation, brain stimulation, prosthetic control, and brain +activity diagnosis. [5]–[8]. A BCI-based closed-loop seizure +detection system that consists of recording, detection, stim- +ulation elements, can heavily support epileptic patients [9]. +Especially for patients with tonic-clonic seizures because +electrical stimulation intervenes can be operated in time prior +to the beginning of convulsions. Hence, an accurate real-time +epileptic seizure onset detection algorithm becomes a critical +factor to alarm the occurrence of seizure promptly [10]. +As typical BCI monitoring modalities, scalp electroen- +cephalogram (EEG) and intracranial EEG (iEEG) are widely +applied to supervise epileptic patients by recording brain activ- +ities. Usually, patients suffer from a couple of seizures per day, +and seizure onsets and endings are identified by an experienced +epileptologist. Clinically the period between seizure onset and +seizure end is defined as ictal period usually lasting 30 s to +2 min, followed by a postictal period usually lasting 5-30 +min [11]–[13]. The other period appearing in the health state +is defined as interictal period. Medical experts annotate the +EEG onset time according to the aberrant signs occurred in +the EEG recordings from epileptic patients. However, patients +do not behave abnormally at EEG onset time immediately, +usually unequivocal EEG onset precedes clinical onset by +arXiv:2301.03465v1 [eess.SP] 4 Jan 2023 + +2 +several seconds. Litt et al. [14], [15] announced this gap +would be 7-10 s. The clinical onset refers to the appearance +of relevant symptoms, such as convulsion and jerking, that +can be obviously reflected on the EEG recordings. Hence, a +reliable seizure detection algorithm is required to obtain short +enough detection latency, thereby being capable of recognizing +the seizure occurred during the period between EEG onset and +clinical onset. As shown in the top of Fig. 1, EEG Recordings +part displays a real EEG recording example of a patient around +the time of EEG onset. +Over the past decades, numerous algorithm-based seizure +detection studies have been published, and most of works an- +nounced they achieved high sensitivity and low false detection +rate (FDR) at the same time. However, high sensitivity alone is +still far away from actual seizure intervention usage. Because +in terms of practical epileptic patients supervision scenarios, +short enough detection latency is crucial to guarantee the risk +alarms promptly and intervenes can be effectively operated +prior to serious onset symptoms. Unfortunately, most previous +studies overlooked this important metric. According to our best +knowledge, most previous seizure detection studies trained the +machine learning (ML)-based or DL-based seizure detection +algorithm as a binary classification model to distinguish the +segmented interictal and ictal samples extracted from corre- +sponding periods, which is shown in Samples part of Fig. 1. +However, this strategy remains a drawback that the trained +binary classification model cannot correctly detect the crossing +samples consisting of partial interictal and ictal components. +As shown in Labelling part of Fig. 1, interictal and ictal +samples are labelled as 0 or 1 in traditional binary way +for training and trained binary model can recognize them as +0 or 1 correctly, but trained binary model would wrongly +detect the crossing samples as 0 or 1 randomly according +to our experiments. The reason is that crossing samples are +significantly different from the major part of ictal period +because they are closed to the interictal period, if crossing +samples are directly considered as ictal periods in binary way, +the binary classification model would only learn the ictal +samples with obvious oscillation characteristics mainly instead +of crossing samples. It should be noted that the time of each +detected dot in the figure is consistent with the end of detected +samples. +An accurate and prompt seizure detection system is ex- +pected to detect crossing samples in linearly increasing prob- +abilistic format according to the corresponding percentage of +ictal component, meanwhile to keep the complete interictal +or ictal samples in binary format. Then, an accumulative +decision-making rule can be used to alarm the seizure oc- +currence in short latency. In Decision part of Fig. 1, gray +dots represent predictive probabilities of samples in real-time, +and the blue line shows accumulative probability. When the +accumulative probability reaches the decision threshold, the +detection system would alarm. Therefore, as shown in Latency +part of Fig. 1, the expected detection latency 𝐿1 can be +shorter than the length of segmented samples, while the binary +classification model can only achieve the latency 𝐿2 at least +longer than the length of segmented samples. +In this manuscript, we propose a novel DL-based framework +intended for real-time detection of epileptic seizures with short +latency. The main innovative contributions aiming to address +the challenges and limitations mentioned above are as follows: +• We pioneer introducing crossing period and convert +seizure detection task from traditional binary classifica- +tion to probabilistic classification. +• We propose a novel DL model based on multiscale 3D +convolution neural networks (M-3D-CNN) to accurately +capture predictive probabilities of samples. +• A rectified probability weighting strategy is proposed to +further enhance the probabilistic results. +• And an accumulative decision-making rule is proposed to +achieve short latency and low FDR simultaneously. +The remaining content of this manuscript is organized as: +We describe in Section II recent related works about seizure +detection field. Section III elaborates on utilized materials, data +processing, and the proposed framework. Section IV illustrates +experimental settings and achieved performance. Sections V +and VI are the objects of discussion and conclusion, respec- +tively. +II. RELATED WORK +Over the past decades, algorithm-based seizure detection +study is a hot topic attracted by numerous researchers. Shoeb +et al. [16] initially detected 131 of 139 seizure events with 8 +s detection latency and 0.25/h FDR on 36 clinical pediatric +subjects in 2004. Then they published significant performance +that 96% detection sensitivity with averaged 4.6 s latency on +24 pediatric subjects in 2010 [17]. They took advantage of +spectral and spatial features combined with support vector +machine (SVM) classifier to achieve advanced results. As +pioneers of this field, Shoeb et al. [17] also published CHB- +MIT scalp dataset, it has become one of the most famous and +important resource intended for seizure-related study. In 2011, +from the same group, researchers applied similar approaches to +the clinical iEEG dataset from 10 patients and achieved 97% +detection sensitivity, 0.03/h FDR and 5 s detection latency +[18]. +In 2017, Vidyaratne et al. [19] used an improved wavelet +method known as harmonic wavelet packet transform to extract +higher frequency information as features, then SVM is used +as classifier. They achieved 96% sensitivity, 0.1/h FDR, and +1.89 s detection latency on CHB-MIT dataset. In [20], authors +applied statistical and morphological features combined with +an adaptive distance-based change point detector to achieve +96% sensitivity, 0.12/h FDR, and 4.21 detection latency, +respectively on CHB-MIT dataset. The empirical mode de- +composition method is a prevalent technique broadly applied +to seizure detection applications, numerous authors utilized +it and its variation to achieve satisfactory performance over +the past decade [21]–[24]. The short-time Fourier transform +(STFT) is an effective method widely used to extract seizure- +related features. Yuan et al. [25] proposed a multi-view +deep learning framework for EEG seizure detection based on +STFT and convolution neural network (CNN), they achieved +94.37% accuracy using 5-fold patient-specific cross validation +on CHB-MIT dataset. Different from traditional convolutional + +3 +operation that considers EEG signal or STFT features as 2D +image-like features, 1D-CNN architecture is also applied to +seizure-related studies to [26]–[29]. In [27], authors achieved +sensitivity, FDR, and detection latency of 99.31%, 0.2/h, and +8.1 s from event-based level on CHB-MIT dataset. Li et al. +[30] proposed a novel channel-embedding spectral-temporal +squeeze-and-excitation network using wavelet features to rec- +ognize epileptic EEG signals, they achieved 92.41% sensitivity +and 96.05% specificity on CHB-MIT dataset. +Burrello et al. [31]–[33] from Swiss research group pub- +lished a seizure-oriented iEEG database, known as SWEC- +ETHZ database. They achieved 94.84% specificity and 95.42% +accuracy on short-term dataset, and detected 79 out of 92 +unseen seizures without any false alarms across all the patients +on long-term dataset. Afterwards, Wang et al. [27] and Sun +et al. [34] obtained sensitivity, FDR, detection latency of +97.52%, 0.07/h FDR, 13.2 s and 97.5%, 0.06/h FDR, 13.7 +s, respectively. +From our point of view, there is a remained controversy in +measuring detection latency metric. As we depicted in Fig. 1, +binary classification model cannot achieve the latency shorter +than the length of segmented samples. According to prior-art +publications, they segmented EEG samples in at least 5 s, +however, most of them announced their proposed algorithms +can achieve less than 5 s detection latency. Hence, we doubt +that previous researchers overlooked the crossing samples +from real-time perspectives, leading to compute the detection +latency by measuring the distance between EEG onset and +begin of detected sample instead of end of detected sample. +In this manuscript, we pioneer introducing crossing period and +probabilistic classification in real-time seizure detection task +to achieve short latency. +III. METHODOLOGY +A. Materials +In this work, EEG and iEEG datasets are both considered to +validate the proposed framework. The CHB-MIT scalp EEG +dataset is one of the most prevalent open access datasets +intended for seizure-related research. There are 23 pediatric +patients monitored by 22 electrode channels with 256 Hz +sampling rate, and each subject obtains various numbers of +seizure and non-seizure EEG recording files in 1h. The earliest +change associated with the clinical seizure is annotated as +EEG onset by clinical experts. We ignored the subjects with +changing electrode placement, thereby 19 subjects are selected +for the following experiments. +The SWEC-ETHZ dataset is an emerging and open-access +iEEG dataset collected from pre-surgical evaluations of pa- +tients with pharmacoresistant epilepsies, it was published in +2018. This database contains two versions according to differ- +ent recording durations – long-term and short-term versions. +In this work, we use short-term version to test our proposed +algorithm. In terms of short-term dataset, each patient obtains +several seizure files, and each file consists of a 3-min interictal +period immediately followed by an ictal period and a 3- +min postictal period. The EEG onset time is identified by +an experienced epileptologist. Because there is insufficient +TABLE I +SUMMARY OF TWO DATASETS USED IN THIS WORK. +dataset +EEG +type +# of +selected +patients +# of +channels +# of +seizures +Interictal +duration +CHB-MIT +sEEG +19 +22 +99 +335.5h +SWEC-ETHZ +iEEG +11 +42∼100 +89 +4.45h +sEEG: scalp EEG +iEEG: intracranial EEG +interictal duration for model training on short-term dataset, we +only select patients with no less than 4 seizures in this dataset, +so that 11 of 16 patients are selected. Furthermore, the number +of implanted electrode channels used in these patients varies +from 42 to 100, and 512 Hz sampling rate is chosen. Table +I summarizes the characteristics of two datasets used in this +work. +B. Definitions of different periods +DL algorithms cannot train the model directly on the suc- +cessive EEG recordings, so that we need to extract successive +segmented samples from the recording in advance. Firstly, +we need to identify the interictal, ictal and crossing periods. +According to the seizure file of CHB-MIT dataset, beginning +and ending time of each seizure is provided, and we manually +set a 30 min postictal period following seizure ending, then +left part is belong to interictal period. In terms of short-term +version of SWEC-ETHZ dataset, each patient obtains several +seizure recording files, each file consists of a 3 min interictal +segment immediately followed by an ictal segment and a 3min +postictal segment. +The schematic figure for definition of crossing period is +shown as Fig. 2, where +𝐿𝑖𝑐𝑡𝑎𝑙 +𝐿𝑐𝑟𝑜𝑠𝑠 denotes the length of ictal +component occupying the whole length of crossing sample. +The crossing period is defined as ending of extracted samples +begins at EEG onset time (the last time for +𝐿𝑖𝑐𝑡𝑎𝑙 +𝐿𝑐𝑟𝑜𝑠𝑠 = 0) and +ends at a duration of extracted sample after EEG onset (the +first time for 𝐿𝑖𝑐𝑡𝑎𝑙 +𝐿𝑐𝑟𝑜𝑠𝑠 = 1). In experiments, samples are extracted +in 5 s and 10 s segments for CHB-MIT dataset and SWEC- +ETHZ dataset, respectively. +C. Data preparation +Furthermore, duration of interictal period is much longer +than duration of ictal period on CHB-MIT dataset, so that we +overcome this unbalanced data issue by extracting interictal +samples without any overlaps, and ictal samples with 80% +overlaps, respectively. And we data-point-wisely extracted +crossing samples. Because the duration of crossing period for +each seizure equals to the length of segmented samples, the +duration of crossing period should be 5 s, and the number +of extracted crossing samples for each seizure is computed +by duration of segmented samples multiplying sampling rate +(5 (s) × 256 (Hz) = 1280 (samples)). As for SWEC-ETHZ +dataset, we directly extracted samples of all three periods with +80% overlaps. + +4 +Fig. 2. Explanation of crossing period. In crossing period, extracted sample +consists of partial interictal and ictal component simultaneously. The duration +of crossing period depends on the length of segmented samples. +D. Probabilistic labelling rule +In terms of traditional binary classification, interictal and +ictal samples are annotated by [1, 0] and [0, 1] according +to the one-hot encoding rule. However, the label of crossing +sample should contain probabilistic information rather than +simple one-hot information. The intuitive operation is to +directly label crossing samples as a single probability as a +regression task required, but this type of labelling would cause +crossing samples cannot be trained with interictal and ictal +samples together. We aim to train the M-3D-CNN model +using interictal, ictal, and crossing samples together, because +crossing samples are comprised of partial interictal and ictal +parts, and M-3D-CNN model is expected to learn the unique +crossing samples characteristics from complete interictal and +ictal samples. +In this work, we keep labelling ictal and interictal sam- +ples as binary format, and take soft-label strategy to an- +notate the crossing samples in probabilistic format [35]. +Compared to traditional regression task training, soft-label +annotation replace the output vector with the shape of +(1, ) (e.g., 𝑃𝑖𝑐𝑡𝑎𝑙 = 0.3, 0.6, 0.8, ... ) with the output vec- +tor with the shape of (1, 2) (e.g., [𝑃𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙, 𝑃𝑖𝑐𝑡𝑎𝑙] += +[0.7, 0.3], [0.4, 0.6], [0.2, 0.8], ...). There are several advan- +tages of soft-label strategy compared to the regression training. +Firstly, the soft-label strategy makes networks can train the +crossing samples, complete interictal and ictal samples simul- +taneously. Secondly, soft-label enables usage of cross entropy +loss function, which takes both interictal and ictal information +into account rather than only ictal probability provided with +simple mean square error loss function in regression task . +In terms of experimental training setting, we keep la- +bels of interictal and ictal samples in the traditional way +(𝐿𝑎𝑏𝑒𝑙𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙 = [1, 0], 𝐿𝑎𝑏𝑒𝑙𝑖𝑐𝑡𝑎𝑙 = [0, 1]) and label cross- +ing samples into 20 probability pairs as following rules: +𝐿𝑎𝑏𝑒𝑙𝑐𝑟𝑜𝑠𝑠 = [𝑃𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙, 𝑃𝑖𝑐𝑡𝑎𝑙] +𝑤ℎ𝑒𝑟𝑒, 𝑃𝑖𝑐𝑡𝑎𝑙 = 0.05𝑝 𝑖 𝑓 +𝐿𝑖𝑐𝑡𝑎𝑙 +𝐿𝑐𝑟𝑜𝑠𝑠 +≤ 0.05𝑝 +𝑃𝑖𝑛𝑡𝑒𝑟𝑖𝑐𝑡𝑎𝑙 = 1 − 𝑃𝑖𝑐𝑡𝑎𝑙, +𝑝 = 0, 1, 2, ...19 +(1) +where, +𝐿𝑖𝑐𝑡𝑎𝑙 +𝐿𝑐𝑟𝑜𝑠𝑠 ranges from 0 to 1. In terms of loss function +utilized for model training, we take binary cross entropy loss +function which is defined as: +L(𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙, ˆ𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙) = − (𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙 · 𝑙𝑜𝑔( ˆ𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙) ++ (1 − 𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙) · 𝑙𝑜𝑔(1 − ˆ𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙)) +(2) +where, ˆ𝑃𝑖𝑐𝑡𝑎𝑙 denotes predictive 𝑃𝑖𝑐𝑡𝑎𝑙. A Sigmoid activation +function is used to scale the +ˆ𝑃𝑖𝑐𝑡𝑎𝑙 from 0 to 1 before +computing the loss. +E. Multiscale 3D convolution neural network architecture +As a multiscale architecture, proposed model aims to ad- +dress the challenge that recognition of target samples in +probabilistic way. Fig. 3 shows the detailed architecture of +M-3D-CNN model. Each segmented sample input is a period +of multichannel EEG signals, we implement multiscale STFT +feature extraction at first. As a prevalent and effective signal +analysis tool in frequency domain, STFT method is widely +used in seizure detection studies [36]–[38]. Different from +previous works, we propose to extract channel-wise STFT +features in different scales simultaneously. The computation +method is elaborated as follows: +𝑋𝑠𝑐𝑎𝑙𝑒[𝜔, 𝑚] = +𝑊 𝐿−1 +∑︁ +𝑘=0 +𝑤[𝑘] · 𝑥[𝑚 × 𝑊𝐿 +2 ++ 𝑘] · 𝑒− 𝑗2𝜋𝜔𝑘 +𝑊𝐿 = 𝐿𝑠𝑖𝑔𝑛𝑎𝑙 +𝑠𝑐𝑎𝑙𝑒 , 𝑠𝑐𝑎𝑙𝑒 = 1, 2, 3, ... +(3) +where 𝑚 and 𝜔 are the index of the sliding window and +frequency coefficient, respectively. 𝑊𝐿 and 𝐿𝑠𝑖𝑔𝑛𝑎𝑙 denote +window length and length of EEG signal, and 𝑊 𝐿 +2 +refers to +the overlapping length is set to half of window length. +Various scales of STFT in M-3D-CNN model stand for +the chosen length of sliding window in STFT indeed, mean- +while we set 50% overlapping for sliding window, thus the +number of detecting windows in time axis of STFT result is +𝑁𝑤𝑖𝑛𝑑𝑜𝑤 = 2𝑛 − 1 at scale 𝑛. Meanwhile, the size of fast +Fourier transform (FFT) is set to 64, so that 32 informative +coefficients are left in the frequency dimension. Fig. 4 displays +STFT result of the single-channel signal at different scales. +In M-3D-CNN, we take 5 different scales from 1 to 5, each +scale would generate a 3D STFT feature tensor (channel × +frequency × time). Previously, most studies considered this +type of feature as a 2D image in the shape of H (time) × +W (frequency) with depth whose size is equal to the number +of channels, then implemented 2D convolution operation [39]– +[41]. It is an intuitive operation, however, this way cannot take +time step information into account. In M-3D-CNN model, we +consider 3D STFT feature as a 2D image in the shape of H +(channel) × W (frequency) with time step Depth, then 3D +convolution operation is utilized. Fig. 5 shows the difference +between 2D and 3D convolution operations for STFT features. +The values from achieved 3D STFT result need to be channel- +wisely normalized from 0 to 1 at each time step. +Then, extracted 3D features in different scales are fed to 3 +same 3D convolution blocks, and each block contains a 3D +convolution module with kernel size in the shape of 3 × 3 × +3 and a max-pooling module with kernel size in 2 × 2 × 1 for +the first two scales, and 2 × 2 × 2 for the rest three scales. The + +5 +Multichannel +Sample +Scale 1 +Scale 2 +Scale 3 +Scale 4 +Scale 5 +3×3×1 +Convolution +ReLU +2×2×1 +Pooling +3× +3×3×3 +Convolution +ReLU +2×2×2 +Pooling +3× +1×512 ++ +5×512 +5×5 +Conv +ReLU +2×2 +Pooling +3× +3D Conv. Blocks +W(Chs.)×H(Freq.)×D(Time) +Multiscale STFT +Feature Extraction +Probabilistic Output: +������������������������������������������������������������������������������������������������������������������������������������ , ������������������������������������������������������������������������ +chs×32×31 +chs×32×1 +chs×32×3 +chs×32×7 +chs×32×15 +Normalization in Frequency Dimension +FC Layer +Flatten +Flatten +Flatten +Flatten +Flatten +FC +FC +FC +Flatten +1024 +256 +64 +Sigmoid +2 +Fig. 3. Architecture of proposed multiscale 3D convolution neural networks. The input is a multichannel EEG sample, we extract short-time Fourier transform +(STFT) features of the input at scale 1 to 5. Then a FreqNorm layer is used to channel-wisely normalize 3D STFT values from 0 to 1 in frequency dimension +at each time step. The 3D STFT feature is fed to 3× same 3D convolution (Conv.) blocks comprising of 3D Conv., ReLU activation function, and 3D +max-pooling. The last layer generated by Conv. block is flattened and connected to a fully connected (FC) layer with 512 nodes. 5 vectors obtained from 5 +different scales are concatenated to a 5×512 matrix, then 3× traditional 2D Conv. blocks and 3× FC layers with ReLU are used to make classification. The +last output FC layer with 2 nodes utilizes a sigmoid activation function to guarantee the output in probability ranged from 0 to 1. +Fig. 4. Multiscale STFT feature extraction scheme for single-channel sample +Fig. 5. Schematic figure for difference between traditional 2D and proposed +3D convolution operation for multichannel STFT feature at each scale. +output generated after 3× convolution blocks is flattened, then +a multilayer perceptron (MLP) layer with a shape of 1 × 512 is +connected to this flattened output to achieve a 1D vector with +the same shape. This operation aims to unify the shape, thereby +eliminating the effects of inconsistent vector dimension due +to input multichannel EEG signals with different numbers of +channels. These 5 vectors originated from 3D feature tensors +at 5 different scales are concatenated together to build a 2D +matrix in the shape of 5 × 512. Then successive three same +2D convolution blocks with 5 × 5 kernel size convolution +operation and 2 × 2 kernel size max-pooling operation, are +connected to the 2D matrix. The output is also flattened to a +vector, and 3 MLP layers are followed. Eventually, we achieve +the output in the last MLP layer in the shape of 1 × 2. It +is important to note that except for the last layer Sigmoid +activation function is used to generate probabilistic output, +ReLU activation function is used in the M-3D-CNN model +everywhere else. +This novel architecture is inspired by the fact that probabilis- +tic crossing samples are comprised of short complete interictal +and ictal periods. 𝑃𝑖𝑐𝑡𝑎𝑙 is the percentage of a complete ictal +period occupying the crossing sample, rather than uniformly +distributed in the crossing sample. However, traditional feature +extraction approaches either implement FFT for the whole +duration or STFT with a single specific scale, which cannot +meet the situation that the probability pairs of crossing samples +are dynamically changing in real-time. Furthermore, 5 scales +setting is applicable for most situations, because we already set +50% overlapping for the STFT time window and the duration +of segmented samples is usually less than 10 s. +F. Rectified weighting strategy +According to the Labelling part of Fig. 1 and Eq. (1), +𝑃𝑖𝑐𝑡𝑎𝑙 of crossing samples is expected to be linearly increased +from 0 to 1 along with the linearly increasing percentage of + +..............6 +ictal period occupying the whole crossing sample ( 𝐿𝑖𝑐𝑡𝑎𝑙 +𝐿𝑐𝑟𝑜𝑠𝑠 ). In +practical experiments, however, predictive ictal probabilities +(PI) cannot be always precise or perfectly linear. Thus, we +propose a rectified weighting strategy to enhance the predic- +tive ictal probability (PIP). The principle of this strategy is +that previously achieved PIPs are used to rectify the current +achieved PIP at time 𝑡. Due to real-time scenario, we store the +PIPs every 0.1 s generated from previous 5 s, then we utilize +PIPs from previous 5 s, 3 s, and 1 s to fit three linear regression +(LR) functions, in order to generate three new PIPs (PIP𝐿𝑅5𝑠, +PIP𝐿𝑅3𝑠, PIP𝐿𝑅1𝑠) only based on previous PIPs from different +durations instead of current PIP𝑡. Eventually, we can achieve +rectified PIP at time 𝑡 (RPIP𝑡) that is computed as Eq. (4). +The weights 𝜆1, 𝜆2, 𝜆3, 𝜆4 are experimentally set to adjust the +weighting of different PIPs. +In short, rectified weighting strategy aims to enhance the +current PIP by rendering it more relevant to previous PIPs, +this can help reduce the impact of abnormal PIPs which are +might be generated by noises, artifacts, or model limitations. +𝑅𝑃𝐼𝑃𝑡 = +� 𝜆1 +𝜆2 +𝜆3 +𝜆4 +� +· +�������� +𝑃𝐼𝑃𝐿𝑅5𝑠 +𝑃𝐼𝑃𝐿𝑅3𝑠 +𝑃𝐼𝑃𝐿𝑅1𝑠 +𝑃𝐼𝑃𝑡 +�������� +(4) +G. Decision-making rule +We do not make decision only based on a single PIP because +it is difficult to significantly shorten the detection latency. The +reason is that if the decision threshold is high, the detection +latency is inevitably longer than the duration of segmented +samples, meanwhile FDR would be high if we set a low +decision threshold. Therefore, in this work, we also propose +an accumulative decision-making rule, whose schematic figure +is shown in the Decision part of Fig. 1. +Same as rectified weighting strategy, we store the RPIPs +from previous 5 s with detection rate 𝑟, which means we +store the RPIPs in a time step of +1 +𝑟 s. Then we compute +the accumulative probability (AP) at current time 𝑡 (𝐴𝑃𝑡) as +Eq. (5). In short, we only accumulate increased RPIPs during +the period of previous 5 s. And the detection system would +alarm at time 𝑡𝑑 when 𝐴𝑃𝑡𝑑 ≥ 𝑇ℎ𝑟., where 𝑇ℎ𝑟. represents +the decision threshold. Eventually, Algorithm 1 illustrates the +detailed decision-making rule of proposed framework intended +to detect seizures. +𝐴𝑃𝑡 = +�𝑡 +𝑖=𝑡−5 𝑅𝑃𝐼𝑃𝑖+1 +𝑟 +(if 𝑅𝑃𝐼𝑃𝑖+1 > 𝑅𝑃𝐼𝑃𝑖) +(5) +H. Performance metrics +In this work, there are four metrics - sensitivity, errors, de- +tection latency, and FDR, used to investigate the performance +of proposed DL-based framework. Eq. (6) reveals how we +compute these four metrics. The sensitivity is computed as the +number of seizures detected during the crossing period over +the total number of seizures in each patient. RPIP errors are +computed as the second equation in Eq. (6), we only consider +RPIP errors of crossing samples to figure out the capacity of +Algorithm +1: Decision-making rule of proposed +framework intended to detect seizures. +Input: Sample at time 𝑡: 𝑆𝑡. +Output: Detection time: 𝑡𝑑. +Initialize: Detection rate: 𝑟; Decision threshold: 𝑇ℎ𝑟.. +Stacking vector with zeros for previous 5s RPIPs: +ℙ = [𝑃𝑡−5, 𝑃𝑡−5+ 1 +𝑟 , ..., 𝑃𝑡− 1 +𝑟 , 𝑃𝑡] = 𝟘. +while True do +[1 − 𝑃𝐼𝑃𝑡, 𝑃𝐼𝑃𝑡] ← M-3D-CNN(𝑆𝑡); +𝑅𝑃𝐼𝑃𝑡 ← 𝑅𝑊𝑆(𝑃𝐼𝑃𝑡); +ℙ ← 𝑎𝑝𝑝𝑒𝑛𝑑(ℙ[𝑃𝑡−5+ 1 +𝑟 : 𝑒𝑛𝑑], 𝑅𝑃𝐼𝑃𝑡) +𝐴𝑃𝑡 ← �𝑡 +𝑖=𝑡−5 ℙ (if 𝑅𝑃𝐼𝑃𝑖+1 > 𝑅𝑃𝐼𝑃𝑖) +if 𝐴𝑃𝑡𝑑 > 𝑇ℎ𝑟. then +Alarm at time 𝑡𝑑; +Refresh ℙ = 𝟘 +end +end +Abbr.: +RPIP: Rectified Predictive Ictal Probability +M-3D-CNN: Multiscale 3D Convolution Neural Networks +RWS: Rectified Weighting Strategy +AP: Accumulative Probability +M-3D-CNN model combined with rectified weighting strategy +to recognize crossing samples in probabilistic way, the 𝑡 +denotes the sample detected at time 𝑡 and the 𝑇 refers to +the duration of crossing period. In terms of detection latency, +we implement Algorithm 1 and mark the time (𝑡𝑑) when +𝐴𝑃𝑡𝑑 larger than and equal to decision threshold (𝑇ℎ𝑟.), then +compute the detection latency by calculating distance between +𝑡𝑑 and EEG onset time (𝑡𝑜𝑛𝑠𝑒𝑡). The last metric is false +detection rate (FDR), we directly count the number of false +detection according to Algorithm 1 during the interictal period +per hour as FDR. +Sensitivity = +𝑁𝐷𝐶 +𝑁𝑇 𝑜𝑡𝑎𝑙 +RPIP Errors = +�𝑇 −1 +𝑡=0 +√︃ +(𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙 − ˆ𝑃(𝑡) +𝑖𝑐𝑡𝑎𝑙)2 +𝑇 +Detection Latency = 𝑡𝑑 − 𝑡𝑜𝑛𝑠𝑒𝑡 +if 𝐴𝑃𝑡𝑑 ≥ 𝑇ℎ𝑟. +FDR = 𝑁𝐹𝐷/ℎ +(6) +IV. EXPERIMENTS +A. Experimental setting +The experiments were implemented by Python with Pytorch +deep learning framework, and M-3D-CNN model training +and inference works are carried out on the single NVIDIA +2080Ti GPU machine. In this work, we trained patient-specific +model training and implemented the leave-one-seizure-out +cross validation (LOSOCV) scheme, which means we select +one seizure for validation, and the rest seizures are used to train +the model. LOSOCV is meaningful from clinical perspective +because the selected validated seizure can be regarded as a +fresh seizure unseen by the model yet, if the model performs +well in this scheme, we can believe that the trained model can +also accurately and promptly alarm seizures in the future. +Experimentally, all interictal, ictal and crossing samples are +used to train the model, and only errors of crossing samples are + +7 +(a) +(b) +(c) +Fig. 6. Performance of rectified probability weighting strategy. Here are 3 representative examples, each figure refers to crossing period of a seizure. Red, +blue and orange dots stand for true, predictive, and rectified probabilities, respectively. The original errors achieved by PIPs and rectified errors achieved by +RPIPs are also highlighted. +TABLE II +PERFORMANCE OF PROPOSED ALGORITHM ON TWO DATASETS IN THE PATIENT-SPECIFIC AND LEAVE-ONE-SEIZURE-OUT CROSS VALIDATION SCHEME. +RECTIFIED PREDICTIVE ICTAL PROBABILITY (RPIP) IS GENERATED BY M-3D-CNN MODEL FOLLOWED BY RECTIFIED WEIGHTING STRATEGY. +DETECTION LATENCY AND FALSE DETECTION RATE ARE (FDR) OBTAINED BY ACCUMULATIVE DECISION-MAKING RULE. THERE ARE 19 AND 11 CASES +IN CHB-MIT AND SWEC-ETHZ DATASETS, RESPECTIVELY. WE PROVIDE MEAN AND STANDARD DEVIATION VALUES FROM EVERY SEIZURE OF EACH +PATIENT FOR RPIP ERRORS, DETECTION LATENCY, AND FDR METRICS. +CHB-MIT dataset +SWEC-ETHZ dataset +Patient +ID +Sensitivity +(NDC / NT) +RPIP Errors of +Crossing Samples +(%) +Detection +Latency +(s) +FDR +(/h) +Patient +ID +Sensitivity +(NDC / NT) +RPIP Errors of +Crossing Samples +(%) +Detection +Latency +(s) +FDR +(/h) +chb01 +7/7 +10.85 ± 6.05 +1.9 ± 0.6 +0 +ID1 +13/13 +11.18 ± 8.27 +3.6 ± 1.0 +0.26 ± 0.44 +chb02 +2/3 +29.76 ± 12.49 +2.9 ± 1.8 +2.10 ± 1.82 +ID2 +4/4 +17.90 ± 6.98 +3.8 ± 1.5 +1.94 ± 2.62 +chb03 +7/7 +9.10 ± 2.74 +2.3 ± 0.4 +0 +ID4 +14/14 +18.35 ± 8.29 +3.9 ± 2.3 +1.03 ± 0.81 +chb04 +3/4 +23.09 ± 12.29 +2.1 ± 1.5 +0.80 ± 1.60 +ID5 +10/10 +13.63 ± 8.30 +4.5 ± 2.1 +0.55 ± 0.94 +chb05 +5/5 +15.33 ± 9.32 +2.3 ± 0.5 +0 +ID6 +4/4 +15.76 ± 10.56 +5.5 ± 2.9 +1.11 ± 1.28 +chb06 +7/7 +8.85 ± 5.22 +2.3 ± 0.3 +0 +ID9 +9/9 +16.42 ± 9.53 +3.1 ± 1.6 +0.74 ± 1.11 +chb07 +3/3 +11.57 ± 3.71 +2.7 ± 0.1 +0 +ID10 +4/5 +10.31 ± 10.46 +4.7 ± 0.5 +0.22 ± 0.50 +chb08 +4/5 +20.44 ± 14.72 +2.5 ± 0.6 +0.80 ± 1.26 +ID12 +10/10 +12.86 ± 7.70 +5.2 ± 1.5 +0.11 ± 0.35 +chb09 +4/4 +11.31 ± 7.51 +2.1 ± 0.6 +1.50 ± 3.02 +ID13 +6/7 +20.96 ± 8.53 +6.2 ± 2.5 +0.79 ± 0.54 +chb10 +7/7 +18.80 ± 10.19 +2.4 ± 0.9 +0.44 ± 1.20 +ID14 +5/7 +28.07 ± 5.95 +7.3 ± 3.0 +2.06 ± 0.76 +chb11 +3/3 +16.82 ± 4.59 +1.7 ± 0.6 +0.48 ± 0.82 +ID16 +5/6 +15.75 ± 6.44 +4.6 ± 3.0 +0.37 ± 0.57 +chb14 +8/8 +10.46 ± 3.97 +2.4 ± 0.3 +0 +chb17 +2/3 +29.86 ± 14.90 +3.2 ± 1.6 +0 +chb18 +5/6 +16.05 ± 15.51 +2.5 ± 0.2 +0 +chb19 +3/3 +9.03 ± 5.11 +2.5 ± 0.4 +0 +chb20 +8/8 +10.15 ± 6.33 +1.9 ± 0.5 +0.72 ± 2.04 +chb21 +4/4 +19.93 ± 8.45 +3.2 ± 0.4 +0 +chb22 +3/3 +23.26 ± 4.01 +2.7 ± 1.2 +1.24 ± 1.92 +chb23 +7/7 +15.60 ± 7.70 +2.1 ± 0.9 +0 +Overall +94/99 +14.84 ± 9.80 +2.3 ± 0.7 +0.34 ± 1.11 +Overall +84/89 +16.17 ± 9.26 +4.7 ± 2.0 +0.75 ± 1.04 +RPIP: Rectified Predictive Ictal Probability +FDR: False Detection Rate +NDC: Number of Seizures Detected during the Crossing Period +NT: Number of Total Seizures +considered as the most crucial metric to quantify the quality of +the model because the trained model can perfectly recognize +accurate probabilities of complete interictal and ictal samples. +During the phase of model training, we set 20 training epochs +and used optimizer is Nesterov-accelerated Adam, known as +Nadam, with 0.0001 learning rate, 𝛽1 = 0.9, 𝛽2 = 0.999 +[42]. We implement the LOSOCV scheme and only save the +best model performing the lowest RPIP errors of crossing +samples on the validated seizure. In terms of rectified weight- +ing strategy, weights 𝜆1, 𝜆2, 𝜆3, 𝜆4 are experimentally set to +0.2, 0.3, 0.3, 0.2. The detection rate and decision threshold +used to make decision are 10 and 0.5 for both datasets. + +Original Errors :-6% +Rectified Errors : 4% +TrueProb. +Predictive Prob +Rectified ProbOriginal Errors : 23% +Rectified Errors : 7% +True Prob. +Predictive Prob +Rectified ProbOriginal Errors :-7% +Rectified Errors : 9% +TrueProb. +Predictive Prob +Rectified Prob8 +Fig. 7. Boxplot for rectified predictive ictal probability of each seizure on two datasets. (RPIP: Rectified Predictive Ictal Probability) +Fig. 8. Boxplot for detection latency of each seizure on two datasets. The averaged latencies of two datasets are 2.3 s and 4.2 s, respectively. +B. Results +At first, we need prove the effectiveness of rectified weight- +ing strategy. Fig. 6 shows 3 representative examples of PIP +and RPIP performance from CHB-MIT dataset, where red, +blue, and orange dots stand for true, predictive, and rectified +probabilities, respectively. The x-axis represents percentage of +ictal period in crossing sample, and y-axis refers to probability. +As mentioned in Section III-B and III-C, there are 1280 +extracted crossing samples in crossing period for each seizure, +and there are divided into 20 probability pairs annotation as +true labels. Thus, every 5% ictal period contains 64 samples. +We can see that even though original PIPs perform well +according to Fig. 4(a), rectified weighting strategy still can +enhance the results from 6% to 4%. In Fig. 4(b), original +PIPs show worse fitting result with 23% errors, while rectified +weighting strategy can significantly decrease the errors to 7%, +and the overall probabilities are increasing more linearly. Fig. +4(c) is another type of representative example that RPIPs seem +to achieve increased errors compared to original PIPs (from +7% to 9%), but we can see that RPIPs increase more linearly +than PIPs, so that we still keep the results of RPIPs. According +to these three representative examples, we can conclude that +rectified weighting strategy is effective to rectify the PIPs +by decreasing errors further and making PIPs increase more +linearly as expected. This operation aims to help detection +system can recognize samples more accurately and meanwhile +decrease FDR. +Table II shows performance of proposed M-3D-CNN model +on CHB-MIT and SWEC-ETHZ datasets. In this table, we +only consider RPIPs after implementing rectified weighting +strategy, then compute detection latency and FDR based on +RPIPs. We achieved overall 94 of 99 and 84 of 89 seizures +detected during the crossing period, 14.84% ± 9.80% and +16.17% ± 9.26% RPIP errors of crossing samples, 2.3 s +± 0.7 s and 4.7 s ± 2.0 s detection latency and 0.34/h ± +1.11/h and 0.75/h ± 1.04/h FDR on CHB-MIT scalp dataset +and SWEC-ETHZ iEEG dataset, respectively. These mean +and standard deviation values are calculated based on results +attained from various numbers of seizures in each patient +according to LOSOCV scheme. In terms of performance on +CHB-MIT dataset, M-3D-CNN model performs well on 8 + +CHB-MIT +SWEC-ETHZ +10 +T +９987 +S +Latency +6５4321 +DB- +cr +PatientIDCHB-MIT +SWEC-ETHZ +50 +45 +40 +(% +535250 +Errors( +RPIP +5 +chl +ch +PatientID9 +TABLE III +PERFORMANCE COMPARISON BETWEEN THIS WORK AND PRIOR-ART STUDIES. +Year +Ref. +Dataset +EEG +type +# of +pat. +Feature +extraction +method +Len. of +Sample +Model +Sen. +(%) +FDR +(/h) +Detection +Latency +LOSO- +CV +scheme +Probabi- +listic +task +2010 +[17] +CHB-MIT +sEEG +24 +FFT +6s +SVM +96 +0.08 +4.6s (+6s) +✓ +– +2011 +[18] +Clinical +iEEG +10 +FFT +3s +SVM +97 +0.03 +5s (+3s) +✓ +– +2017 +[19] +CHB-MIT +sEEG +22 +Wavelet +6s +SVM +96 +0.1 +1.89s (+6s) +✓ +– +2018 +[20] +CHB-MIT +sEEG +22 +Statistical +6s +ADCD +96 +0.12 +4.21s (+3s) +✓ +– +2019 +[25] +CHB-MIT +sEEG +22 +STFT +3s +2D-CNN +93.9 +– +– +– +– +2019 +[26] +CHB-MIT +sEEG +23 +Raw +2s +2D-CNN +90 +– +– +– +– +2020 +[30] +CHB-MIT +sEEG +21 +Wavelet +– +2D-CNN +92.4 +– +– +– +– +2020 +[33] +SWEC-ETHZ +iEEG +11 +LBP ++ LGP +6-bit +SVM +MLP +94.8 +0 +15.9s +– +– +2021 +[24] +CHB-MIT +sEEG +24 +Wavelet ++ EMD +4s +SVM +97.3 +0.64 +– +– +– +2021 +[27] +CHB-MIT +SWEC-ETHZ +sEEG +iEEG +24 +18 +Raw +2s +1D-CNN +88.1 +90.1 +0.2 +0.07 +8.1s (+2s) +13.2s (+2s) +– +– +2021 +[43] +SWEC-ETHZ +iEEG +16 +Statistical +2s +1D-CNN +96.4 +– +8.8s (+2s) +– +– +2023 +This +work +CHB-MIT +SWEC-ETHZ +sEEG +iEEG +19 +11 +Multiscale +STFT +5s +10s +3D-CNN +95.0 +97.7 +0.12 +0.75 +2.3s +4.7s +� +� +–: N/A +Len.: Length +Sen.: Sensitivity +FDR: False Detection Rate +LOSOCV: Leave-One-Seizure-Out Cross Validation +sEEG: scalp EEG +iEEG: intracranial EEG +FFT: Fast Fourier Transform +SVM: Support Vector Machine +STFT: Short-Time Fourier Transform +ADCD: Adaptive Distance-based Change Point Detector +LBP: Local Binary Pattern +LGP: Local Gradient Pattern +MLP: Multilayer Perception +EMD: Empirical Mode Decomposition +cases (chb01, chb03, chb05, chb06, chb07, chb14, chb19, +chb23), all seizures are detected during the crossing period, +low RPIP errors of crossing samples (≤15%), short detection +latency (≤2.7 s) and none false detection are attained on these +patients. The rest subjects show slight drawbacks on one or +two performance metrics. There are 5 patients (chb02, chb04, +chb08, chb17, chb18) containing 1 seizure do not be detected +by M-3D-CNN model during the crossing period. However, +these miss-detected seizures still can be detected after crossing +period, so that we set the detection latency of them to 5 s and +10 s which equals to the length of crossing period for CHB- +MIT and SWEC-ETHZ datasets, respectively. We can see from +table that 1 miss-detected seizure leads to high RPIP and +FDR, except for chb18, M-3D-CNN model obtains larger than +20% even close to 30% mean and larger than 10% standard +deviation RPIP errors, and larger than 0.8/h FDR on the rest 4 +patients (chb02, chb04, chb08, chb17). As for chb22 patient, +even though there is no miss-detected seizure, we still achieved +slightly higher RPIP errors and FDR. In terms of SWEC- +ETHZ dataset, except for ID13 and ID14 subjects, proposed +model performs well on the rest 8 patients, where ≤20% +RPIP errors of crossing samples and ≤5 s detection latency +are achieved. There are 4 patients (ID10, ID13, ID14, ID16) +containing miss-detected seizures during the crossing period, +correspondingly worse performance metrics are achieved on +these patients. Especially for ID14 case where M-3D-CNN +model performs worst, there are two miss-detected seizures +and the highest RPIP errors, detection latency, and FDR are +attained. +Fig. 7 and Fig. 8 display boxplots specifying every seizure +performance of RPIP errors and detection latency on two +datasets, respectively. According to Fig. 7, it is obvious that +most seizures achieved less than 30% RPIP errors and the +majority is less than 20%, meanwhile there only 8 out of 99 +seizures achieved larger than 30%, and 4 of them achieved +larger than 40% on CHB-MIT dataset. Each of these seizures +obtaining higher RPIP errors appears in different subjects, +this phenomenon indicates that worse RPIP errors are not +generated by the poor model or abnormal patient, this may be +caused by a single abnormal seizure among these correspond- +ing patients. And these possible abnormal seizures lead to a +large standard deviation as shown in reverent cases from Table +II. In terms of SWEC-ETHZ dataset, all seizures obtained less +than 30% RPIP errors except for the aforementioned worst- +performing case ID14, we suspect ID14 is an abnormal patient +different from others. As for Fig. 8, we can see from CHB- +MIT dataset that except for 5 miss-detected seizures during +the crossing period set to 5 s latency, all rest seizures achieve +less than 4 s latency, and the averaged latency is 2.3 s. From +SWEC-ETHZ dataset, there are also 5 miss-detected seizures +set to 10 s latency, among the rest seizures, around 10% +seizures are larger than 8 s, and the majority is less than 6s. +The averaged detection latency among 89 seizures is 4.2 s. +Table III shows performance comparisons between prior-art +publications and this work, we list several characteristics of the +used dataset and proposed method to make comparisons. There + +10 +are 10 previous highly cited works with prior-art performance +selected to prove the advantages of our work. It should be +noted that in terms of detection latency item, we add the +length of detected sample to previously reported latencies +because we doubt previous works did not compute the latency +by measuring the distance between EEG onset and the end +of detected sample. According prior publications, the state- +of-the-art detection latencies on two datasets are 4.2 s and +8.1 s respectively. The latencies obtained by our proposed +algorithm are 2.3 s and 4.7 s which are significantly faster +than prior-art results. And we can see that several previous +studies even if utilized the naive FFT feature extraction method +and SVM classifier, they still achieved good performance. +However, numerous recent emerging studies took advantage +of advanced feature extraction methods and deep learning +models, they cannot significantly enhance the seizure detection +performance. Furthermore, less recent studies focused on +testing meaningful metrics from clinical perspectives, such +as FDR, detection latency, and LOSOCV scheme. In the last +three columns of the table, we highlight three innovations of +this work, which are whether or not implementing LOSOCV +scheme and probabilistic classification task. +V. DISCUSSION +In this section, we discuss several issues, including further +clarification of achieved performance, model comparison from +hardware and performance perspectives. +A. Performance clarification +Firstly, we will clarify the sensitivity metric. In this work, +we only consider the number of seizures detected during +the crossing period as effectively detected seizures instead +of seizures detected at any time as previous works did [16], +[17]. Because we think the seizure can be detected during the +crossing period means detection latency of this seizure is short +enough to guarantee detection time can precede clinical onset. +Experimentally, M-3D-CNN model can detect all seizures after +crossing period (during the ictal period), so that sensitivity +would be 100% as the way previous researchers measured. +But we think such 100% sensitivity would not be clinically +beneficial for epileptic patients. +Secondly, in Table III we only displayed RPIP errors +of crossing samples instead of all three kinds of samples +(interictal, ictal, and crossing samples). The reason is that +proposed M-3D-CNN model is capable of recognizing com- +plete interictal and ictal samples perfectly, and the errors +are less than 3% stably. We believe that such good results +on complete samples cannot lead to short detection latency, +only accurately predicted interictal samples can help us to +reduce FDR. Therefore, we showed errors of crossing samples, +detection latency, and FDR to investigate the performance of +proposed M-3D-CNN model. +Thirdly, the lengths of extracted samples for two datasets +are different. We empirically and experimentally set 5 s and 10 +s for CHB-MIT and SWEC-ETHZ datasets, respectively. We +initially learn from previous studies that achieved detection +latency of two datasets were around 5 s and larger than 10 +TABLE IV +PERFORMANCE COMPARISONS OF VARIOUS MODELS ON CHB03 SUBJECT +WITH 7 SEIZURES FROM CHB-MIT DATASET. +Model Name +PIP Errors of +Crossing Samples +(%) +Number of +Parameters +Model +Size +M-3D-CNN +7.14 ± 3.38 +3.8M +14.46MB +M-2D-CNN +12.41 ± 8.36 +5.6M +21.42MB +5-2D-CNN +16.07 ± 5.34 +1.2M +4.65MB +M-LSTM +8.53 ± 5.61 +5.1M +19.56MB +5-LSTM +9.13 ± 5.20 +5.1M +19.56MB +M-ViT +8.87 ± 4.28 +7.9M +30.32MB +5-ViT +9.65 ± 8.00 +7.9M +30.32MB +PIP: Predictive Ictal Probability +M: Multiscale +5: Scale 5 +ViT: Vision Transformer +CNN: Convolution Neural Network +LSTM: Long Short-Term Memory +s. Then we experimentally tuned this parameter, we found +that longer extracted samples would bring longer detection +latency and lower FDR, while shorter extracted samples would +cause shorter detection latency and higher FDR. Thus, this +is a kind of trade-off parameter tuning work, eventually the +length of extracted sample making the first ictal sample (or +the last crossing sample) just contain the obvious EEG signal +oscillations is expected, thereby we set 5 s and 10 s for two +datasets. +Fourthly, achieved detection latency on SWEC-ETHZ is ob- +viously longer than the latency achieved on CHB-MIT dataset. +There are two possible reasons to answer this phenomenon. +The first is that the criterion of annotating EEG onset time is +implemented by different clinical experts. The second is that +iEEG modality is more sensitive to the abnormal discharging +inside brain, thereby can detect the slight EEG abnormality +more earlier than scalp EEG [44], [45]. However, such earlier +slight abnormality appearing in EEG signal is difficult to be +detected by the algorithm. +B. Model comparison +In this manuscript, we proposed a novel M-3D-CNN model +deep learning architecture based on multiscale STFT features +and 3D convolution operation in CNN backbone. In Table +IV, we change three innovative parameters - multiscale or +not, 3D or 2D, CNN or other DL backbone architectures, +then generate several variant DL models to make comparisons +with proposed M-3D-CNN model. The original PIP errors +of crossing samples and the model size achieved on the +chb03 patient from CHB-MIT dataset are used to compare the +performance. According to the table, M-3D-CNN architecture +performs best among all variant models. We can also conclude +that multiscale is better than single-scale STFT features, and +3D-CNN outperforms 2D-CNN. Although 5-2D-CNN model +shows advantages in model size, it obtains the worst PIP er- +rors. Furthermore, long short-term memory (LSTM) recurrent +neural networks and vision transformer (ViT) also achieve +satisfactory performance, but their model sizes are quite large +compared to the M-3D-CNN model. ViT is an emerging and + +11 +powerful deep learning model applied to various applications, +but ViT model is too large to fit this real-time seizure detection +application even if we already tried our best to shrink the +model parameters, and eventually model size is still doubled as +M-3D-CNN architecture. It should be noted that we flatten the +time axis of all STFT features as various time steps for both +LSTM and ViT models, which makes multiscale or single- +scale features only change the length of time steps, and would +not change the number of trainable parameters. +VI. CONCLUSION +The proposed framework uses a deep M-3D-CNN model +intended to address several challenges and limitations in the +field of seizure detection study. It consists of a novel proba- +bilistic classification concept to accurately recognize crossing +samples simultaneously containing partial interictal and ictal +components. Then we also propose rectified weighting strategy +and accumulative decision-making rule to significantly shorten +the detection latency of seizure onset. +Furthermore, although proposed framework is intended for +seizure detection application, the concept of probabilistic +classification, rectified weighting strategy and accumulative +decision-making rule can be applied to other electrophys- +iological signal based real-time BCI applications. Also, it +can benefit other detection systems to make decisions more +promptly and accurately. +REFERENCES +[1] N. Kissani, Y. T. M. Lengan´e, V. Patterson, B. Mesraoua, E. Dawn, +C. Ozkara, G. Shears, H. Riphagen, A. A. Asadi-Pooya, A. Bogacz, +I. E. Aarrouni, and P. P. Nair, “Telemedicine in epilepsy: How can we +improve care, teaching, and awareness?” Epilepsy & Behavior, vol. 103, +p. 106854, 2020. +[2] Y. Xu, J. Yang, and M. Sawan, “Trends and challenges of processing +measurements from wearable devices intended for epileptic seizure +prediction,” Journal of signal processing systems, vol. 94, no. 6, pp. +527–542, 2022. +[3] S. Ramgopal, S. Thome-Souza, M. Jackson, N. E. Kadish, I. S´anchez +Fern´andez, J. Klehm, W. Bosl, C. Reinsberger, S. Schachter, and +T. Loddenkemper, “Seizure detection, seizure prediction, and closed- +loop warning systems in epilepsy,” Epilepsy & Behavior, vol. 37, pp. +291–307, 2014. +[4] E. B. Assi, D. K. Nguyen, S. Rihana, and M. Sawan, “Towards accurate +prediction of epileptic seizures: A review,” Biomedical Signal Processing +and Control, vol. 34, pp. 144–157, 2017. +[5] X. Gu, Z. Cao, A. Jolfaei, P. Xu, D. Wu, T.-P. Jung, and C.-T. Lin, “EEG- +based brain-computer interfaces (bcis): A survey of recent studies on +signal sensing technologies and computational intelligence approaches +and their applications,” IEEE/ACM Transactions on Computational +Biology and Bioinformatics, vol. 18, no. 5, pp. 1645–1666, 2021. +[6] M. Zhuang, Q. Wu, F. Wan, and Y. Hu, “State-of-the-art non-invasive +brain–computer interface for neural rehabilitation: A review,” Journal of +Neurorestoratology, vol. 8, no. 1, pp. 12–25, 2020. +[7] M. Vilela and L. R. Hochberg, “Chapter 8 - applications of brain- +computer interfaces to the control of robotic and prosthetic arms,” in +Brain-Computer Interfaces, ser. Handbook of Clinical Neurology, N. F. +Ramsey and J. del R. Mill´an, Eds. +Elsevier, 2020, vol. 168, pp. 87–99. +[8] M. S. AL-Quraishi, I. Elamvazuthi, S. A. Daud, S. Parasuraman, and +A. Borboni, “EEG-based control for upper and lower limb exoskeletons +and prostheses: A systematic review,” Sensors, vol. 18, no. 10, 2018. +[9] J. Yang and M. Sawan, “From seizure detection to smart and fully +embedded seizure prediction engine: A review,” IEEE Transactions on +Biomedical Circuits and Systems, vol. 14, no. 5, pp. 1008–1023, 2020. +[10] A. Ulate-Campos, F. Coughlin, M. Ga´ınza-Lein, I. S. Fern´andez, P. Pearl, +and T. Loddenkemper, “Automated seizure detection systems and their +effectiveness for each type of seizure,” Seizure, vol. 40, pp. 88–101, +2016. +[11] M. S. Scher, M. Y. Hamid, D. A. Steppe, M. E. Beggarly, and M. J. +Painter, “Ictal and interictal electrographic seizure durations in preterm +and term neonates,” Epilepsia, vol. 34, no. 2, pp. 284–288, 1993. +[12] R. S. Fisher, H. E. Scharfman, and M. deCurtis, How Can We Identify Ic- +tal and Interictal Abnormal Activity? Dordrecht: Springer Netherlands, +2014, pp. 3–23. +[13] R. S. Fisher and J. J. Engel Jr, “Definition of the postictal state: when +does it start and end?” Epilepsy & Behavior, vol. 19, no. 2, pp. 100–104, +2010. +[14] B. Litt, R. Esteller, J. Echauz, M. D’Alessandro, R. Shor, T. Henry, +P. Pennell, C. Epstein, R. Bakay, M. Dichter, and G. Vachtsevanos, +“Epileptic seizures may begin hours in advance of clinical onset: A +report of five patients,” Neuron, vol. 30, no. 1, pp. 51–64, 2001. +[15] B. Litt and J. Echauz, “Prediction of epileptic seizures,” The Lancet +Neurology, vol. 1, no. 1, pp. 22–30, 2002. +[16] A. Shoeb, H. Edwards, J. Connolly, B. Bourgeois, S. Ted Treves, +and J. Guttag, “Patient-specific seizure onset detection,” Epilepsy & +Behavior, vol. 5, no. 4, pp. 483–498, 2004. +[17] A. H. Shoeb and J. V. Guttag, “Application of machine learning to +epileptic seizure detection,” in ICML, 2010, pp. 975–982. +[18] A. Kharbouch, A. Shoeb, J. Guttag, and S. S. Cash, “An algorithm for +seizure onset detection using intracranial EEG,” Epilepsy & Behavior, +vol. 22, pp. S29–S35, 2011, the Future of Automated Seizure Detection +and Prediction. +[19] L. S. Vidyaratne and K. M. Iftekharuddin, “Real-time epileptic seizure +detection using EEG,” IEEE Transactions on Neural Systems and +Rehabilitation Engineering, vol. 25, no. 11, pp. 2146–2156, 2017. +[20] S. Khanmohammadi and C.-A. Chou, “Adaptive seizure onset detection +framework using a hybrid pca–csp approach,” IEEE Journal of Biomed- +ical and Health Informatics, vol. 22, no. 1, pp. 154–160, 2018. +[21] A. R. Hassan, A. Subasi, and Y. Zhang, “Epilepsy seizure detection +using complete ensemble empirical mode decomposition with adaptive +noise,” Knowledge-Based Systems, vol. 191, p. 105333, 2020. +[22] E. Alickovic, J. Kevric, and A. Subasi, “Performance evaluation of +empirical mode decomposition, discrete wavelet transform, and wavelet +packed decomposition for automated epileptic seizure detection and +prediction,” Biomedical Signal Processing and Control, vol. 39, pp. 94– +102, 2018. +[23] H. G. Daoud, A. M. Abdelhameed, and M. Bayoumi, “Automatic +epileptic seizure detection based on empirical mode decomposition and +deep neural network,” in 2018 IEEE 14th International Colloquium on +Signal Processing & Its Applications (CSPA), 2018, pp. 182–186. +[24] C. Li, W. Zhou, G. Liu, Y. Zhang, M. Geng, Z. Liu, S. Wang, and +W. Shang, “Seizure onset detection using empirical mode decomposition +and common spatial pattern,” IEEE Transactions on Neural Systems and +Rehabilitation Engineering, vol. 29, pp. 458–467, 2021. +[25] Y. Yuan, G. Xun, K. Jia, and A. Zhang, “A multi-view deep learning +framework for EEG seizure detection,” IEEE Journal of Biomedical and +Health Informatics, vol. 23, no. 1, pp. 83–94, 2019. +[26] M. S. Hossain, S. U. Amin, M. Alsulaiman, and G. Muhammad, “Ap- +plying deep learning for epilepsy seizure detection and brain mapping +visualization,” ACM Trans. Multimedia Comput. Commun. Appl., vol. 15, +no. 1s, feb 2019. +[27] X. Wang, X. Wang, W. Liu, Z. Chang, T. K¨arkk¨ainen, and F. Cong, “One +dimensional convolutional neural networks for seizure onset detection +using long-term scalp and intracranial EEG,” Neurocomputing, vol. 459, +pp. 212–222, 2021. +[28] Y. Xu, J. Yang, and M. Sawan, “Multichannel synthetic preictal EEG +signals to enhance the prediction of epileptic seizures,” IEEE Transac- +tions on Biomedical Engineering, vol. 69, no. 11, pp. 3516–3525, 2022. +[29] G. C. Jana, R. Sharma, and A. Agrawal, “A 1d-cnn-spectrogram based +approach for seizure detection from EEG signal,” Procedia Computer +Science, vol. 167, pp. 403–412, 2020, international Conference on +Computational Intelligence and Data Science. +[30] Y. Li, Y. Liu, W.-G. Cui, Y.-Z. Guo, H. Huang, and Z.-Y. Hu, “Epileptic +seizure detection in EEG signals using a unified temporal-spectral +squeeze-and-excitation network,” IEEE Transactions on Neural Systems +and Rehabilitation Engineering, vol. 28, no. 4, pp. 782–794, 2020. +[31] A. Burrello, K. Schindler, L. Benini, and A. Rahimi, “One-shot learning +for iEEG seizure detection using end-to-end binary operations: Lo- +cal binary patterns with hyperdimensional computing,” in 2018 IEEE +Biomedical Circuits and Systems Conference (BioCAS), 2018, pp. 1–4. +[32] A. Burrello, L. Cavigelli, K. Schindler, L. Benini, and A. Rahimi, +“Laelaps: An energy-efficient seizure detection algorithm from long- +term human iEEG recordings without false alarms,” in 2019 Design, +Automation & Test in Europe Conference & Exhibition (DATE), 2019, +pp. 752–757. + +12 +[33] A. Burrello, K. Schindler, L. Benini, and A. Rahimi, “Hyperdimensional +computing with local binary patterns: One-shot learning of seizure onset +and identification of ictogenic brain regions using short-time iEEG +recordings,” IEEE Transactions on Biomedical Engineering, vol. 67, +no. 2, pp. 601–613, 2020. +[34] Y. Sun, W. Jin, X. Si, X. Zhang, J. Cao, L. Wang, S. Yin, and D. Ming, +“Continuous seizure detection based on transformer and long-term +iEEG,” IEEE Journal of Biomedical and Health Informatics, vol. 26, +no. 11, pp. 5418–5427, 2022. +[35] Q. Nguyen, H. Valizadegan, and M. Hauskrecht, “Learning classification +models with soft-label information,” Journal of the American Medical +Informatics Association, vol. 21, no. 3, pp. 501–508, 11 2013. +[36] K. Samiee, P. Kov´acs, and M. Gabbouj, “Epileptic seizure classification +of eeg time-series using rational discrete short-time fourier transform,” +IEEE Transactions on Biomedical Engineering, vol. 62, no. 2, pp. 541– +552, 2015. +[37] H. Wang, W. Shi, and C.-S. Choy, “Hardware design of real time +epileptic seizure detection based on stft and svm,” IEEE Access, vol. 6, +pp. 67 277–67 290, 2018. +[38] M. Sameer, A. K. Gupta, C. Chakraborty, and B. Gupta, “Epileptical +seizure detection: Performance analysis of gamma band in eeg signal +using short-time fourier transform,” in 2019 22nd International Sympo- +sium on Wireless Personal Multimedia Communications (WPMC), 2019, +pp. 1–6. +[39] A. Shoeibi, M. Khodatars, N. Ghassemi, M. Jafari, P. Moridian, R. Al- +izadehsani, M. Panahiazar, F. Khozeimeh, A. Zare, H. Hosseini-Nejad, +A. Khosravi, A. F. Atiya, D. Aminshahidi, S. Hussain, M. Rouhani, +S. Nahavandi, and U. R. Acharya, “Epileptic seizures detection using +deep learning techniques: A review,” International Journal of Environ- +mental Research and Public Health, vol. 18, no. 11, 2021. +[40] I. Ahmad, X. Wang, M. Zhu, C. Wang, Y. Pi, J. A. Khan, S. Khan, O. W. +Samuel, S. Chen, and G. Li, “Eeg-based epileptic seizure detection via +machine/deep learning approaches: A systematic review,” Computational +Intelligence and Neuroscience, vol. 2022, 2022. +[41] A. Shoeibi, P. Moridian, M. Khodatars, N. Ghassemi, M. Jafari, R. Al- +izadehsani, Y. Kong, J. M. Gorriz, J. Ram´ırez, A. Khosravi, S. Naha- +vandi, and U. R. Acharya, “An overview of deep learning techniques +for epileptic seizures detection and prediction based on neuroimaging +modalities: Methods, challenges, and future works,” Computers in Biol- +ogy and Medicine, vol. 149, p. 106053, 2022. +[42] T. Dozat, “Incorporating nesterov momentum into adam,” ICLR Work- +shop, 2016. +[43] A. Burrello, S. Benatti, K. Schindler, L. Benini, and A. Rahimi, +“An ensemble of hyperdimensional classifiers: Hardware-friendly short- +latency seizure detection with automatic ieeg electrode selection,” IEEE +Journal of Biomedical and Health Informatics, vol. 25, no. 4, pp. 935– +946, 2021. +[44] T. Ball, M. Kern, I. Mutschler, A. Aertsen, and A. Schulze-Bonhage, +“Signal quality of simultaneously recorded invasive and non-invasive +EEG,” NeuroImage, vol. 46, no. 3, pp. 708–716, 2009. +[45] H. Stefan, P. Sch¨uler, K. Abraham-Fuchs, S. Schneider, M. Gebhardt, +U. Neubauer, C. Hummel, W. J. Huk, and P. Thierauf, “Magnetic source +localization and morphological changes in temporal lobe epilepsy: +comparison of MEG/EEG, ECoG and volumetric MRI in presurgical +evaluation of operated patients,” Acta Neurologica Scandinavica, vol. 89, +no. S152, pp. 83–88, 1994. + diff --git a/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf b/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..dd7b4795a0cdc67b9c79430cca72845743301d63 --- /dev/null +++ b/mdAyT4oBgHgl3EQfYvcB/content/2301.00207v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3a1f108602253ac43a290380069e266e86e58e474d7448cb3c24e8d9511b3b72 +size 10003807 diff --git a/mdAyT4oBgHgl3EQfYvcB/vector_store/index.faiss b/mdAyT4oBgHgl3EQfYvcB/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8f2c03afadd078c689485d24f45f8b7c2359a378 --- /dev/null +++ b/mdAyT4oBgHgl3EQfYvcB/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:91189162c03bf97f846907ad9852ce6839be031d71cea1c0597ec722b7e7041b +size 3473453 diff --git a/mdAyT4oBgHgl3EQfYvcB/vector_store/index.pkl b/mdAyT4oBgHgl3EQfYvcB/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..ebc0ad2d3e23aac53891f09a50ab70e2ac60418c --- /dev/null +++ b/mdAyT4oBgHgl3EQfYvcB/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f07b20bd93d322a282c5b9dc18f6c08750a9baeaf8c69a8c7d4aa1c85a1172c7 +size 130312 diff --git a/mtE2T4oBgHgl3EQfJQaw/content/tmp_files/2301.03690v1.pdf.txt b/mtE2T4oBgHgl3EQfJQaw/content/tmp_files/2301.03690v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8b04e8bded40ef00519368ecbbaf259b3df73b9f --- /dev/null +++ b/mtE2T4oBgHgl3EQfJQaw/content/tmp_files/2301.03690v1.pdf.txt @@ -0,0 +1,779 @@ +arXiv:2301.03690v1 [cs.CR] 9 Jan 2023 +Quantifying User Password Exposure to +Third-Party CDNs +Rui Xin, Shihan Lin, Xiaowei Yang +Duke University +Abstract. Web services commonly employ Content Distribution Net- +works (CDNs) for performance and security. As web traffic is becoming +100% HTTPS, more and more websites allow CDNs to terminate their +HTTPS connections. This practice may expose a website’s user sensitive +information such as a user’s login password to a third-party CDN. In this +paper, we measure and quantify the extent of user password exposure to +third-party CDNs. We find that among Alexa top 50K websites, at least +12,451 of them use CDNs and contain user login entrances. Among those +websites, 33% of them expose users’ passwords to the CDNs, and a pop- +ular CDN may observe passwords from more than 40% of its customers. +This result suggests that if a CDN infrastructure has a security vulnera- +bility or an insider attack, many users’ accounts will be at risk. A simple +fix to this security vulnerability is for a website to encrypt a user’s pass- +word inside the HTTPS request. Our measurement shows that less than +17% of the websites adopt this solution. +Keywords: HTTPS · CDN · password · security · measurement +1 +Introduction +Content Distribution Networks (CDNs) [32,40] play an important role in im- +proving the performance and security of web services. A CDN caches web pages +at servers near end users to reduce retrieval latency. It also blocks malicious +requests to defend a web server against various attacks [14]. Currently, many +websites employ CDNs provided by third-party companies such as Akamai [1], +Cloudflare [3], and Fastly [4]. +However, third-party CDNs introduce a considerable security and privacy risk +when they serve websites that enables HTTPS [9,11]. HTTPS uses a certificate +to certify the domain name of a website. Thus, to make the web pages appear +as if they come from the original site, a website has to share its TLS private key +[9] or TLS session keys[46] with the CDN. In both cases, a third-party CDN can +observe the content of all connections between a website and its users. +In this work, we aim to raise awareness of this security and privacy risk +and quantify its severeness from a user’s perspective. We choose to measure the +extent to which users’ website login passwords are exposed to CDNs due to the +HTTPS key sharing practice. Although prior research has shown that private +key sharing is prevalent on the Internet [9] and HTTPS termination weakens + +2 +Anonymous Authors +connection security of a great portion of the Internet [11], it is not clear whether a +website has taken simple countermeasures such as client-side encryption (see§ 2) +to protect users’ passwords. +We conduct a measurement on Alexa top 50K sites [2] to quantify password +exposure to CDNs during the user login procedures. We also measure the de- +ployment of client-side password encryption on websites to understand websites’ +treatment of users’ passwords. Such a large-scale measurement is technically +non-trivial, because we need to automate the login procedures on websites with +diverse structures to inspect login requests. Thus, we design and implement a +framework for automatic login. The framework can detect login elements on a +website and collect login requests when it submits credentials to websites. +Our main contributions and findings can be concluded as the following: +– We propose an open-source framework for automatic login 1, which can be +applied for other research such as measurement of authentication methods. +– Our measurement presents that 33.0% of websites that employ CDNs and +contain login entrances expose users’ passwords in plaintext to their CDNs. +– We find that two popular CDN providers, Cloudflare and Akamai, can ob- +serve users’ passwords from 44% and 25% of their customers in our dataset, +respectively. +– We find prevalent password exposure in most website categories, including +websites whose user accounts should be carefully protected, such as websites +related to finance and health. +– Our result shows that less than 17% of the websites encrypt users’ passwords +when transferring login requests to CDNs, and the top 1K websites are more +likely to adopt password encryption. +Overall, our measurement points out potential security issues caused by pass- +word exposure to CDNs. Even though websites choose to trust CDNs, users may +concern about their privacy when CDNs can monitor their private data including +passwords during transmission. Moreover, CDNs have never been secure enough. +Prior work already showed that an attacker can trick some CDNs to cache and +reveal other users’ private data [13,33,34]. Thus, private data leakage to CDNs +may turn into a disaster when attackers or malicious insiders exploit vulnerabil- +ities of CDNs. +2 +Background +In this section, we briefly introduce CDNs and HTTPS, and we analyze the +security issues when a website with HTTPS employs a CDN. We also discuss +two countermeasures adopted by websites in practice to address such issues. +1 The source code can be accessed by https://anonymous.4open.science/r/PAM2023-Anonymous-Submission-863D + +Quantifying User Password Exposure to Third-Party CDNs +3 +2.1 +HTTPS on CDNs +A CDN reduces web retrieval time by directing a client’s request to an edge +server which is hosted by the CDN and geographically close to the user. The +edge server responds to the client with cached content. If the requested content +is not cached, the edge server may fetch the content from the origin server which +is hosted by the website (the CDN’s customer) and is the initial source of all +content. CDNs do not cache private data, as they are usually dynamic. +Modern CDNs are used not only to speed up page loading but also to pro- +vide an effective shield against attacks such as DDoS and code injections [14]. +A CDN enlarges the serving capability of its customers to prevent volumetric +DDoS attacks. It also applies techniques such as IP blocking and rate limiting +to block attacks when DDoS happens. For example, Akamai protected its cus- +tomers from 38,905 separate DDoS attacks from 2014 to 2019 [45]. CDNs also +inspect the content of requests and use Web Application Firewall (WAF) to filter +out malicious requests such as XSS injection [54] and SQL injection [18]. +Unfortunately, CDNs have become a source of vulnerabilities in the HTTPS +ecosystem in recent years [9,11]. The security of HTTPS relies on certificates and +private keys generated by totally trusted certification authorities (CAs). How- +ever, since HTTPS requires server authentication by private keys in HTTPS +handshakes, if a website employs a CDN to represent it to respond to clients’ +HTTPS requests, it has to share its private key to the CDN. With the private key, +the CDN can build HTTPS connections with clients, and clients cannot differen- +tiate between the CDN and the origin server. When a client requests for private +data, the CDN will forward the request by terminating the HTTPS connection +and building another HTTPS connection with the origin server. Therefore, the +CDN becomes a man in the middle when a user’s private data are transmitted +between the client and the origin server [9]. +2.2 +Countermeasures in Practice +Two instant but imperfect countermeasures have been deployed by some web- +sites. First, a website can bypass the CDN and send the private requests to the +origin server directly. In this countermeasure, a website should use a separate +domain or subdomain for the private data, because the CDN possesses the pri- +vate key of the original domain. We refer to this method as “CDN bypassing” +in this paper. This method will not affect CDNs’ benefit of page loading accel- +eration, since the private data are not cached by CDNs. However, it eliminates +the benefit of having the origin server shielded against DDoS attacks, because +the IP address of the origin server is exposed to the public. When attackers can +connect to the origin server directly, it is much easier to launch DDoS attacks +since the origin server usually cannot construct a DDoS defense as effectively +as CDNs [16,49]. Besides DDoS, the CDN cannot inspect the private content to +filter out malicious requests, and thus the origin server may suffer from attacks +such as code injections. + +4 +Anonymous Authors +Another countermeasure is to encrypt private data inside HTTPS connec- +tions. The website generates another key pair and delivers the new public key to +the client. The client uses the public key to encrypt the private data to be sent +out. Therefore, when a CDN forwards the request, the private data are invisible +to the CDN. We refer to this method as “client-side encryption” in our paper. +We observe some websites use this method to protect users’ passwords only, as +encrypting all private data may introduce too much overhead. However, a secure +public key delivery is non-trivial when HTTPS connections are already inter- +cepted by a CDN [30]. Delivering another certificate differing from the HTTPS +certificate is useless, because a website has to use JavaScript to conduct en- +cryption in current browsers, and the JavaScript code cannot obtain the root +certificates of a client to verify a certificate. Without a certificate, if the public +key is delivered by a CDN, an active CDN can launch the man-in-the-middle +attacks by replacing the public key. If the public key is delivered by the origin +server, the origin server is exposed to the public and under the threat of DDoS. +Therefore, the client-side encryption only defends against password leakage to a +certain extent. +Despite the defects of these two methods, they preserve users’ privacy to +some extent. Moreover, if the origin server builds its own DDoS defense or a +CDN is assumed to be passive, these two countermeasures can provide sufficient +protection. However, it is unclear about the deployment of these two counter- +measures on websites. Thus, we investigate the password exposure to provide a +profile of their deployment. Our measurement will show that few websites adopt +the client-side encryption for passwords. +3 +Threat Model +We use the threat model proposed by the prior work [30]. We consider the pri- +vate data in a website as the data can only be accessed by a authenticated user. +The users can be authenticated by the traditional password, one-time password +(OTP), OAuth [20], certificates, etc. The credentials for authentication are con- +sidered as private data as well. We focus on the measurement of the traditional +password in this paper. +We considered two types of attackers defined in the prior work [30]. +– Passive attacker: A CDN behaves honestly to serve the requests but an +attacker inside the CDN may eavesdrop on the transmitted messages. For +example, a malicious administer of a CDN cannot change the CDN’s behavior +but may peek at the transmitted traffic and record users’ passwords. +– Active attacker: An attacker insider CDN may launch arbitrary attacks +including eavesdropping and tampering. For example, a CDN may modify or +corrupt the cached HTML to disable the front-end password encryption so +that it can observe users’ passwords in the login requests. This may happen +when attackers exploit a vulnerability of a CDN. + +Quantifying User Password Exposure to Third-Party CDNs +5 +4 +Method +To detect the password exposure, we should inspect a website’s login request and +the destination. Thus, we need a framework for automatic login in a large-scale +measurement. Currently, a website may adopt multiple authentication methods, +such as text passwords, OAuth [20], one-time password (OTP). In our measure- +ment, we only consider the method of text passwords. +Based on the existing frameworks [42,26,43], we designed and implemented +an automatic login framework that copes with more web pages with diverse +structures. Browsers such as Chrome and Firefox can help users automatically +fill the credentials in some web pages. We do not use this function because it relies +on the existence of the “autocomplete” attribute in HTML elements, and thus it +cannot handle the websites that do not enable this attribute in HTML. Besides +the automation of browsers, Peng et al. implemented a framework to log into +phishing websites automatically [42]. Our framework can handle issues that are +common in legitimate sites but rare in phishing sites, such as confusion caused +by sign up forms and pop-ups. Jonker et al.. also proposed a framework for post- +login security analysis [26]. Our framework shares many similarities with theirs, +but adds the capability to operate in the presence of HTTP Authentication and +reCAPTCHA. In our work, we do not need to successfully log into a website, so +the framework merely triggers a failed login and collects the login request. +Besides the automatic login framework, we use the method in the prior +work [9,22,27,28,30] to discover the CDN usage of a website. This method also +help to inspect the destination of the collected login requests to determine +whether the requests are sent to a CDN server. +Some cloud providers will provide both hosting service and CDN service, +such as AWS and Azure. In our method, when a request is sent to such a cloud +provider, we cannot determine whether the website is using the CDN service +or the hosting service. If the password is sent to a hosting service, it should +not be considered as an exposure to a CDN. Since our goal is to provide an +underestimation of password exposure, our CDN list does not include a CDN +service provider that also provides hosting service. As a result, our CDN list +contains 9 popular CDNs. +Ethical concerns: We respect user privacy and our work does not raise eth- +ical concerns. The method of CDN discovery only used public data from the +Internet, such as Registration Data Access Protocol (RDAP) [38]. As for the +automatic login framework, since we do not require a successful login, we use +a randomly generated fake account that is nearly impossible to coincide with +existing ones. We skip the websites that require a test of the account existence +before submitting the login credentials. We only conduct the login trial once for +each website, so we do not overload the websites in our test. +5 +Password Exposure +We only consider HTTPS-enabled websites because a website without HTTPS +apparently contains major vulnerabilities. In Alexa top 50K sites [2], 42,502 + +6 +Anonymous Authors +0 +1 +2 +3 +4 +5 +Ranking +104 +0 +0.2 +0.4 +0.6 +0.8 +1 +CDF +(a) CDF +I0 +I20 +I40 +I60 +I80 +I100 +Ranking Intervals +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Percentage +(b) Percentage +Fig. 1: (a) Distribution of login-detected websites. (b) The percentage of +password-exposed websites among CDN-enabled websites across different rank- +ing intervals. We divide 50K websites into 100 ranking intervals. Each interval +contains 500 websites. The x-axis ticks at every 20 intervals. +of them enable HTTPS. We run the framework to automatically log into the +websites with HTTPS. If the framework submits the fake credentials to a website, +we consider it performs a login. The framework performs 17,111 logins in total. +In this paper, we focus on these 17,111 websites and call them “login-detected +websites”. +We detect CDNs employed by these websites according to § 4. Our result +shows that 12,451 websites employ CDN service, and we call them “CDN-enabled +websites” in this paper. By inspecting their login procedures, we find that 4,114 +websites send the login requests with users’ passwords in plaintext or Base64 +encoding to CDNs. We denote these websites as “password-exposed websites”. +We discovered that 33.0% of CDN-enabled websites expose users’ passwords to +CDNs, demonstrating a potential privacy issue. In this section, we present the +results in detail. +5.1 +Distribution over Rankings +Since our framework may fail to detect the login forms of some websites, the +dataset of login-detected websites can be considered as a sample set of all web- +sites that enable logins. We first investigate the distribution of these samples +over rankings. +Figure 1a shows the distribution of login-detected websites. A linear relation- +ship between the CDF and ranking shows a uniform distribution of the websites. +Therefore, the logins detected by our framework are unbiased in the rankings. +To investigate the relationship between websites’ rankings and their prefer- +ence for password exposure, we divide the rankings into 100 intervals. For an +interval Ij, it contains 500 websites ranking in the range of [1+500∗(j−1), 500∗j]. + +Quantifying User Password Exposure to Third-Party CDNs +7 +44% +25% +18% +5% +7% +66% +6% +9% +17% +Cloudflare +Akamai +Fastly +Highwinds +Edgecast +Incapsula +Quantil +CDNetworks +Limelight +100 +101 +102 +103 +104 +# of Websites +CDN-enabled websites +Password-exposed websites +Fig. 2: Distribution across CDN providers. The y-axis is in log scale. The number +above the bar denotes the percentage of password-exposed websites in CDN- +enabled websites. +58% +30% 32% 36% +34% +38% 27% 10% +31% 43% +36% 31% 40% 36% 19% 44% 67% +Retail +Internet +Business +Entertainment +News +Finance +Technology +Education +Society +Travel +Science +Sports +Health +Reference +Government +Recreation +Home +0 +50 +100 +150 +200 +250 +300 +350 +# of Websites +CDN-enabled websites +Password-exposed websites +Fig. 3: Distribution across website categories. The number above the bar denotes +the percentage of password-exposed websites in CDN-enabled websites. +For each interval, we count the password-exposed websites and the CDN-enabled +websites, and we compute the percentage of password-exposed websites in CDN- +enabled websites. +Figure 1b presents the percentage variation across the intervals. Given the +result of unbiased detection in Figure 1a, we can examine the distribution of +password exposure on website rankings through Figure 1b. Even though some +fluctuations exist, the percentages are overall above 20%, meaning that the pass- +word exposure is common across all rankings. We can also find that the percent- +ages of password-exposed websites with a higher ranking are slightly larger than +those with a lower ranking. The average percentage from the I1 to I50 and from +the I51 to I100 are 34.6% and 30.8%, respectively. The reason for the difference +may be that websites with higher rankings should handle more traffic, and they +have a stronger preference for adopting CDNs to filter out the malicious requests. +Thus, those websites tend to expose the private requests destined at the origin +server to CDNs for inspection. Besides, we can find that the most popular web- +sites in the first two intervals have relatively low password exposure percentage. +It is because that the top websites are more likely to deploy defense mechanisms, +which can be justified by our analysis in § 6. + +8 +Anonymous Authors +5.2 +Distribution over CDN Providers +We also consider how password-exposed websites are distributed among the CDN +providers. Figure 2 presents the number of password-exposed websites in each +CDN provider. As shown in the figure, Cloudflare and Akamai are the two most +popular CDNs in the world, and they observe the most users’ passwords from +their customers’ requests. More than 40% of Cloudflare’s customer websites in +our dataset share users’ passwords to Cloudflare, and Akamai observes passwords +from 25% of its customers. Besides, 66% of websites who use Incapsula expose +passwords to the CDN. Some CDNs only observe a small fraction of sensitive +traffic, such as Highwinds and Edgecast. +It is reasonable for websites to trust famous CDN providers and employ their +defense against attacks. However, it does not necessarily mean users should also +trust CDNs. From the users’ perspective, they may be concerned about their +private or sensitive data when it is shared with a third-party CDN. The results +also imply a risk of the single point failure of popular CDNs: a malicious insider +in a popular CDN may divulge the users’ passwords of more than 40% of its +customer websites, leading to a large-scale user data leakage. +5.3 +Distribution over Website Categories +We investigate the practice of exposing password among different website cat- +egories. We collect the website categories data from Alexa Top Sites by Cate- +gory [2]. In 12,451 CDN-enabled websites, 2,010 of them can be classified by the +Alexa data. We use these 2,010 sites for analysis in this section. +Figure 3 presents the statistics of CDN usage and password exposure across +17 website categories. As we can see, retail websites employ most CDNs. It may +be because retail websites need to display many photos of their products, which +would obtain substantial benefit from using a CDN service. The retail websites +also expose the most passwords to CDNs. Besides retail websites, more than +40% of websites in several categories expose users’ passwords. We note that a +large portion (38%) of finance websites which are usually considered to require +sophisticated defense divulges users’ password to CDNs. Moreover, education +websites have the least percentage of password exposure. Our results point out +that password exposure is prevalent within a wide range of categories. +6 +Countermeasures +In this section, we first present the measurement of the countermeasures against +password exposure used by current websites. We also discuss possible counter- +measures can be adopted by websites and users. +6.1 +Client-side Encryption and CDN Bypassing +In our measurement, we observe that some websites indeed adopt the method +of client-side encryption discussed in § 2.2 to protect users’ passwords. For ex- +ample, baidu.com, dropbox.com, and chase.com deliver public keys by their + +Quantifying User Password Exposure to Third-Party CDNs +9 +I0 +I20 +I40 +I60 +I80 +I100 +Ranking Intervals +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +Percentage +Fig. 4: This figure shows the percentage of password-encrypted websites among +CDN-enabled websites across different ranking intervals. We divide 50K websites +into 100 ranking intervals. Each interval contains 500 websites. The x-axis ticks +at every 20 intervals. +origin servers. However, such a solution is rarely adopted by the websites. In our +measurement, if our framework submits the credentials but cannot find the pass- +word in plain text or Base64 encoding in the login request, we consider that the +website encrypts the password. Since our framework may fail to login, we have +an upper-bound estimation of the deployment of password encryption. There- +fore, in our dataset, at most 2,057 (16.5%) out of 12,451 CDN-enabled websites +adopt such a solution. We call these websites as “password-encrypted websites”. +This result demonstrates that password encryption is a rare practice on the web. +We investigate the relationship between a website’s ranking and password +encryption deployment. We used the same method and intervals in Figure 1b, and +the results are shown in Figure 4. As we can see, even for the top websites, less +than 30% of them encrypt users’ passwords. Nevertheless, when compared with +other websites, they have a relatively higher percentage of password encryption. +However, an outstandingly high percentage exists around the intervals of quite +low ranking. We manually inspected websites located in that interval. We found +13 websites of all 20 password-encrypted websites are subdomains of tmall.com +for different retailers, such as www.kfc.tmall.com and www.lenovo.tmall.com. +Once a user attempts to sign into these subdomain sites, they all direct the user +to tmall.com. This website is one of the top websites for electronic shopping, +and it adopts client-side password encryption. Overall, we can conclude that the +top ranking websites are more likely to encrypt users’ passwords in transmission. +CDN bypassing can protect users’ privacy, even though it exposes servers’ IP +addresses and leave servers at the risk of DDoS. In our measurement, we cannot +verify whether the destination of a login request is the origin server through +RDAP. We leave the further measurement of CDN bypassing as future work. + +10 +Anonymous Authors +6.2 +Other Countermeasures +Besides password encryption and CDN bypassing, Password Authenticated Key +Exchange (PAKE) [8,15] also prevents password exposure. PAKE protocols, such +as SRP [55] and OPAQUE [24], authenticate users without the requirement of +revealing passwords in login requests. Moreover, it is proven to be secure during +login even when CDNs can launch active attacks. However, PAKE protocols re- +quire trust on first use (TOFU), meaning that a secure channel is required during +account registration. Therefore, PAKE solves the password exposure problem for +web services that do not allow online registration. For example, it can be used +in banking industry, as users are required to open a bank account physically at +branches. Nevertheless, PAKE is almost never used by websites [15]. The reason +may be the difficulty of understanding and implementing PAKE protocols for de- +velopers. It may also be because developers usually trust third-party CDNs and +lack of awareness of security issues caused by insufficient password protection. +From the users’ perspective, a user can use OAuth [20] such as using a Google +account to sign in to other websites. Because leading tech companies such as +Google and Facebook have built their own CDNs, a user’s password will not +be exposed to a third party during the login. However, more OAuth practice +may lead to a severe single-point failure if a user’s password of Google account +is leaked. Besides OAuth, users can also adopt two-factor authentication. Even +though two-factor authentication cannot prevent passwords from being exposed +to third-party CDNs, it prevents accounts from being compromised even when +the passwords are exposed to attackers. +These countermeasures can only protect users’ passwords. However, our re- +sults also suggest a potential privacy threat. Users’ private data stored on a web- +site may also be divulged to a CDN during the transmission. As private data is +much more complicated and diverse than the passwords, developing countermea- +sures would be harder. Thus, private data leakage may be much more prevalent +than password leakage. We leave the measurement of private data leakage as +future work. +7 +Discussion and Future Work +Our measurement quantifies password exposure to CDNs and suggests potential +security issues in current web ecosystem. In this section, we provide suggestions +to the security community, users, and the industry. +We need further research on the solutions. As presented in § 2.2, the pre- +liminary strategies of CDN bypassing and client-side encryption can be eas- +ily deployed but contain vulnerabilities. Proposed techniques such as Keyless +SSL [46,12,35], certificate delegation [29], and mcTLS [37] are ineffective in pre- +serving user privacy. The SGX-based [36,21] solutions can provide comprehen- +sive protection, but it is hard to be deployed on CDNs. InviCloak [30], a solution +proposed just recently can achieve the goal of securing servers from DDoS, pro- +tecting users’ privacy, and allowing instant deployment simultaneously. However, + +Quantifying User Password Exposure to Third-Party CDNs +11 +InviCloak disable the Web Application Firewall (WAF) of CDNs, and it is still +unclear whether website developers is willing to adopt InviCloak. Therefore, +further research on this area is critical to a more secure Internet. +We recommend users to adopt OAuth and two-factor authentication. As dis- +cussed in § 6, signing into a website with existing accounts of leading tech com- +panies such as Google and Facebook could be safer than creating a new account. +Besides, two-factor authentication is also recommended, as it provides additional +protection for an account even when the password is stolen by a hacker. +Websites should adopt preliminary defense. The results shows that many +websites do not apply the minimal defense against password exposure. Despite +the preliminary strategies are vulnerable to some attacks, they provide basic +protection for users’ privacy. Since it is acceptable to assume a passive CDNs in +most cases, the client-side encryption usually provides a sufficient protection. +CDN providers should involve in developing and deploying advanced solu- +tions. The widespread of Keyless SSL on Cloudflare demonstrates that a CDN +provider plays an important role in the security community [46] Cooperation +from CDN providers can validate researchers’ ideas and advance further research. +CDNs can also guide their customers to deploy a defense mechanism. +This paper presents the preliminary results of password sharing to third-party +CDNS. We propose the following directions as the future work. +1. Augment the existing CDN discovery method to differentiate the hosting +service and the CDN service of a cloud provider, as mentioned in § 4. +2. Quantify the adopted or available countermeasures besides the client-side en- +cryption in websites, including CDN bypassing, OAuth, one-time password, +two-factor authentication, etc, as mentioned in § 6. +3. Measure private data leakage in websites to understand the security impact +of TLS private key sharing from users’ perspectives, as mentioned in § 6. +4. Survey the users and website developers to understand their awareness of +private data leakage to thrid-party CDNs. Such a survey helps to figure out +the reason why countermeasures are not widespread. +8 +Related Work +Password security. Password security has attracted attention from many re- +searchers. Lu et al. analyzed how websites deploy measures to prevent online +password cracking [31]. Wang et al. manually inspected 188 websites to char- +acterize the login process and built an extension to inform users of potential +password leakage caused by the lack of HTTPS [51]. Acker et al. studied the +security of password input fields among the Alexa top 100K sites, and they +found that 62.8% of the websites with a login page are vulnerable to basic man- +in-the-middle attacks [48]. Bonneau et al. surveyed the proposals for replacing +passwords and pointed out the difficulty of replacing passwords [7]. Peng et al. +explored how passwords are spread after they are divulged by phishing sites [42]. +In addition, many prior works investigated the prevalence of the password reuse +problem [41,23,52,44] and its countermeasures [50]. + +12 +Anonymous Authors +CDN security. Researchers have shown the existence of a wide range of vul- +nerabilities in CDNs. Nguyen et al. presented an attack of poisoning CDN cache +with error pages, and five CDN services were vulnerable to such an attack [39]. +Mirheidari et al.’s measurement shows that private data can be divulged by +CDNs through web cache deception [13,33,34]. Besides CDN cache, researchers +also presented approaches to disclosing the IP addresses of origin servers hid- +den behind CDNs, demonstrating insufficient DDoS protection of CDNs [49,25]. +Moreover, attackers may utilize a CDN to launch DoS to an origin server or +to the CDN itself. Triukose et al. presented an amplify method to launch DoS +to an origin server through the CDN [47]. The forwarding loop discovered by +Chen et al. can lead to resource-consuming DoS to CDNs [10]. Guo introduced +three attacks to break CDN DoS protection, including HTTP/2 amplification, +pre-post slow HTTP, and availability degradation [17]. In addition, Hao et al.’s +research demonstrated that attackers can hijack the DNS redirection used by +a CDN to downgrade the content delivery performance [19]. Durumeric et al.’s +measurement shows that the HTTPS interception on CDNs may downgrade the +TLS version or cipher suites and thus reduce connection security [11]. +Solutions to TLS key sharing. A line of the research focuses on building +keyless CDNs. Cloudflare, Akamai, and Modadugu et al. proposed similar solu- +tions called “Keyless SSL”, respectively [46,12,35]. Certificate delegation [29] and +mcTLS [37] enables a client to recognize the CDN as a delegation of the website. +Wei et al. [53] and Ahmed et al. [5] adopted Trust Executive Environment (TEE) +on CDNs for private key management. However, these strategies only prevent +the TLS private key sharing, while users’ private data are still visible to CDNs. +Phoenix [21] and mbTLS [36] extend TEE solutions to fully protect users’ private +data. However, deploying TEE-based solutions on CDNs may take a long time as +it requires upgrades of hardware and operating systems. InviCloak [30] protects +users’ private data with an additional encryption channel and low overhead, but +its adoption by websites in the future remains unclear. +9 +Conclusion +In this paper, we conduct a large-scale measurement to quantify user password +exposure to third-party CDNs in the web ecosystem. Our results show that 4,114 +of 12,451 (33.0%) HTTPS-enabled websites that employ third-party CDNs ex- +pose users’ passwords to the CDNs during the login procedures. Besides, as a +popular CDN, Cloudflare sees users’ passwords from more than 40% of its cus- +tomers. In addition, password encryption is rarely adopted by websites, even +though it is simple and effective to a certain extent. Overall, our results sug- +gest that current websites excessively trust CDNs, leading to potential security +issues when CDNs’ vulnerabilities are exploited by attackers. As HTTPS and +CDNs becomes more popular, we encourage further research on the privacy is- +sues caused by HTTPS termination on CDNs. We publicly release the code of +the auto-login framework to facilitate future research [6]. + +Quantifying User Password Exposure to Third-Party CDNs +13 +References +1. Akamai (2020), https://www.akamai.com/ +2. Alexa Top Sites (2020), https://www.alexa.com/topsites +3. Cloudflare (2020), https://www.cloudflare.com/ +4. Fastly (2020), https://www.fastly.com/ +5. Ahmed, R., Zaheer, Z., Li, R., Ricci, R.: Harpocrates: Giving Out Your Secrets +and Keeping Them Too. In: Proc. of IEEE/ACM Symposium on Edge Computing +(SEC). pp. 103–114. IEEE (2018) +6. Anonymous: +Code +of +pam +2023 +anonymous +submission +(2022), +https://anonymous.4open.science/r/PAM2023-Anonymous-Submission-863D +7. Bonneau, J., Herley, C., Van Oorschot, P.C., Stajano, F.: The quest to replace +passwords: A framework for comparative evaluation of web authentication schemes. +In: Proc. of S&P. pp. 553–567. IEEE (2012) +8. Boyko, V., MacKenzie, P., Patel, S.: Provably secure password-authenticated key +exchange using diffie-hellman. In: Proc. of EUROCRYPT. pp. 156–171. Springer +(2000) +9. Cangialosi, F., Chung, T., Choffnes, D., Levin, D., Maggs, B.M., Mislove, A., Wil- +son, C.: Measurement and Analysis of Private Key Sharing in the HTTPS Ecosys- +tem. In: Proc. of CCS. pp. 628–640. ACM (2016) +10. Chen, J., Zheng, X., Duan, H.X., Liang, J., Jiang, J., Li, K., Wan, T., Paxson, V.: +Forwarding-loop attacks in content delivery networks. In: Proc. of NDSS. ISOC +(2016) +11. Durumeric, Z., Ma, Z., Springall, D., Barnes, R., Sullivan, N., Bursztein, E., Bailey, +M., Halderman, J.A., Paxson, V.: The security impact of https interception. In: +Proc. of NDSS. ISOC (2017) +12. Gero, C.E., Shapiro, J.N., Burd, D.J.: Terminating SSL Connections without +Locally-Accessible Private Keys (2013), u.S. Patents, No. 9,647,835 +13. Gil, +O.: +Web +cache +deception +attack +(2017), +https://omergil.blogspot.com/2017/02/web-cache-deception-attack.html +14. Gillman, D., Lin, Y., Maggs, B., Sitaraman, R.K.: Protecting Websites from Attack +with Secure Delivery Networks. Computer 48(4), 26–34 (2015) +15. Green, +M.: +Let’s +Talk +About +PAKE +(2018), +https://blog.cryptographyengineering.com/2018/10/19/lets-talk-about-pake/ +16. Guo, R., Chen, J., Liu, B., Zhang, J., Zhang, C., Duan, H., Wan, T., Jiang, J., +Hao, S., Jia, Y.: Abusing cdns for fun and profit: Security issues in cdns’ origin +validation. In: Proc. of SRDS. pp. 1–10. IEEE (2018) +17. Guo, R., Li, W., Liu, B., Hao, S., Zhang, J., Duan, H., Shen, K., Chen, J., Liu, Y.: +Cdn judo: Breaking the cdn dos protection with itself. In: Proc. of NDSS. ISOC +(2020) +18. Halfond, W.G., Viegas, J., Orso, A., et al.: A Classification of SQL-Injection At- +tacks and Countermeasures. In: Proc. of International Symposium on Secure Soft- +ware Engineering. vol. 1, pp. 13–15. IEEE (2006) +19. Hao, S., Zhang, Y., Wang, H., Stavrou, A.: End-users get maneuvered: Empirical +analysis of redirection hijacking in content delivery networks. In: Proc. of Security +Symposium. pp. 1129–1145. USENIX (2018) +20. Hardt, D.: The oauth 2.0 authorization framework. Internet Engineering Task Force +(IETF) (2012) +21. Herwig, S., Garman, C., Levin, D.: Achieving Keyless CDNs with Conclaves. In: +Proc. of Security Symposium. pp. 735–751. USENIX (2020) + +14 +Anonymous Authors +22. Huang, C., Wang, A., Li, J., Ross, K.W.: Measuring and evaluating large-scale +cdns. In: Prof. of IMC. pp. 15–29. ACM (2008) +23. Ion, I., Reeder, R., Consolvo, S.: “... no one can hack my mind”: Comparing expert +and non-expert security practices. In: Proc. of SOUPS. pp. 327–346. USENIX +(2015) +24. Jarecki, S., Krawczyk, H., Xu, J.: Opaque: an asymmetric pake protocol secure +against pre-computation attacks. In: Proc. of EUROCRYPT. pp. 456–486. Springer +(2018) +25. Jin, L., Hao, S., Wang, H., Cotton, C.: Your remnant tells secret: Residual resolu- +tion in ddos protection services. In: Proc. of DSN. pp. 362–373. IEEE (2018) +26. Jonker, H., Karsch, S., Krumnow, B., Sleegers, M.: Shepherd: A generic approach to +automating website login. In: Proc. of NDSS Workshop on Measurements, Attacks, +and Defenses for the Web. ISOC (2021) +27. Krishnamurthy, B., Wills, C., Zhang, Y.: On the use and performance of content +distribution networks. In: Proc. of IMC. pp. 169–182. ACM (2001) +28. Levy, A.: CDNs and Privacy Threats: A Measurement Study. Ph.D. thesis, Prince- +ton University (2017) +29. Liang, J., Jiang, J., Duan, H., Li, K., Wan, T., Wu, J.: When HTTPS meets CDN: +A Case of Authentication in Delegated Service. In: Proc. of S&P. pp. 67–82. IEEE +(2014) +30. Lin, S., Xin, R., Goel, A., Yang, X.: InviCloak: An End-to-End Approach to Privacy +and Performance in Web Content Distribution. In: Proc. of CCS. ACM (2022) +31. Lu, B., Zhang, X., Ling, Z., Zhang, Y., Lin, Z.: A measurement study of authen- +tication rate-limiting mechanisms of modern websites. In: Proc. of ACSAC. pp. +89–100 (2018) +32. Maggs, B.M., Sitaraman, R.K.: Algorithmic Nuggets in Content Delivery. ACM +SIGCOMM CCR 45(3), 52–66 (2015) +33. Mirheidari, S.A., Arshad, S., Onarlioglu, K., Crispo, B., Kirda, E., Robertson, +W.: Cached and confused: Web cache deception in the wild. In: Proc. of Security +Symposium. pp. 665–682. USENIX (2020) +34. Mirheidari, S.A., Golinelli, M., Onarlioglu, K., Kirda, E., Crispo, B.: Web cache de- +ception escalates! In: Proc. of Security Symposium. pp. 179–196. USENIX, Boston, +MA (2022) +35. Modadugu, N., Goh, E.J.: The Design and Implementation of WASP: A Wide-Area +Secure Proxy. Tech. rep., Stanford University (2002) +36. Naylor, D., Li, R., Gkantsidis, C., Karagiannis, T., Steenkiste, P.: And then there +were more: Secure communication for more than two parties. In: Proc. of CoNEXT. +pp. 88–100 (2017) +37. Naylor, D., Schomp, K., Varvello, M., Leontiadis, I., Blackburn, J., Lopez, D.R., +Papagiannaki, K., Rodriguez Rodriguez, P., Steenkiste, P.: Multi-context tls +(mctls): Enabling secure in-network functionality in tls. In: Proc. of SIGCOMM. +pp. 199–212. ACM (2015) +38. Newton, A., Hollenbeck, S.: Rfc7482: Registration data access protocol (rdap) +query format. Internet Engineering Task Force (IETF) (2015) +39. Nguyen, H.V., Iacono, L.L., Federrath, H.: Your cache has fallen: Cache-poisoned +denial-of-service attack. In: Proc. of CCS. pp. 1915–1936. ACM (2019) +40. Nygren, E., Sitaraman, R.K., Sun, J.: The Akamai Network: a Platform for High- +Performance Internet Applications. SIGOPS OSR 44(3), 2–19 (2010) +41. Pearman, S., Thomas, J., Naeini, P.E., Habib, H., Bauer, L., Christin, N., Cranor, +L.F., Egelman, S., Forget, A.: Let’s go in for a closer look: Observing passwords in +their natural habitat. In: Proc. of CCS. pp. 295–310. ACM (2017) + +Quantifying User Password Exposure to Third-Party CDNs +15 +42. Peng, P., Xu, C., Quinn, L., Hu, H., Viswanath, B., Wang, G.: What happens +after you leak your password: Understanding credential sharing on phishing sites. +In: Proc. of AsiaCCS. pp. 181–192 (2019) +43. Senol, A., Acar, G., Humbert, M., Borgesius, F.Z.: Leaky Forms: A Study of Email +and Password Exfiltration Before Form Submission. In: Proc. of Security Sympo- +sium. pp. 1813–1830. USENIX, Boston, MA (2022) +44. Shay, R., Komanduri, S., Kelley, P.G., Leon, P.G., Mazurek, M.L., Bauer, L., +Christin, N., Cranor, L.F.: Encountering stronger password requirements: User +attitudes and behaviors. In: Proc. of SOUPS. pp. 1–20. USENIX (2010) +45. Sparling, C.: 5 Years of Fighting DDoS with the Power of Akamai (2019), +https://blogs.akamai.com/2019/07/5-years-of-fighting-ddos-with-the-power-of-akamai.html +46. Sullivan, +N.: +Keyless +SSL: +The +Nitty +Gritty +Technical +Details +(2014), +https://blog.cloudflare.com/keyless-ssl-the-nitty-gritty-technical-details/ +47. Triukose, S., Al-Qudah, Z., Rabinovich, M.: Content delivery networks: Protection +or threat? In: Proc. of ESORICS. pp. 371–389. Springer (2009) +48. Van Acker, S., Hausknecht, D., Sabelfeld, A.: Measuring login webpage security. +In: Proc. of the SAC. pp. 1753–1760. ACM (2017) +49. Vissers, T., Van Goethem, T., Joosen, W., Nikiforakis, N.: Maneuvering around +clouds: Bypassing cloud-based security providers. In: Proc. of CCS. pp. 1530–1541. +ACM (2015) +50. Wang, K.C., Reiter, M.K.: How to end password reuse on the web. In: Proc. of +NDSS. ISOC (2019) +51. Wang, X.S., Choffnes, D., Gage Kelley, P., Greenstein, B., Wetherall, D.: Measuring +and predicting web login safety. In: Proc. of SIGCOMM Workshop on Measure- +ments up the Stack. pp. 55–60 (2011) +52. Wash, R., Rader, E., Berman, R., Wellmer, Z.: Understanding password choices: +How frequently entered passwords are re-used across websites. In: Proc. of SOUPS. +pp. 175–188. USENIX (2016) +53. Wei, C., Li, J., Li, W., Yu, P., Guan, H.: STYX: A Trusted and Accelerated +Hierarchical SSL Key Management and Distribution System for Cloud Based CDN +Application. In: Proc. of SoCC. pp. 201–213. ACM (2017) +54. Weinberger, J., Saxena, P., Akhawe, D., Finifter, M., Shin, R., Song, D.: A System- +atic Analysis of XSS Sanitization in Web Application Frameworks. In: ESORICS. +pp. 150–171. Springer (2011) +55. Wu, T.D., et al.: The secure remote password protocol. In: NDSS. vol. 98, pp. +97–111. Citeseer (1998) + diff --git a/mtE2T4oBgHgl3EQfJQaw/content/tmp_files/load_file.txt b/mtE2T4oBgHgl3EQfJQaw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..9cb092e27ed84703fa55f7f7db014edf9e8d3ebd --- /dev/null +++ b/mtE2T4oBgHgl3EQfJQaw/content/tmp_files/load_file.txt @@ -0,0 +1,858 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf,len=857 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='03690v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='CR] 9 Jan 2023 Quantifying User Password Exposure to Third-Party CDNs Rui Xin, Shihan Lin, Xiaowei Yang Duke University Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Web services commonly employ Content Distribution Net- works (CDNs) for performance and security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As web traffic is becoming 100% HTTPS, more and more websites allow CDNs to terminate their HTTPS connections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This practice may expose a website’s user sensitive information such as a user’s login password to a third-party CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In this paper, we measure and quantify the extent of user password exposure to third-party CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We find that among Alexa top 50K websites, at least 12,451 of them use CDNs and contain user login entrances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Among those websites, 33% of them expose users’ passwords to the CDNs, and a pop- ular CDN may observe passwords from more than 40% of its customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This result suggests that if a CDN infrastructure has a security vulnera- bility or an insider attack, many users’ accounts will be at risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' A simple fix to this security vulnerability is for a website to encrypt a user’s pass- word inside the HTTPS request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our measurement shows that less than 17% of the websites adopt this solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Keywords: HTTPS · CDN · password · security · measurement 1 Introduction Content Distribution Networks (CDNs) [32,40] play an important role in im- proving the performance and security of web services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' A CDN caches web pages at servers near end users to reduce retrieval latency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' It also blocks malicious requests to defend a web server against various attacks [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Currently, many websites employ CDNs provided by third-party companies such as Akamai [1], Cloudflare [3], and Fastly [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, third-party CDNs introduce a considerable security and privacy risk when they serve websites that enables HTTPS [9,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' HTTPS uses a certificate to certify the domain name of a website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, to make the web pages appear as if they come from the original site, a website has to share its TLS private key [9] or TLS session keys[46] with the CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In both cases, a third-party CDN can observe the content of all connections between a website and its users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In this work, we aim to raise awareness of this security and privacy risk and quantify its severeness from a user’s perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We choose to measure the extent to which users’ website login passwords are exposed to CDNs due to the HTTPS key sharing practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Although prior research has shown that private key sharing is prevalent on the Internet [9] and HTTPS termination weakens 2 Anonymous Authors connection security of a great portion of the Internet [11], it is not clear whether a website has taken simple countermeasures such as client-side encryption (see§ 2) to protect users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We conduct a measurement on Alexa top 50K sites [2] to quantify password exposure to CDNs during the user login procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We also measure the de- ployment of client-side password encryption on websites to understand websites’ treatment of users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Such a large-scale measurement is technically non-trivial, because we need to automate the login procedures on websites with diverse structures to inspect login requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, we design and implement a framework for automatic login.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The framework can detect login elements on a website and collect login requests when it submits credentials to websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our main contributions and findings can be concluded as the following: – We propose an open-source framework for automatic login 1, which can be applied for other research such as measurement of authentication methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' – Our measurement presents that 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='0% of websites that employ CDNs and contain login entrances expose users’ passwords in plaintext to their CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' – We find that two popular CDN providers, Cloudflare and Akamai, can ob- serve users’ passwords from 44% and 25% of their customers in our dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' – We find prevalent password exposure in most website categories, including websites whose user accounts should be carefully protected, such as websites related to finance and health.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' – Our result shows that less than 17% of the websites encrypt users’ passwords when transferring login requests to CDNs, and the top 1K websites are more likely to adopt password encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Overall, our measurement points out potential security issues caused by pass- word exposure to CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Even though websites choose to trust CDNs, users may concern about their privacy when CDNs can monitor their private data including passwords during transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Moreover, CDNs have never been secure enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Prior work already showed that an attacker can trick some CDNs to cache and reveal other users’ private data [13,33,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, private data leakage to CDNs may turn into a disaster when attackers or malicious insiders exploit vulnerabil- ities of CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 2 Background In this section, we briefly introduce CDNs and HTTPS, and we analyze the security issues when a website with HTTPS employs a CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We also discuss two countermeasures adopted by websites in practice to address such issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1 The source code can be accessed by https://anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='4open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='science/r/PAM2023-Anonymous-Submission-863D Quantifying User Password Exposure to Third-Party CDNs 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='1 HTTPS on CDNs A CDN reduces web retrieval time by directing a client’s request to an edge server which is hosted by the CDN and geographically close to the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The edge server responds to the client with cached content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' If the requested content is not cached, the edge server may fetch the content from the origin server which is hosted by the website (the CDN’s customer) and is the initial source of all content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' CDNs do not cache private data, as they are usually dynamic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Modern CDNs are used not only to speed up page loading but also to pro- vide an effective shield against attacks such as DDoS and code injections [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' A CDN enlarges the serving capability of its customers to prevent volumetric DDoS attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' It also applies techniques such as IP blocking and rate limiting to block attacks when DDoS happens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For example, Akamai protected its cus- tomers from 38,905 separate DDoS attacks from 2014 to 2019 [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' CDNs also inspect the content of requests and use Web Application Firewall (WAF) to filter out malicious requests such as XSS injection [54] and SQL injection [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Unfortunately, CDNs have become a source of vulnerabilities in the HTTPS ecosystem in recent years [9,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The security of HTTPS relies on certificates and private keys generated by totally trusted certification authorities (CAs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' How- ever, since HTTPS requires server authentication by private keys in HTTPS handshakes, if a website employs a CDN to represent it to respond to clients’ HTTPS requests, it has to share its private key to the CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' With the private key, the CDN can build HTTPS connections with clients, and clients cannot differen- tiate between the CDN and the origin server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' When a client requests for private data, the CDN will forward the request by terminating the HTTPS connection and building another HTTPS connection with the origin server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Therefore, the CDN becomes a man in the middle when a user’s private data are transmitted between the client and the origin server [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 Countermeasures in Practice Two instant but imperfect countermeasures have been deployed by some web- sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' First, a website can bypass the CDN and send the private requests to the origin server directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In this countermeasure, a website should use a separate domain or subdomain for the private data, because the CDN possesses the pri- vate key of the original domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We refer to this method as “CDN bypassing” in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This method will not affect CDNs’ benefit of page loading accel- eration, since the private data are not cached by CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, it eliminates the benefit of having the origin server shielded against DDoS attacks, because the IP address of the origin server is exposed to the public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' When attackers can connect to the origin server directly, it is much easier to launch DDoS attacks since the origin server usually cannot construct a DDoS defense as effectively as CDNs [16,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides DDoS, the CDN cannot inspect the private content to filter out malicious requests, and thus the origin server may suffer from attacks such as code injections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 4 Anonymous Authors Another countermeasure is to encrypt private data inside HTTPS connec- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The website generates another key pair and delivers the new public key to the client.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The client uses the public key to encrypt the private data to be sent out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Therefore, when a CDN forwards the request, the private data are invisible to the CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We refer to this method as “client-side encryption” in our paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We observe some websites use this method to protect users’ passwords only, as encrypting all private data may introduce too much overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, a secure public key delivery is non-trivial when HTTPS connections are already inter- cepted by a CDN [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Delivering another certificate differing from the HTTPS certificate is useless, because a website has to use JavaScript to conduct en- cryption in current browsers, and the JavaScript code cannot obtain the root certificates of a client to verify a certificate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Without a certificate, if the public key is delivered by a CDN, an active CDN can launch the man-in-the-middle attacks by replacing the public key.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' If the public key is delivered by the origin server, the origin server is exposed to the public and under the threat of DDoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Therefore, the client-side encryption only defends against password leakage to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Despite the defects of these two methods, they preserve users’ privacy to some extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Moreover, if the origin server builds its own DDoS defense or a CDN is assumed to be passive, these two countermeasures can provide sufficient protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, it is unclear about the deployment of these two counter- measures on websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, we investigate the password exposure to provide a profile of their deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our measurement will show that few websites adopt the client-side encryption for passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 3 Threat Model We use the threat model proposed by the prior work [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We consider the pri- vate data in a website as the data can only be accessed by a authenticated user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The users can be authenticated by the traditional password, one-time password (OTP), OAuth [20], certificates, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The credentials for authentication are con- sidered as private data as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We focus on the measurement of the traditional password in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We considered two types of attackers defined in the prior work [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' – Passive attacker: A CDN behaves honestly to serve the requests but an attacker inside the CDN may eavesdrop on the transmitted messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For example, a malicious administer of a CDN cannot change the CDN’s behavior but may peek at the transmitted traffic and record users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' – Active attacker: An attacker insider CDN may launch arbitrary attacks including eavesdropping and tampering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For example, a CDN may modify or corrupt the cached HTML to disable the front-end password encryption so that it can observe users’ passwords in the login requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This may happen when attackers exploit a vulnerability of a CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Quantifying User Password Exposure to Third-Party CDNs 5 4 Method To detect the password exposure, we should inspect a website’s login request and the destination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, we need a framework for automatic login in a large-scale measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Currently, a website may adopt multiple authentication methods, such as text passwords, OAuth [20], one-time password (OTP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In our measure- ment, we only consider the method of text passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Based on the existing frameworks [42,26,43], we designed and implemented an automatic login framework that copes with more web pages with diverse structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Browsers such as Chrome and Firefox can help users automatically fill the credentials in some web pages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We do not use this function because it relies on the existence of the “autocomplete” attribute in HTML elements, and thus it cannot handle the websites that do not enable this attribute in HTML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides the automation of browsers, Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' implemented a framework to log into phishing websites automatically [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our framework can handle issues that are common in legitimate sites but rare in phishing sites, such as confusion caused by sign up forms and pop-ups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Jonker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='. also proposed a framework for post- login security analysis [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our framework shares many similarities with theirs, but adds the capability to operate in the presence of HTTP Authentication and reCAPTCHA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In our work, we do not need to successfully log into a website, so the framework merely triggers a failed login and collects the login request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides the automatic login framework, we use the method in the prior work [9,22,27,28,30] to discover the CDN usage of a website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This method also help to inspect the destination of the collected login requests to determine whether the requests are sent to a CDN server.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Some cloud providers will provide both hosting service and CDN service, such as AWS and Azure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In our method, when a request is sent to such a cloud provider, we cannot determine whether the website is using the CDN service or the hosting service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' If the password is sent to a hosting service, it should not be considered as an exposure to a CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Since our goal is to provide an underestimation of password exposure, our CDN list does not include a CDN service provider that also provides hosting service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As a result, our CDN list contains 9 popular CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Ethical concerns: We respect user privacy and our work does not raise eth- ical concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The method of CDN discovery only used public data from the Internet, such as Registration Data Access Protocol (RDAP) [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As for the automatic login framework, since we do not require a successful login, we use a randomly generated fake account that is nearly impossible to coincide with existing ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We skip the websites that require a test of the account existence before submitting the login credentials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We only conduct the login trial once for each website, so we do not overload the websites in our test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 5 Password Exposure We only consider HTTPS-enabled websites because a website without HTTPS apparently contains major vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In Alexa top 50K sites [2], 42,502 6 Anonymous Authors 0 1 2 3 4 5 Ranking 104 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='8 1 CDF (a) CDF I0 I20 I40 I60 I80 I100 Ranking Intervals 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='5 Percentage (b) Percentage Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1: (a) Distribution of login-detected websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' (b) The percentage of password-exposed websites among CDN-enabled websites across different rank- ing intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We divide 50K websites into 100 ranking intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Each interval contains 500 websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The x-axis ticks at every 20 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of them enable HTTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We run the framework to automatically log into the websites with HTTPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' If the framework submits the fake credentials to a website, we consider it performs a login.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The framework performs 17,111 logins in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In this paper, we focus on these 17,111 websites and call them “login-detected websites”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We detect CDNs employed by these websites according to § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our result shows that 12,451 websites employ CDN service, and we call them “CDN-enabled websites” in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' By inspecting their login procedures, we find that 4,114 websites send the login requests with users’ passwords in plaintext or Base64 encoding to CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We denote these websites as “password-exposed websites”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We discovered that 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='0% of CDN-enabled websites expose users’ passwords to CDNs, demonstrating a potential privacy issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In this section, we present the results in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='1 Distribution over Rankings Since our framework may fail to detect the login forms of some websites, the dataset of login-detected websites can be considered as a sample set of all web- sites that enable logins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We first investigate the distribution of these samples over rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Figure 1a shows the distribution of login-detected websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' A linear relation- ship between the CDF and ranking shows a uniform distribution of the websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Therefore, the logins detected by our framework are unbiased in the rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' To investigate the relationship between websites’ rankings and their prefer- ence for password exposure, we divide the rankings into 100 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For an interval Ij, it contains 500 websites ranking in the range of [1+500∗(j−1), 500∗j].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Quantifying User Password Exposure to Third-Party CDNs 7 44% 25% 18% 5% 7% 66% 6% 9% 17% Cloudflare Akamai Fastly Highwinds Edgecast Incapsula Quantil CDNetworks Limelight 100 101 102 103 104 # of Websites CDN-enabled websites Password-exposed websites Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 2: Distribution across CDN providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The y-axis is in log scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The number above the bar denotes the percentage of password-exposed websites in CDN- enabled websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 58% 30% 32% 36% 34% 38% 27% 10% 31% 43% 36% 31% 40% 36% 19% 44% 67% Retail Internet Business Entertainment News Finance Technology Education Society Travel Science Sports Health Reference Government Recreation Home 0 50 100 150 200 250 300 350 # of Websites CDN-enabled websites Password-exposed websites Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 3: Distribution across website categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The number above the bar denotes the percentage of password-exposed websites in CDN-enabled websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For each interval, we count the password-exposed websites and the CDN-enabled websites, and we compute the percentage of password-exposed websites in CDN- enabled websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Figure 1b presents the percentage variation across the intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Given the result of unbiased detection in Figure 1a, we can examine the distribution of password exposure on website rankings through Figure 1b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Even though some fluctuations exist, the percentages are overall above 20%, meaning that the pass- word exposure is common across all rankings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We can also find that the percent- ages of password-exposed websites with a higher ranking are slightly larger than those with a lower ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The average percentage from the I1 to I50 and from the I51 to I100 are 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='6% and 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='8%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The reason for the difference may be that websites with higher rankings should handle more traffic, and they have a stronger preference for adopting CDNs to filter out the malicious requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, those websites tend to expose the private requests destined at the origin server to CDNs for inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides, we can find that the most popular web- sites in the first two intervals have relatively low password exposure percentage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' It is because that the top websites are more likely to deploy defense mechanisms, which can be justified by our analysis in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 8 Anonymous Authors 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 Distribution over CDN Providers We also consider how password-exposed websites are distributed among the CDN providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Figure 2 presents the number of password-exposed websites in each CDN provider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As shown in the figure, Cloudflare and Akamai are the two most popular CDNs in the world, and they observe the most users’ passwords from their customers’ requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' More than 40% of Cloudflare’s customer websites in our dataset share users’ passwords to Cloudflare, and Akamai observes passwords from 25% of its customers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides, 66% of websites who use Incapsula expose passwords to the CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Some CDNs only observe a small fraction of sensitive traffic, such as Highwinds and Edgecast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' It is reasonable for websites to trust famous CDN providers and employ their defense against attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, it does not necessarily mean users should also trust CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' From the users’ perspective, they may be concerned about their private or sensitive data when it is shared with a third-party CDN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The results also imply a risk of the single point failure of popular CDNs: a malicious insider in a popular CDN may divulge the users’ passwords of more than 40% of its customer websites, leading to a large-scale user data leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='3 Distribution over Website Categories We investigate the practice of exposing password among different website cat- egories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We collect the website categories data from Alexa Top Sites by Cate- gory [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In 12,451 CDN-enabled websites, 2,010 of them can be classified by the Alexa data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We use these 2,010 sites for analysis in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Figure 3 presents the statistics of CDN usage and password exposure across 17 website categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As we can see, retail websites employ most CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' It may be because retail websites need to display many photos of their products, which would obtain substantial benefit from using a CDN service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The retail websites also expose the most passwords to CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides retail websites, more than 40% of websites in several categories expose users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We note that a large portion (38%) of finance websites which are usually considered to require sophisticated defense divulges users’ password to CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Moreover, education websites have the least percentage of password exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our results point out that password exposure is prevalent within a wide range of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 6 Countermeasures In this section, we first present the measurement of the countermeasures against password exposure used by current websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We also discuss possible counter- measures can be adopted by websites and users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='1 Client-side Encryption and CDN Bypassing In our measurement, we observe that some websites indeed adopt the method of client-side encryption discussed in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 to protect users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For ex- ample, baidu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com, dropbox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com, and chase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com deliver public keys by their Quantifying User Password Exposure to Third-Party CDNs 9 I0 I20 I40 I60 I80 I100 Ranking Intervals 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='6 Percentage Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 4: This figure shows the percentage of password-encrypted websites among CDN-enabled websites across different ranking intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We divide 50K websites into 100 ranking intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Each interval contains 500 websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The x-axis ticks at every 20 intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' origin servers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, such a solution is rarely adopted by the websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In our measurement, if our framework submits the credentials but cannot find the pass- word in plain text or Base64 encoding in the login request, we consider that the website encrypts the password.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Since our framework may fail to login, we have an upper-bound estimation of the deployment of password encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' There- fore, in our dataset, at most 2,057 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='5%) out of 12,451 CDN-enabled websites adopt such a solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We call these websites as “password-encrypted websites”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This result demonstrates that password encryption is a rare practice on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We investigate the relationship between a website’s ranking and password encryption deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We used the same method and intervals in Figure 1b, and the results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As we can see, even for the top websites, less than 30% of them encrypt users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Nevertheless, when compared with other websites, they have a relatively higher percentage of password encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, an outstandingly high percentage exists around the intervals of quite low ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We manually inspected websites located in that interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We found 13 websites of all 20 password-encrypted websites are subdomains of tmall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com for different retailers, such as www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='kfc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='tmall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com and www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='lenovo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='tmall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Once a user attempts to sign into these subdomain sites, they all direct the user to tmall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This website is one of the top websites for electronic shopping, and it adopts client-side password encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Overall, we can conclude that the top ranking websites are more likely to encrypt users’ passwords in transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' CDN bypassing can protect users’ privacy, even though it exposes servers’ IP addresses and leave servers at the risk of DDoS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In our measurement, we cannot verify whether the destination of a login request is the origin server through RDAP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We leave the further measurement of CDN bypassing as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 10 Anonymous Authors 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2 Other Countermeasures Besides password encryption and CDN bypassing, Password Authenticated Key Exchange (PAKE) [8,15] also prevents password exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' PAKE protocols, such as SRP [55] and OPAQUE [24], authenticate users without the requirement of revealing passwords in login requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Moreover, it is proven to be secure during login even when CDNs can launch active attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, PAKE protocols re- quire trust on first use (TOFU), meaning that a secure channel is required during account registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Therefore, PAKE solves the password exposure problem for web services that do not allow online registration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' For example, it can be used in banking industry, as users are required to open a bank account physically at branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Nevertheless, PAKE is almost never used by websites [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The reason may be the difficulty of understanding and implementing PAKE protocols for de- velopers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' It may also be because developers usually trust third-party CDNs and lack of awareness of security issues caused by insufficient password protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' From the users’ perspective, a user can use OAuth [20] such as using a Google account to sign in to other websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Because leading tech companies such as Google and Facebook have built their own CDNs, a user’s password will not be exposed to a third party during the login.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, more OAuth practice may lead to a severe single-point failure if a user’s password of Google account is leaked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides OAuth, users can also adopt two-factor authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Even though two-factor authentication cannot prevent passwords from being exposed to third-party CDNs, it prevents accounts from being compromised even when the passwords are exposed to attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' These countermeasures can only protect users’ passwords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, our re- sults also suggest a potential privacy threat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Users’ private data stored on a web- site may also be divulged to a CDN during the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As private data is much more complicated and diverse than the passwords, developing countermea- sures would be harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Thus, private data leakage may be much more prevalent than password leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We leave the measurement of private data leakage as future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 7 Discussion and Future Work Our measurement quantifies password exposure to CDNs and suggests potential security issues in current web ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In this section, we provide suggestions to the security community, users, and the industry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We need further research on the solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As presented in § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='2, the pre- liminary strategies of CDN bypassing and client-side encryption can be eas- ily deployed but contain vulnerabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Proposed techniques such as Keyless SSL [46,12,35], certificate delegation [29], and mcTLS [37] are ineffective in pre- serving user privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The SGX-based [36,21] solutions can provide comprehen- sive protection, but it is hard to be deployed on CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' InviCloak [30], a solution proposed just recently can achieve the goal of securing servers from DDoS, pro- tecting users’ privacy, and allowing instant deployment simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, Quantifying User Password Exposure to Third-Party CDNs 11 InviCloak disable the Web Application Firewall (WAF) of CDNs, and it is still unclear whether website developers is willing to adopt InviCloak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Therefore, further research on this area is critical to a more secure Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We recommend users to adopt OAuth and two-factor authentication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As dis- cussed in § 6, signing into a website with existing accounts of leading tech com- panies such as Google and Facebook could be safer than creating a new account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides, two-factor authentication is also recommended, as it provides additional protection for an account even when the password is stolen by a hacker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Websites should adopt preliminary defense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The results shows that many websites do not apply the minimal defense against password exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Despite the preliminary strategies are vulnerable to some attacks, they provide basic protection for users’ privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Since it is acceptable to assume a passive CDNs in most cases, the client-side encryption usually provides a sufficient protection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' CDN providers should involve in developing and deploying advanced solu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The widespread of Keyless SSL on Cloudflare demonstrates that a CDN provider plays an important role in the security community [46] Cooperation from CDN providers can validate researchers’ ideas and advance further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' CDNs can also guide their customers to deploy a defense mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' This paper presents the preliminary results of password sharing to third-party CDNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We propose the following directions as the future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Augment the existing CDN discovery method to differentiate the hosting service and the CDN service of a cloud provider, as mentioned in § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Quantify the adopted or available countermeasures besides the client-side en- cryption in websites, including CDN bypassing, OAuth, one-time password, two-factor authentication, etc, as mentioned in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Measure private data leakage in websites to understand the security impact of TLS private key sharing from users’ perspectives, as mentioned in § 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Survey the users and website developers to understand their awareness of private data leakage to thrid-party CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Such a survey helps to figure out the reason why countermeasures are not widespread.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 8 Related Work Password security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Password security has attracted attention from many re- searchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' analyzed how websites deploy measures to prevent online password cracking [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' manually inspected 188 websites to char- acterize the login process and built an extension to inform users of potential password leakage caused by the lack of HTTPS [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Acker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' studied the security of password input fields among the Alexa top 100K sites, and they found that 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='8% of the websites with a login page are vulnerable to basic man- in-the-middle attacks [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Bonneau et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' surveyed the proposals for replacing passwords and pointed out the difficulty of replacing passwords [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' explored how passwords are spread after they are divulged by phishing sites [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In addition, many prior works investigated the prevalence of the password reuse problem [41,23,52,44] and its countermeasures [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 12 Anonymous Authors CDN security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Researchers have shown the existence of a wide range of vul- nerabilities in CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Nguyen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' presented an attack of poisoning CDN cache with error pages, and five CDN services were vulnerable to such an attack [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Mirheidari et al.’s measurement shows that private data can be divulged by CDNs through web cache deception [13,33,34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides CDN cache, researchers also presented approaches to disclosing the IP addresses of origin servers hid- den behind CDNs, demonstrating insufficient DDoS protection of CDNs [49,25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Moreover, attackers may utilize a CDN to launch DoS to an origin server or to the CDN itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Triukose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' presented an amplify method to launch DoS to an origin server through the CDN [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' The forwarding loop discovered by Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' can lead to resource-consuming DoS to CDNs [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Guo introduced three attacks to break CDN DoS protection, including HTTP/2 amplification, pre-post slow HTTP, and availability degradation [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In addition, Hao et al.’s research demonstrated that attackers can hijack the DNS redirection used by a CDN to downgrade the content delivery performance [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Durumeric et al.’s measurement shows that the HTTPS interception on CDNs may downgrade the TLS version or cipher suites and thus reduce connection security [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Solutions to TLS key sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' A line of the research focuses on building keyless CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Cloudflare, Akamai, and Modadugu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' proposed similar solu- tions called “Keyless SSL”, respectively [46,12,35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Certificate delegation [29] and mcTLS [37] enables a client to recognize the CDN as a delegation of the website.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wei et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' [53] and Ahmed et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' [5] adopted Trust Executive Environment (TEE) on CDNs for private key management.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, these strategies only prevent the TLS private key sharing, while users’ private data are still visible to CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Phoenix [21] and mbTLS [36] extend TEE solutions to fully protect users’ private data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' However, deploying TEE-based solutions on CDNs may take a long time as it requires upgrades of hardware and operating systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' InviCloak [30] protects users’ private data with an additional encryption channel and low overhead, but its adoption by websites in the future remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 9 Conclusion In this paper, we conduct a large-scale measurement to quantify user password exposure to third-party CDNs in the web ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Our results show that 4,114 of 12,451 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='0%) HTTPS-enabled websites that employ third-party CDNs ex- pose users’ passwords to the CDNs during the login procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Besides, as a popular CDN, Cloudflare sees users’ passwords from more than 40% of its cus- tomers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In addition, password encryption is rarely adopted by websites, even though it is simple and effective to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Overall, our results sug- gest that current websites excessively trust CDNs, leading to potential security issues when CDNs’ vulnerabilities are exploited by attackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' As HTTPS and CDNs becomes more popular, we encourage further research on the privacy is- sues caused by HTTPS termination on CDNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' We publicly release the code of the auto-login framework to facilitate future research [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Quantifying User Password Exposure to Third-Party CDNs 13 References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Akamai (2020), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='akamai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Alexa Top Sites (2020), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='alexa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/topsites 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Cloudflare (2020), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='cloudflare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Fastly (2020), https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='fastly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Ahmed, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zaheer, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Ricci, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Harpocrates: Giving Out Your Secrets and Keeping Them Too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of IEEE/ACM Symposium on Edge Computing (SEC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 103–114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' IEEE (2018) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Anonymous: Code of pam 2023 anonymous submission (2022), https://anonymous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='4open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='science/r/PAM2023-Anonymous-Submission-863D 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Bonneau, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Herley, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Van Oorschot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Stajano, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': The quest to replace passwords: A framework for comparative evaluation of web authentication schemes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of S&P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 553–567.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' IEEE (2012) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Boyko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', MacKenzie, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Patel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Provably secure password-authenticated key exchange using diffie-hellman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of EUROCRYPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 156–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Springer (2000) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Cangialosi, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Chung, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Choffnes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Levin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Maggs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Mislove, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wil- son, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Measurement and Analysis of Private Key Sharing in the HTTPS Ecosys- tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of CCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 628–640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2016) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zheng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Paxson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Forwarding-loop attacks in content delivery networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of NDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ISOC (2016) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Durumeric, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Ma, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Springall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Barnes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sullivan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Bursztein, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Bailey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Halderman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Paxson, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': The security impact of https interception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of NDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ISOC (2017) 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Gero, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Shapiro, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Burd, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : Terminating SSL Connections without Locally-Accessible Private Keys (2013), u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Patents, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 9,647,835 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Gil, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Web cache deception attack (2017), https://omergil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='blogspot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/2017/02/web-cache-deception-attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='html 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Gillman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Lin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Maggs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sitaraman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Protecting Websites from Attack with Secure Delivery Networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Computer 48(4), 26–34 (2015) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Green, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Let’s Talk About PAKE (2018), https://blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='cryptographyengineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/2018/10/19/lets-talk-about-pake/ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Guo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Jia, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Abusing cdns for fun and profit: Security issues in cdns’ origin validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SRDS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' IEEE (2018) 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Guo, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Liu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Shen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Chen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Cdn judo: Breaking the cdn dos protection with itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of NDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ISOC (2020) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Halfond, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Viegas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Orso, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : A Classification of SQL-Injection At- tacks and Countermeasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of International Symposium on Secure Soft- ware Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 13–15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' IEEE (2006) 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Stavrou, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': End-users get maneuvered: Empirical analysis of redirection hijacking in content delivery networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1129–1145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX (2018) 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Hardt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': The oauth 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='0 authorization framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Internet Engineering Task Force (IETF) (2012) 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Herwig, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Garman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Levin, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Achieving Keyless CDNs with Conclaves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 735–751.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX (2020) 14 Anonymous Authors 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Huang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Ross, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : Measuring and evaluating large-scale cdns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 15–29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2008) 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Ion, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Reeder, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Consolvo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': “.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' no one can hack my mind”: Comparing expert and non-expert security practices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SOUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 327–346.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX (2015) 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Jarecki, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Krawczyk, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Opaque: an asymmetric pake protocol secure against pre-computation attacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of EUROCRYPT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 456–486.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Springer (2018) 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Hao, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Cotton, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Your remnant tells secret: Residual resolu- tion in ddos protection services.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of DSN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 362–373.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' IEEE (2018) 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Jonker, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Karsch, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Krumnow, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sleegers, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Shepherd: A generic approach to automating website login.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of NDSS Workshop on Measurements, Attacks, and Defenses for the Web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ISOC (2021) 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Krishnamurthy, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wills, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': On the use and performance of content distribution networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of IMC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 169–182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2001) 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Levy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': CDNs and Privacy Threats: A Measurement Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' thesis, Prince- ton University (2017) 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Liang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Duan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wan, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': When HTTPS meets CDN: A Case of Authentication in Delegated Service.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of S&P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 67–82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' IEEE (2014) 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Lin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Xin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Goel, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Yang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': InviCloak: An End-to-End Approach to Privacy and Performance in Web Content Distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of CCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2022) 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Lu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Ling, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Lin, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': A measurement study of authen- tication rate-limiting mechanisms of modern websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of ACSAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 89–100 (2018) 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Maggs, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sitaraman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Algorithmic Nuggets in Content Delivery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM SIGCOMM CCR 45(3), 52–66 (2015) 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Mirheidari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Arshad, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Onarlioglu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Crispo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Kirda, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Robertson, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Cached and confused: Web cache deception in the wild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 665–682.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX (2020) 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Mirheidari, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Golinelli, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Onarlioglu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Kirda, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Crispo, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Web cache de- ception escalates!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of Security Symposium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 179–196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX, Boston, MA (2022) 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Modadugu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Goh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : The Design and Implementation of WASP: A Wide-Area Secure Proxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Stanford University (2002) 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Naylor, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Gkantsidis, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Karagiannis, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Steenkiste, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': And then there were more: Secure communication for more than two parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of CoNEXT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 88–100 (2017) 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Naylor, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Schomp, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Varvello, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Leontiadis, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Blackburn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Lopez, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Papagiannaki, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Rodriguez Rodriguez, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Steenkiste, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Multi-context tls (mctls): Enabling secure in-network functionality in tls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SIGCOMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 199–212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2015) 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Newton, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Hollenbeck, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Rfc7482: Registration data access protocol (rdap) query format.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Internet Engineering Task Force (IETF) (2015) 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Nguyen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Iacono, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Federrath, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Your cache has fallen: Cache-poisoned denial-of-service attack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of CCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1915–1936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2019) 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Nygren, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sitaraman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': The Akamai Network: a Platform for High- Performance Internet Applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' SIGOPS OSR 44(3), 2–19 (2010) 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Pearman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Thomas, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Naeini, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Habib, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Bauer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Christin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Cranor, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Egelman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Forget, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Let’s go in for a closer look: Observing passwords in their natural habitat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of CCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 295–310.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2017) Quantifying User Password Exposure to Third-Party CDNs 15 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Peng, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Xu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Quinn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Hu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Viswanath, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': What happens after you leak your password: Understanding credential sharing on phishing sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of AsiaCCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 181–192 (2019) 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Senol, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Acar, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Humbert, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Borgesius, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : Leaky Forms: A Study of Email and Password Exfiltration Before Form Submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of Security Sympo- sium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1813–1830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX, Boston, MA (2022) 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Shay, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Komanduri, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Kelley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Leon, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Mazurek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Bauer, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Christin, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Cranor, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Encountering stronger password requirements: User attitudes and behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SOUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1–20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX (2010) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Sparling, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': 5 Years of Fighting DDoS with the Power of Akamai (2019), https://blogs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='akamai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/2019/07/5-years-of-fighting-ddos-with-the-power-of-akamai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='html 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Sullivan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Keyless SSL: The Nitty Gritty Technical Details (2014), https://blog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='cloudflare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='com/keyless-ssl-the-nitty-gritty-technical-details/ 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Triukose, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Al-Qudah, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Rabinovich, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Content delivery networks: Protection or threat?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of ESORICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 371–389.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Springer (2009) 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Van Acker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Hausknecht, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Sabelfeld, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Measuring login webpage security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of the SAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1753–1760.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2017) 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Vissers, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Van Goethem, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Joosen, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Nikiforakis, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Maneuvering around clouds: Bypassing cloud-based security providers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of CCS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 1530–1541.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2015) 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Reiter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : How to end password reuse on the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of NDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ISOC (2019) 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Choffnes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Gage Kelley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Greenstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wetherall, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Measuring and predicting web login safety.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SIGCOMM Workshop on Measure- ments up the Stack.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 55–60 (2011) 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wash, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Rader, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Berman, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Wellmer, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': Understanding password choices: How frequently entered passwords are re-used across websites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SOUPS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 175–188.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' USENIX (2016) 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wei, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Li, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Yu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Guan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': STYX: A Trusted and Accelerated Hierarchical SSL Key Management and Distribution System for Cloud Based CDN Application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' of SoCC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 201–213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' ACM (2017) 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Weinberger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Saxena, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Akhawe, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Finifter, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Shin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', Song, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=': A System- atic Analysis of XSS Sanitization in Web Application Frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: ESORICS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 150–171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Springer (2011) 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Wu, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' : The secure remote password protocol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' In: NDSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 98, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' 97–111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} +page_content=' Citeseer (1998)' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/mtE2T4oBgHgl3EQfJQaw/content/2301.03690v1.pdf'} diff --git a/n9E3T4oBgHgl3EQfLAl3/content/tmp_files/2301.04359v1.pdf.txt b/n9E3T4oBgHgl3EQfLAl3/content/tmp_files/2301.04359v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..315ea24fc813ca97bfb09855e1acde91ee3c4f6b --- /dev/null +++ b/n9E3T4oBgHgl3EQfLAl3/content/tmp_files/2301.04359v1.pdf.txt @@ -0,0 +1,787 @@ +DECIPHERING PANCHARATNAM’S DISCOVERY OF GEOMETRIC +PHASE +PREPRINT, COMPILED JANUARY 12, 2023 +Luis Garza-Soto1, Nathan Hagen1∗, and Dorilian Lopez-Mago2 +1Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585 Japan +2Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias Ave. Eugenio Garza Sada 2501, Monterrey, N.L., México, +64849 +ABSTRACT +While Pancharatnam discovered the geometric phase in 1956, his work was not widely recognized +until its endorsement by Berry in 1987, after which it received wide appreciation. However, because +Pancharatnam’s paper is unusually difficult to follow, his work has often been misinterpreted as +referring to an evolution of states of polarization, just as Berry’s work focused on a cycle of states, +even though this consideration does not appear in Pancharatnam’s work. We walk the reader through +Pancharatnam’s original derivation and show how Pancharatnam’s approach connects to recent work +in geometric phase. It is our hope to make this widely cited classic paper more accessible and better +understood. +1 +HISTORICAL INTRODUCTION +Sivaramakrishnan Pancharatnam was born in Calcutta +in 1934 in a family of remarkable scientists. He joined +his brother S. Ramaseshan at Bangalore in the Raman Re- +search Institute in 1953 when he was 19 years old.[1, 2] At +that time Ramaseshan’s research supervisor was C. V. Ra- +man, their uncle, and who received the Nobel prize in +physics in 1930. Under C. V. Raman’s instruction, Pan- +charatnam studied the behavior of absorbing bi-axial crys- +tals, as he points out in the opening of his now-famous +paper, “Generalized theory of interference and its applica- +tions, part I. Coherent pencils”.[3] This impressive work +was published in 1956 when Pancharatnam was 22. In it, +Pancharatnam proposes a definition for two beams of dif- +ferent states of polarization to be in phase: “The phase ad- +vance of one polarised beam over another (not necessarily +in the same state of polarization) is the amount by which +its phase must be retarded relative to the second, in order +that the intensity resulting from their mutual interference +may be maximum.”[3] Berry named this Pancharatnam’s +connection.[4] +After providing expressions for the phase shift — what +we now refer to as the geometric phase — Pancharatnam +goes on to relate the expressions to the subtended solid +angle on the Poincaré sphere. In Sections 2 & 3 below, +we retrace Pancharatnam’s derivation using a more mod- +ern approach that should be easier for modern readers to +follow. Along the way, it becomes clear that Pancharat- +nam nowhere considers the case that is widely attributed +to him, that the phase of polarized light changes after a +cyclic evolution of its polarization.[4, 5, 6, 7, 8, 9, 10, 11] +This misconception of his actual work appears to be a +result of the difficulty one encounters when reading Pan- +charatnam’s paper, which often feels like it belongs to the +19th century, and also of confounding Pancharatnam’s +work with the closely related work of Berry. +Almost 30 years after Pancharatnam’s original work, +Michael Berry at the University of Bristol, unaware of Pan- +charatnam’s work, discovered that an unexpected phase +emerges after the adiabatic evolution of a quantum state +around a cycle in parameter space.[12] In 1983, before his +initial paper was published, Berry introduced the geo- +metric phase to Barry Simon, who immediately coined it +as Berry’s phase.[13] By the end of 1986 Ramaseshan and +Nityananda revived Pancharatnam’s work and presented +it as an example of Berry’s phase.[14] Berry himself re- +ceived Nityananda’s manuscript and read it, but mentions +that it was not until he visited Bangalore in July 1987 that +he came to appreciate Pancharatnam’s work. One can sur- +mise that Nityananda’s interpretation of Pancharatnam’s +phase was likely greatly influenced by Berry’s personal +approach to the geometric phase via cycles of states. This +seems likely because Berry, on his return from India, pre- +pared a new manuscript that revealed the connections be- +tween his and Pancharatnam’s work, in which he explains +Pancharatnam’s work in terms of cycles of states.[15, 16] +Subsequent researchers have almost invariably followed +this interpretation, with the result that Pancharatnam’s ac- +tual achievement is obscured beneath the misconception. +As Berry himself pointed out, there have been several +anticipations to geometric phase that arise before Pan- +charatnam’s work.[16] Vinitskii et al., for example, men- +tion the work of Rytov (1938) and Vladimirskii (1941) as +precursors.[17] Oriol Arteaga has also pointed out that +Fresnel and Arago in 1816 developed their “fifth” law +of optical interference in such a way that a geometric +phase term (at the time not well understood) had to be +included in the interference equations.[11] However, as +is commonly the case in science, every discovery can be +traced to its anticipations, and we focus on Pancharat- +nam because his achievement is widely recognized by the +scientific community.[18] +arXiv:2301.04359v1 [physics.optics] 11 Jan 2023 + +S2 +S3 +A′ +S1 +A +C +B +b +Figure 1: Polarization state C with respect to the two orthogonal +states A and A′, represented on the Poincaré sphere. Point B +represents an SOP lying between A and A′. +2 +PANCHARATNAM’S STARTING POINT +At the beginning of his manuscript, after explaining briefly +some of the properties of the Poincaré sphere and Stokes +parameters, Pancharatnam introduces the following theo- +rem: +Theorem 1. When [an electromagnetic] vi- +bration of intensity I in the state of po- +larisation C is decomposed into two vi- +brations in the opposite states of polar- +isation A and A′, the intensities of the +A-component and the A′-component are +I cos2(AC/2) and I sin2(AC/2) respec- +tively. +(See Fig. 1 for an illustration of the geometry.) Here Pan- +charatnam makes use of the fact that the angle ∠A′C, be- +tween points A′ and C, subtended from the center of the +Poincaré sphere, is complementary to angle ∠AC, and +so writes both components in terms of ∠AC alone. Since +we will be making use of these angles in many of the +equations below, we follow Pancharatnam and define the +individual angles a, b, and c as +c/2 = ∠AB +b/2 = ∠AC +a/2 = ∠BC +� +� +� +(1) +Note that these angles a, b, and c on the left hand side of +the equations are defined in terms of the electric fields in +Cartesian space, whereas the arcs on the right hand side +are defined in Poincaré space, hence the division by two +in each definition. +Since this theorem is the basis for much of Pancharatnam’s +subsequent results, we show how one can derive it using +a modern approach. Let us consider a monochromatic +electromagnetic wave propagating along the z axis, i.e., +E = EC exp[i(kz − ωt)]. The electric field amplitude EC +can be written using elliptical polarization states {ˆeA, ˆeA′} +as a basis, with the properties |ˆeA,A′| = 1, and ˆeA · ˆe∗ +A′ = 0 +(where ∗ represents the complex conjugate). Therefore, +EC = E cos(α)ˆeA + E exp(iβ) sin(α)ˆeA′, +(2) +where E is a real amplitude (thus, I = E2), α controls +the projection over the basis vectors, and β is the relative +phase. With the above equation, we can represent any +state of polarization (SOP) by modifying α and β. +For the common definition of the Stokes parameters, we +would choose ˆeA = ˆx and ˆeA′ = ˆy, so that +S0 = |Ex|2 + |Ey|2 , +S1 = |Ex|2 − |Ey|2 , +S2 = 2Re[ExE∗ +y] , +S3 = 2Im[ExE∗ +y] , +� +� +� +� +� +� +� +� +� +� +� +(3) +However, this is only one specific choice. An equally +valid choice is the following general form of the Stokes +parameters[19] +S0 = |EA|2 + |EA′|2 , +S1 = |EA|2 − |EA′|2 , +S2 = 2Re[EAE∗ +A′] , +S3 = 2Im[EAE∗ +A′] , +� +� +� +� +� +� +� +� +� +� +� +(4) +where Ej, with j ≡ A, A′, are the components of EC, i.e., +Ej = EC · ˆej. Let us consider C = {S1, S2, S3} as the +Stokes vector representing the SOP of EC. C lies on the +surface of the Poincaré sphere of radius S0 with main axes +{S1, S2, S3}. These definitions are illustrated in Fig. 1, +where the Si are with respect to a general elliptical basis +and need not be with respect to the x-y basis. By using (2), +the Stokes parameters for EC are +S0 = E2 = I , +S1 = I cos 2α , +S2 = I sin 2α cos β , +S3 = −I sin 2α sin β . +� +� +� +� +� +� +� +� +� +(5) +Here, 2α and β are the polar and azimuthal angles, re- +spectively, in a spherical coordinate system with S1 being +the polar axis, and {S2, S3} the equatorial plane. Hence, +if α = [0, π/2], and β = [0, 2π], we can cover the entire +surface of the sphere. +From (5), we see that the Stokes vector of ˆeA (shown in +Fig. 1 as A) is equal to S1. Therefore, 2α is the angle in +Poincaré space that the SOP of EC makes with the basis +vector ˆeA (cf. Fig. 1). Moreover, if we can define ∠AC = +2α, we can use (2) to conclude that the intensity of the ˆeA- +component is I cos2(∠AC/2) and the intensity of the ˆeA′- +component is I sin2(∠AC/2), which is Pancharatnam’s +Theorem 1. +Pancharatnam next considers the problem of explaining +interference between two non-orthogonal states of polar- +ization. It was already well known that the interference of +two beams of the same state of polarization (SOP) results +in an intensity +I(I1, I2, δ) = I1 + I2 + 2 +� +I1I2 cos(δ) , +(6) +2 + +where I1 and I2 are the individual intensities, and δ is the +phase difference between the two beams. We can see that +changing I1 or I2 only modifies the modulation contrast +(visibility), but not the phase of the sum wave I.[20] +In order to see what happens when two beams of different +SOP are combined, Pancharatnam considers the superpo- +sition of beams in states A and B. He considers one of +them, e.g., A, and its orthogonal A′, as the polarization +basis, and finds the intensity of the sum by applying the +following steps: +1. Coherently adding A to the A-component of B, +2. Incoherently adding the A′-component of B to the +intensity obtained in step 1. +That is, the projection of A onto B gives the component +of A that directly interferes with A. The component of A +orthogonal to B does not interfere, and so only adds a bias +to the intensity. +Considering Pancharatnam’s Theorem 1, we therefore de- +compose B into two beams of orthogonal polarizations, +yielding the intensities +IB,A = IB cos2(c/2) , +(7) +IB,A′ = IB sin2(c/2) , +(8) +where IB is the intensity of B, and c is the angle between +the two states on the Poincaré sphere, and is thus twice +the angle between the states in Cartesian space. For step +1, we coherently combine IA (the intensity of A) with IB,A +using (6), giving +I(IA, IB,A, δ) = +IA + IB cos2(c/2) + 2 +� +IAIB cos2(c/2) +�1/2 cos(δ) . +(9) +This expression represents the intensity that results from +the addition of the A-component of both beams. +Finally, with step 2, we add IB,A′ to obtain the intensity +I = IA + IB cos2(c/2) + IB sin2(c/2) ++ 2 +� +IAIB cos2(c/2) +�1/2 cos(δ) , +(10) +which simplifies to +I = IA + IB + 2 +� +IAIB cos(c/2) cos(δ). +(11) +The above equation is Pancharatnam’s Eq. (1) in Ref. [3]. +This expression has the advantage that the factor contain- +ing the phase δ between the beams and cos(c) — what +Pancharatnam calls their “similarity factor” — are sepa- +rated as a product. +Equation 11 is a useful result because it makes two prop- +erties clear. First, while δ is the phase delay between input +beams A and B, the equation shows that the intensity of +the beams’ superposition varies sinusoidally, with δ as the +phase of I, independent of what polarization states are in- +volved. This is illustrated in Fig. 2. Second, if either of the +two intensities IA or IB changes, this changes I but does +not change the value of δ. Here we see that c determines +the fringe visibility of the interferogram. Therefore, this +0 +2 +3 +4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +interferogram intensity +attenuated wave +original wave +interferogram OPD, δ (radians) +Figure 2: The “original wave” interferogram is obtained in (11) +when the two intensities IA and IB are equal and held fixed as +we vary the relative phase δ between them. The “attenuated +wave” occurs when we reduce the intensity of one beam, and +again vary the phase between the two beams. +one of the results that Pancharatnam has in mind when he +titles his paper the “generalized theory of interference”. +To compare with more modern approaches, we can ana- +lyze the same case using the Jones calculus. In order to +simplify the math, we will simply define the state of the +first beam to be horizontal, while that of the second beam +is oriented at an angle of θ with respect to the horizontal. +We also allow there to be a propagation phase delay δ be- +tween the two beams. Therefore, we have the two electric +field vectors +EA = |EA| ˆx, +and +EB = |EB|e−iδ (cos θ ˆx + sin θ ˆy) . +(12) +The amplitude of their sum is +EA + EB = +� +|EA| + |EB| cos θ e−iδ� +ˆx + |EB| sin θ e−iδ ˆy, +(13) +so that intensity of the two beams’ interference is +I = +� +EA + EB +� · +� +EA + EB +�∗ += +�|EA| + |EB| cos θ e+iδ��|EA| + |EB| cos θ e−iδ� ++ +�|EB| sin θ e+iδ��|EB| sin θ e−iδ� += |EA|2 + |EB|2 + 2|EA||EB| cos θ cos δ , +(14) +which is exactly Pancharatnam’s expression [given as (11) +above], since c = 2θ. +This can be generalized to an arbitrary pair of elliptical +states, considering two waves with electric field ampli- +tudes EA and EB, and a phase difference δ. The intensity +of their superposition is +I = (EA + eiδEB) · (E∗ +A + e−iδE∗ +B) += |EA|2 + |EB|2 + 2Re +� +e−iδ(EA · E∗ +B) +� +. +(15) +The result of EA · E∗ +B can be inferred from the discus- +sion about Pancharatnam’s Theorem 1, which gives +EAEB cos(c/2), where EA, EB are the real amplitudes of +the electric fields. Therefore, we conclude that +I = |EA|2 + |EB|2 + 2EAEB cos(c/2) cos(δ) += IA + IB + 2 +� +IAIB cos(c/2) cos(δ), +(16) +which agrees with Pancharatnam’s result. +3 + +S2 +S1 +S3 +0 +A +C +B +C +S +S2 +S1 +3 +B +0 +A +C +C +S2 +S1 +S3 +0 +A +C +B +C +(a) +(b) +(c) +Figure 3: The increasing area (drawn in orange) on the surface of the sphere, formed by the spherical triangle between points A, B, +and C. +3 +THE RELATIONSHIP BETWEEN INTENSITIES +AND LOCATIONS ON THE POINCARÉ SPHERE +Following his expression for the interference of nonorthog- +onal beams, Pancharatnam requires only a few short steps +(Sec. 4 of his paper) to develop his famous solid angle +formula for the geometric phase. Here he considers the de- +composition of a beam of state of polarization C into two +nonorthogonal beams of polarization states A and B. If +(11) is a general equation for the interference of two beams +with different SOPs, then one should be able to re-arrange +the equation to solve for the phase delay δ between them +via the intensities of A and B (but which Pancharatnam +labels I1 and I2): +cos(δ) = +I − (IA + IB) +2 √IAIB cos(c/2) . +(17) +We can interpret this result as saying that “whatever this +phase δ may be, it maintains a specific relationship be- +tween the intensities of the beam being decomposed (C) +and the intensities of the beams that result from the de- +composition (A and B).” +While he could have finished with this numerical formula +for δ, he went a step further and realized that this equation +expresses a solid angle relationship between the SOPs of +the three beams. To explain this he uses electric field +vectors EA and EB as components of a sum vector EC = +EA + EB. From vector analysis, it is easy to see that the +part of EB perpendicular to EA also has to be equal to the +part of EC perpendicular to EA. Writing this in terms of +the angles a, b, and c from (1), we have +� +IB sin(c/2) = +√ +I sin(b/2) . +(18) +Taking the square of the components to find the intensities +results in the proportions +IA = I sin2(a/2) +sin2(c/2) +and +IB = I sin2(b/2) +sin2(c/2) . +(19) +By expressing the intensities IA and IB in terms of the total +intensity I, together with the angles between states on the +Poincaré sphere, we can rewrite (17) as +cos(δ) = sin2(c/2) − sin2(a/2) − sin2(b/2) +2 sin(a/2) sin(b/2) cos(c/2) +, +(20) +which is Pancharatnam’s Eq. 4. Now the equation is en- +tirely in terms of the angles between states, rather than +their intensities, and he recognizes that the form of this +expression is almost the same as that of a solid angle for- +mula. In fact, if we re-express the above relationships in +terms of states A, B, and C′ rather than A, B, and C, where +C′ is the antipodal point to C on the Poincaré sphere, we +obtain +cos(δ) = 1 − cos2(c/2) − cos2(a′/2) − cos2(b′/2) +2 cos(a′/2) cos(b′/2) cos(c/2) +, +(21) +which does have the recognizable form of a solid angle +formula. The primes indicate that the angles are to be +taken with respect to C′ rather than C. That is, b′/2 = +∠AC′, and a′/2 = ∠BC′. Making use of the solid angle +formula,[21] +δ = Ω′/2 +(22) +when Ω′ is the angle subtended by the spherical triangle +A, B, C′ from the center of the sphere. (Pancharatnam +actually expresses this as δ = π − 1 +2E′, where E′ is the +spherical excess of the triangle.) The sign of δ given here +corresponds to describing the sequence of states A → B → +C′ in a clockwise sense. If the direction of the sequence is +reversed, then the sign flips. +These last statements, in which Pancharatnam works out +the correct sign of the solid angle, is the only location in the +paper where he talks about a sequence of states. However, +it is clear from context that he is not referring to a cycle +of polarization states but rather to the phase relationships +between the two output states A and B, and the state C +from which they were decomposed. +4 +EXTENDING THE REASONING TO THE +ORTHOGONAL CASE +Considering that Pancharatnam found the phase between +two beams using intensity of interference, his definition +4 + +does not apply for the case in which states A and B are +orthogonal. However, he provides a geometric argument +to show that the formula can still be applied in the limit +as the states approach orthogonality. Figure 3a shows an +initial situation with states of polarization A and B, and +the state C obtained by their sum. This is the therefore the +inverse of the case treated in Sec. 3 above, but which is de- +scribed by (11) in Sec. 2. A point C0 lying on the geodesic +arc AB describes a state of polarization that results from +adding the beams of polarization A and B with no phase +between them. Recalling that the solid angle used by Pan- +charatnam is the one subtended by triangle ABC′ rather +than ABC, this situation gives a solid angle of Ω′ = π. +Next we modify the polarization state of B so that it moves +further away from A along the equator of the Poincaré +sphere, as shown in Fig. 3b, and then Fig. 3c. As B moves +away from A, the enclosed solid angle subtended by ABC +increases. As B approaches the point orthogonal to A (this +point is labelled by Pancharatnam as A′), the geodesic +ACB becomes half a great circle, as in Fig. 4. One might +argue that in this case the enclosed area becomes unde- +fined, since we can draw the geodesic connecting A and +B in either a clockwise or an anticlockwise sense. How- +ever, if we note that the geodesic from A to B has until +this point always passed through the intermediate point +C0, then it will for this limit case as well. The solid angle +that relates the phase between the two beams is therefore +the one enclosed between the two geodesic arcs AC0A′ +and AC′A′ (where A′ coincides with the point written as +B in Fig. 4). This is a spherical lune (drawn in green in +the figure) whose solid angle subtended from the origin +is exactly twice the value of the angle α formed between +states C0, A, and C′ at the surface of the sphere. +C' +S2 +S3 +B +S1 +0 +C +A +C +α +Figure 4: If we continue the evolution shown in Fig. 3 until +state B becomes orthogonal to A, then we form a spherical lune +(drawn in orange) between geodesic arcs AC0B and ACB. +Using (22) finally allows us to equate the phase difference +between beams A and B to the angle α: δ = α. That is, the +angle α (or half the solid angle Ω) is equal to the phase δ +that one must retard state A′ from A in order that both be +correctly decomposed from input state C. +5 +INTERFERENCE OF THE COMPONENTS +TRANSMITTED BY AN ANALYZER +It was of practical importance for Pancharatnam to have +an expression for the phase of two arbitrarily-polarized +beams transmitted through an analyzer because this was +the configuration has was using to measure the pattern +transmitted by biaxial crystals such as iolite. Finding such +an expression is the purpose of Section 8 of Pancharat- +nam’s manuscript. +Pancharatnam first defines D as the state of polarization +transmitted by the analyzer. Unfortunately, Pancharatnam +once again reuses the symbol C here. We write this as D +to avoid confusion with the states defined in the earlier +sections. He also uses I1 and I2 in place of IA and IB, but +we have retained the latter for consistency. Pancharatnam +also reuses the symbols a, b, and c for the angles between +states, but since the definition of C has changed, we in- +stead define θij to represent the angle between points i and +j on the surface of the Poincaré sphere, as subtended from +the center of the sphere. His goal is to relate the phase of +the interference transmitted by the analyzer to the phase +difference δ between the two input states (see Fig. 5). +When beams A and B are incident on the analyzer, the +transmitted intensity ID will be the component of state +A along direction D, plus the component of state B also +along D, while incorporating an unknown phase differ- +ence δ′ between A and B: +ID = IA cos2(θAD/2) + IB cos2(θBD/2) ++ +� +IAIB cos(θAD/2) cos(θBD/2) cos(δ′) . +(23) +If we use the analyzer oriented at angle D′ orthogonal +to D, then we would get a different intensity ID′ and an +unknown phase difference δ′′: +ID′ = IA sin2(θAD′/2) + IB sin2(θBD′/2) ++ +� +IAIB sin(θAD′/2) sin(θBD′/2) cos(δ′′) . (24) +From Fig. 5, we can see that the area enclosed by DAD′BD +is a spherical lune. The phase δ′ needed to generate state +D from adding A and B is given by δ′ = π − 1 +2E′, where +E′ is the solid angle subtended by the spherical triangle +ABD′ (drawn in green in the figure). In a similar fashion, +the phase δ′′ needed when adding A and B to get D′ is +given by δ′′ = π − 1 +2E′′, where E′′ is the solid angle of +ABD (drawn in orange). +Since both δ′ and δ′′ are equivalent to corresponding solid +angles, if we subtract the two, we obtain the solid angle +subtended by quadrangular area F in the sphere[22, 8] +F = ±(δ′′ − δ′) . +(25) +Confusingly, Pancharatnam yet again reuses the symbol +C to indicate this quadrangular area. We will instead use +F, as Fig. 5 does. The sign of F is determined by whether +the sequence of points ABD proceeds in a clockwise or +anticlockwise fashion. Pancharatnam then takes both ID +and ID′ and adds them together to get the total intensity, +5 + +D' +D +S2 +S1 +S3 +F +A +B +C +Figure 5: The spherical surface elements used to calculate the +intensity transmitted by an arbitrary analyzer (state D) when +two states A and B are incident upon it. +equal to the intensity of C that depends on δ but now +expressed in terms of δ′ and F: +I = IA + IB + 2 +� +IAIB +� +cos(θAD/2) cos(θBD/2) cos(δ′) ++ sin(θAD′/2) sin(θBD′/2) cos(δ′ ± F) +� +. +(26) +Now that we have the same angle δ′ in both terms inside +the square brackets, we can recognize that this is a stan- +dard expression for the spherical excess of a triangle, so +that the equation simplifies to [21] +I = IA + IB + 2 +� +IAIB cos(c/2) cos(δ′ + 1 +2E) . +(27) +This equation is now identical in form to (11), from which +we can then say that δ′ = δ − 1 +2E. Now we have a means +of calculating δ′ for a given pair of input states A and B, +together with the analyzer orientation giving D. Using +this in (23), we can now calculate the intensity transmitted +by the analyzer depending on the original phase difference +δ between the two input beams. +6 +CONCLUSION +Through an impressive set of spherical trigonometry ma- +nipulations on the Poincaré sphere, Pancharatnam found +that polarization states have specific phase relationships +that are generally not taken into account, unless one is +performing interferometric measurements. This was the +case for him, since he was analyzing the light transmitted +through dichroic biaxial crystals. The correct analysis of +these measurements required that he incorporate this new +phase, what we now refer to as the geometric phase, into +his equations. +Among his less-known results is Equation 17, which shows +that if one knows the intensities of two input beams as +well as the intensity of their interference, one can infer +the phase difference δ between the two input beams from +these simple intensity measurements alone. Also notable +is (27), which demonstrates that the phase of transmitted +by an analyzer is not in general equal to the phase of a +wave incident upon it. +The recently published “wave description of geometric +phase” interprets the geometric phase as a shift in the wave +peak location away from the midpoint between the peaks +of the two input waves.[23] One can see the close connec- +tion between the wave description and Pancharatnam’s ap- +proach by considering (11) (Pancharatnam’s Eq. 1), which +expresses the intensity produced by adding two waves +as being a bias value (I1 + I2) plus a single cosine term +with amplitude √I1I2 cos(c/2). Thus, Pancharatnam is +also considering the case of two cosine waves summing +together into a single cosine output wave. +In (14), we were also able to show how Pancharatnam’s +results are derived using the familiar modern approach, +not available to Pancharatnam, of the Jones calculus. The +Jones calculus explicitly forms a pair of 2D electric field +vectors, and adding these two waves produces a cosine +factor cos(δ) that includes the same phase delay that Pan- +charatnam obtained, but which does not seem to have +been replicated until Michael Berry’s work in 1987.[4] +Berry naturally interpreted Pancharatnam’s work through +the lens of his own work, which approached the geometric +phase as a shift resulting from the evolution of a quantum +state around a cycle in parameter space. Upon reading +Pancharatnam’s approach, he saw that this could easily be +a cycle of polarization states generating this phase shift. +Pancharatnam, however, was focused on interferometric +measurement and not on modeling the evolution of po- +larization states. As it turns out, the two are equivalent, +and so the misunderstanding is not at all a serious one. In +the literature, many authors actually refer to this shift due +to polarization state evolution as a “Pancharatnam-Berry +phase”. This seems the ideal choice, since “Pancharatnam +phase” should perhaps be more narrowly defined as a +shift resulting from adding polarized waves. +REFERENCES +1. S. Ramaseshan, “The Poincare sphere and the Pan- +charatnam phase — some historical remarks,” Current +Science 59 (1990). +2. R. Nityananda, K. Ramaseshan, N. Madhusudana, +and G. Series, “S pancharatnam (1934–1969): three +phases,” Resonance 18, 301–305 (2013). +3. S. Pancharatnam, “Generalized theory of interference +and its applications, part I. Coherent pencils,” Pro- +ceedings of the Indian Academy of Sciences—Section A 44, +398–417 (1956). +4. M. V. Berry, “The adiabatic phase and Pancharatnam’s +phase for polarized light,” J. Mod. Opt. 34, 1401–1407 +(1987). +5. P. K. Aravind, “A simple proof of pancharatnam’s +theorem,” Opt. Comm. 094, 191–196 (1992). +6. S. C. Tiwari, “Geometric phase in optics: quantal or +classical?” J. Mod. Opt. 39, 1097–1104 (1992). +6 + +7. M. Roy, P. Svahn, L. Cherel, and C. J. R. Sheppard, +“Geometric phase-shifting for low-coherence interfer- +ence microscopy,” Optics and Lasers in Engineering 37, +631–641 (2002). +8. P. Kurzynowski, W. A. Wo´zniak, and M. Szarycz, “Ge- +ometric phase: two triangles on the Poincaré sphere,” +J. Opt. Soc. Am. A 28, 475–482 (2011). +9. J. Lages, R. Giust, and J.-M. Vigoureux, “Geometric +phase and Pancharatnam phase induced by light wave +polarization,” Physica E 59, 6–14 (2014). +10. E. Cohen, H. Larocque, F. Bouchard, F. Nejadsat- +tari, Y. Gefen, and E. Karimi, “Geometric phase from +Aharonov-Bohm to Pancharatnam-Berry and beyond,” +Nature Rev. Phys. 1, 437–449 (2019). +11. O. Arteaga, “Fresnel-Arago fifth law of interference: +the first description of a geometric phase in optics,” J. +Mod. Opt. 68, 350–357 (2021). +12. M. V. Berry, “Quantal phase factors accompanying +adiabatic changes,” Proc. Roy. Soc. London A 392, 45–54 +(1984). +13. B. Simon, “Holonomy, the quantum adiabatic theo- +rem, and Berry’s phase,” Phys. Rev. Lett. 51, 2167–2170 +(1983). +14. S. Ramaseshan and R. Nityananda, “The interference +of polarized light as an early example of Berry’s phase,” +Current Science 55, 1225–1226 (1986). +15. M. Berry, “Pancharatnam, virtuoso of the Poincaré +sphere: an appreciation,” Current Science 67, 220–223 +(1994). +16. M. Berry, “Geometric phase memories,” Nature Physics +6, 148–150 (2010). +17. S. I. Vinitski˘ı, V. L. Derbov, V. M. Dubovik, B. L. +Markovski, and Y. P. Stepanovski˘ı, “Topological +phases in quantum mechanics and polarization op- +tics,” Soviet Physics Uspekhi 33, 403 (1990). +18. J. D. Jackson, “Examples of the zeroth theorem of the +history of science,” Am. J. Phys. 76, 704–719 (2008). +19. E. Collett, Field Guide to Polarization, Field Guide Series +(Society of Photo Optical, 2005). +20. L. Garza-Soto, A. De-Luna-Pamanes, I. Melendez- +Montoya, N. Sanchez-Soria, D. Gonzalez-Hernandez, +and D. Lopez-Mago, “Geometric-phase polarimetry,” +J. Optics 22, 125606–125615 (2020). +21. I. Todhunter, Spherical Trigonometry: For the Use of Col- +leges and Schools, with Numerous Examples (Macmillan, +1863). +22. J. C. Gutiérrez-Vega, “Pancharatnam-Berry phase of +optical systems,” Opt. Lett. 36, 1143–1145 (2011). +23. L. Garza-Soto, N. Hagen, D. Lopez-Mago, and +Y. Otani, “Wave description of geometric phase,” Sub- +mitted to JOSA-A, 2022. +7 + diff --git a/n9E3T4oBgHgl3EQfLAl3/content/tmp_files/load_file.txt b/n9E3T4oBgHgl3EQfLAl3/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..990ec42e301146312a7206cdbe44cbbf603f8f97 --- /dev/null +++ b/n9E3T4oBgHgl3EQfLAl3/content/tmp_files/load_file.txt @@ -0,0 +1,345 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf,len=344 +page_content='DECIPHERING PANCHARATNAM’S DISCOVERY OF GEOMETRIC PHASE PREPRINT, COMPILED JANUARY 12, 2023 Luis Garza-Soto1, Nathan Hagen1∗, and Dorilian Lopez-Mago2 1Department of Optical Engineering, Utsunomiya University, 7-1-2 Yoto, Utsunomiya, Tochigi 321-8585 Japan 2Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias Ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Eugenio Garza Sada 2501, Monterrey, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=', México, 64849 ABSTRACT While Pancharatnam discovered the geometric phase in 1956, his work was not widely recognized until its endorsement by Berry in 1987, after which it received wide appreciation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' However, because Pancharatnam’s paper is unusually difficult to follow, his work has often been misinterpreted as referring to an evolution of states of polarization, just as Berry’s work focused on a cycle of states, even though this consideration does not appear in Pancharatnam’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' We walk the reader through Pancharatnam’s original derivation and show how Pancharatnam’s approach connects to recent work in geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' It is our hope to make this widely cited classic paper more accessible and better understood.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1 HISTORICAL INTRODUCTION Sivaramakrishnan Pancharatnam was born in Calcutta in 1934 in a family of remarkable scientists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' He joined his brother S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Ramaseshan at Bangalore in the Raman Re- search Institute in 1953 when he was 19 years old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [1, 2] At that time Ramaseshan’s research supervisor was C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Ra- man, their uncle, and who received the Nobel prize in physics in 1930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Under C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Raman’s instruction, Pan- charatnam studied the behavior of absorbing bi-axial crys- tals, as he points out in the opening of his now-famous paper, “Generalized theory of interference and its applica- tions, part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Coherent pencils”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [3] This impressive work was published in 1956 when Pancharatnam was 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In it, Pancharatnam proposes a definition for two beams of dif- ferent states of polarization to be in phase: “The phase ad- vance of one polarised beam over another (not necessarily in the same state of polarization) is the amount by which its phase must be retarded relative to the second, in order that the intensity resulting from their mutual interference may be maximum.”[3] Berry named this Pancharatnam’s connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [4] After providing expressions for the phase shift — what we now refer to as the geometric phase — Pancharatnam goes on to relate the expressions to the subtended solid angle on the Poincaré sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In Sections 2 & 3 below, we retrace Pancharatnam’s derivation using a more mod- ern approach that should be easier for modern readers to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Along the way, it becomes clear that Pancharat- nam nowhere considers the case that is widely attributed to him, that the phase of polarized light changes after a cyclic evolution of its polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [4, 5, 6, 7, 8, 9, 10, 11] This misconception of his actual work appears to be a result of the difficulty one encounters when reading Pan- charatnam’s paper, which often feels like it belongs to the 19th century, and also of confounding Pancharatnam’s work with the closely related work of Berry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Almost 30 years after Pancharatnam’s original work, Michael Berry at the University of Bristol, unaware of Pan- charatnam’s work, discovered that an unexpected phase emerges after the adiabatic evolution of a quantum state around a cycle in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [12] In 1983, before his initial paper was published, Berry introduced the geo- metric phase to Barry Simon, who immediately coined it as Berry’s phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [13] By the end of 1986 Ramaseshan and Nityananda revived Pancharatnam’s work and presented it as an example of Berry’s phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [14] Berry himself re- ceived Nityananda’s manuscript and read it, but mentions that it was not until he visited Bangalore in July 1987 that he came to appreciate Pancharatnam’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' One can sur- mise that Nityananda’s interpretation of Pancharatnam’s phase was likely greatly influenced by Berry’s personal approach to the geometric phase via cycles of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This seems likely because Berry, on his return from India, pre- pared a new manuscript that revealed the connections be- tween his and Pancharatnam’s work, in which he explains Pancharatnam’s work in terms of cycles of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [15, 16] Subsequent researchers have almost invariably followed this interpretation, with the result that Pancharatnam’s ac- tual achievement is obscured beneath the misconception.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' As Berry himself pointed out, there have been several anticipations to geometric phase that arise before Pan- charatnam’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [16] Vinitskii et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=', for example, men- tion the work of Rytov (1938) and Vladimirskii (1941) as precursors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [17] Oriol Arteaga has also pointed out that Fresnel and Arago in 1816 developed their “fifth” law of optical interference in such a way that a geometric phase term (at the time not well understood) had to be included in the interference equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [11] However, as is commonly the case in science, every discovery can be traced to its anticipations, and we focus on Pancharat- nam because his achievement is widely recognized by the scientific community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [18] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='04359v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='optics] 11 Jan 2023 S2 S3 A′ S1 A C B b Figure 1: Polarization state C with respect to the two orthogonal states A and A′, represented on the Poincaré sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Point B represents an SOP lying between A and A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 2 PANCHARATNAM’S STARTING POINT At the beginning of his manuscript, after explaining briefly some of the properties of the Poincaré sphere and Stokes parameters, Pancharatnam introduces the following theo- rem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' When [an electromagnetic] vi- bration of intensity I in the state of po- larisation C is decomposed into two vi- brations in the opposite states of polar- isation A and A′, the intensities of the A-component and the A′-component are I cos2(AC/2) and I sin2(AC/2) respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1 for an illustration of the geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=') Here Pan- charatnam makes use of the fact that the angle ∠A′C, be- tween points A′ and C, subtended from the center of the Poincaré sphere, is complementary to angle ∠AC, and so writes both components in terms of ∠AC alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Since we will be making use of these angles in many of the equations below, we follow Pancharatnam and define the individual angles a, b, and c as c/2 = ∠AB b/2 = ∠AC a/2 = ∠BC � � � (1) Note that these angles a, b, and c on the left hand side of the equations are defined in terms of the electric fields in Cartesian space, whereas the arcs on the right hand side are defined in Poincaré space, hence the division by two in each definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Since this theorem is the basis for much of Pancharatnam’s subsequent results, we show how one can derive it using a modern approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Let us consider a monochromatic electromagnetic wave propagating along the z axis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=', E = EC exp[i(kz − ωt)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The electric field amplitude EC can be written using elliptical polarization states {ˆeA, ˆeA′} as a basis, with the properties |ˆeA,A′| = 1, and ˆeA · ˆe∗ A′ = 0 (where ∗ represents the complex conjugate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Therefore, EC = E cos(α)ˆeA + E exp(iβ) sin(α)ˆeA′, (2) where E is a real amplitude (thus, I = E2), α controls the projection over the basis vectors, and β is the relative phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' With the above equation, we can represent any state of polarization (SOP) by modifying α and β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' For the common definition of the Stokes parameters, we would choose ˆeA = ˆx and ˆeA′ = ˆy, so that S0 = |Ex|2 + |Ey|2 , S1 = |Ex|2 − |Ey|2 , S2 = 2Re[ExE∗ y] , S3 = 2Im[ExE∗ y] , � � � � � � � � � � � (3) However, this is only one specific choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' An equally valid choice is the following general form of the Stokes parameters[19] S0 = |EA|2 + |EA′|2 , S1 = |EA|2 − |EA′|2 , S2 = 2Re[EAE∗ A′] , S3 = 2Im[EAE∗ A′] , � � � � � � � � � � � (4) where Ej, with j ≡ A, A′, are the components of EC, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=', Ej = EC · ˆej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Let us consider C = {S1, S2, S3} as the Stokes vector representing the SOP of EC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' C lies on the surface of the Poincaré sphere of radius S0 with main axes {S1, S2, S3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' These definitions are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1, where the Si are with respect to a general elliptical basis and need not be with respect to the x-y basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' By using (2), the Stokes parameters for EC are S0 = E2 = I , S1 = I cos 2α , S2 = I sin 2α cos β , S3 = −I sin 2α sin β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' � � � � � � � � � (5) Here, 2α and β are the polar and azimuthal angles, re- spectively, in a spherical coordinate system with S1 being the polar axis, and {S2, S3} the equatorial plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Hence, if α = [0, π/2], and β = [0, 2π], we can cover the entire surface of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' From (5), we see that the Stokes vector of ˆeA (shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1 as A) is equal to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Therefore, 2α is the angle in Poincaré space that the SOP of EC makes with the basis vector ˆeA (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Moreover, if we can define ∠AC = 2α, we can use (2) to conclude that the intensity of the ˆeA- component is I cos2(∠AC/2) and the intensity of the ˆeA′- component is I sin2(∠AC/2), which is Pancharatnam’s Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Pancharatnam next considers the problem of explaining interference between two non-orthogonal states of polar- ization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' It was already well known that the interference of two beams of the same state of polarization (SOP) results in an intensity I(I1, I2, δ) = I1 + I2 + 2 � I1I2 cos(δ) , (6) 2 where I1 and I2 are the individual intensities, and δ is the phase difference between the two beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' We can see that changing I1 or I2 only modifies the modulation contrast (visibility), but not the phase of the sum wave I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [20] In order to see what happens when two beams of different SOP are combined, Pancharatnam considers the superpo- sition of beams in states A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' He considers one of them, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=', A, and its orthogonal A′, as the polarization basis, and finds the intensity of the sum by applying the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Coherently adding A to the A-component of B, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Incoherently adding the A′-component of B to the intensity obtained in step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' That is, the projection of A onto B gives the component of A that directly interferes with A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The component of A orthogonal to B does not interfere, and so only adds a bias to the intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Considering Pancharatnam’s Theorem 1, we therefore de- compose B into two beams of orthogonal polarizations, yielding the intensities IB,A = IB cos2(c/2) , (7) IB,A′ = IB sin2(c/2) , (8) where IB is the intensity of B, and c is the angle between the two states on the Poincaré sphere, and is thus twice the angle between the states in Cartesian space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' For step 1, we coherently combine IA (the intensity of A) with IB,A using (6), giving I(IA, IB,A, δ) = IA + IB cos2(c/2) + 2 � IAIB cos2(c/2) �1/2 cos(δ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (9) This expression represents the intensity that results from the addition of the A-component of both beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Finally, with step 2, we add IB,A′ to obtain the intensity I = IA + IB cos2(c/2) + IB sin2(c/2) + 2 � IAIB cos2(c/2) �1/2 cos(δ) , (10) which simplifies to I = IA + IB + 2 � IAIB cos(c/2) cos(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (11) The above equation is Pancharatnam’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (1) in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This expression has the advantage that the factor contain- ing the phase δ between the beams and cos(c) — what Pancharatnam calls their “similarity factor” — are sepa- rated as a product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Equation 11 is a useful result because it makes two prop- erties clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' First, while δ is the phase delay between input beams A and B, the equation shows that the intensity of the beams’ superposition varies sinusoidally, with δ as the phase of I, independent of what polarization states are in- volved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Second, if either of the two intensities IA or IB changes, this changes I but does not change the value of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Here we see that c determines the fringe visibility of the interferogram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Therefore, this 0 2 3 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='0 interferogram intensity attenuated wave original wave interferogram OPD, δ (radians) Figure 2: The “original wave” interferogram is obtained in (11) when the two intensities IA and IB are equal and held fixed as we vary the relative phase δ between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The “attenuated wave” occurs when we reduce the intensity of one beam, and again vary the phase between the two beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' one of the results that Pancharatnam has in mind when he titles his paper the “generalized theory of interference”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' To compare with more modern approaches, we can ana- lyze the same case using the Jones calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In order to simplify the math, we will simply define the state of the first beam to be horizontal, while that of the second beam is oriented at an angle of θ with respect to the horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' We also allow there to be a propagation phase delay δ be- tween the two beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Therefore, we have the two electric field vectors EA = |EA| ˆx, and EB = |EB|e−iδ (cos θ ˆx + sin θ ˆy) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (12) The amplitude of their sum is EA + EB = � |EA| + |EB| cos θ e−iδ� ˆx + |EB| sin θ e−iδ ˆy, (13) so that intensity of the two beams’ interference is I = � EA + EB � · � EA + EB �∗ = �|EA| + |EB| cos θ e+iδ��|EA| + |EB| cos θ e−iδ� + �|EB| sin θ e+iδ��|EB| sin θ e−iδ� = |EA|2 + |EB|2 + 2|EA||EB| cos θ cos δ , (14) which is exactly Pancharatnam’s expression [given as (11) above], since c = 2θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This can be generalized to an arbitrary pair of elliptical states, considering two waves with electric field ampli- tudes EA and EB, and a phase difference δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The intensity of their superposition is I = (EA + eiδEB) · (E∗ A + e−iδE∗ B) = |EA|2 + |EB|2 + 2Re � e−iδ(EA · E∗ B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (15) The result of EA · E∗ B can be inferred from the discus- sion about Pancharatnam’s Theorem 1, which gives EAEB cos(c/2), where EA, EB are the real amplitudes of the electric fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Therefore, we conclude that I = |EA|2 + |EB|2 + 2EAEB cos(c/2) cos(δ) = IA + IB + 2 � IAIB cos(c/2) cos(δ), (16) which agrees with Pancharatnam’s result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3 S2 S1 S3 0 A C B C S S2 S1 3 B 0 A C C S2 S1 S3 0 A C B C (a) (b) (c) Figure 3: The increasing area (drawn in orange) on the surface of the sphere, formed by the spherical triangle between points A, B, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3 THE RELATIONSHIP BETWEEN INTENSITIES AND LOCATIONS ON THE POINCARÉ SPHERE Following his expression for the interference of nonorthog- onal beams, Pancharatnam requires only a few short steps (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 4 of his paper) to develop his famous solid angle formula for the geometric phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Here he considers the de- composition of a beam of state of polarization C into two nonorthogonal beams of polarization states A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' If (11) is a general equation for the interference of two beams with different SOPs, then one should be able to re-arrange the equation to solve for the phase delay δ between them via the intensities of A and B (but which Pancharatnam labels I1 and I2): cos(δ) = I − (IA + IB) 2 √IAIB cos(c/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (17) We can interpret this result as saying that “whatever this phase δ may be, it maintains a specific relationship be- tween the intensities of the beam being decomposed (C) and the intensities of the beams that result from the de- composition (A and B).” While he could have finished with this numerical formula for δ, he went a step further and realized that this equation expresses a solid angle relationship between the SOPs of the three beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' To explain this he uses electric field vectors EA and EB as components of a sum vector EC = EA + EB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' From vector analysis, it is easy to see that the part of EB perpendicular to EA also has to be equal to the part of EC perpendicular to EA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Writing this in terms of the angles a, b, and c from (1), we have � IB sin(c/2) = √ I sin(b/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (18) Taking the square of the components to find the intensities results in the proportions IA = I sin2(a/2) sin2(c/2) and IB = I sin2(b/2) sin2(c/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (19) By expressing the intensities IA and IB in terms of the total intensity I, together with the angles between states on the Poincaré sphere, we can rewrite (17) as cos(δ) = sin2(c/2) − sin2(a/2) − sin2(b/2) 2 sin(a/2) sin(b/2) cos(c/2) , (20) which is Pancharatnam’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Now the equation is en- tirely in terms of the angles between states, rather than their intensities, and he recognizes that the form of this expression is almost the same as that of a solid angle for- mula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In fact, if we re-express the above relationships in terms of states A, B, and C′ rather than A, B, and C, where C′ is the antipodal point to C on the Poincaré sphere, we obtain cos(δ) = 1 − cos2(c/2) − cos2(a′/2) − cos2(b′/2) 2 cos(a′/2) cos(b′/2) cos(c/2) , (21) which does have the recognizable form of a solid angle formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The primes indicate that the angles are to be taken with respect to C′ rather than C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' That is, b′/2 = ∠AC′, and a′/2 = ∠BC′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Making use of the solid angle formula,[21] δ = Ω′/2 (22) when Ω′ is the angle subtended by the spherical triangle A, B, C′ from the center of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (Pancharatnam actually expresses this as δ = π − 1 2E′, where E′ is the spherical excess of the triangle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=') The sign of δ given here corresponds to describing the sequence of states A → B → C′ in a clockwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' If the direction of the sequence is reversed, then the sign flips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' These last statements, in which Pancharatnam works out the correct sign of the solid angle, is the only location in the paper where he talks about a sequence of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' However, it is clear from context that he is not referring to a cycle of polarization states but rather to the phase relationships between the two output states A and B, and the state C from which they were decomposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 4 EXTENDING THE REASONING TO THE ORTHOGONAL CASE Considering that Pancharatnam found the phase between two beams using intensity of interference, his definition 4 does not apply for the case in which states A and B are orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' However, he provides a geometric argument to show that the formula can still be applied in the limit as the states approach orthogonality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Figure 3a shows an initial situation with states of polarization A and B, and the state C obtained by their sum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This is the therefore the inverse of the case treated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3 above, but which is de- scribed by (11) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' A point C0 lying on the geodesic arc AB describes a state of polarization that results from adding the beams of polarization A and B with no phase between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Recalling that the solid angle used by Pan- charatnam is the one subtended by triangle ABC′ rather than ABC, this situation gives a solid angle of Ω′ = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Next we modify the polarization state of B so that it moves further away from A along the equator of the Poincaré sphere, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3b, and then Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' As B moves away from A, the enclosed solid angle subtended by ABC increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' As B approaches the point orthogonal to A (this point is labelled by Pancharatnam as A′), the geodesic ACB becomes half a great circle, as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' One might argue that in this case the enclosed area becomes unde- fined, since we can draw the geodesic connecting A and B in either a clockwise or an anticlockwise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' How- ever, if we note that the geodesic from A to B has until this point always passed through the intermediate point C0, then it will for this limit case as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The solid angle that relates the phase between the two beams is therefore the one enclosed between the two geodesic arcs AC0A′ and AC′A′ (where A′ coincides with the point written as B in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This is a spherical lune (drawn in green in the figure) whose solid angle subtended from the origin is exactly twice the value of the angle α formed between states C0, A, and C′ at the surface of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=" C' S2 S3 B S1 0 C A C α Figure 4: If we continue the evolution shown in Fig." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3 until state B becomes orthogonal to A, then we form a spherical lune (drawn in orange) between geodesic arcs AC0B and ACB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Using (22) finally allows us to equate the phase difference between beams A and B to the angle α: δ = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' That is, the angle α (or half the solid angle Ω) is equal to the phase δ that one must retard state A′ from A in order that both be correctly decomposed from input state C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 5 INTERFERENCE OF THE COMPONENTS TRANSMITTED BY AN ANALYZER It was of practical importance for Pancharatnam to have an expression for the phase of two arbitrarily-polarized beams transmitted through an analyzer because this was the configuration has was using to measure the pattern transmitted by biaxial crystals such as iolite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Finding such an expression is the purpose of Section 8 of Pancharat- nam’s manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Pancharatnam first defines D as the state of polarization transmitted by the analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Unfortunately, Pancharatnam once again reuses the symbol C here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' We write this as D to avoid confusion with the states defined in the earlier sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' He also uses I1 and I2 in place of IA and IB, but we have retained the latter for consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Pancharatnam also reuses the symbols a, b, and c for the angles between states, but since the definition of C has changed, we in- stead define θij to represent the angle between points i and j on the surface of the Poincaré sphere, as subtended from the center of the sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' His goal is to relate the phase of the interference transmitted by the analyzer to the phase difference δ between the two input states (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' When beams A and B are incident on the analyzer, the transmitted intensity ID will be the component of state A along direction D, plus the component of state B also along D, while incorporating an unknown phase differ- ence δ′ between A and B: ID = IA cos2(θAD/2) + IB cos2(θBD/2) + � IAIB cos(θAD/2) cos(θBD/2) cos(δ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (23) If we use the analyzer oriented at angle D′ orthogonal to D, then we would get a different intensity ID′ and an unknown phase difference δ′′: ID′ = IA sin2(θAD′/2) + IB sin2(θBD′/2) + � IAIB sin(θAD′/2) sin(θBD′/2) cos(δ′′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (24) From Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 5, we can see that the area enclosed by DAD′BD is a spherical lune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The phase δ′ needed to generate state D from adding A and B is given by δ′ = π − 1 2E′, where E′ is the solid angle subtended by the spherical triangle ABD′ (drawn in green in the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In a similar fashion, the phase δ′′ needed when adding A and B to get D′ is given by δ′′ = π − 1 2E′′, where E′′ is the solid angle of ABD (drawn in orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Since both δ′ and δ′′ are equivalent to corresponding solid angles, if we subtract the two, we obtain the solid angle subtended by quadrangular area F in the sphere[22, 8] F = ±(δ′′ − δ′) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (25) Confusingly, Pancharatnam yet again reuses the symbol C to indicate this quadrangular area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' We will instead use F, as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 5 does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The sign of F is determined by whether the sequence of points ABD proceeds in a clockwise or anticlockwise fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=" Pancharatnam then takes both ID and ID′ and adds them together to get the total intensity, 5 D' D S2 S1 S3 F A B C Figure 5: The spherical surface elements used to calculate the intensity transmitted by an arbitrary analyzer (state D) when two states A and B are incident upon it." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' equal to the intensity of C that depends on δ but now expressed in terms of δ′ and F: I = IA + IB + 2 � IAIB � cos(θAD/2) cos(θBD/2) cos(δ′) + sin(θAD′/2) sin(θBD′/2) cos(δ′ ± F) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (26) Now that we have the same angle δ′ in both terms inside the square brackets, we can recognize that this is a stan- dard expression for the spherical excess of a triangle, so that the equation simplifies to [21] I = IA + IB + 2 � IAIB cos(c/2) cos(δ′ + 1 2E) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' (27) This equation is now identical in form to (11), from which we can then say that δ′ = δ − 1 2E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Now we have a means of calculating δ′ for a given pair of input states A and B, together with the analyzer orientation giving D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Using this in (23), we can now calculate the intensity transmitted by the analyzer depending on the original phase difference δ between the two input beams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 6 CONCLUSION Through an impressive set of spherical trigonometry ma- nipulations on the Poincaré sphere, Pancharatnam found that polarization states have specific phase relationships that are generally not taken into account, unless one is performing interferometric measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This was the case for him, since he was analyzing the light transmitted through dichroic biaxial crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The correct analysis of these measurements required that he incorporate this new phase, what we now refer to as the geometric phase, into his equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Among his less-known results is Equation 17, which shows that if one knows the intensities of two input beams as well as the intensity of their interference, one can infer the phase difference δ between the two input beams from these simple intensity measurements alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Also notable is (27), which demonstrates that the phase of transmitted by an analyzer is not in general equal to the phase of a wave incident upon it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The recently published “wave description of geometric phase” interprets the geometric phase as a shift in the wave peak location away from the midpoint between the peaks of the two input waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [23] One can see the close connec- tion between the wave description and Pancharatnam’s ap- proach by considering (11) (Pancharatnam’s Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1), which expresses the intensity produced by adding two waves as being a bias value (I1 + I2) plus a single cosine term with amplitude √I1I2 cos(c/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Thus, Pancharatnam is also considering the case of two cosine waves summing together into a single cosine output wave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In (14), we were also able to show how Pancharatnam’s results are derived using the familiar modern approach, not available to Pancharatnam, of the Jones calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' The Jones calculus explicitly forms a pair of 2D electric field vectors, and adding these two waves produces a cosine factor cos(δ) that includes the same phase delay that Pan- charatnam obtained, but which does not seem to have been replicated until Michael Berry’s work in 1987.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' [4] Berry naturally interpreted Pancharatnam’s work through the lens of his own work, which approached the geometric phase as a shift resulting from the evolution of a quantum state around a cycle in parameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Upon reading Pancharatnam’s approach, he saw that this could easily be a cycle of polarization states generating this phase shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Pancharatnam, however, was focused on interferometric measurement and not on modeling the evolution of po- larization states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' As it turns out, the two are equivalent, and so the misunderstanding is not at all a serious one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' In the literature, many authors actually refer to this shift due to polarization state evolution as a “Pancharatnam-Berry phase”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' This seems the ideal choice, since “Pancharatnam phase” should perhaps be more narrowly defined as a shift resulting from adding polarized waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' REFERENCES 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Ramaseshan, “The Poincare sphere and the Pan- charatnam phase — some historical remarks,” Current Science 59 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Nityananda, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Ramaseshan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Madhusudana, and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Series, “S pancharatnam (1934–1969): three phases,” Resonance 18, 301–305 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Pancharatnam, “Generalized theory of interference and its applications, part I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Coherent pencils,” Pro- ceedings of the Indian Academy of Sciences—Section A 44, 398–417 (1956).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Berry, “The adiabatic phase and Pancharatnam’s phase for polarized light,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 34, 1401–1407 (1987).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Aravind, “A simple proof of pancharatnam’s theorem,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Comm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 094, 191–196 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Tiwari, “Geometric phase in optics: quantal or classical?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 39, 1097–1104 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Roy, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Svahn, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Cherel, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Sheppard, “Geometric phase-shifting for low-coherence interfer- ence microscopy,” Optics and Lasers in Engineering 37, 631–641 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Kurzynowski, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Wo´zniak, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Szarycz, “Ge- ometric phase: two triangles on the Poincaré sphere,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' A 28, 475–482 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Lages, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Giust, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content='-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Vigoureux, “Geometric phase and Pancharatnam phase induced by light wave polarization,” Physica E 59, 6–14 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Cohen, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Larocque, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Bouchard, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Nejadsat- tari, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Gefen, and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Karimi, “Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond,” Nature Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 1, 437–449 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Arteaga, “Fresnel-Arago fifth law of interference: the first description of a geometric phase in optics,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 68, 350–357 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Berry, “Quantal phase factors accompanying adiabatic changes,” Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' London A 392, 45–54 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Simon, “Holonomy, the quantum adiabatic theo- rem, and Berry’s phase,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 51, 2167–2170 (1983).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Ramaseshan and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Nityananda, “The interference of polarized light as an early example of Berry’s phase,” Current Science 55, 1225–1226 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Berry, “Pancharatnam, virtuoso of the Poincaré sphere: an appreciation,” Current Science 67, 220–223 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Berry, “Geometric phase memories,” Nature Physics 6, 148–150 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Vinitski˘ı, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Derbov, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Dubovik, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Markovski, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Stepanovski˘ı, “Topological phases in quantum mechanics and polarization op- tics,” Soviet Physics Uspekhi 33, 403 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Jackson, “Examples of the zeroth theorem of the history of science,” Am.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 76, 704–719 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Collett, Field Guide to Polarization, Field Guide Series (Society of Photo Optical, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Garza-Soto, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' De-Luna-Pamanes, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Melendez- Montoya, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Sanchez-Soria, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Gonzalez-Hernandez, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Lopez-Mago, “Geometric-phase polarimetry,” J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Optics 22, 125606–125615 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Todhunter, Spherical Trigonometry: For the Use of Col- leges and Schools, with Numerous Examples (Macmillan, 1863).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Gutiérrez-Vega, “Pancharatnam-Berry phase of optical systems,” Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 36, 1143–1145 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Garza-Soto, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Hagen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Lopez-Mago, and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' Otani, “Wave description of geometric phase,” Sub- mitted to JOSA-A, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} +page_content=' 7' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/n9E3T4oBgHgl3EQfLAl3/content/2301.04359v1.pdf'} diff --git a/o9AzT4oBgHgl3EQfAfpB/vector_store/index.pkl b/o9AzT4oBgHgl3EQfAfpB/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..c92c083a67a887db799730244fe1e21269950b7b --- /dev/null +++ b/o9AzT4oBgHgl3EQfAfpB/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a4412ded5029383c5d6ce19224a21fcac86a4f78f653a2b1c1976c4756942c3d +size 87580 diff --git a/o9FPT4oBgHgl3EQfLTSh/vector_store/index.faiss b/o9FPT4oBgHgl3EQfLTSh/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..775f5cd82abf586dbf5655594e559906a8e45c59 --- /dev/null +++ b/o9FPT4oBgHgl3EQfLTSh/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:49dfaa5349968c339e0a0818437698825b410592abcdb700f869c649dc5850f9 +size 9502765 diff --git a/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf b/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..e16dd2cded50bfa4874accf7c9be253fcd35b626 --- /dev/null +++ b/otE1T4oBgHgl3EQf1wXJ/content/2301.03472v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:dfbbf6f70bd85f2f7f479b1abb2c4be0aae0f012d968c883a499937f1807af8e +size 1049296 diff --git a/otE1T4oBgHgl3EQf1wXJ/vector_store/index.pkl b/otE1T4oBgHgl3EQf1wXJ/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..685a90442fef6eda73d47d030440ea3b6f11a0e1 --- /dev/null +++ b/otE1T4oBgHgl3EQf1wXJ/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:3d4db6719c8a82928e72f8179965d64905efaec338261412e60ab64142e7db68 +size 161349 diff --git a/pNE2T4oBgHgl3EQfKQZI/content/tmp_files/2301.03700v1.pdf.txt b/pNE2T4oBgHgl3EQfKQZI/content/tmp_files/2301.03700v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee58a82c6a78cf052531427e706a3ee7e1c05e8f --- /dev/null +++ b/pNE2T4oBgHgl3EQfKQZI/content/tmp_files/2301.03700v1.pdf.txt @@ -0,0 +1,16688 @@ +Multi-Differential Cross Section Measurements of νµ-Argon +Quasielastic-like Reactions with the MicroBooNE Detector +P. Abratenko,35 O. Alterkait,35 D. Andrade Aldana,15 J. Anthony,5 L. Arellano,20 J. Asaadi,34 A. Ashkenazi,32 +S. Balasubramanian,12 B. Baller,12 G. Barr,25 J. Barrow,21, 32 V. Basque,12 O. Benevides Rodrigues,31 +S. Berkman,12 A. Bhanderi,20 M. Bhattacharya,12 M. Bishai,3 A. Blake,17 B. Bogart,22 T. Bolton,16 J. Y. Book,14 +L. Camilleri,10 D. Caratelli,4 I. Caro Terrazas,9 F. Cavanna,12 G. Cerati,12 Y. Chen,28 J. M. Conrad,21 +M. Convery,28 L. Cooper-Troendle,39 J. I. Crespo-Anad´on,6 M. Del Tutto,12 S. R. Dennis,5 P. Detje,5 A. Devitt,17 +R. Diurba,2 Z. Djurcic,1 R. Dorrill,15 K. Duffy,25 S. Dytman,26 B. Eberly,30 A. Ereditato,2 J. J. Evans,20 +R. Fine,18 O. G. Finnerud,20 W. Foreman,15 B. T. Fleming,39 N. Foppiani,14 D. Franco,39 A. P. Furmanski,23 +D. Garcia-Gamez,13 S. Gardiner,12 G. Ge,10 S. Gollapinni,33, 18 O. Goodwin,20 E. Gramellini,12 P. Green,20, 25 +H. Greenlee,12 W. Gu,3 R. Guenette,20 P. Guzowski,20 L. Hagaman,39 O. Hen,21 R. Hicks,18 C. Hilgenberg,23 +G. A. Horton-Smith,16 B. Irwin,23 R. Itay,28 C. James,12 X. Ji,3 L. Jiang,37 J. H. Jo,3, 39 R. A. Johnson,8 +Y.-J. Jwa,10 D. Kalra,10 N. Kamp,21 G. Karagiorgi,10 W. Ketchum,12 M. Kirby,12 T. Kobilarcik,12 I. Kreslo,2 +M. B. Leibovitch,4 I. Lepetic,27 J.-Y. Li,11 K. Li,39 Y. Li,3 K. Lin,27 B. R. Littlejohn,15 W. C. Louis,18 +X. Luo,4 C. Mariani,37 D. Marsden,20 J. Marshall,38 N. Martinez,16 D. A. Martinez Caicedo,29 K. Mason,35 +A. Mastbaum,27 N. McConkey,20, 36 V. Meddage,16 K. Miller,7 J. Mills,35 A. Mogan,9 T. Mohayai,12 +M. Mooney,9 A. F. Moor,5 C. D. Moore,12 L. Mora Lepin,20 J. Mousseau,22 S. Mulleriababu,2 D. Naples,26 +A. Navrer-Agasson,20 N. Nayak,3 M. Nebot-Guinot,11 J. Nowak,17 N. Oza,10, 18 O. Palamara,12 N. Pallat,23 +V. Paolone,26 A. Papadopoulou,1, 21 V. Papavassiliou,24 H. B. Parkinson,11 S. F. Pate,24 N. Patel,17 Z. Pavlovic,12 +E. Piasetzky,32 I. D. Ponce-Pinto,39 I. Pophale,17 S. Prince,14 X. Qian,3 J. L. Raaf,12 V. Radeka,3 A. Rafique,1 +M. Reggiani-Guzzo,20 L. Ren,24 L. Rochester,28 J. Rodriguez Rondon,29 M. Rosenberg,35 M. Ross-Lonergan,18 +C. Rudolf von Rohr,2 G. Scanavini,39 D. W. Schmitz,7 A. Schukraft,12 W. Seligman,10 M. H. Shaevitz,10 +R. Sharankova,12 J. Shi,5 E. L. Snider,12 M. Soderberg,31 S. S¨oldner-Rembold,20 J. Spitz,22 M. Stancari,12 +J. St. John,12 T. Strauss,12 S. Sword-Fehlberg,24 A. M. Szelc,11 W. Tang,33 N. Taniuchi,5 K. Terao,28 C. Thorpe,17 +D. Torbunov,3 D. Totani,4 M. Toups,12 Y.-T. Tsai,28 J. Tyler,16 M. A. Uchida,5 T. Usher,28 B. Viren,3 M. Weber,2 +H. Wei,19 A. J. White,39 Z. Williams,34 S. Wolbers,12 T. Wongjirad,35 M. Wospakrik,12 K. Wresilo,5 N. Wright,21 +W. Wu,12 E. Yandel,4 T. Yang,12 L. E. Yates,12 H. W. Yu,3 G. P. Zeller,12 J. Zennamo,12 and C. Zhang3 +(The MicroBooNE Collaboration)∗ +1Argonne National Laboratory (ANL), Lemont, IL, 60439, USA +2Universit¨at Bern, Bern CH-3012, Switzerland +3Brookhaven National Laboratory (BNL), Upton, NY, 11973, USA +4University of California, Santa Barbara, CA, 93106, USA +5University of Cambridge, Cambridge CB3 0HE, United Kingdom +6Centro de Investigaciones Energ´eticas, Medioambientales y Tecnol´ogicas (CIEMAT), Madrid E-28040, Spain +7University of Chicago, Chicago, IL, 60637, USA +8University of Cincinnati, Cincinnati, OH, 45221, USA +9Colorado State University, Fort Collins, CO, 80523, USA +10Columbia University, New York, NY, 10027, USA +11University of Edinburgh, Edinburgh EH9 3FD, United Kingdom +12Fermi National Accelerator Laboratory (FNAL), Batavia, IL 60510, USA +13Universidad de Granada, Granada E-18071, Spain +14Harvard University, Cambridge, MA 02138, USA +15Illinois Institute of Technology (IIT), Chicago, IL 60616, USA +16Kansas State University (KSU), Manhattan, KS, 66506, USA +17Lancaster University, Lancaster LA1 4YW, United Kingdom +18Los Alamos National Laboratory (LANL), Los Alamos, NM, 87545, USA +19Louisiana State University, Baton Rouge, LA, 70803, USA +20The University of Manchester, Manchester M13 9PL, United Kingdom +21Massachusetts Institute of Technology (MIT), Cambridge, MA, 02139, USA +22University of Michigan, Ann Arbor, MI, 48109, USA +23University of Minnesota, Minneapolis, MN, 55455, USA +24New Mexico State University (NMSU), Las Cruces, NM, 88003, USA +25University of Oxford, Oxford OX1 3RH, United Kingdom +26University of Pittsburgh, Pittsburgh, PA, 15260, USA +27Rutgers University, Piscataway, NJ, 08854, USA +28SLAC National Accelerator Laboratory, Menlo Park, CA, 94025, USA +29South Dakota School of Mines and Technology (SDSMT), Rapid City, SD, 57701, USA +arXiv:2301.03700v1 [hep-ex] 9 Jan 2023 + +2 +30University of Southern Maine, Portland, ME, 04104, USA +31Syracuse University, Syracuse, NY, 13244, USA +32Tel Aviv University, Tel Aviv, Israel, 69978 +33University of Tennessee, Knoxville, TN, 37996, USA +34University of Texas, Arlington, TX, 76019, USA +35Tufts University, Medford, MA, 02155, USA +36University College London, London WC1E 6BT, United Kingdom +37Center for Neutrino Physics, Virginia Tech, Blacksburg, VA, 24061, USA +38University of Warwick, Coventry CV4 7AL, United Kingdom +39Wright Laboratory, Department of Physics, Yale University, New Haven, CT, 06520, USA +(Dated: January 11, 2023) +We report on a flux-integrated multi-differential measurement of charged-current muon neutrino +scattering on argon with one muon and one proton in the final state using the Booster Neutrino +Beam and MicroBooNE detector at Fermi National Accelerator Laboratory. The data are studied +as a function of various kinematic imbalance variables and of a neutrino energy estimator, and are +compared to a number of event generator predictions. We find that the measured cross sections in +different phase-space regions are sensitive to nuclear effects. Our results provide precision data to test +and improve the neutrino-nucleus interaction models needed to perform high-accuracy oscillation +analyses. Specific regions of phase-space are identified where further model refinements are most +needed. +I. +INTRODUCTION +High-precision measurements of the neutrino mix- +ing angles, mass differences, and charge-parity violat- +ing phase, and the search for physics beyond the Stan- +dard Model are the primary physics goals of many cur- +rently operating as well as next-generation neutrino ex- +periments [1–6]. +These measurements require reliable +comparisons of measured and theoretically-expected neu- +trino interaction rates in the corresponding detectors. +Thus, understanding the neutrino-nucleus scattering pro- +cesses in detail is a prerequisite for these experiments to +reach their discovery potential. +A number of neutrino +oscillation experiments employ liquid argon time projec- +tion chambers (LArTPCs) [3–5, 7–9] to detect the par- +ticles produced in neutrino interactions. +The ultimate +goal of these efforts is both to reconstruct the energy +of the neutrino based on the kinematics of the outgo- +ing particles and to enable few-percent-level modeling of +neutrino-argon interaction rates [10]. +Therefore, high- +accuracy modeling of neutrino-argon interactions is of +the utmost importance [11–13]. +This work presents the first measurement of flux- +integrated single- and double-differential cross sections +for muon-neutrino-argon (νµ-Ar) charged-current (CC) +quasielastic (QE)-like scattering reactions as a func- +tion of kinematic imbalance variables [14–18]. Double- +differential measurements as a function of a neutrino en- +ergy estimator are further reported for the first time in +kinematic imbalance bins on argon. Motivated by a pre- +vious analysis with a similar signal event topology [19], +we focus on reactions where a single muon-proton pair is +reconstructed with no additional detected particles. The +results reported here use the MicroBooNE detector [20] +∗ microboone info@fnal.gov +with an exposure of 6.79 × 1020 protons on target from +the Booster Neutrino Beam (BNB) [21] at Fermi National +Accelerator Laboratory. +The experimental setup is presented in Sec. II, followed +by the signal definition and event selection in Sec. III. +The observables of interest are defined in Sec. IV. Sec- +tion V describes the cross section extraction and system- +atics procedure and Sec. VI outlines the modeling config- +urations used for comparison to the data. The results are +reported in Sec. VII and the conclusions are discussed in +Sec. VIII. +II. +EXPERIMENTAL SETUP +The MicroBooNE LArTPC has an active volume that +contains 85 tonnes of argon. It is exposed to BNB neutri- +nos, with an energy spectrum that peaks around 0.8 GeV +and extends to 2 GeV. +Charged particles are produced after the primary neu- +trino interaction with the argon nuclei in the LArTPC +active volume. Scintillation light and electron ionization +trails are produced while these charged particles travel +through the liquid argon. In the presence of an electric +field of 273 V/cm, the ionization electrons drift towards a +system of three anode wire planes. Photomultiplier tubes +(PMTs) are used to measure the scintillation light. +If the PMT signals are in time coincidence with the +beam arrival time, then events are recorded. +Trigger +hardware and software selection criteria are designed +to minimize the contribution from background events, +which are primarily cosmic muons. After these are ap- +plied, enriched data samples are obtained in which a +neutrino interaction occurs in ≈ 15% of selected beam +spills [22]. +Individual +particle +tracks +are +reconstructed +with +Pandora pattern recognition algorithms based on the +measured ionization signals in the enriched data sam- + +3 +ples [23]. Particles are identified based on the measured +track energy deposition profile, while the particle mo- +menta are obtained based on the track length [24, 25]. +III. +SIGNAL DEFINITION & EVENT +SELECTION +The QE-like signal definition used in this analysis in- +cludes all νµ-Ar scattering events with a final-state muon +with momentum 0.1 < pµ < 1.2 GeV/c, and exactly one +proton with 0.3 < pp < 1 GeV/c. Events with final-state +neutral pions at any momentum are excluded. +Signal +events may contain any number of protons below 300 +MeV/c or above 1 GeV/c, neutrons at any momentum, +and charged pions with momentum lower than 70 MeV/c. +We refer to the events passing this definition as CC1p0π. +This signal consists predominantly of QE events. More +complex interactions, namely meson exchange currents +(MEC), resonance interactions (RES) and deep inelastic +scattering events (DIS), can mimic the experimental sig- +nature of true QE events due to final-state interactions +(FSI) or particles not satisfying the signal definition as +defined above. +Candidate muon-proton pairs are isolated by requiring +the existence of precisely two track-like and no shower- +like objects, as classified by Pandora using a track-score +variable [26, 27]. The log-likelihood ratio (LLR) parti- +cle identification (PID) score [28] is used to identify the +muon and proton candidates. Muons tend to have higher +LLR PID score values than protons, thus the track with +the highest score is tagged as the candidate muon. Mean- +while, the track with the lower score is treated as the +candidate proton. +Cosmic muon and non-CC1p0π contamination back- +grounds were significantly reduced by applying a require- +ment on the candidate proton LLR PID score. We stud- +ied the effect of cutting on different values of this quan- +tity, which has a strong discrimination power for rejecting +MC non-CC1p0π background, out-of-cryostat and cosmic +events. That yielded an optimal cut on the proton candi- +date LLR score of < 0.05. To further minimize the con- +tribution of mis-reconstructed track directions, we took +advantage of two muon momentum reconstruction meth- +ods available for contained tracks, namely the momen- +tum from range [29] and the momentum from Multiple +Coulomb Scattering (MCS) [30]. +The range and MCS +muon momenta needed to be in agreement within 25%. +We required that the distance between the track start +points and the vertex is smaller than the corresponding +distance between the track end points and the vertex. +We also demanded that the distance between the start +points of the two candidate tracks is smaller than the +distance between the two end points. More details are +provided in the Supplemental Material. +Further reduction of the cosmic tracks and minimiza- +tion of bin-migration effects is achieved by considering +only fully contained candidate muon-proton pairs within +a fiducial volume of 10 cm inside the edge of the detector +active volume. We retain 9051 data events that satisfy +all event selection criteria. +In order to provide an accurate description of the +dominant cosmic backgrounds pertinent to surface de- +tectors, the full Monte Carlo (MC) simulation consists +of a combination of simulated neutrino interactions over- +laid on top of beam-off background data. This approach +has been extensively used by MicroBooNE [19, 31–33]. +The GENIE v3.0.6 event generator is used to simulate +neutrino interactions with the G18 10a 02 11a configu- +ration [34, 35]. +The CCQE and CCMEC predictions +have been additionally tuned to T2K νµ-carbon CC0π +data [36, 37]. We refer to the corresponding prediction +as G18. All the final state particles following the primary +neutrino interaction are generated by GENIE. They are +further propagated in GENIE through the nucleus to ac- +count for FSI. The propagation of the particles outside +the nucleus is simulated using GEANT4 [38]. The Micro- +BooNE detector response is modeled using the LArSoft +framework [39, 40]. +Based on this MC prediction, we +obtain a purity of ≈ 70% and an efficiency for selecting +CC1p0π events of ≈ 10%. +IV. +OBSERVABLES +In neutrino-nucleus scattering events, there is an im- +balance between the true initial neutrino momentum and +the true sum of final-state lepton and hadron momenta +as a result of nuclear effects [14]. A schematic represen- +tation of the kinematic imbalance variables of interest in +this work is shown in Fig. 1. +FIG. 1. +Schematic representation of the kinematic imbalance +variables on the plane transverse to the beam direction using +CC1p0π events. +Using the CC1p0π candidate muon-proton pair kine- +matics, the missing momentum in the plane transverse +to the beam direction is defined as +δpT = |⃗pT µ + ⃗pT p|, +(1) +where ⃗pT µ and ⃗pT p are the projections of the momenta of +the outgoing lepton and proton on the transverse plane, +respectively. In the absence of nuclear effects, purely QE +interactions would yield δpT = 0. In the presence of the + +SPTX +o0 +T +SPTy +-ph +SPT +SPT +Sα.4 +dense nuclear medium, this variable encapsulates infor- +mation related to the Fermi motion, but it is smeared +due to FSI and non-QE interactions, as can be seen in +Fig. 2. Further discussion on the FSI smearing effects +can be found in the Supplemental Material. +BNB Data +Cosmic (8%) +MC QE (58%) +MC MEC (19%) +MC RES (13%) +MC DIS (2%) +0 +500 +1000 +1500 +Number of events / bin +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +Reconstructed +0.8 +1 +1.2 +Prediction +Data +Prediction Uncertainty + POT +20 + 10 +× +MicroBooNE 6.79 +FIG. 2. +Distribution of the selected CC1p0π events as a +function of the transverse missing momentum δpT . Only sta- +tistical uncertainties are shown on the data. +The interac- +tion contributions are obtained from simulation. The bottom +panel shows the ratio of data to prediction. +The direction of the transverse momentum imbalance +δpT is described by the angle +δαT = arccos +� − ⃗pT µ · δ⃗pT +pT µ δpT +� +, +(2) +which is uniformly distributed in the absence of FSI due +to the isotropic nature of the Fermi motion. In the pres- +ence of FSI, the proton momentum is generally reduced +and the δαT distribution becomes weighted towards 180◦, +as can be seen in Fig. 3. +The opening angle δφT between the correlated candi- +date muon-proton pair on the transverse plane is given +by +δφT = arccos +� − ⃗pT µ · ⃗pT p +pT µ pT p +� +. +(3) +In the absence of nuclear effects, QE events would be con- +centrated at δφT = 0. When nuclear effects are present, +QE events can occupy wider angles. At the same time, +non-QE events are dominant in the high δφT part of the +tail and their contribution is fairly flat across all angles, +as can be seen in Fig. 4. +The muon-proton momentum imbalances transverse +and longitudinal to the transverse lepton momentum [17] +are defined as +δpT,x = (ˆpν × ˆpµ +T ) · δ⃗pT +δpT,y = −ˆpµ +T · δ⃗pT , +(4) +BNB Data +Cosmic (8%) +MC QE (58%) +MC MEC (19%) +MC RES (13%) +MC DIS (2%) +0 +500 +1000 +1500 +2000 +Number of events / bin +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 + [deg] +T +α +δ +Reconstructed +0.8 +1 +1.2 +Prediction +Data +Prediction Uncertainty + POT +20 + 10 +× +MicroBooNE 6.79 +FIG. 3. +Distribution of the selected CC1p0π events as a +function of the transverse missing momentum direction δαT . +Only statistical uncertainties are shown on the data. +The +interaction contributions are obtained from simulation. The +bottom panel shows the ratio of data to prediction. +BNB Data +Cosmic (8%) +MC QE (58%) +MC MEC (19%) +MC RES (13%) +MC DIS (2%) +0 +1000 +2000 +3000 +Number of events / bin +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 + [deg] +T +φ +δ +Reconstructed +0.8 +1 +1.2 +Prediction +Data +Prediction Uncertainty + POT +20 + 10 +× +MicroBooNE 6.79 +FIG. 4. +Distribution of the selected CC1p0π events as a +function of the muon-proton transverse opening angle δφT . +Only statistical uncertainties are shown on the data. +The +interaction contributions are obtained from simulation. The +bottom panel shows the ratio of data to prediction. +and can also be written as +δpT,x = δpT · sin δαT +δpT,y = δpT · cos δαT . +(5) +These distributions can be seen in Fig. 5 and Fig. 6, re- +spectively. The δpT,x distribution is symmetric around +0 GeV/c due to the presence of the sin δαT factor in Eq. 5 +and the fact that δαT ranges from 0o to 180o. The width +of the distribution is driven by the Fermi motion that +affects the δpT magnitude. Unlike δpT,x, the δpT,y dis- +tribution is asymmetric with an enhanced contribution +from negative values. The asymmetry is driven by the + +5 +presence of the cos δαT factor in Eq. 5 and the fact that +δαT is mainly peaked around 180o. Given that the for- +ward δαT peak is driven by FSI, the size of the δpT,y +asymmetry is also sensitive to the FSI strength. +BNB Data +Cosmic (8%) +MC QE (58%) +MC MEC (19%) +MC RES (13%) +MC DIS (2%) +0 +500 +1000 +1500 +2000 +2500 +Number of events / bin +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +Reconstructed +0.8 +1 +1.2 +Prediction +Data +Prediction Uncertainty + POT +20 + 10 +× +MicroBooNE 6.79 +FIG. 5. +Distribution of the selected CC1p0π events as a +function of the perpendicular component of the transverse +missing momentum δpT,x. Only statistical uncertainties are +shown on the data. +The interaction contributions are ob- +tained from simulation. The bottom panel shows the ratio of +data to prediction. +BNB Data +Cosmic (8%) +MC QE (58%) +MC MEC (19%) +MC RES (13%) +MC DIS (2%) +0 +500 +1000 +1500 +2000 +Number of events / bin +0.6 +− +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,y +p +δ +Reconstructed +0.8 +1 +1.2 +Prediction +Data +Prediction Uncertainty + POT +20 + 10 +× +MicroBooNE 6.79 +FIG. 6. +Distribution of the selected CC1p0π events as a func- +tion of the longitudinal component of the transverse missing +momentum δpT,y. +Only statistical uncertainties are shown +on the data. The interaction contributions are obtained from +simulation. The bottom panel shows the ratio of data to pre- +diction. +Finally, the calorimetric energy reconstruction +ECal = Eµ + Tp + BE +(6) +is investigated, where Eµ is the muon energy, Tp is the +proton kinetic energy and BE = 0.04 GeV/c is the aver- +age binding energy for argon [41]. This energy estimator, +shown in Fig. 7, is an approximation for the true energy of +the incoming neutrino and is used in oscillation searches. +BNB Data +Cosmic (8%) +MC QE (58%) +MC MEC (19%) +MC RES (13%) +MC DIS (2%) +0 +1000 +2000 +3000 +Number of events / bin +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +Reconstructed E +0.8 +1 +1.2 +Prediction +Data +Prediction Uncertainty + POT +20 + 10 +× +MicroBooNE 6.79 +FIG. 7. +Distribution of the selected CC1p0π events as a +function of the calorimetric energy reconstruction ECal. Only +statistical uncertainties are shown on the data. The interac- +tion contributions are obtained from simulation. The bottom +panel shows the ratio of data to prediction. +V. +CROSS SECTION EXTRACTION & +SYSTEMATICS +The flux-averaged differential event rate as a function +of a given variable x in bin i is obtained by +dR +dxi += +Ni − Bi +T · Φν · ∆i +(7) +where Ni and Bi are the number of measured events and +the expected background events, respectively. T is the +number of target argon nuclei in the fiducial volume of in- +terest. Φν corresponds to the integrated BNB flux and ∆i +corresponds to the i-th bin width or area for the single- +and double-differential results, respectively. +We report the extracted cross sections for CC1p0π in- +teractions using the Wiener singular value decomposition +(Wiener-SVD) unfolding technique as a function of un- +folded kinematic variables [42]. This unfolding procedure +corrects a measured event rate for inefficiency and resolu- +tion effects. This is achieved by performing a minimiza- +tion of a χ2 score that compares data to a prediction and +allows for a regularization term. A Wiener filter deter- +mines the level of regularization that is required to mini- +mize the mean square error between the variance and bias +of the result. In addition to the measured event rate, the +method uses a covariance matrix calculated from simu- +lated events accounting for the statistical and systematic +uncertainties on the measurement as input. It also re- +quires the construction of a response matrix describing + +6 +the expected detector smearing and reconstruction effi- +ciency. +The output of the method is an unfolded differential +cross section, a covariance matrix describing the total +uncertainty on the unfolded result, and an additional +smearing matrix that we refer to as AC. The latter con- +tains information about the regularization and bias of +the measurement. The corresponding AC matrices have +been applied to all the cross section predictions included +in this work when a comparison to the unfolded data is +performed. The AC matrix should be applied to any in- +dependent theoretical prediction when a comparison is +performed to the data reported in this paper. The data +release, the unfolded covariance matrices, and the ad- +ditional matrices AC can be found in the Supplemental +Material. +The total covariance matrix Eij = Estat +ij ++ Esyst +ij +in- +cludes the statistical and systematic uncertainties on the +differential event rate associated with our measurement. +Estat +ij +is a diagonal covariance matrix with the statisti- +cal uncertainties and Esyst +ij +is a covariance matrix that +incorporates the total systematic uncertainties detailed +below. +The neutrino flux is predicted using the flux simula- +tion of the MiniBooNE collaboration that used the same +beam line [43]. Neutrino cross section modeling uncer- +tainties were estimated using the GENIE framework of +event reweighting [34, 35, 37]. The rescattering uncer- +tainties were obtained using GEANT4 and the relevant +reweighting package [44]. +For each of these sources of +uncertainty, we use a multisim technique [45], which con- +sists of generating a large number of MC replicas, each +one called a “universe”, where model parameters are var- +ied within their uncertainties. The simultaneous varying +of many model parameters provides a correct treatment +of their correlations. A total of n such universes are used +to construct a covariance matrix corresponding to each +source of uncertainty, +Eij = 1 +n +k=n +� +k=1 +� +Rk +i − RCV +i +� +· +� +Rk +j − RCV +j +� +(8) +where RCV +i +(RCV +j +) and Rk +i (Rk +j ) are the flux-averaged +event rates for the central value and systematic universe +k in a measured bin i (j), respectively. +The resulting +covariance matrices are summed together to estimate the +relevant uncertainty from each source. +An additional cross section uncertainty using the +NuWro v19.02.2 event generator prediction [46] as an +alternative universe has been added. The relevant mod- +eling is detailed in section VI. The flux-integrated NuWro +cross sections are obtained using Eq. 7 and the corre- +sponding covariance matrices are constructed using Eq. 8 +and a single universe (n = 1). +For detector model systematic uncertainties, one de- +tector parameter is varied each time by 1σ and is re- +ferred to as a “unisim”. These include variations in the +light yield, the ionization electron recombination model, +space-charge effects, and waveform deconvolution [47]. +We then examine the impact of each parameter varia- +tion on the MC event rates by obtaining the differences +with respect to the central value on a bin-by-bin basis. +We define the total detector 1σ systematic uncertainty +by summing in quadrature the effect of m detector vari- +ations using the formalism introduced in Eq. 8, +Eij = +k=m +� +k=1 +� +Rk +i − RCV +i +� +· +� +Rk +j − RCV +j +� +. +(9) +The full fractional uncertainty on the integrated to- +tal cross section is 11% and includes contributions from +the neutrino flux prediction (7.3%), neutrino interaction +cross section modeling (5.3%), detector response mod- +eling (4.9%), beam exposure (2.3%), statistics (1.5%), +number-of-scattering-targets (1.2%), reinteractions (1%), +and out-of-cryostat interaction modeling (0.2%). +In the results presented below, the inner error bars +on the reported cross sections correspond to the statisti- +cal uncertainties. The systematic uncertainties were de- +composed into shape- and normalization-related sources +following the procedure outlined in [48]. +The cross- +term uncertainties were incorporated in the normaliza- +tion part. +The outer error bars on the reported cross +sections correspond to statistical and shape uncertain- +ties added in quadrature. The normalization uncertain- +ties are presented with the gray band at the bottom of +each plot. Overflow (underflow) values are included in +the last (first) bin. +VI. +MODELING CONFIGURATIONS +The nominal MC neutrino interaction prediction +(G18) uses the local Fermi gas (LFG) model [49], the +Nieves CCQE scattering prescription [50] which includes +Coulomb corrections for the outgoing muon [51] and ran- +dom phase approximation (RPA) corrections [52]. Addi- +tionally, it uses the Nieves MEC model [53], the KLN- +BS RES [54–57] and Berger-Sehgal coherent (COH) [58] +scattering models, the hA2018 FSI model [59], and +MicroBooNE-specific tuning of model parameters [37]. +Our results are also compared to a number of alterna- +tive event generators. +GiBUU 2021 (GiBUU) uses sim- +ilar models, but they are implemented in a coherent +way by solving the Boltzmann-Uehling-Uhlenbeck trans- +port equation [60]. +The modeling includes the LFG +model [49], a standard CCQE expression [61], an em- +pirical MEC model and a dedicated spin dependent res- +onance amplitude calculation following the MAID anal- +ysis [60]. The DIS model is from PYTHIA [62]. GiBUU’s +FSI treatment propagates the hadrons through the resid- +ual nucleus in a nuclear potential which is consistent +with the initial state. +NuWro v19.02.2 (NuWro) uses +the LFG model [49], the Llewellyn Smith model for +QE events [63], the Nieves model for MEC events [64], +the Adler-Rarita-Schwinger formalism to calculate the + +7 +∆ resonance explicitly [57], the BS COH [58] scat- +tering model and an intranuclear cascade model for +FSI [64]. NEUT v5.4.0 (NEUT) uses the LFG model [49], +the Nieves CCQE scattering prescription [50], the Nieves +MEC model [53], the BS RES [54–57] and BS COH [58] +scattering models, and FSI with Oset medium corrections +for pions [34, 35]. +In +addition +to +the +alternative +event +generators, +our +results +are +compared +to +a +number +of +differ- +ent GENIE configurations. +These include an older +version, GENIE v2.12.10 (Gv2) [34, 35], which uses +the Bodek-Ritchie Fermi Gas model, +the Llewellyn +Smith CCQE scattering prescription [63], +the em- +pirical +MEC +model +[65], +a +Rein-Sehgal +RES +and +COH scattering model [66], and a data driven FSI +model +denoted +as +“hA” +[67]. +Another +model, +“Untuned”, +uses +the +GENIE v3.0.6 G18 10a 02 11a +configuration without additional MicroBooNE-specific +tuning. +Finally, +the +newly +added +theory-driven +GENIE v3.2.0 G21 11b 00 000 configuration (G21) is +shown. +This includes the SuSAv2 prediction for the +QE and MEC scattering parts [68] and the hN2018 FSI +model [69]. +The modeling options for RES, DIS, and +COH interactions are the same as for G18. +The χ2/bins data comparison for each generator shown +on all the figures takes into account the total covariance +matrix. Theoretical uncertainties on the models them- +selves are not included. +VII. +RESULTS +Along with the aforementioned kinematic imbalance +and energy estimator results, the data are also pre- +sented as a function of the lepton angular orientation +(Fig. 8). Previous MicroBooNE measurements using dif- +ferent signal definitions [19, 70, 71] showed discrepan- +cies in that quantity, primarily in the forward direc- +tion. These analyses used an older simulation prediction, +namely GENIE v2.12.2, to account for the efficiency cor- +rections and beam-induced backgrounds. This work illus- +trates that all generator (Fig. 8a) and GENIE configura- +tion (Fig. 8b) predictions are in good agreement with the +data when reported as a function of cosθµ. +Figures 9 and 10 show the measured single-differential +cross sections as a function of δpT using all the events +(panel a), as well as the double-differential results as a +function of the same kinematic variable in δαT bins (pan- +els b-e). In the presence of FSI, the proton can rescat- +ter or be absorbed, yielding larger kinematic imbalances +on the transverse plane and δpT values that extend be- +yond the Fermi momentum. Furthermore, the same ex- +tended tail can be obtained when pions produced due to +multi-nucleon effects (MEC or RES) are either absorbed +or below the detection threshold. The single-differential +result shows such a high-momentum tail that extends +above 0.8 GeV/c. +This picture is consistent with the +results reported by the T2K and MINERvA collabora- +tions [15, 16, 72]. +Unlike the single-differential result, +the double differential results with low δαT extend only +slightly above 0.4 GeV/c. That indicates that this region +contains minimal FSI and multi-nucleon effects and the +δpT distribution is driven by the nucleon Fermi motion. +On the other hand, the higher δαT values correspond to +δpT distributions that extend beyond 0.8 GeV/c. This +behavior is indicative of the presence of FSI and multi- +nucleon effects that smear the δpT distribution to higher +values. Future multi-differential results can help further +disentangle the contributions from these effects. Figure 9 +shows the comparisons to a number of available neutrino +event generators with NuWro and G18 showing the best +agreement over all events. +Figure 10 shows the same +results compared to a number of GENIE configurations +illustrating that Gv2 is disfavored, an observation that +is driven by the Gv2 low δpT behavior. +Furthermore, +Untuned shows a good χ2/bins performance across all +slices but predicts lower values than data. +Figure 11 shows the double-differential results as a +function of δpT in cosθµ bins. +In a factorized nuclear +model such as the LFG, the Fermi motion part of δpT +should stay constant in terms of the shape as a function +of the outgoing lepton kinematics, since in such models +the initial state nucleon momentum is a property of the +nucleus that cannot be affected by the interaction mo- +mentum or energy transfer. That is indeed the observed +behavior in the reported results across all event genera- +tors and configurations, where no evidence of the inad- +equacy of the factorization approach is observed. Fig- +ure 11 shows the comparisons to a number of available +neutrino event generators, where the G18 prediction is +favored based on the χ2/ndf results. +Apart from the +factorization, a better separation between QE and non- +QE can be gained depending on the cosθµ region. +As +can be seen in Fig. 12 for G18, MEC events play a more +pronounced role for forward muon scattering and in the +high δpT tail, as opposed to backward scattering angles, +which are much more strongly populated by QE events. +Furthermore, the G18 cross section prediction falls be- +low the data in the -1 < cosθµ < 0.5 region, as seen in +Fig. 12a and Fig. 12b. That could indicate that addi- +tional contribution from the QE part of the G18 predic- +tion is needed beyond the MicroBooNE tune. Figure 13 +shows the same interaction breakdown for GiBUU. Unlike +G18, GiBUU illustrates a peak shift to the right, which be- +comes more pronounced in the backward direction. This +shift is driven by the enhanced MEC contribution in +higher δpT values and the reduced QE contribution at +smaller values. In the backward direction, GiBUU further +shows a cross section excess driven by the MEC contri- +bution. Figure 14 shows the same results compared to a +number of GENIE configurations illustrating that Gv2 is +disfavored due to the low δpT bin behavior. +Figures 15 and 16 show the double-differential cross +section as a function of δpT in cosθp bins. The factoriza- +tion of the nuclear motion is mostly preserved in cosθp +bins, analogously to the previous result in cosθµ. Fig- + +8 +1 +− +0.5 +− +0 +0.5 +1 +µ +θ +cos +0 +5 +10 +15 +20 +Ar +2 +cm + +-38 +10 + +µ +θ +dcos +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (18.0/18) +GiBUU (10.2/18) +NEUT (10.7/18) +NuWro (16.1/18) +(a) +1 +− +0.5 +− +0 +0.5 +1 +µ +θ +cos +0 +5 +10 +15 +20 +Ar +2 +cm + +-38 +10 + +µ +θ +dcos +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (18.0/18) +Untuned (14.0/18) +G21 (30.0/18) +Gv2 (19.7/18) +(b) +FIG. 8. +The flux-integrated single-differential cross sections as a function of cosθµ. (a) Generator and (b) GENIE configuration +predictions are compared to data. Inner and outer error bars show the statistical and total (statistical and shape systematic) +uncertainty at the 1σ, or 68%, confidence level. The gray band shows the normalization systematic uncertainty. The numbers +in parentheses show the χ2/bins calculation for each one of the predictions. +ure 15 shows the comparisons to a number of available +neutrino event generators. The GiBUU prediction is sig- +nificantly lower than the data in the backward proton +angle for low δpT values, as shown in Fig. 15a. +Fig- +ure 16 shows the same results compared to a number of +GENIE configurations illustrating that Gv2 is disfavored +across all cosθp bins. +As can be seen in Fig. 17, this +particularly poor performance is driven by the QE con- +tribution. For backward scattering events (panel a), the +QE contribution predicted by Llewellyn Smith is signifi- +cantly overestimated. For intermediate angles (0 < cosθp +< 0.5), the same QE model results in an unphysical dou- +ble peak. For forward scattering (0.5 < cosθp < 1), the +Gv2 QE prediction yields a pronounced contribution at +lower values of δpT compared to the data. +Figures 18 and 19 show the single-differential cross sec- +tions as a function of δαT using all the events (panel +a), as well as the double-differential results in the same +kinematic variable in δpT bins (panels b-d). The single- +differential results shown in panel a yield some inter- +esting observations when compared to the relevant T2K +and MINERvA results [15, 16, 72]. Our distribution il- +lustrates a slightly asymmetric behavior, similar to the +one reported by the T2K collaboration at a compara- +ble energy with MicroBooNE. Within the precision of +the data sets, the mass-number dependence of the nu- +clear effects seems to be reasonably well-modeled. Un- +like our result, the measurement by MINERvA reports +a more pronounced asymmetry on hydrocarbon. +The +breakdown plots in Fig. 18 in Ref. [72] show that this be- +havior is driven by enhanced pion-production rates due +to the higher average beam energy. Low δpT values re- +sult in a fairly uniform δαT distribution indicative of the +absence of FSI effects in that part of the phase-space. On +the other hand, higher δpT values result in a highly asym- +metric δαT distribution, which is driven by the strength +of the FSI interactions. +Figure 18 shows the compar- +isons to a number of available neutrino event generators, +where NuWro is the generator with the most conservative +FSI strength. +Figure 19 shows the same results com- +pared to a number of GENIE configurations, where Gv2 +yields the highest χ2/bins result, especially in the lowest +δpT region. As shown in Fig. 20, this is driven by the +Gv2 QE performance, which results in peaks at the edges +of the distribution, unlike the data result. +Figures 21 and 22 show the double-differential results +as a function of δαT in cosθµ bins. All the bins illustrate +an asymmetric δαT distribution, with the exception of +the region where cosθµ ≈ 1, with the latter implying that +this part of phase-space includes events with minimal FSI +effects. Figure 21 shows the comparisons to a number of +available neutrino event generators with GiBUU giving the +best performance. Figure 22 shows the same results com- +pared to a number of GENIE configurations, illustrating +that Gv2 is disfavored in the region where cosθµ < 0.75. +Figures 23 and 24 show the double-differential cross +sections as a function of δαT in cosθp bins. The results +in the region with 0 < cosθp < 0.75 show a fairly flat +distribution. The cross section distributions correspond- +ing to forward and backward proton scattering exhibit +an FSI-driven asymmetric behavior. Figure 23 shows the +comparisons to a number of available neutrino event gen- +erators, where NuWro yields a prediction that is disfavored +for forward scattering. Figure 24 shows the same results +compared to a number of GENIE configurations, illustrat- +ing that Gv2 is disfavored across all cosθp bins. In the -1 +< cosθp < 0 region shown in Fig. 24a, all the predictions +illustrate a peak close to 180◦ with the exception of Gv2. +The driving force for this difference is the Gv2 QE con- +tribution, as can be seen in Fig. 25. This is indicative +of potential modeling issues in the Llewellyn Smith QE +cross section and of the hA FSI performance used in the + +9 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.0/13) +GiBUU (21.5/13) +NEUT (20.0/13) +NuWro (12.5/13) +(a) All events +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (10.1/11) +GiBUU (3.2/11) +NEUT (18.4/11) +NuWro (6.2/11) +o + < 45 +T +α +δ +(b) +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.9/12) +GiBUU (18.3/12) +NEUT (16.8/12) +NuWro (15.9/12) +o + < 90 +T +α +δ + < +o +(c) 45 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.5/13) +GiBUU (27.7/13) +NEUT (19.7/13) +NuWro (19.0/13) +o + < 135 +T +α +δ + < +o +(d) 90 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (14.7/13) +GiBUU (9.4/13) +NEUT (20.6/13) +NuWro (24.5/13) +o + < 180 +T +α +δ + < +o +(e) 135 +FIG. 9. +The flux-integrated (a) single- and (b-e) double- (in δαT bins) differential cross sections as a function of δpT . Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. +The gray band shows the normalization systematic uncertainty. +Colored lines show the results of theoretical cross +section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers +in parentheses show the χ2/bins calculation for each one of the predictions. +Gv2 prediction. Unlike Gv2, the theory-driven GENIE v3 +family of predictions (G18, Untuned, and G21) closely fol- +low the data. +Figures 26 and 27 show the single-differential cross sec- +tions as a function of δφT using all the events (panel +a), as well as the double-differential results as a func- + +10 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.0/13) +Untuned (11.3/13) +G21 (6.3/13) +Gv2 (70.4/13) +(a) All events +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (10.1/11) +Untuned (11.5/11) +G21 (8.3/11) +Gv2 (64.8/11) +o + < 45 +T +α +δ +(b) +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.9/12) +Untuned (10.9/12) +G21 (19.7/12) +Gv2 (31.7/12) +o + < 90 +T +α +δ + < +o +(c) 45 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.5/13) +Untuned (15.3/13) +G21 (13.9/13) +Gv2 (54.8/13) +o + < 135 +T +α +δ + < +o +(d) 90 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (14.7/13) +Untuned (17.9/13) +G21 (19.1/13) +Gv2 (84.2/13) +o + < 180 +T +α +δ + < +o +(e) 135 +FIG. 10. +The flux-integrated (a) single- and (b-e) double- (in δαT bins) differential cross sections as a function of δpT . Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section +calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. +The +numbers in parentheses show the χ2/bins calculation for each one of the predictions. +tion of the same kinematic variable in δpT bins (panels +b-d). Figure 26 shows the comparisons to a number of +available neutrino event generators, with all the genera- +tors illustrating a fairly good performance. This result is +consistent with the one reported by the T2K collabora- +tion [15, 72]. In the lowest δpT region shown in panel b, + +11 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +1 +2 +3 +4 +5 +6 +7 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.3/13) +GiBUU (16.4/13) +NEUT (20.3/13) +NuWro (19.6/13) + < 0 +µ +θ +(a) -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +5 +10 +15 +20 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (18.0/13) +GiBUU (10.3/13) +NEUT (34.0/13) +NuWro (32.4/13) + < 0.5 +µ +θ +(b) 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.7/13) +GiBUU (19.6/13) +NEUT (16.0/13) +NuWro (11.7/13) + < 0.75 +µ +θ +(c) 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.1/13) +GiBUU (15.7/13) +NEUT (6.5/13) +NuWro (4.9/13) + < 1 +µ +θ +(d) 0.75 < cos +FIG. 11. +The flux-integrated double-differential cross sections as a function of δpT in cosθµ bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers in parentheses show the +χ2/bins calculation for each one of the predictions. +NuWro is the generator with the best performance. Fig- +ure 27 shows the same results compared to a number +of GENIE configurations, where Gv2 is disfavored in all +regions. At small δpT values the cross section is decreas- +ing and zero above ≈ 40◦ which indicates the absence of +multi-nucleon and FSI effects. Higher δpT values lead to +δφT cross sections that extend up to 180◦. This behavior +is primarily driven by multi-body effects with hadrons +below the detection threshold that introduce large kine- +matic imbalances, as can be seen in panels c-d of Fig. 28. +Figures 29 and 30 show the single-differential cross sec- +tions as a function of δpT,x using all the events (panel +a), as well as the double-differential results in the same +kinematic variable in δpT,y slices (panels b-c). +Fig- +ure 29 shows the comparisons to a number of avail- +able neutrino event generators. The central region with +|δpT,y| < 0.15 GeV/c is dominated by QE interactions, +while the broader distributions with |δpT,y| > 0.15 GeV/c +are mainly driven by MEC events, as can be seen in +Fig. 31. +In the MEC dominated region of δpT,y < +-0.15 GeV/c, all the generators, apart from GiBUU, seem +to be lacking in terms of the peak strength. +GiBUU +seems to be overestimating that MEC contribution in the +δpT,y < -0.15 GeV/c bin. With the exception of NEUT, all +the event generators illustrate a good performance in the +|δpT,y| < 0.15 GeV/c region. Figure 30 shows the same +results compared to a number of GENIE configurations, +where Gv2 shows the worst performance. +The aforementioned results in kinematic imbalance +variables illustrate significant differences across the event +generators and configurations used for comparison, espe- +cially in the case of the double-differential studies. Yet, +the quantity that enters the oscillation probability is the +true neutrino energy. Neutrino energy estimators, such +as the calorimetric energy ECal defined in Eq. 6, are used +as a proxy for the true quantity. The studies reported +next present the results as a function of ECal in bins of +the kinematic imbalance variables. + +12 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +1 +2 +3 +4 +5 +6 +7 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0 +µ +θ +(a) G18, -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +5 +10 +15 +20 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.5 +µ +θ +(b) G18, 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.75 +µ +θ +(c) G18, 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 1 +µ +θ +(d) G18, 0.75 < cos +FIG. 12. +Comparison between the flux-integrated double-differential cross sections as a function of δpT for data and the +G18 GENIE prediction in cosθµ bins. Inner and outer error bars show the statistical and total (statistical and shape systematic) +uncertainty at the 1σ, or 68%, confidence level. +The gray band shows the normalization systematic uncertainty. +Colored +stacked histograms show the results of theoretical cross section calculations using the G18 GENIE prediction for QE (blue), +MEC (orange), RES (green), and DIS (red) interactions. +Figures 32 and 33 show the single-differential cross sec- +tions as a function of ECal using all the events (panel a), +as well as the double-differential results in the same kine- +matic variable in δpT bins (panels b-d). Figure 32 shows +the comparisons to a number of available neutrino event +generators, where the ECal distribution covers the same +energy spectrum across all bins. All the event generators +illustrate an equally good performance in the lowest δpT +bin. NEUT and NuWro show a deficit relative to the data in +the highest δpT bins. Figure 33 shows the same results +compared to a number of GENIE configurations, where +G18 illustrates the best performance. +Interestingly, all +the alternative GENIE configurations illustrate a plateau +in the highest δpT bin. +Figures 34 and 35 show the double-differential results +as a function of ECal in δαT bins. Figure 34 shows the +comparisons to a number of available neutrino event gen- +erators. +Once again, the ECal distribution covers the +same energy spectrum across all of our results and all +the event generators show fairly good behavior. Figure 35 +shows the same results compared to a number of GENIE +configurations, where all the GENIE configurations except +for G18 illustrate shape and strength differences. +Figures 36 and 37 show the double-differential results +as a function of ECal in δpT,y bins. Figure 36 shows the +comparisons to a number of available neutrino event gen- +erators. All event generators predict very similar cross +sections for -0.15 < δpT,y < 0.15 GeV/c (panel a). Unlike +this central region, the |δpT,y| > 0.15 GeV/c results yield +a wide spread across the generator predictions (panels +b-c). Furthermore, apart from GiBUU, all the predictions +lack strength in the δpT,y < -0.15 GeV/c bin (panel b). +Additionally, NEUT illustrates the same deficit in the δpT,y +> 0.15 GeV/c bin (panel c). Figure 37 shows the same +results compared to a number of GENIE configurations, +where all the GENIE configurations but G18 illustrate a +poor performance due to shape and strength issues. + +13 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +1 +2 +3 +4 +5 +6 +7 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0 +µ +θ +(a) GiBUU, -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +5 +10 +15 +20 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.5 +µ +θ +(b) GiBUU, 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.75 +µ +θ +(c) GiBUU, 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 1 +µ +θ +(d) GiBUU, 0.75 < cos +FIG. 13. +Comparison between the flux-integrated double-differential cross sections as a function of δpT for data and the +GiBUU prediction in cosθµ bins. Inner and outer error bars show the statistical and total (statistical and shape systematic) +uncertainty at the 1σ, or 68%, confidence level. +The gray band shows the normalization systematic uncertainty. +Colored +stacked histograms show the results of theoretical cross section calculations using the GiBUU prediction for QE (blue), MEC +(orange), RES (green), and DIS (red) interactions. +VIII. +CONCLUSIONS +This work reports on measurements of flux-integrated +differential cross sections for event topologies with a sin- +gle muon and a single proton detected in the final state +using the Booster Neutrino Beam at Fermi National Ac- +celerator Laboratory and the MicroBooNE detector. The +data were studied for the first time in the form of single- +differential cross sections in kinematic imbalance vari- +ables on argon. Furthermore, the first double-differential +cross sections in these variables were reported on the +same nucleus. +Additionally, novel double-differential +cross section measurements of a neutrino energy esti- +mator in bins of these variables were presented. +The +results were compared to a number of event genera- +tors and model configurations. +The predictions as a +function of the energy estimator across all generators +and model configurations remain mostly unchanged re- +gardless of the kinematic variable used for the double- +differential measurements. The good agreement observed +across the calorimetric energy distributions suggests that +the energy dependence is largely well-modeled across +all predictions. Unlike the energy estimator results, we +found that the measured kinematic imbalance cross sec- +tions in different phase-space regions are sensitive to nu- +clear effects. The performance of the event generators +and configurations varies depending on the observable +of interest. Overall, the GENIE v3.0.6 G18 10a 02 11a +cross section predictions with the MicroBooNE-specific +tuning (G18) fit the data well. +On the other hand, +the GENIE v2.12.10 (Gv2) cross section predictions +are systematically a poor fit to data with significant +shape differences across all variables of interest. +The +GENIE v3.0.6 G18 10a 02 11a +configuration +without +additional tuning (Untuned) shows a systematic deficit of +∼ 20% which necessitated the development of the afore- +mentioned tune. +The GENIE v3.2.0 G21 11b 00 000 +configuration (G21) serves as an example of a theory- + +14 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +1 +2 +3 +4 +5 +6 +7 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.3/13) +Untuned (9.9/13) +G21 (16.0/13) +Gv2 (70.9/13) + < 0 +µ +θ +(a) -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +5 +10 +15 +20 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (18.0/13) +Untuned (21.4/13) +G21 (21.8/13) +Gv2 (65.7/13) + < 0.5 +µ +θ +(b) 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.7/13) +Untuned (11.8/13) +G21 (8.1/13) +Gv2 (34.6/13) + < 0.75 +µ +θ +(c) 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.1/13) +Untuned (11.7/13) +G21 (8.8/13) +Gv2 (24.5/13) + < 1 +µ +θ +(d) 0.75 < cos +FIG. 14. +The flux-integrated double-differential cross sections as a function of δpT in cosθµ bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. The numbers in parentheses +show the χ2/bins calculation for each one of the predictions. +driven GENIE configuration that shows good agreement +with data in most variables without the need for addi- +tional tuning. +GiBUU 2021 (GiBUU) shows good agree- +ment with data in most kinematic variables, with the +exception of δpT , where a systematic shift to higher val- +ues of δpT has been identified. +A potential source of +this shift is due to the GiBUU MEC modeling. +The +NuWro v19.02.2 (NuWro) prediction falls bellow the data +due to poor FSI modeling and shows significant shape +differences in FSI-dominated parts of the phase-space. +NEUT v5.4.0 (NEUT) also results in predictions mostly +falling below the data points. This mismodeling remains +largely unnoticed when combined into the calorimetric +energy estimator. Yet, future neutrino oscillation mea- +surements will rely on accurate cross section predictions +and a precise mapping between measured and true neu- +trino energies. Therefore, such mismodeling effects might +impact their experimental sensitivity. The reported re- +sults both provide precision data to benchmark neutrino- +nucleus interaction models and establish phase-space re- +gions where precise reaction modeling is still needed. +IX. +ACKNOWLEDGMENTS +This document was prepared by the MicroBooNE col- +laboration using the resources of the Fermi National Ac- +celerator Laboratory (Fermilab), a U.S. Department of +Energy, Office of Science, HEP User Facility. Fermilab is +managed by Fermi Research Alliance, LLC (FRA), act- +ing under Contract No. DE-AC02-07CH11359. Micro- +BooNE is supported by the following: the U.S. Depart- +ment of Energy, Office of Science, Offices of High En- +ergy Physics and Nuclear Physics; the U.S. National Sci- +ence Foundation; the Swiss National Science Foundation; +the Science and Technology Facilities Council (STFC), +part of the United Kingdom Research and Innovation; +the Royal Society (United Kingdom); the UK Research + +15 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.5 +1 +1.5 +2 +2.5 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.6/8) +GiBUU (11.5/8) +NEUT (6.3/8) +NuWro (2.9/8) + < 0 +p +θ +(a) -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (17.1/13) +GiBUU (18.5/13) +NEUT (12.9/13) +NuWro (7.3/13) + < 0.5 +p +θ +(b) 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.6/13) +GiBUU (17.0/13) +NEUT (11.9/13) +NuWro (7.1/13) + < 0.75 +p +θ +(c) 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +70 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.3/12) +GiBUU (15.8/12) +NEUT (7.2/12) +NuWro (10.1/12) + < 1 +p +θ +(d) 0.75 < cos +FIG. 15. +The flux-integrated double-differential cross sections as a function of δpT in cosθp bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers in parentheses show the +χ2/bins calculation for each one of the predictions. +and Innovation (UKRI) Future Leaders Fellowship; and +The European Union’s Horizon 2020 Marie Sklodowska- +Curie Actions. Additional support for the laser calibra- +tion system and cosmic ray tagger was provided by the +Albert Einstein Center for Fundamental Physics, Bern, +Switzerland. We also acknowledge the contributions of +technical and scientific staff to the design, construction, +and operation of the MicroBooNE detector as well as the +contributions of past collaborators to the development of +MicroBooNE analyses, without whom this work would +not have been possible. For the purpose of open access, +the authors have applied a Creative Commons Attribu- +tion (CC BY) license to any Author Accepted Manuscript +version arising from this submission. + +16 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.6/8) +Untuned (3.7/8) +G21 (2.1/8) +Gv2 (28.4/8) + < 0 +p +θ +(a) -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (17.1/13) +Untuned (21.0/13) +G21 (9.5/13) +Gv2 (58.1/13) + < 0.5 +p +θ +(b) 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.6/13) +Untuned (12.1/13) +G21 (5.2/13) +Gv2 (70.9/13) + < 0.75 +p +θ +(c) 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +70 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.3/12) +Untuned (6.8/12) +G21 (7.9/12) +Gv2 (39.9/12) + < 1 +p +θ +(d) 0.75 < cos +FIG. 16. +The flux-integrated double-differential cross sections as a function of δpT in cosθp bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. The numbers in parentheses +show the χ2/bins calculation for each one of the predictions. + +17 +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +0.5 +1 +1.5 +2 +2.5 +3 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0 +p +θ +(a) Gv2 , -1 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +2 +4 +6 +8 +10 +12 +14 +16 +18 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.5 +p +θ +(b) Gv2 , 0 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.75 +p +θ +(c) Gv2 , 0.5 < cos +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +60 +70 +GeV/c Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 1 +p +θ +(d) Gv2 , 0.75 < cos +FIG. 17. +Comparison between the flux-integrated double-differential cross sections as a function of δpT for data and the Gv2 +GENIE prediction in cosθp bins. Inner and outer error bars show the statistical and total (statistical and shape systematic) +uncertainty at the 1σ, or 68%, confidence level. +The gray band shows the normalization systematic uncertainty. +Colored +stacked histograms show the results of theoretical cross section calculations using the Gv2 GENIE prediction for QE (blue), +MEC (orange), RES (green), and DIS (red) interactions. + +18 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +deg Ar +2 +cm + +-38 +10 + +T +α +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.8/7) +GiBUU (2.0/7) +NEUT (4.1/7) +NuWro (24.4/7) +(a) All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.1 +0.2 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.4/7) +GiBUU (2.6/7) +NEUT (3.2/7) +NuWro (13.2/7) + < 0.2 GeV/c +T +p +δ +(b) +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +0.2 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.3/7) +GiBUU (4.4/7) +NEUT (5.4/7) +NuWro (13.5/7) + < 0.4 GeV/c +T +p +δ +(c) 0.2 < +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.01 +0.02 +0.03 +0.04 +0.05 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.9/7) +GiBUU (3.6/7) +NEUT (4.9/7) +NuWro (13.6/7) + > 0.4 GeV/c +T +p +δ +(d) +FIG. 18. +The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δαT . Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. +The gray band shows the normalization systematic uncertainty. +Colored lines show the results of theoretical cross +section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers +in parentheses show the χ2/bins calculation for each one of the predictions. + +19 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +deg Ar +2 +cm + +-38 +10 + +T +α +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.8/7) +Untuned (7.5/7) +G21 hN (10.9/7) +Gv2 (12.6/7) +(a) All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.1 +0.2 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.4/7) +Untuned (8.2/7) +G21 hN (7.0/7) +Gv2 (84.1/7) + < 0.2 GeV/c +T +p +δ +(b) +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +0.2 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.3/7) +Untuned (7.6/7) +G21 hN (15.9/7) +Gv2 (9.8/7) + < 0.4 GeV/c +T +p +δ +(c) 0.2 < +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.01 +0.02 +0.03 +0.04 +0.05 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.9/7) +Untuned (6.4/7) +G21 hN (10.0/7) +Gv2 (17.5/7) + > 0.4 GeV/c +T +p +δ +(d) +FIG. 19. +The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δαT . Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section +calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. +The +numbers in parentheses show the χ2/bins calculation for each one of the predictions. + +20 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.2 GeV/c +T +p +δ +(a) G18, +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.2 GeV/c +T +p +δ +(b) Gv2, +FIG. 20. +Comparison between the data flux-integrated double-differential cross section as a function of δαT for events in the +region δpT < 0.2 GeV/c region against the G18 and Gv2 GENIE predictions. Inner and outer error bars show the statistical and +total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band shows the normalization +systematic uncertainty. Colored stacked histograms show the results of theoretical cross section calculations using the (a) G18 +and (b) Gv2 GENIE predictions for QE (blue), MEC (orange), RES (green), and DIS (red) interactions. + +21 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.005 +0.01 +0.015 +0.02 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.3/7) +GiBUU (2.9/7) +NEUT (6.7/7) +NuWro (13.4/7) + < 0 +µ +θ +(a) -1 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.01 +0.02 +0.03 +0.04 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.8/7) +GiBUU (4.0/7) +NEUT (8.8/7) +NuWro (12.5/7) + < 0.5 +µ +θ +(b) 0 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.7/7) +GiBUU (4.0/7) +NEUT (5.9/7) +NuWro (14.9/7) + < 0.75 +µ +θ +(c) 0.5 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (1.1/7) +GiBUU (9.8/7) +NEUT (3.4/7) +NuWro (6.2/7) + < 1 +µ +θ +(d) 0.75 < cos +FIG. 21. +The flux-integrated double-differential cross sections as a function of δαT in cosθµ bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers in parentheses show the +χ2/bins calculation for each one of the predictions. + +22 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.005 +0.01 +0.015 +0.02 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.3/7) +Untuned (9.7/7) +G21 (11.5/7) +Gv2 (22.3/7) + < 0 +µ +θ +(a) -1 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.01 +0.02 +0.03 +0.04 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.8/7) +Untuned (11.9/7) +G21 (10.7/7) +Gv2 (13.1/7) + < 0.5 +µ +θ +(b) 0 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.7/7) +Untuned (7.6/7) +G21 (9.1/7) +Gv2 (41.9/7) + < 0.75 +µ +θ +(c) 0.5 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +deg Ar +2 +cm + +-38 +10 + +µ +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (1.1/7) +Untuned (4.3/7) +G21 (3.2/7) +Gv2 (26.7/7) + < 1 +µ +θ +(d) 0.75 < cos +FIG. 22. +The flux-integrated double-differential cross sections as a function of δαT in cosθµ bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. The numbers in parentheses +show the χ2/bins calculation for each one of the predictions. + +23 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.6/7) +GiBUU (2.4/7) +NEUT (5.7/7) +NuWro (4.7/7) + < 0 +p +θ +(a) -1 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.01 +0.02 +0.03 +0.04 +0.05 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (5.6/7) +GiBUU (1.3/7) +NEUT (6.5/7) +NuWro (14.5/7) + < 0.5 +p +θ +(b) 0 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.6/7) +GiBUU (0.5/7) +NEUT (2.8/7) +NuWro (11.2/7) + < 0.75 +p +θ +(c) 0.5 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +0.2 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (3.3/7) +GiBUU (1.0/7) +NEUT (3.7/7) +NuWro (24.6/7) + < 1 +p +θ +(d) 0.75 < cos +FIG. 23. +The flux-integrated double-differential cross sections as a function of δαT in cosθp bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers in parentheses show the +χ2/bins calculation for each one of the predictions. + +24 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.6/7) +Untuned (9.3/7) +G21 (5.4/7) +Gv2 (29.6/7) + < 0 +p +θ +(a) -1 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.01 +0.02 +0.03 +0.04 +0.05 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (5.6/7) +Untuned (10.4/7) +G21 (9.7/7) +Gv2 (23.2/7) + < 0.5 +p +θ +(b) 0 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.6/7) +Untuned (4.0/7) +G21 (4.5/7) +Gv2 (27.9/7) + < 0.75 +p +θ +(c) 0.5 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.05 +0.1 +0.15 +0.2 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (3.3/7) +Untuned (6.7/7) +G21 (8.6/7) +Gv2 (41.8/7) + < 1 +p +θ +(d) 0.75 < cos +FIG. 24. +The flux-integrated double-differential cross sections as a function of δαT in cosθp bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. The numbers in parentheses +show the χ2/bins calculation for each one of the predictions. + +25 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0 +p +θ +(a) G18, -1 < cos +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.002 +0.004 +0.006 +0.008 +0.01 +0.012 +deg Ar +2 +cm + +-38 +10 + +p +θ +dcos +T +α +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0 +p +θ +(b) Gv2, -1 < cos +FIG. 25. +Comparison between the data flux-integrated double-differential cross section as a function of δαT for events in the +region -1 < cosθp < 0 region against the G18 and Gv2 GENIE predictions. Inner and outer error bars show the statistical and +total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band shows the normalization +systematic uncertainty. Colored stacked histograms show the results of theoretical cross section calculations using the (a) G18 +and (b) Gv2 GENIE predictions for QE (blue), MEC (orange), RES (green), and DIS (red) interactions. + +26 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.1 +0.2 +0.3 +deg Ar +2 +cm + +-38 +10 + +T +φ +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.2/12) +GiBUU (8.5/12) +NEUT (11.9/12) +NuWro (13.9/12) +(a) All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.5 +1 +1.5 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.5/4) +GiBUU (6.4/4) +NEUT (6.9/4) +NuWro (5.1/4) + < 0.2 GeV/c +T +p +δ +(b) +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.1 +0.2 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (9.1/9) +GiBUU (9.3/9) +NEUT (14.0/9) +NuWro (13.6/9) + < 0.4 GeV/c +T +p +δ +(c) 0.2 < +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.01 +0.02 +0.03 +0.04 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.7/6) +GiBUU (1.9/6) +NEUT (6.3/6) +NuWro (9.4/6) + > 0.4 GeV/c +T +p +δ +(d) +FIG. 26. +The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δφT . Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. +The gray band shows the normalization systematic uncertainty. +Colored lines show the results of theoretical cross +section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers +in parentheses show the χ2/bins calculation for each one of the predictions. + +27 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.1 +0.2 +0.3 +deg Ar +2 +cm + +-38 +10 + +T +φ +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.2/12) +Untuned (13.7/12) +G21 (8.3/12) +Gv2 (41.6/12) +(a) All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.5 +1 +1.5 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (8.5/4) +Untuned (13.5/4) +G21 (9.3/4) +Gv2 (45.7/4) + < 0.2 GeV/c +T +p +δ +(b) +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.1 +0.2 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (9.1/9) +Untuned (14.1/9) +G21 (10.1/9) +Gv2 (50.5/9) + < 0.4 GeV/c +T +p +δ +(c) 0.2 < +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.01 +0.02 +0.03 +0.04 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (2.7/6) +Untuned (4.1/6) +G21 (4.6/6) +Gv2 (28.4/6) + > 0.4 GeV/c +T +p +δ +(d) +FIG. 27. +The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δφT . Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section +calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. +The +numbers in parentheses show the χ2/bins calculation for each one of the predictions. + +28 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +deg Ar +2 +cm + +-38 +10 + +T +φ +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS +(a) G18, All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.2 GeV/c +T +p +δ +(b) G18, +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < 0.4 GeV/c +T +p +δ +(c) G18, 0.2 < +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.01 +0.02 +0.03 +0.04 +deg GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +T +φ +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + > 0.4 GeV/c +T +p +δ +(d) G18, +FIG. 28. +Comparison between the flux-integrated double- (in δpT bins) differential cross sections as a function of δφT for +data and the G18 GENIE prediction. Inner and outer error bars show the statistical and total (statistical and shape systematic) +uncertainty at the 1σ, or 68%, confidence level. The gray band shows the normalization systematic uncertainty. Colored stacked +histograms show the results of theoretical cross section calculations using the G18 prediction for QE (blue), MEC (orange), +RES (green), and DIS (red) interactions. + +29 +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +5 +10 +15 +20 +25 +30 +35 +GeV/c Ar +2 +cm + +-38 +10 + +T,x +p +δ +d +σ +d +(a) All events +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (5.1/11) +GiBUU (9.5/11) +NEUT (9.9/11) +NuWro (9.8/11) +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +2 +4 +6 +8 +10 +12 +14 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d + < -0.15 GeV/c +T,y +p +δ +(b) +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.8/11) +GiBUU (15.9/11) +NEUT (25.2/11) +NuWro (22.2/11) +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +20 +40 +60 +80 +100 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +| < 0.15 GeV/c +T,y +p +δ +(c) | +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.5/11) +GiBUU (18.3/11) +NEUT (24.4/11) +NuWro (19.1/11) +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +2 +4 +6 +8 +10 +12 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d + > 0.15 GeV/c +T,y +p +δ +(d) +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (5.5/9) +GiBUU (9.9/9) +NEUT (10.2/9) +NuWro (4.4/9) +FIG. 29. +The flux-integrated (a) single- and (b-d) double- (in δpT,y bins) differential cross sections as a function of δpT,x. +Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, +confidence level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical +cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. +The +numbers in parentheses show the χ2/bins calculation for each one of the predictions. + +30 +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +5 +10 +15 +20 +25 +30 +35 +GeV/c Ar +2 +cm + +-38 +10 + +T,x +p +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (5.1/11) +Untuned (8.9/11) +G21 (6.3/11) +Gv2 (74.6/11) +(a) All events +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +2 +4 +6 +8 +10 +12 +14 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.8/11) +Untuned (16.5/11) +G21 (14.5/11) +Gv2 (79.7/11) + < -0.15 GeV/c +T,y +p +δ +(b) +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +20 +40 +60 +80 +100 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.4/11) +Untuned (18.5/11) +G21 (13.8/11) +Gv2 (107.4/11) +| < 0.15 GeV/c +T,y +p +δ +(c) | +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +2 +4 +6 +8 +10 +12 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (5.5/9) +Untuned (7.6/9) +G21 (4.8/9) +Gv2 (74.1/9) + > 0.15 GeV/c +T,y +p +δ +(d) +FIG. 30. +The flux-integrated (a) single- and (b-d) double- (in δpT,y bins) differential cross sections as a function of δpT,x. +Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, +confidence level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical +cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. +The numbers in parentheses show the χ2/bins calculation for each one of the predictions. + +31 +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +5 +10 +15 +20 +25 +30 +35 +GeV/c Ar +2 +cm + +-38 +10 + +T,x +p +δ +d +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS +(a) All events +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +2 +4 +6 +8 +10 +12 +14 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + < -0.15 GeV/c +T,y +p +δ +(b) +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +20 +40 +60 +80 +100 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS +| < 0.15 GeV/c +T,y +p +δ +(c) | +0.4 +− +0.2 +− +0 +0.2 +0.4 + [GeV/c] +T,x +p +δ +0 +2 +4 +6 +8 +10 +12 + Ar +2 +/c +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +T,x +p +δ +d +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +QE +MEC +RES +DIS + > 0.15 GeV/c +T,y +p +δ +(d) +FIG. 31. +Comparison between the flux-integrated double- (in δpT,y bins) differential cross sections as a function of δpT,x for +data and the G18 GENIE prediction. Inner and outer error bars show the statistical and total (statistical and shape systematic) +uncertainty at the 1σ, or 68%, confidence level. +The gray band shows the normalization systematic uncertainty. +Colored +stacked histograms show the results of theoretical cross section calculations using the G18 GENIE prediction for QE (blue), +MEC (orange), RES (green), and DIS (red) interactions. + +32 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +5 +10 +15 +20 +GeV Ar +2 +cm + +-38 +10 + +Cal +dE +σ +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (9.9/9) +GiBUU (6.6/9) +NEUT (10.2/9) +NuWro (9.9/9) +(a) All events +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +20 +40 +60 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (14.8/9) +GiBUU (9.3/9) +NEUT (8.4/9) +NuWro (7.6/9) + < 0.2 GeV/c +T +p +δ +(b) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +10 +20 +30 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (6.1/9) +GiBUU (4.2/9) +NEUT (6.2/9) +NuWro (7.5/9) + < 0.4 GeV/c +T +p +δ +(c) 0.2 < +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +2 +4 +6 +8 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (4.6/9) +GiBUU (4.9/9) +NEUT (8.2/9) +NuWro (12.2/9) + > 0.4 GeV/c +T +p +δ +(d) +FIG. 32. +The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of ECal. +Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, +confidence level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical +cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. +The +numbers in parentheses show the χ2/bins calculation for each one of the predictions. + +33 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +5 +10 +15 +20 +GeV Ar +2 +cm + +-38 +10 + +Cal +dE +σ +d +MicroBooNE Data +6.79e+20 POT +Shape +⊕ +Stat +Norm +G18 (9.9/9) +Untuned (51.2/9) +G21 (51.1/9) +Gv2 (63.0/9) +(a) All events +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +20 +40 +60 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data +6.79e+20 POT +Shape +⊕ +Stat +Norm +G18 (14.8/9) +Untuned (71.8/9) +G21 (76.8/9) +Gv2 (93.5/9) + < 0.2 GeV/c +T +p +δ +(b) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +10 +20 +30 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data +6.79e+20 POT +Shape +⊕ +Stat +Norm +G18 (6.1/9) +Untuned (18.0/9) +G21 (16.7/9) +Gv2 (18.3/9) + < 0.4 GeV/c +T +p +δ +(c) 0.2 < +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +2 +4 +6 +8 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data +6.79e+20 POT +Shape +⊕ +Stat +Norm +G18 (4.6/9) +Untuned (18.5/9) +G21 (23.4/9) +Gv2 (22.0/9) + > 0.4 GeV/c +T +p +δ +(d) +FIG. 33. +The flux-integrated (a) single- and (b-d) double- in δpT bins differential cross sections as a function of ECal. Inner +and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence +level. The gray band shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section +calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. + +34 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.02 +0.04 +0.06 +0.08 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.9/8) +GiBUU (4.6/8) +NEUT (7.5/8) +NuWro (8.2/8) +o + < 45 +T +α +δ +(a) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.02 +0.04 +0.06 +0.08 +0.1 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (15.0/9) +GiBUU (7.1/9) +NEUT (15.7/9) +NuWro (9.6/9) +o + < 90 +T +α +δ + < +o +(b) 45 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.05 +0.1 +0.15 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (12.1/9) +GiBUU (2.6/9) +NEUT (13.0/9) +NuWro (14.1/9) +o + < 135 +T +α +δ + < +o +(c) 90 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.05 +0.1 +0.15 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.9/8) +GiBUU (2.7/8) +NEUT (8.7/8) +NuWro (16.8/8) +o + < 180 +T +α +δ + < +o +(d) 135 +FIG. 34. +The flux-integrated double-differential cross sections as a function of ECal in δαT bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers in parentheses show the +χ2/bins calculation for each one of the predictions. + +35 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.02 +0.04 +0.06 +0.08 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.9/8) +Untuned (40.4/8) +G21 (29.2/8) +Gv2 (36.2/8) +o + < 45 +T +α +δ +(a) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.02 +0.04 +0.06 +0.08 +0.1 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (15.0/9) +Untuned (45.0/9) +G21 (37.2/9) +Gv2 (73.5/9) +o + < 90 +T +α +δ + < +o +(b) 45 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.05 +0.1 +0.15 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (12.1/9) +Untuned (53.0/9) +G21 (49.4/9) +Gv2 (88.1/9) +o + < 135 +T +α +δ + < +o +(c) 90 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +0.05 +0.1 +0.15 +deg GeV Ar +2 +cm + +-38 +10 + +T +α +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (7.9/8) +Untuned (28.9/8) +G21 (33.5/8) +Gv2 (30.3/8) +o + < 180 +T +α +δ + < +o +(d) 135 +FIG. 35. +The flux-integrated double-differential cross sections as a function of ECal in δαT bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. The numbers in parentheses +show the χ2/bins calculation for each one of the predictions. + +36 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +10 +20 +30 +40 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.0/9) +GiBUU (5.8/9) +NEUT (8.6/9) +NuWro (6.7/9) +| < 0.15 GeV/c +T,y +p +δ +(a) | +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +2 +4 +6 +8 +10 +12 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (10.0/9) +GiBUU (4.1/9) +NEUT (16.1/9) +NuWro (28.5/9) + < -0.15 GeV/c +T,y +p +δ +(b) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +2 +4 +6 +8 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (12.8/9) +GiBUU (3.6/9) +NEUT (20.3/9) +NuWro (11.2/9) + > 0.15 GeV/c +T,y +p +δ +(c) +FIG. 36. +The flux-integrated double-differential cross sections as a function of ECal in δpT,y bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators. The numbers in parentheses show the +χ2/bins calculation for each one of the predictions. + +37 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +10 +20 +30 +40 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (11.0/9) +Untuned (67.0/9) +G21 (58.3/9) +Gv2 (93.6/9) +| < 0.15 GeV/c +T,y +p +δ +(a) | +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +2 +4 +6 +8 +10 +12 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (10.0/9) +Untuned (42.0/9) +G21 (59.1/9) +Gv2 (69.2/9) + < -0.15 GeV/c +T,y +p +δ +(b) +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +1.6 + [GeV] +Cal +E +0 +2 +4 +6 +8 +/c Ar +2 +GeV +2 +cm + +-38 +10 + +T,y +p +δ + d +Cal +dE +σ +2 +d +MicroBooNE Data + POT +20 + 10 +× +6.79 +Shape +⊕ +Stat +Norm +G18 (12.8/9) +Untuned (22.4/9) +G21 (8.7/9) +Gv2 (15.0/9) + > 0.15 GeV/c +T,y +p +δ +(c) +FIG. 37. +The flux-integrated double-differential cross sections as a function of ECal in δpT,y bins. Inner and outer error bars +show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level. The gray band +shows the normalization systematic uncertainty. Colored lines show the results of theoretical cross section calculations using +the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations. The numbers in parentheses +show the χ2/bins calculation for each one of the predictions. + +38 +[1] M. Tanabashi et al. (Particle Data Group), Phys. Rev. +D 98, 030001 (2018). +[2] K. Abe et al. (T2K Collaboration), Nature 580, 339 +(2020). +[3] B. Abi et al. (DUNE Collaboration), arXiv +(2018), +1807.10334 [physics.ins-det]. +[4] B. Abi et al. (DUNE Collaboration), arXiv +(2018), +1807.10327 [physics.ins-det]. +[5] B. Abi et al. (DUNE Collaboration), arXiv +(2018), +1807.10340 [physics.ins-det]. +[6] K. Abe et al. (Hyper-Kamiokande Collaboration), arXiv +(2018), 1805.04163 [physics.ins-det]. +[7] M. Antonello et al. (MicroBooNE, LAr1-ND, ICARUS- +WA104), arXiv (2015), 1503.01520 [physics.ins-det]. +[8] F. Tortorici, V. Bellini, and C. Sutera (ICARUS Collab- +oration), J. Phys. Conf. Ser. 1056, 012057 (2018). +[9] B. Abi et al. (DUNE Collaboration), arXiv +(2018), +1807.10334 [physics.ins-det]. +[10] B. Abi et al. (DUNE Collaboration), arXiv +(2020), +2002.03005 [hep-ex]. +[11] S. Dolan, U. Mosel, K. Gallmeister, L. Pickering, +and +S. Bolognesi, Phys. Rev. C 98, 045502 (2018). +[12] N. Rocco, A. Lovato, and O. Benhar, Phys. Rev. Lett. +116, 192501 (2016). +[13] N. Rocco, Frontiers in Physics 8, 116 (2020). +[14] X.-G. Lu, L. Pickering, S. Dolan, G. Barr, D. Coplowe, +Y. Uchida, D. Wark, M. O. Wascko, A. Weber, +and +T. Yuan, Phys. Rev. C 94, 015503 (2016). +[15] K. Abe et al. (T2K Collaboration), Phys. Rev. D 98, +032003 (2018). +[16] X.-G. Lu et al. (MINERvA Collaboration), Phys. Rev. +Lett. 121, 022504 (2018). +[17] T. Cai et al. (MINERvA Collaboration), Phys. Rev. D +101, 092001 (2020). +[18] L. Bathe-Peters, S. Gardiner, +and R. Guenette, arXiv +(2022), 2201.04664 [hep-ph]. +[19] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. Lett. 125, 201803 (2020). +[20] R. Acciarri et al. (MicroBooNE Collaboration), J. In- +strum. 12, P02017 (2017). +[21] A. Aguilar-Arevalo et al. (MiniBooNE Collaboration), +Phys. Rev. D 79, 072002 (2009). +[22] D. Kaleko, J. Instrum. 8, C09009 (2013). +[23] R. Acciarri et al. (MicroBooNE Collaboration), Eur. +Phys. J. C 78, 82 (2018). +[24] “Table +289: +Muons +in +liquid +argon +(Ar),” +http://pdg.lbl.gov/2012/AtomicNuclearProperties/ +MUON_ELOSS_TABLES/muonloss_289.pdf (2012). +[25] S. K. H. Bichsel, D. E. Groom, “Passage of particles +through matter,” PDG Chapter 27, Figure 27.1 http: +//pdg.lbl.gov/2005/reviews/passagerpp.pdf (2005). +[26] W. Van De Pontseele, Search for Electron Neutrino +Anomalies with the MicroBooNE Detector, Ph.D. thesis, +Oxford U. (2020). +[27] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. D 105, 112004 (2022). +[28] P. Abratenko et al. (MicroBooNE Collaboration), J. High +Energ. Phys. 2021, 153 (2021). +[29] M. Berger, J. Coursey, M. Zucker, and J. Chang, “Stop- +ping powers and ranges for protons and alpha particles,” +NIST Standard Reference Database 124 (2017). +[30] P. Abratenko et al. (MicroBooNE Collaboration), J. In- +strum. 12, P10010 (2017). +[31] C. Adams et al. (MicroBooNE Collaboration), Eur. Phys. +J. C 79, 673 (2019). +[32] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. Lett. 128, 151801 (2022). +[33] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. D 105, L051102 (2022). +[34] C. Andreopoulos et al., Nucl. Instrum. Meth. A 614, 87 +(2010). +[35] C. Andreopoulos et al., arXiv (2015), 1510.05494 [hep- +ph]. +[36] K. Abe et al. (T2K Collaboration), Phys. Rev. D 93, +112012 (2016). +[37] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. D 105, 072001 (2022). +[38] S. Agostinelli et al. (GEANT4), Nucl. Instrum. Meth. A +506, 250 (2003). +[39] R. Pordes and E. Snider, PoS ICHEP2016, 182 (2016). +[40] E. Snider and G. Petrillo, J. Phys. Conf. Ser. 898, 042057 +(2017). +[41] A. Bodek and T. Cai, Eur. Phys. J. C 79, 293 (2019). +[42] W. Tang, X. Li, X. Qian, H. Wei, +and C. Zhang, J. +Instrum. 12, P10002–P10002 (2017). +[43] A. Aguilar-Arevalo et al. (MiniBooNE Collaboration), +Phys. Rev. D 88 (2013), 10.1103/PhysRevD.88.032001. +[44] J. Calcutt, C. Thorpe, K. Mahn, +and L. Fields, J. In- +strum. 16, P08042 (2021). +[45] B. Roe, Nucl. Instrum. Meth. A 570, 159 (2007). +[46] T. Golan, J. Sobczyk, and J. ˙Zmuda, Nucl. Phys. B Proc. +Suppl. 229-232, 499 (2012), Neutrino 2010. +[47] P. Abratenko et al. (MicroBooNE Collaboration), Eur. +Phys. J. C 82, 454 (2022). +[48] K. Mahn, A search for muon neutrino and antineutrino +disappearance in the Booster Neutrino Beam, Ph.D. the- +sis, Columbia University (2009). +[49] R. Carrasco and E. Oset, Nucl. Phys. A 536, 445 (1992). +[50] J. Nieves, F. Sanchez, I. Ruiz Simo, and M. Vicente Va- +cas, Phys. Rev. D 85, 113008 (2012). +[51] J. Engel, Phys. Rev. C 57, 2004 (1998). +[52] J. Nieves, J. E. Amaro, and M. Valverde, Phys. Rev. C +70, 055503 (2004). +[53] J. Schwehr, D. Cherdack, +and R. Gran, arXiv (2016), +1601.02038 [hep-ph]. +[54] J. A. Nowak (MiniBooNE Collaboration), AIP Conf. +Proc. 1189, 243 (2009). +[55] K. Kuzmin, V. Lyubushkin, and V. Naumov, Phys. Part. +Nucl. 35, S133 (2004). +[56] C. Berger and L. Sehgal, Phys. Rev. D 76, 113004 (2007). +[57] K. M. Graczyk and J. T. Sobczyk, Phys. Rev. D +77, 053001 (2008), [Erratum: +Phys.Rev.D 79, 079903 +(2009)]. +[58] C. Berger and L. Sehgal, Phys. Rev. D 79, 053003 (2009). +[59] D. Ashery, I. Navon, G. Azuelos, H. Walter, H. Pfeiffer, +and F. Schleputz, Phys. Rev. C 23, 2173 (1981). +[60] U. Mosel, Phys. Rev. G 46, 113001 (2019). +[61] T. Leitner, L. Alvarez-Ruso, and U. Mosel, Phys. Rev. +C 73, 065502 (2006). +[62] T. Sjostrand, S. Mrenna, and P. Z. Skands, J. High En- +erg. Phys. 05, 026 (2006). +[63] C. Llewellyn Smith, Phys. Rept. 3, 261 (1972). + +39 +[64] J. Nieves, I. R. Simo, and M. J. V. Vacas, Phys. Rev. C +83, 045501 (2011). +[65] T. Katori, AIP Conf. Proc. 1663, 030001 (2015). +[66] D. Rein and L. Sehgal, Annals Phys. 133, 79 (1981). +[67] S. Mashnik, A. Sierk, K. Gudima, +and M. Baznat, J. +Phys. Conf. Ser. 41, 340 (2006). +[68] S. Dolan, G. D. Megias, and S. Bolognesi, Phys. Rev. D +101, 033003 (2020). +[69] S. Dytman, Y. Hayato, R. Raboanary, J. T. Sobczyk, +J. Tena-Vidal, and N. Vololoniaina, Phys. Rev. D 104, +053006 (2021). +[70] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. Lett. 123, 131801 (2019). +[71] P. Abratenko et al. (MicroBooNE Collaboration), Phys. +Rev. D 102, 112013 (2020). +[72] M. B. Avanzini et al., Phys. Rev. D 105, 092004 (2022). + +1 +Multi-Differential Cross-Section Measurements in νμ-Argon +Quasielastic-like Reactions with the MicroBooNE Detector +(Dated: January 11, 2023) +DATA RELEASE +Overflow (underflow) values are included in the last (first) bin. The additional smearing matrix +AC should be applied to an independent theoretical prediction when a comparison is performed to +the data reported herein. The AC matrices are dimensionless. +Cross Section δpT, All events +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +17.619713 +2.5914076 +2 +0.05 +0.1 +31.267059 +3.4668375 +3 +0.1 +0.15 +38.993846 +3.8932224 +4 +0.15 +0.2 +36.591946 +3.6680086 +5 +0.2 +0.25 +26.613702 +2.9077564 +6 +0.25 +0.3 +15.443932 +2.2848366 +7 +0.3 +0.35 +11.568607 +2.2445447 +8 +0.35 +0.4 +10.864394 +2.1026524 +9 +0.4 +0.47 +9.5845512 +1.6582198 +10 +0.47 +0.55 +7.7027679 +1.3718027 +11 +0.55 +0.65 +4.6962171 +0.97287168 +12 +0.65 +0.75 +2.5539428 +0.77973266 +13 +0.75 +0.9 +1.1167688 +0.50570624 +Unfolded Covariance Matrix δpT, All events +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +6.71539 +8.25146 +7.17662 +6.39751 +6.01335 +4.85967 +4.31444 +3.44092 +2.05538 +1.37278 +1.03097 0.897909 0.568438 +Bin 2 +8.25146 +12.019 +12.3112 +11.0706 +8.9652 +6.14157 +5.26953 +4.68817 +3.33499 +2.44135 +1.61673 +1.12272 0.621567 +Bin 3 +7.17662 +12.3112 +15.1572 +13.9752 +9.81664 +5.72466 +4.60095 +4.58379 +4.05767 +3.19748 +1.9267 +1.04027 0.451845 +Bin 4 +6.39751 +11.0706 +13.9752 +13.4543 +9.72979 +5.64056 +4.50451 +4.55976 +4.1161 +3.32919 +2.0333 +1.10892 0.487379 +Bin 5 +6.01335 +8.9652 +9.81664 +9.72979 +8.45505 +5.92495 +5.1107 +4.68871 +3.5945 +2.82418 +1.90884 +1.29809 0.703471 +Bin 6 +4.85967 +6.14157 +5.72466 +5.64056 +5.92495 +5.22048 +4.9238 +4.23195 +2.82268 +2.06617 +1.54814 +1.26196 0.766047 +Bin 7 +4.31444 +5.26953 +4.60095 +4.50451 +5.1107 +4.9238 +5.03798 +4.4844 +2.92434 +2.07059 +1.57546 +1.33828 0.832317 +Bin 8 +3.44092 +4.68817 +4.58379 +4.55976 +4.68871 +4.23195 +4.4844 +4.42115 +3.19164 +2.35671 +1.67218 +1.29864 0.771476 +Bin 9 +2.05538 +3.33499 +4.05767 +4.1161 +3.5945 +2.82268 +2.92434 +3.19164 +2.74969 +2.19561 +1.45701 +0.97354 0.521087 +Bin 10 +1.37278 +2.44135 +3.19748 +3.32919 +2.82418 +2.06617 +2.07059 +2.35671 +2.19561 +1.88184 +1.27118 0.811564 0.413841 +Bin 11 +1.03097 +1.61673 +1.9267 +2.0333 +1.90884 +1.54814 +1.57546 +1.67218 +1.45701 +1.27118 0.946479 0.68629 0.383166 +Bin 12 +0.897909 1.12272 +1.04027 +1.10892 +1.29809 +1.26196 +1.33828 +1.29864 +0.97354 0.811564 0.68629 0.607983 0.382896 +Bin 13 +0.568438 0.621567 0.451845 0.487379 0.703471 0.766047 0.832317 0.771476 0.521087 0.413841 0.383166 0.382896 0.255739 +arXiv:2301.03700v1 [hep-ex] 9 Jan 2023 + +2 +Additional Smearing Matrix (AC) δpT, All events +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.608391 +0.218619 +-0.00823167 0.00556978 0.0206322 +0.0337794 +0.0201499 +0.133722 +-0.158207 -0.0523749 -0.00223226 0.146027 +-0.0347278 +Bin 2 +0.518315 +0.317154 +0.153494 +0.128394 +0.0423483 -0.0342697 +-0.0715514 +0.11802 +-0.0997334 -0.0152325 +0.0625475 +0.139085 +-0.127988 +Bin 3 +0.19863 +0.199797 +0.338918 +0.343725 +0.0663399 -0.0799824 +-0.160458 +-0.0312012 0.0656408 +0.0537008 +0.157131 +0.0733254 +-0.230515 +Bin 4 +0.0797768 +0.0737304 +0.254057 +0.40585 +0.200673 -0.00453236 +-0.13551 +-0.0487452 0.0285734 +0.0869409 +0.186015 +0.0752719 +-0.205579 +Bin 5 +0.118485 +0.0370381 +0.0532791 +0.20873 +0.300893 +0.194744 +0.00465611 +0.0522884 +-0.113457 +0.047862 +0.125362 +0.140973 +-0.0611183 +Bin 6 +0.149492 +0.0107943 +-0.0375769 +0.033779 +0.193534 +0.272525 +0.130546 +0.152064 +-0.116554 -0.0210045 +0.0405033 +0.185986 +0.0473373 +Bin 7 +0.135886 +0.00896947 +-0.0535465 +-0.0131631 +0.108855 +0.205572 +0.186408 +0.256089 +-0.0521243 -0.0106936 +0.0032654 +0.203121 +0.0925498 +Bin 8 +0.0653087 +0.0112558 +-0.0217705 0.00246232 0.0457396 +0.0988458 +0.103481 +0.2535 +0.0808979 +0.11681 +0.0204205 +0.155759 +0.0809313 +Bin 9 -0.00863876 -0.00860406 +0.0381422 +0.0412628 0.0152354 +0.0405864 +-0.0271734 +0.0944415 +0.24761 +0.327082 +0.150844 +0.100864 +0.0283769 +Bin 10 -0.0382239 +-0.0104248 +0.0388503 +0.0401655 0.0314258 +0.0221658 +-0.0990147 -0.0341562 +0.179956 +0.389518 +0.333004 +0.118609 0.00375582 +Bin 11 -0.0192523 -0.00391206 0.00749073 0.00898804 0.0399121 +0.043164 +-0.0655587 -0.0432441 0.0350014 +0.247309 +0.402844 +0.269755 +0.101832 +Bin 12 +0.0146406 +0.00557305 +-0.0253563 +-0.0254045 0.0338288 +0.0575915 +-0.00973714 0.0123258 -0.0433998 0.0470516 +0.243597 +0.383929 +0.215191 +Bin 13 +0.0297581 +0.0108119 +-0.0397229 +-0.0414738 +0.033123 +0.062826 +0.0117404 +0.0414886 -0.0638342 -0.0383003 +0.161767 +0.42382 +0.269655 +Cross Section δpT, δαT < 45o +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +0.05 +0.092457234 +0.017843801 +2 +0.05 +0.1 +0.1564988 +0.020582334 +3 +0.1 +0.15 +0.18951511 +0.021746966 +4 +0.15 +0.2 +0.17474694 +0.022869393 +5 +0.2 +0.25 +0.12335155 +0.018393756 +6 +0.25 +0.3 +0.054899234 +0.01141765 +7 +0.3 +0.35 +0.028157619 +0.010923284 +8 +0.35 +0.4 +0.013531823 +0.0082138729 +9 +0.4 +0.47 +0.0076007037 +0.0046983808 +10 +0.47 +0.55 +0.0070376678 +0.0029152099 +11 +0.55 +0.9 +0.0013784316 +0.00037879095 +Unfolded Covariance Matrix δpT, δαT < 45o +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.000318401 0.000325794 0.000165729 4.51192e-05 3.76839e-05 6.70784e-05 9.46573e-05 8.35268e-05 4.42137e-05 1.56596e-05 9.68585e-07 +Bin 2 +0.000325794 0.000423632 0.000341819 0.00021417 0.000140789 9.94733e-05 0.000106398 9.25507e-05 5.59273e-05 3.00488e-05 +2.659e-06 +Bin 3 +0.000165729 0.000341819 0.000472931 0.000443965 0.000292367 0.00012278 +7.46405e-05 6.18356e-05 5.32803e-05 3.87653e-05 4.4326e-06 +Bin 4 +4.51192e-05 +0.00021417 0.000443965 0.000523009 0.000384776 0.000151883 6.24282e-05 4.12025e-05 4.72727e-05 3.94327e-05 5.19496e-06 +Bin 5 +3.76839e-05 0.000140789 0.000292367 0.000384776 0.00033833 0.000171181 9.62622e-05 5.76139e-05 4.21197e-05 2.96973e-05 4.02607e-06 +Bin 6 +6.70784e-05 9.94733e-05 +0.00012278 0.000151883 0.000171181 0.000130363 0.000109843 7.10706e-05 3.24314e-05 1.43608e-05 1.71258e-06 +Bin 7 +9.46573e-05 0.000106398 7.46405e-05 6.24282e-05 9.62622e-05 0.000109843 0.000119318 8.47709e-05 3.45946e-05 1.03844e-05 8.10519e-07 +Bin 8 +8.35268e-05 9.25507e-05 6.18356e-05 4.12025e-05 5.76139e-05 7.10706e-05 8.47709e-05 6.74677e-05 3.23546e-05 1.05961e-05 6.99402e-07 +Bin 9 +4.42137e-05 5.59273e-05 5.32803e-05 4.72727e-05 4.21197e-05 3.24314e-05 3.45946e-05 3.23546e-05 2.20748e-05 1.05918e-05 9.46038e-07 +Bin 10 +1.56596e-05 3.00488e-05 3.87653e-05 3.94327e-05 2.96973e-05 1.43608e-05 1.03844e-05 1.05961e-05 1.05918e-05 8.49845e-06 9.80001e-07 +Bin 11 +9.68585e-07 +2.659e-06 +4.4326e-06 +5.19496e-06 4.02607e-06 1.71258e-06 8.10519e-07 6.99402e-07 9.46038e-07 9.80001e-07 1.43483e-07 + +3 +Additional Smearing Matrix (AC) δpT, δαT < 45o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.334149 +0.234164 +0.0223846 +-0.0620358 +-0.108251 +-0.0284375 +0.0494776 +0.149627 +0.0593212 +0.172473 +0.0774406 +Bin 2 +0.270181 +0.292076 +0.102139 +0.0198281 +-0.055425 +-0.0788581 +-0.0293701 +0.137963 +0.0602708 +0.476025 +0.239922 +Bin 3 +0.0625865 +0.168462 +0.179062 +0.199135 +0.1086 +-0.0651717 +-0.179045 -0.0530697 0.219075 +0.589391 +0.297954 +Bin 4 -0.0430749 0.0400979 +0.128311 +0.280637 +0.266466 +0.0705921 +-0.220829 +-0.190817 +0.341118 +0.453966 +0.209747 +Bin 5 -0.0288676 -0.0010926 +0.0377225 +0.178237 +0.267568 +0.229462 +-0.0362133 -0.0608711 0.244083 +0.174797 +0.00947498 +Bin 6 +0.0194727 0.00610772 -0.00391155 0.0344214 +0.0993643 +0.232419 +0.171242 +0.143508 +0.0834889 -0.0419699 +-0.126522 +Bin 7 +0.0412064 +0.0233472 -0.00687408 -0.0241392 0.00733942 +0.190745 +0.260493 +0.248838 +0.101021 -0.0811857 +-0.153201 +Bin 8 +0.0396195 +0.0282355 -0.00404456 -0.0240852 -0.0165453 +0.097616 +0.166702 +0.184636 +0.210901 +0.0161929 -0.0703993 +Bin 9 +0.035856 +0.026787 +-0.00465383 0.00251156 0.00149926 +0.0282931 +0.0383342 +0.0516738 +0.332516 +0.162915 +0.0485199 +Bin 10 0.0176183 +0.025551 +-0.0038694 +0.0136261 +0.0126307 +-0.00168902 -0.0313676 -0.0241266 0.179637 +0.227336 +0.129974 +Bin 11 0.0123295 +0.0138957 -0.00243578 0.00920375 0.00908874 -0.000293633 -0.0268065 -0.0291088 0.056371 +0.105126 +0.0689814 +Cross Section δpT, 45o < δαT < 90o +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +0.05 +0.080836614 +0.016092518 +2 +0.05 +0.1 +0.16504577 +0.023783968 +3 +0.1 +0.15 +0.21304334 +0.023374406 +4 +0.15 +0.2 +0.19022336 +0.021716322 +5 +0.2 +0.25 +0.11467602 +0.01723157 +6 +0.25 +0.3 +0.053371488 +0.01148308 +7 +0.3 +0.35 +0.040890294 +0.010114624 +8 +0.35 +0.4 +0.038336569 +0.0094371064 +9 +0.4 +0.47 +0.025850733 +0.0062116622 +10 +0.47 +0.55 +0.016096871 +0.0044190856 +11 +0.55 +0.65 +0.0093470892 +0.0027906058 +12 +0.65 +0.9 +0.0014248176 +0.00037616895 +Unfolded Covariance Matrix δpT, 45o < δαT < 90o +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 1 +0.000258969 0.000344307 0.000215312 7.22798e-05 4.44531e-05 6.94356e-05 +9.1977e-05 +9.87235e-05 6.07598e-05 1.73143e-05 1.61582e-06 2.09881e-07 +Bin 2 +0.000344307 0.000565677 +0.0004613 +0.000238555 0.00010029 +7.58054e-05 0.000105136 0.000126494 8.24862e-05 2.38332e-05 1.52389e-06 2.77052e-07 +Bin 3 +0.000215312 +0.0004613 +0.000546363 0.000424386 0.000210141 8.61126e-05 8.38352e-05 9.87849e-05 +6.9264e-05 +3.3406e-05 1.27006e-05 1.48542e-06 +Bin 4 +7.22798e-05 0.000238555 0.000424386 0.000471599 0.000315713 0.000132261 6.91192e-05 5.39685e-05 4.84039e-05 4.26593e-05 2.46019e-05 3.10894e-06 +Bin 5 +4.44531e-05 +0.00010029 0.000210141 0.000315713 0.000296927 0.000166211 8.13575e-05 4.16139e-05 3.74452e-05 3.72486e-05 2.35031e-05 3.19913e-06 +Bin 6 +6.94356e-05 7.58054e-05 8.61126e-05 0.000132261 0.000166211 0.000131861 9.28267e-05 5.89367e-05 3.31562e-05 2.09889e-05 1.20397e-05 1.70845e-06 +Bin 7 +9.1977e-05 +0.000105136 8.38352e-05 6.91192e-05 8.13575e-05 9.28267e-05 0.000102306 8.57727e-05 3.91643e-05 1.39093e-05 4.42752e-06 5.45229e-07 +Bin 8 +9.87235e-05 0.000126494 9.87849e-05 5.39685e-05 4.16139e-05 5.89367e-05 8.57727e-05 +8.9059e-05 +4.79009e-05 1.58739e-05 3.07243e-06 1.95662e-07 +Bin 9 +6.07598e-05 8.24862e-05 +6.9264e-05 +4.84039e-05 3.74452e-05 3.31562e-05 3.91643e-05 4.79009e-05 3.85847e-05 2.01415e-05 7.36753e-06 7.27451e-07 +Bin 10 +1.73143e-05 2.38332e-05 +3.3406e-05 +4.26593e-05 3.72486e-05 2.09889e-05 1.39093e-05 1.58739e-05 2.01415e-05 1.95283e-05 1.08504e-05 1.26302e-06 +Bin 11 +1.61582e-06 1.52389e-06 1.27006e-05 2.46019e-05 2.35031e-05 1.20397e-05 4.42752e-06 3.07243e-06 7.36753e-06 1.08504e-05 7.78748e-06 9.93848e-07 +Bin 12 +2.09881e-07 2.77052e-07 1.48542e-06 3.10894e-06 3.19913e-06 1.70845e-06 5.45229e-07 1.95662e-07 7.27451e-07 1.26302e-06 9.93848e-07 1.41503e-07 + +4 +Additional Smearing Matrix (AC) δpT, 45o < δαT < 90o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 1 +0.144258 +0.22337 +0.074978 +-0.0829661 +-0.0977091 +0.0384619 +0.0783802 +0.196463 +0.258146 +0.0805772 +-0.363502 +-0.197379 +Bin 2 +0.18186 +0.341253 +0.193214 +-0.00315365 +-0.118033 +-0.0762631 +0.0791543 +0.311593 +0.358342 +0.0809632 +-0.523708 +-0.214217 +Bin 3 +0.0918317 +0.200268 +0.241828 +0.184341 +-0.000146056 +-0.113627 +0.0507077 +0.230081 +0.219773 +0.140314 +-0.199698 +-0.129839 +Bin 4 +0.0251096 +0.0207458 +0.121708 +0.279568 +0.196315 +0.0758035 +0.0472131 +0.0142602 +0.0122951 +0.259761 +0.0499588 -0.0741814 +Bin 5 +0.0160966 +-0.0442857 +-0.0268192 +0.160552 +0.270119 +0.335156 +0.106804 +-0.0721694 -0.0675044 +0.197528 +0.0446691 -0.0836594 +Bin 6 +0.0157009 +-0.0162483 +-0.0581193 +0.0182767 +0.137846 +0.33848 +0.184832 +0.0559395 -0.0238669 +0.0328982 +-0.0415853 -0.0846101 +Bin 7 +0.0100619 +0.0198443 +-0.0257014 +-0.0343673 +0.0183917 +0.205315 +0.22408 +0.230904 +0.0662662 -0.00497163 -0.142736 +-0.10248 +Bin 8 +0.0107271 +0.040376 +0.00979932 +-0.0439613 +-0.041259 +0.0715393 +0.149359 +0.266534 +0.204833 +0.0802027 +-0.172233 +-0.104415 +Bin 9 +0.0145792 +0.0388795 +0.0202758 +-0.027765 +-0.0384119 +0.0151164 +0.0295031 +0.126504 +0.276856 +0.32618 +-0.0518135 -0.0787813 +Bin 10 0.0055833 +0.0130169 +0.0109926 +0.0039168 +-0.00709286 0.000215394 -0.0273113 -0.0330379 0.0781422 +0.448427 +0.208775 +0.00179196 +Bin 11 0.00931318 -0.000119097 +0.00507384 +0.00942823 +0.00328942 +0.00886275 -0.0291231 -0.0615924 -0.0182208 +0.260284 +0.307959 +0.0560094 +Bin 12 0.00765898 +0.00148581 +-0.000489381 0.00190886 +0.00282892 +0.00532749 -0.0094215 -0.0212541 -0.0134666 +0.0629965 +0.0958337 +0.0234679 +Cross Section δpT, 90o < δαT < 135o +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +0.05 +0.09521135 +0.01745665 +2 +0.05 +0.1 +0.18478331 +0.021506156 +3 +0.1 +0.15 +0.22497344 +0.021801294 +4 +0.15 +0.2 +0.20256678 +0.022339224 +5 +0.2 +0.25 +0.1579132 +0.019450883 +6 +0.25 +0.3 +0.11380471 +0.015790683 +7 +0.3 +0.35 +0.099324443 +0.014449218 +8 +0.35 +0.4 +0.093102719 +0.01471589 +9 +0.4 +0.47 +0.070194885 +0.010694 +10 +0.47 +0.55 +0.044756146 +0.0077854456 +11 +0.55 +0.65 +0.01945489 +0.0048989672 +12 +0.65 +0.75 +0.0082820388 +0.0031612306 +13 +0.75 +0.9 +0.0034413958 +0.0013179012 +Unfolded Covariance Matrix δpT, 90o < δαT < 135o +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.000304735 0.000327676 0.000209902 0.000125501 0.000115852 0.000122321 0.000137878 0.000139792 9.96145e-05 6.62393e-05 3.50149e-05 1.8124e-05 6.85779e-06 +Bin 2 +0.000327676 0.000462515 0.000393721 0.000257562 0.000216126 0.000199646 0.000191164 0.000172847 0.000135942 8.83516e-05 3.97423e-05 2.23294e-05 1.0287e-05 +Bin 3 +0.000209902 0.000393721 0.000475296 0.000413123 0.000314666 0.000227026 0.000194574 0.000174989 0.000131558 8.6443e-05 4.21309e-05 2.19954e-05 9.06881e-06 +Bin 4 +0.000125501 0.000257562 0.000413123 0.000499041 0.00038842 0.000237391 0.000196556 0.000183499 0.00010499 7.28512e-05 5.06394e-05 2.36578e-05 6.93751e-06 +Bin 5 +0.000115852 0.000216126 0.000314666 0.00038842 0.000378337 0.000276951 0.000209686 0.000157816 0.000101404 7.59041e-05 4.99723e-05 2.9417e-05 1.12336e-05 +Bin 6 +0.000122321 0.000199646 0.000227026 0.000237391 0.000276951 0.000249346 0.000203314 0.000148675 0.000103954 7.34626e-05 4.30704e-05 2.83436e-05 1.16739e-05 +Bin 7 +0.000137878 0.000191164 0.000194574 0.000196556 0.000209686 0.000203314 0.00020878 0.000191563 0.000119474 7.54265e-05 4.52392e-05 2.57002e-05 8.78634e-06 +Bin 8 +0.000139792 0.000172847 0.000174989 0.000183499 0.000157816 0.000148675 0.000191563 0.000216557 0.000134291 8.22813e-05 4.82534e-05 2.16058e-05 5.46107e-06 +Bin 9 +9.96145e-05 0.000135942 0.000131558 0.00010499 0.000101404 0.000103954 0.000119474 0.000134291 0.000114362 7.78858e-05 3.65907e-05 1.67683e-05 5.71225e-06 +Bin 10 +6.62393e-05 8.83516e-05 +8.6443e-05 +7.28512e-05 7.59041e-05 7.34626e-05 7.54265e-05 8.22813e-05 7.78858e-05 6.06132e-05 3.20014e-05 1.53492e-05 5.19316e-06 +Bin 11 +3.50149e-05 3.97423e-05 4.21309e-05 5.06394e-05 4.99723e-05 4.30704e-05 4.52392e-05 4.82534e-05 3.65907e-05 3.20014e-05 2.39999e-05 1.33174e-05 4.0908e-06 +Bin 12 +1.8124e-05 +2.23294e-05 2.19954e-05 2.36578e-05 +2.9417e-05 +2.83436e-05 2.57002e-05 2.16058e-05 1.67683e-05 1.53492e-05 1.33174e-05 9.99338e-06 3.79228e-06 +Bin 13 +6.85779e-06 +1.0287e-05 +9.06881e-06 6.93751e-06 1.12336e-05 1.16739e-05 8.78634e-06 5.46107e-06 5.71225e-06 5.19316e-06 4.0908e-06 3.79228e-06 1.73686e-06 + +5 +Additional Smearing Matrix (AC) δpT, 90o < δαT < 135o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.213302 +0.143403 +0.0472538 +-0.0294652 -0.00355256 +0.019776 +0.0380359 +0.100017 +0.105642 +0.0579156 +-0.121656 +-0.17691 +-0.239484 +Bin 2 +0.203703 +0.230434 +0.170586 +0.0142559 +0.030142 +0.0542944 +0.0720635 +0.0824348 +0.128112 +0.0305376 +-0.206831 +-0.271931 -0.0250846 +Bin 3 +0.060762 +0.150003 +0.235882 +0.166201 +0.129761 +0.0704369 +0.0611954 +0.049026 +0.0652899 +-0.0338124 +-0.164223 +-0.159276 +0.0668586 +Bin 4 +0.00206288 +0.0155551 +0.150401 +0.288712 +0.272921 +0.102726 +0.0401011 +0.0469898 +-0.0498202 +-0.0760499 -0.0298625 -0.0154717 -0.0935439 +Bin 5 +-0.0134587 +-0.0210431 +0.0613301 +0.185078 +0.341238 +0.211772 +0.0633331 +-0.0319431 +-0.0520368 +-0.0111069 -0.0231149 -0.0129635 +0.176306 +Bin 6 -0.00943557 -0.0015439 +0.0216698 +0.0559392 +0.233232 +0.213554 +0.0911468 +0.00640754 +0.0245296 +0.0296632 -0.0548393 -0.0197375 +0.26659 +Bin 7 +0.0119003 +0.0103708 +0.00503933 +0.0229602 +0.111555 +0.12026 +0.0980477 +0.162117 +0.107933 +0.0344193 -0.0620902 -0.0111392 0.0917068 +Bin 8 +0.0258947 +0.0118838 +-0.0004686 +0.0179612 +0.0119837 +0.0132319 +0.0646848 +0.268814 +0.188252 +0.0718173 -0.0452721 -0.019704 -0.0979195 +Bin 9 +0.00982194 +0.020523 +0.0126429 +-0.0268634 +-0.0209681 +-0.00252312 +0.0302897 +0.16741 +0.277828 +0.197119 +-0.0445011 -0.0544066 0.0904394 +Bin 10 0.00745678 +0.0157421 -0.000976458 -0.0337072 -0.00361503 +0.00472199 +-0.00500756 0.0532215 +0.167208 +0.210571 +0.0942449 +0.0500002 +0.0759872 +Bin 11 +0.02363 +0.00700856 +-0.0175687 +-0.00748022 +0.0127108 +-0.00241449 +-0.0172558 +0.0235768 +0.0257031 +0.104843 +0.194305 +0.236822 +0.045016 +Bin 12 +0.0253061 +0.00841863 -0.00909259 -0.00441502 +0.0150895 +-0.00102935 +-0.0085354 +-0.0120066 -0.00858064 +0.023581 +0.0924414 +0.211867 +0.24741 +Bin 13 +0.0210545 +0.010603 +-0.00218508 -0.00561258 +0.0107864 +-0.000436053 -0.00365227 -0.0207961 -0.00377806 0.00778932 0.0226378 +0.0974349 +0.247006 +Cross Section δpT, 135o < δαT < 180o +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +0.05 +0.066581268 +0.014820438 +2 +0.05 +0.1 +0.16062795 +0.021214869 +3 +0.1 +0.15 +0.22961992 +0.025079325 +4 +0.15 +0.2 +0.23266351 +0.024402602 +5 +0.2 +0.25 +0.17853598 +0.022262933 +6 +0.25 +0.3 +0.11627078 +0.019363331 +7 +0.3 +0.35 +0.10123569 +0.019007703 +8 +0.35 +0.4 +0.1049638 +0.018703819 +9 +0.4 +0.47 +0.09874567 +0.016174564 +10 +0.47 +0.55 +0.091015041 +0.013985124 +11 +0.55 +0.65 +0.067935328 +0.010618221 +12 +0.65 +0.75 +0.049079696 +0.0090362775 +13 +0.75 +0.9 +0.027273377 +0.006218243 +Unfolded Covariance Matrix δpT, 135o < δαT < 180o +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.000219645 0.000265891 0.000226961 0.000210609 0.000202423 0.000160416 0.000138227 0.00011654 +8.90464e-05 8.46404e-05 8.13283e-05 7.92613e-05 5.26661e-05 +Bin 2 +0.000265891 0.000450071 0.000478877 0.000420906 0.000320775 0.000214812 0.000195546 0.000200779 0.000180606 0.000168301 0.000134319 0.00010283 5.95601e-05 +Bin 3 +0.000226961 0.000478877 0.000628973 0.000575656 0.000391684 0.000237633 0.000228029 0.00026615 0.000257532 0.000225661 0.000155179 0.000105935 5.79933e-05 +Bin 4 +0.000210609 0.000420906 0.000575656 0.000595487 0.000472681 0.000319052 0.000295682 0.000312616 0.000279274 0.000231367 0.00015546 0.000109746 6.18038e-05 +Bin 5 +0.000202423 0.000320775 0.000391684 0.000472681 0.000495638 0.000403406 0.000366668 0.000328443 0.000246268 0.000194413 0.000141749 0.000109205 6.48968e-05 +Bin 6 +0.000160416 0.000214812 0.000237633 0.000319052 0.000403406 0.000374939 0.000354332 0.000301474 0.000202591 0.000152555 0.000116983 +9.5053e-05 +5.77128e-05 +Bin 7 +0.000138227 0.000195546 0.000228029 0.000295682 0.000366668 0.000354332 0.000361293 0.000331578 0.000233655 0.000172412 0.000124508 9.74835e-05 5.81115e-05 +Bin 8 +0.00011654 0.000200779 0.00026615 0.000312616 0.000328443 0.000301474 0.000331578 0.000349833 0.000281757 0.000214005 0.000141829 0.000101832 5.84934e-05 +Bin 9 +8.90464e-05 0.000180606 0.000257532 0.000279274 0.000246268 0.000202591 0.000233655 0.000281757 0.000261617 0.000214167 0.000138578 9.22713e-05 5.01577e-05 +Bin 10 +8.46404e-05 0.000168301 0.000225661 0.000231367 0.000194413 0.000152555 0.000172412 0.000214005 0.000214167 0.000195584 0.000137531 9.15925e-05 4.80177e-05 +Bin 11 +8.13283e-05 0.000134319 0.000155179 0.00015546 0.000141749 0.000116983 0.000124508 0.000141829 0.000138578 0.000137531 0.000112747 +8.5784e-05 +4.88272e-05 +Bin 12 +7.92613e-05 +0.00010283 0.000105935 0.000109746 0.000109205 +9.5053e-05 +9.74835e-05 0.000101832 9.22713e-05 9.15925e-05 +8.5784e-05 +8.16543e-05 5.37207e-05 +Bin 13 +5.26661e-05 5.95601e-05 5.79933e-05 6.18038e-05 6.48968e-05 5.77128e-05 5.81115e-05 5.84934e-05 5.01577e-05 4.80177e-05 4.88272e-05 5.37207e-05 3.86665e-05 + +6 +Additional Smearing Matrix (AC) δpT, 135o < δαT < 180o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.204247 +0.148664 +0.0407981 +0.00752731 +0.0361286 +0.0224066 +-0.0312236 -0.0461844 -0.0663142 +-0.0110903 +0.0784301 +0.0593238 -0.0241906 +Bin 2 +0.181438 +0.247664 +0.169545 +0.0748076 +0.0519574 +-0.00292192 +-0.044334 +-0.0445275 +-0.053145 +-0.000647254 +0.104797 +0.0514641 -0.0577072 +Bin 3 +0.0222758 +0.185426 +0.27361 +0.171169 +0.0794506 +-6.02916e-05 -0.0349125 -0.0165781 +0.0101567 +0.0276959 +0.0559446 +0.0139131 -0.0742549 +Bin 4 -0.00791078 +0.0839733 +0.190064 +0.203107 +0.180861 +0.105023 +0.00815828 +0.01593 +0.032659 +0.0236999 +0.0177592 0.00371927 -0.0937358 +Bin 5 +0.0460174 +0.0177146 +0.0377159 +0.129112 +0.253117 +0.246198 +0.0846931 +0.0577698 +0.00290768 +-0.0112792 +0.0264449 0.00507141 -0.0990717 +Bin 6 +0.0465907 +-0.0106864 +-0.0261667 +0.0508792 +0.202435 +0.259719 +0.131823 +0.095473 +-0.00442573 +-0.0229454 +0.0225262 +0.0168466 -0.0814339 +Bin 7 +0.0133032 +-0.0170859 +-0.0211366 +0.034229 +0.151653 +0.207841 +0.153136 +0.139234 +0.0389739 +-0.00508968 0.00841402 +0.026358 +-0.0697206 +Bin 8 +-0.0272604 +-0.0142441 +0.0103163 +0.0467388 +0.0977147 +0.113145 +0.113715 +0.148295 +0.109927 +0.0449523 +0.00611837 0.0315543 -0.0517233 +Bin 9 +-0.0477319 -0.00224557 +0.0455374 +0.0703165 +0.0643685 +0.0361518 +0.0508241 +0.115317 +0.174227 +0.136218 +0.057052 +0.0575197 -0.0463255 +Bin 10 -0.00509726 +0.0326052 +0.0471358 +0.0475981 +0.0312654 +0.00784017 +-0.00872553 0.0278992 +0.105212 +0.185923 +0.184552 +0.108761 +-0.0353218 +Bin 11 +0.0386023 +0.0774963 +0.021036 +-0.00033048 +0.0119997 +0.0241553 +-0.0364747 -0.0311501 0.00835365 +0.139077 +0.282291 +0.203206 +0.0196984 +Bin 12 +0.0543867 +0.0738844 +0.000826599 -0.0205318 -0.00770406 +0.0270939 +-0.0421725 -0.0440621 -0.0180303 +0.0589739 +0.186857 +0.268149 +0.10793 +Bin 13 +0.057112 +0.0658198 +-0.00132882 +-0.0282947 +-0.0146056 +0.024635 +-0.048399 +-0.0420272 -0.0155515 +0.0216731 +0.127765 +0.281048 +0.173459 +Cross Section δpT, –1 < cosθμ < 0 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +2.8969562 +0.47028307 +2 +0.05 +0.1 +4.8791802 +0.59171167 +3 +0.1 +0.15 +6.2153662 +0.69443198 +4 +0.15 +0.2 +6.2967538 +0.68637835 +5 +0.2 +0.25 +4.8681037 +0.60571957 +6 +0.25 +0.3 +2.8610855 +0.48818439 +7 +0.3 +0.35 +2.0740716 +0.39655185 +8 +0.35 +0.4 +2.0155115 +0.35083279 +9 +0.4 +0.47 +1.5788478 +0.28480735 +10 +0.47 +0.55 +1.2206318 +0.22960154 +11 +0.55 +0.65 +0.7124747 +0.15544671 +12 +0.65 +0.75 +0.40178047 +0.10230988 +13 +0.75 +0.9 +0.21327361 +0.049330314 + +7 +Unfolded Covariance Matrix δpT, –1 < cosθμ < 0 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.221166 +0.244079 +0.172794 +0.153918 +0.164153 +0.133361 +0.100666 0.0731202 0.0462467 +0.0449851 +0.041953 +0.0328209 +0.0159533 +Bin 2 +0.244079 +0.350123 +0.350368 +0.305853 +0.237719 +0.162494 +0.1309 +0.118078 +0.0872878 +0.0723317 +0.052663 +0.0356552 +0.0181032 +Bin 3 +0.172794 +0.350368 +0.482236 +0.441593 +0.28956 +0.160342 +0.131128 +0.145446 +0.117437 +0.0859852 +0.0469973 +0.0236109 +0.0137991 +Bin 4 +0.153918 +0.305853 +0.441593 +0.471115 +0.359526 +0.208785 +0.15682 +0.157666 +0.119095 +0.0818173 +0.0437763 +0.0248323 +0.0141402 +Bin 5 +0.164153 +0.237719 +0.28956 +0.359526 +0.366896 +0.266861 +0.195187 +0.147844 +0.0891233 +0.0640818 +0.0447763 +0.0312267 +0.0162131 +Bin 6 +0.133361 +0.162494 +0.160342 +0.208785 +0.266861 +0.238324 +0.184317 +0.120398 +0.0591081 +0.0467065 +0.0401737 +0.0287793 +0.0140729 +Bin 7 +0.100666 +0.1309 +0.131128 +0.15682 +0.195187 +0.184317 +0.157253 +0.116961 +0.0653398 +0.0499785 +0.0379558 +0.0257813 +0.0121777 +Bin 8 +0.0731202 0.118078 +0.145446 +0.157666 +0.147844 +0.120398 +0.116961 +0.123084 +0.0908358 +0.064964 +0.0389044 +0.0230249 +0.0105873 +Bin 9 +0.0462467 0.0872878 0.117437 +0.119095 0.0891233 0.0591081 0.0653398 0.0908358 0.0811152 +0.0606037 +0.0336237 +0.0169986 0.00690663 +Bin 10 +0.0449851 0.0723317 0.0859852 0.0818173 0.0640818 0.0467065 0.0499785 0.064964 +0.0606037 +0.0527169 +0.0322509 +0.0159149 0.00577918 +Bin 11 +0.041953 +0.052663 0.0469973 0.0437763 0.0447763 0.0401737 0.0379558 0.0389044 0.0336237 +0.0322509 +0.0241637 +0.0137844 0.00528131 +Bin 12 +0.0328209 0.0356552 0.0236109 0.0248323 0.0312267 0.0287793 0.0257813 0.0230249 0.0169986 +0.0159149 +0.0137844 +0.0104673 0.00464599 +Bin 13 +0.0159533 0.0181032 0.0137991 0.0141402 0.0162131 0.0140729 0.0121777 0.0105873 0.00690663 0.00577918 0.00528131 0.00464599 0.00243348 +Additional Smearing Matrix (AC) δpT, –1 < cosθμ < 0 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.413364 +0.274925 +0.0374968 +-0.0434767 +-0.0265977 +0.0272224 +0.00289372 -0.0801879 +-0.0658809 -0.00357411 0.0784738 +0.280037 +0.0311748 +Bin 2 +0.341194 +0.362092 +0.173483 +0.0426272 +-0.0332632 +-0.0219173 +-0.0162116 -0.0474379 +-0.0093844 -0.00420526 0.0756319 +0.237034 +-0.119043 +Bin 3 +0.0607942 +0.244363 +0.295547 +0.203646 +0.0533021 +-0.0416542 +-0.0445716 +0.0367766 +0.0785281 +-0.00949505 0.0423003 -0.00943888 -0.241983 +Bin 4 -0.0628411 +0.0873893 +0.186788 +0.291781 +0.24518 +0.073805 +0.00738086 0.0497339 +0.0841751 +-0.0258297 -0.0481042 +0.0252116 +-0.229496 +Bin 5 -0.0380659 +-0.0139799 +0.0219465 +0.206075 +0.34003 +0.259976 +0.162726 +0.0112417 -0.00497227 -0.0277828 -0.0379502 +0.144641 +-0.0907513 +Bin 6 -0.0240416 +-0.0311056 +-0.0273816 +0.0853475 +0.226957 +0.292168 +0.262563 +0.0242655 +-0.0460384 -0.00998375 0.0424374 +0.182585 +-0.0402536 +Bin 7 -0.0529068 +-0.0165569 +-0.00735209 +0.0453947 +0.128104 +0.200156 +0.252875 +0.0946865 +0.0319989 +0.0307269 +0.0795483 +0.198964 +-0.0553282 +Bin 8 -0.0847243 -0.000492521 +0.0295183 +0.0455743 +0.0674227 +0.0654574 +0.134996 +0.177848 +0.177316 +0.0979346 +0.12303 +0.179165 +-0.0716237 +Bin 9 -0.0877119 +0.0200718 +0.0525439 +0.0391908 +0.0312688 +-0.0310312 +0.0336011 +0.185513 +0.280695 +0.192863 +0.250589 +0.190092 +-0.135464 +Bin 10 -0.032451 +0.0348986 +0.0418936 +0.0032057 +-0.0014641 +-0.035941 +0.00988257 +0.102974 +0.19567 +0.236121 +0.38627 +0.22863 +-0.144808 +Bin 11 0.0258231 +0.0429484 +0.0108961 +-0.0245312 +-0.0150959 -0.00208762 +0.027851 +0.0316845 +0.0825304 +0.156264 +0.393631 +0.304011 +-0.0625754 +Bin 12 0.0375132 +0.0355029 +-0.0146268 +-0.0190569 -0.00308078 0.00734749 +0.0247791 +0.0100358 +0.0195876 +0.0522745 +0.154564 +0.284765 +0.0260269 +Bin 13 0.0260829 +0.0236288 +-0.00885009 -0.00920575 0.00306917 +0.00713788 +0.0116325 0.00869197 0.00432296 +0.0111207 +0.0484439 +0.163055 +0.0546125 + +8 +Cross Section δpT, 0 < cosθμ < 0.5 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +6.4072537 +1.0364815 +2 +0.05 +0.1 +13.534754 +1.488025 +3 +0.1 +0.15 +18.328263 +1.8050866 +4 +0.15 +0.2 +17.583625 +1.7108738 +5 +0.2 +0.25 +12.715974 +1.4129169 +6 +0.25 +0.3 +7.2660878 +1.059323 +7 +0.3 +0.35 +6.1185826 +0.89294996 +8 +0.35 +0.4 +6.6069672 +0.85032139 +9 +0.4 +0.47 +5.6332932 +0.74466871 +10 +0.47 +0.55 +4.2549493 +0.5943019 +11 +0.55 +0.65 +2.5113146 +0.4710171 +12 +0.65 +0.75 +1.4982949 +0.42198144 +13 +0.75 +0.9 +0.94577651 +0.25023832 +Unfolded Covariance Matrix δpT, 0 < cosθμ < 0.5 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +1.07429 +1.28147 0.994964 0.834872 0.854299 0.699379 0.570867 0.397501 0.241534 +0.235622 +0.256108 +0.255389 +0.165957 +Bin 2 +1.28147 +2.21422 +2.33875 +1.95335 +1.36991 0.796529 0.688155 0.688667 0.555966 +0.419068 +0.276657 +0.215381 +0.156787 +Bin 3 +0.994964 2.33875 +3.25834 +2.91462 +1.72745 0.746269 0.614595 0.821877 0.792989 +0.573593 +0.277342 +0.130297 +0.101644 +Bin 4 +0.834872 1.95335 +2.91462 +2.92709 +2.03556 +1.00675 0.735775 0.780957 0.717699 +0.573808 +0.353517 +0.222774 +0.143515 +Bin 5 +0.854299 1.36991 +1.72745 +2.03556 +1.99633 +1.32676 0.960586 0.672993 0.459362 +0.436117 +0.408744 +0.36646 +0.226405 +Bin 6 +0.699379 0.796529 0.746269 1.00675 +1.32676 +1.12217 0.867808 0.54662 +0.297389 +0.283356 +0.314284 +0.313813 +0.193063 +Bin 7 +0.570867 0.688155 0.614595 0.735775 0.960586 0.867808 0.79736 0.631608 0.396269 +0.305907 +0.2568 +0.224494 +0.137193 +Bin 8 +0.397501 0.688667 0.821877 0.780957 0.672993 0.54662 0.631608 0.723046 0.580104 +0.404737 +0.222078 +0.121994 +0.067166 +Bin 9 +0.241534 0.555966 0.792989 0.717699 0.459362 0.297389 0.396269 0.580104 0.554531 +0.406636 +0.204905 0.0864203 0.0365171 +Bin 10 +0.235622 0.419068 0.573593 0.573808 0.436117 0.283356 0.305907 0.404737 0.406636 +0.353195 +0.233011 +0.140715 0.0656834 +Bin 11 +0.256108 0.276657 0.277342 0.353517 0.408744 0.314284 +0.2568 +0.222078 0.204905 +0.233011 +0.221857 +0.18129 +0.0946957 +Bin 12 +0.255389 0.215381 0.130297 0.222774 0.36646 0.313813 0.224494 0.121994 0.0864203 0.140715 +0.18129 +0.178068 +0.100845 +Bin 13 +0.165957 0.156787 0.101644 0.143515 0.226405 0.193063 0.137193 0.067166 0.0365171 0.0656834 0.0946957 0.100845 0.0626192 + +9 +Additional Smearing Matrix (AC) δpT, 0 < cosθμ < 0.5 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.357572 +0.197384 +0.0104737 +-0.066634 +-0.0650186 +-0.019629 +0.0392803 +0.0133809 +-0.0532685 +-0.0771741 0.00408395 +0.118994 +-0.0309646 +Bin 2 +0.357927 +0.298237 +0.12495 +0.0128014 +-0.0583549 +-0.091999 -0.0106305 0.0505337 0.000757665 -0.0701626 -0.0316451 +0.0118552 +-0.048686 +Bin 3 +0.0842722 +0.175349 +0.245482 +0.183601 +0.0268115 +-0.0936062 -0.0890018 0.0427286 +0.066392 +-0.0172387 -0.0268197 +-0.0919518 +-0.0846838 +Bin 4 +-0.0995551 +0.0389345 +0.180183 +0.24089 +0.171486 +0.0623978 -0.0277434 0.00645468 +0.0142712 +-0.01469 +0.00602731 -0.00858007 -0.0964352 +Bin 5 +-0.0886072 +-0.0165199 +0.0204467 +0.130513 +0.24317 +0.265468 +0.148308 +0.0317295 +-0.0765981 +-0.0602305 0.0189137 +0.135173 +-0.0518218 +Bin 6 -0.00397283 -0.0240503 +-0.0542346 +0.0138125 +0.140421 +0.289379 +0.233652 +0.113248 +-0.0520921 +-0.0629133 0.00653457 +0.143348 +-0.0376774 +Bin 7 +0.00176019 +-0.0127556 +-0.0560215 +-0.0265764 +0.0629988 +0.188978 +0.268896 +0.227696 +0.0481723 +-0.0204946 0.00366693 +0.0695446 +-0.0509405 +Bin 8 +-0.0319358 -0.00840165 +-0.020982 +-0.0194119 -0.00206434 0.0628278 +0.182657 +0.297747 +0.193456 +0.0667404 +0.0166855 +-0.0150684 +-0.0697172 +Bin 9 +-0.0706499 +-0.0139773 +0.00430313 0.00333253 -0.0340493 -0.0149317 0.0830574 +0.289421 +0.325345 +0.167031 +0.0697084 -0.00941066 -0.0847108 +Bin 10 -0.0520724 +-0.01505 +-0.00707522 0.00745826 -0.0198208 -0.0128821 0.0220114 +0.163499 +0.251086 +0.18429 +0.144608 +0.118185 +-0.0497093 +Bin 11 0.00588436 -0.00557116 -0.0368843 +-0.0115992 +0.0075801 +0.0427189 +0.0305049 +0.0389183 +0.105222 +0.114596 +0.191996 +0.277246 +0.00164327 +Bin 12 +0.0496212 +0.0134001 +-0.044596 +-0.0281246 +0.0190006 +0.0756961 +0.0468722 -0.0286461 -0.00793796 0.0294047 +0.130229 +0.298471 +0.0425192 +Bin 13 +0.0695293 +0.0302512 +-0.0309085 +-0.0308317 +0.0151455 +0.0672554 +0.0524307 -0.0316419 +-0.0426197 +-0.0121328 0.0731626 +0.224195 +0.0637115 +Cross Section δpT, 0.5 < cosθμ < 0.75 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +15.379123 +2.670063 +2 +0.05 +0.1 +29.849612 +3.460739 +3 +0.1 +0.15 +37.658288 +3.9965989 +4 +0.15 +0.2 +34.430432 +3.8611275 +5 +0.2 +0.25 +24.702258 +3.2700023 +6 +0.25 +0.3 +14.658558 +2.32953 +7 +0.3 +0.35 +11.73851 +2.2388642 +8 +0.35 +0.4 +10.081407 +2.2761151 +9 +0.4 +0.47 +7.2445367 +1.779744 +10 +0.47 +0.55 +5.4507555 +1.4475703 +11 +0.55 +0.65 +4.2613597 +1.1132975 +12 +0.65 +0.75 +4.0109188 +0.84604385 +13 +0.75 +0.9 +3.1824419 +0.49126105 + +10 +Unfolded Covariance Matrix δpT, 0.5 < cosθμ < 0.75 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +7.12924 +7.58474 +5.48054 +4.27361 +4.0579 +3.26568 +3.01388 +2.80377 +2.22939 +1.88815 +1.47779 +1.19179 0.759175 +Bin 2 +7.58474 +11.9767 +12.2655 +10.0005 +6.97839 +4.5272 +4.31907 +4.32399 +3.30219 +2.51253 +1.8325 +1.41025 0.917322 +Bin 3 +5.48054 +12.2655 +15.9728 +14.2992 +9.67621 +5.51433 +4.66386 +4.41488 +3.34696 +2.54665 +1.86364 +1.43893 0.917689 +Bin 4 +4.27361 +10.0005 +14.2992 +14.9083 +11.4896 +6.64198 +4.80614 +3.87122 +2.90659 +2.42538 +1.88893 +1.41695 0.877689 +Bin 5 +4.0579 +6.97839 +9.67621 +11.4896 +10.6929 +6.99471 +5.0572 +3.74327 +2.65522 +2.20813 +1.73062 +1.33865 0.831861 +Bin 6 +3.26568 +4.5272 +5.51433 +6.64198 +6.99471 +5.42671 +4.66446 +3.79722 +2.49175 +1.78439 +1.29141 +1.01912 0.643628 +Bin 7 +3.01388 +4.31907 +4.66386 +4.80614 +5.0572 +4.66446 +5.01251 +4.76323 +3.14527 +1.99638 +1.26072 +0.92724 0.579584 +Bin 8 +2.80377 +4.32399 +4.41488 +3.87122 +3.74327 +3.79722 +4.76323 +5.1807 +3.75465 +2.43922 +1.42866 0.909718 0.524239 +Bin 9 +2.22939 +3.30219 +3.34696 +2.90659 +2.65522 +2.49175 +3.14527 +3.75465 +3.16749 +2.36357 +1.44847 0.865411 0.438761 +Bin 10 +1.88815 +2.51253 +2.54665 +2.42538 +2.20813 +1.78439 +1.99638 +2.43922 +2.36357 +2.09546 +1.4778 +0.939208 0.446597 +Bin 11 +1.47779 +1.8325 +1.86364 +1.88893 +1.73062 +1.29141 +1.26072 +1.42866 +1.44847 +1.4778 +1.23943 0.879134 0.429904 +Bin 12 +1.19179 +1.41025 +1.43893 +1.41695 +1.33865 +1.01912 +0.92724 0.909718 0.865411 0.939208 0.879134 0.71579 +0.38796 +Bin 13 +0.759175 0.917322 0.917689 0.877689 0.831861 0.643628 0.579584 0.524239 0.438761 0.446597 0.429904 0.38796 0.241337 +Additional Smearing Matrix (AC) δpT, 0.5 < cosθμ < 0.75 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.338247 +0.181055 +0.0557819 +-0.0444002 -0.0182889 -0.054567 +-0.0395433 0.00988355 -0.0266796 +0.0177639 +-0.0350631 +0.00037824 +-0.042754 +Bin 2 +0.325456 +0.285417 +0.202114 +0.0430611 -0.0305097 -0.132838 +-0.0468179 +0.0562853 +-0.0294053 -0.0187443 +-0.0337263 +-0.0355299 +-0.0512976 +Bin 3 +0.114788 +0.211031 +0.291007 +0.168938 +0.0517172 -0.0878593 -0.0438666 +0.0295244 +-0.0386872 -0.0419223 +-0.0106127 +-0.0632565 +-0.0382287 +Bin 4 +-0.0314852 +0.0705993 +0.200231 +0.237079 +0.19722 +0.0789847 -0.0243366 +-0.0310157 -0.0551673 -0.0260121 +0.0181269 +-0.108121 +-0.0558271 +Bin 5 +-0.0516305 +-0.0411916 +0.0699132 +0.170586 +0.263477 +0.217895 +0.03525 +-0.00288842 -0.0337281 -0.0105199 0.000285898 +-0.126818 +-0.0645719 +Bin 6 +-0.021555 +-0.057608 +0.00662162 +0.0675663 +0.159821 +0.185605 +0.102479 +0.112643 +0.0161758 -0.00474702 +-0.024706 +-0.113309 +-0.0544455 +Bin 7 7.75665e-05 -0.0374968 +0.00259733 +0.0115782 +0.058263 +0.10652 +0.153241 +0.243657 +0.0769511 +0.0181891 +-0.0199415 +-0.128699 +-0.0538951 +Bin 8 +0.0233856 +-0.0132565 +0.00890178 -0.0222703 -0.0169091 0.0185733 +0.137774 +0.290818 +0.131081 +0.086702 +0.0464809 +-0.140356 +-0.0477687 +Bin 9 +0.0489283 +-0.00537801 0.00622217 -0.0370408 -0.0441174 -0.0359678 0.0788245 +0.235723 +0.160682 +0.218017 +0.191374 +-0.127919 +-0.0327569 +Bin 10 +0.0487309 +-0.00293825 -0.0052965 -0.0305208 -0.0345482 -0.049295 0.00253759 +0.086631 +0.0954233 +0.282867 +0.329908 +-0.0365455 -0.00938233 +Bin 11 +0.0390696 +-0.00198104 -0.00948851 -0.0260064 -0.0244436 -0.0443374 -0.0323653 -0.00459105 0.0220972 +0.243841 +0.403501 +0.0624616 +0.0037073 +Bin 12 +0.0301862 +-0.00337277 -0.0054649 -0.0243847 -0.0171204 -0.0281379 -0.0235188 +-0.0255029 -0.0142924 +0.137466 +0.260613 +0.0989811 +0.0193207 +Bin 13 +0.0320992 +-0.00155677 -0.00628957 -0.0246112 -0.0149941 -0.0239619 -0.0138482 +-0.0127776 -0.0143898 +0.0735423 +0.14412 +0.0707074 +0.0209479 + +11 +Cross Section δpT, 0.75 < cosθμ < 1 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +27.024279 +3.2181418 +2 +0.05 +0.1 +46.708221 +4.4406778 +3 +0.1 +0.15 +53.810979 +4.8134699 +4 +0.15 +0.2 +49.485573 +4.7402503 +5 +0.2 +0.25 +36.507298 +4.2066832 +6 +0.25 +0.3 +22.374607 +3.6765154 +7 +0.3 +0.35 +17.07327 +3.6889473 +8 +0.35 +0.4 +14.475107 +3.3116438 +9 +0.4 +0.47 +11.281648 +2.4681369 +10 +0.47 +0.55 +8.5689957 +1.8988171 +11 +0.55 +0.65 +5.2056831 +1.2528841 +12 +0.65 +0.75 +3.0276347 +0.90117192 +13 +0.75 +0.9 +1.3485493 +0.50604703 +Unfolded Covariance Matrix δpT, 0.75 < cosθμ < 1 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +10.3564 12.9318 10.9987 8.86061 +7.37093 +5.84204 +5.39126 4.60981 3.20114 2.49122 2.10149 +1.92183 +1.13277 +Bin 2 +12.9318 19.7196 19.2693 15.7595 +11.662 +7.86436 +6.86699 6.24425 4.91836 4.21191 3.17774 +2.41868 +1.26071 +Bin 3 +10.9987 19.2693 23.1695 21.3263 +15.259 +9.25441 +7.59276 7.60727 6.9277 +5.8794 +3.89052 +2.35565 +1.0187 +Bin 4 +8.86061 15.7595 21.3263 +22.47 +17.9645 +11.7067 +10.0128 +9.8524 8.39735 6.56404 3.99395 +2.26321 0.939984 +Bin 5 +7.37093 11.662 +15.259 +17.9645 +17.6962 +14.0497 +13.0273 11.6709 +8.368 +5.80895 3.39959 +2.07478 0.962426 +Bin 6 +5.84204 7.86436 9.25441 11.7067 +14.0497 +13.5168 +13.2687 11.2135 7.15019 4.43723 2.57686 +1.72074 0.881016 +Bin 7 +5.39126 6.86699 7.59276 10.0128 +13.0273 +13.2687 +13.6083 +11.758 7.43027 4.5269 +2.59107 +1.74543 0.900989 +Bin 8 +4.60981 6.24425 7.60727 +9.8524 +11.6709 +11.2135 +11.758 +10.967 7.59913 4.97607 2.84478 +1.77212 +0.84793 +Bin 9 +3.20114 4.91836 6.9277 +8.39735 +8.368 +7.15019 +7.43027 7.59913 6.0917 4.43575 2.60782 +1.5091 +0.65104 +Bin 10 +2.49122 4.21191 5.8794 +6.56404 +5.80895 +4.43723 +4.5269 +4.97607 4.43575 3.60551 2.25839 +1.3149 +0.55603 +Bin 11 +2.10149 3.17774 3.89052 3.99395 +3.39959 +2.57686 +2.59107 2.84478 2.60782 2.25839 1.56972 +1.02221 0.478498 +Bin 12 +1.92183 2.41868 2.35565 2.26321 +2.07478 +1.72074 +1.74543 1.77212 1.5091 +1.3149 +1.02221 0.812111 0.435865 +Bin 13 +1.13277 1.26071 1.0187 0.939984 0.962426 0.881016 0.900989 0.84793 0.65104 0.55603 0.478498 0.435865 0.256084 + +12 +Additional Smearing Matrix (AC) δpT, 0.75 < cosθμ < 1 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.384657 +0.185765 +0.0553495 +0.0241078 +0.0296859 +0.0793514 0.0198712 +-0.0403141 +-0.111393 +-0.0829512 0.0150641 +0.194456 +0.194276 +Bin 2 +0.451664 +0.302344 +0.177426 +0.109447 +0.0518724 +0.0551758 0.00303118 -0.0504308 +-0.116692 +-0.0243592 0.0759207 +0.207352 +0.065638 +Bin 3 +0.288352 +0.235377 +0.264871 +0.232834 +0.0887745 +0.0460066 -0.0268092 +-0.0587237 -0.0189124 0.0766786 +0.117213 +0.096092 +-0.139785 +Bin 4 +0.132118 +0.127199 +0.216745 +0.262679 +0.150871 +0.1366 +0.0206973 +-0.0249262 +0.046453 +0.113436 +0.0519168 -0.000114072 +-0.158453 +Bin 5 +0.0723883 +0.0627335 +0.0897787 +0.155561 +0.178182 +0.295976 +0.135435 +0.0341738 +0.0394512 +0.068061 +-0.0575902 +-0.0267137 +-0.00932699 +Bin 6 +0.0582858 +0.025271 +0.00138134 +0.0451777 +0.129119 +0.363188 +0.198199 +0.0702425 +0.0244084 0.00948488 -0.100322 -0.000996697 +0.113323 +Bin 7 +0.037793 +0.00892294 +-0.0265868 +0.0130587 +0.109277 +0.341985 +0.210836 +0.109209 +0.0314291 +0.0131758 -0.0927936 +0.0524476 +0.150021 +Bin 8 -0.00487896 -0.00922399 -0.0110112 +0.0298237 +0.0880319 +0.234799 +0.137492 +0.104608 +0.0649594 +0.0674215 -0.0214187 +0.106725 +0.117802 +Bin 9 +-0.0471862 +-0.0253277 +0.0262304 +0.0663415 +0.070356 +0.146863 +0.0467986 +0.0495879 +0.111813 +0.157712 +0.0946817 +0.188989 +0.0931032 +Bin 10 -0.0489527 +-0.0101504 +0.0425763 +0.0628849 +0.0395834 +0.0546652 -0.0229053 -0.00655114 0.0726846 +0.188842 +0.196471 +0.269323 +0.104566 +Bin 11 0.00936115 +0.00929629 +0.0286803 +0.0281961 +0.00912331 +0.023517 +-0.0343762 +-0.0280602 +0.0157977 +0.1263 +0.235906 +0.35949 +0.185503 +Bin 12 +0.0648266 +0.0219428 +-0.00328263 -0.0108632 -0.00564771 0.0230018 -0.0115905 +-0.0226026 -0.0231344 0.0450974 +0.156235 +0.358325 +0.242847 +Bin 13 +0.0807166 +0.0237072 +-0.0182799 -0.0260993 -0.00891949 0.0286955 0.00153088 -0.0172138 +-0.034022 +0.0100094 +0.107282 +0.300882 +0.263973 +Cross Section δpT, –1 < cosθp < 0 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.1 +1.3785276 +0.33040672 +2 +0.1 +0.2 +2.0322901 +0.42979616 +3 +0.2 +0.3 +1.9818081 +0.35210571 +4 +0.3 +0.4 +1.6100049 +0.30139374 +5 +0.4 +0.55 +1.000457 +0.22411673 +6 +0.55 +0.65 +0.80073041 +0.20208985 +7 +0.65 +0.75 +0.54891848 +0.15741415 +8 +0.75 +0.9 +0.36990176 +0.10005425 +Unfolded Covariance Matrix δpT, –1 < cosθp < 0 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 1 +0.109169 +0.140158 +0.098696 0.0434857 0.0107221 0.013197 +0.019217 0.0141377 +Bin 2 +0.140158 +0.184725 +0.139168 0.0729882 0.0267644 0.0253857 0.028319 0.0197072 +Bin 3 +0.098696 +0.139168 +0.123978 0.0884386 0.0468497 0.0369666 0.0281636 0.0174466 +Bin 4 +0.0434857 0.0729882 0.0884386 0.0908382 0.0618047 0.0468257 0.0266923 0.0138663 +Bin 5 +0.0107221 0.0267644 0.0468497 0.0618047 0.0502283 0.0411323 0.0230141 0.0107688 +Bin 6 +0.013197 0.0253857 0.0369666 0.0468257 0.0411323 0.0408403 0.0278607 0.0141436 +Bin 7 +0.019217 +0.028319 0.0281636 0.0266923 0.0230141 0.0278607 0.0247792 0.0146779 +Bin 8 +0.0141377 0.0197072 0.0174466 0.0138663 0.0107688 0.0141436 0.0146779 0.0100109 + +13 +Additional Smearing Matrix (AC) δpT, –1 < cosθp < 0 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 1 +0.0565382 +0.0850707 +0.133679 +0.0160541 +-0.108007 -0.0416427 +0.131782 +0.174316 +Bin 2 +0.0518873 +0.103606 +0.182727 +0.0370454 -0.0887635 -0.033744 +0.145365 +0.192806 +Bin 3 0.00628258 +0.0484866 +0.146595 +0.0675518 +0.0374136 +0.0443475 +0.0400332 +0.0739565 +Bin 4 +-0.035414 +-0.00642723 +0.060506 +0.0816017 +0.151835 +0.171406 +-0.0437941 -0.0468351 +Bin 5 -0.0630991 +-0.0299789 +0.00880477 +0.0245643 +0.212651 +0.328881 +0.0359311 -0.0938149 +Bin 6 -0.0326063 +-0.0123868 +-0.00587622 -0.0306397 0.0589522 +0.262663 +0.172128 +0.0106106 +Bin 7 -0.0165222 -0.000979674 0.00380812 -0.0443815 -0.0151857 +0.139617 +0.233177 +0.0988266 +Bin 8 -0.00986243 +-0.001694 +0.00970362 -0.0327057 -0.0362059 0.0577859 +0.177943 +0.142234 +Cross Section δpT, 0 < cosθp < 0.5 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +6.2035597 +1.1218257 +2 +0.05 +0.1 +12.611311 +1.4923618 +3 +0.1 +0.15 +16.317929 +1.7121838 +4 +0.15 +0.2 +15.722065 +1.737097 +5 +0.2 +0.25 +11.970913 +1.4026135 +6 +0.25 +0.3 +7.623598 +1.0930818 +7 +0.3 +0.35 +6.6757667 +1.2204584 +8 +0.35 +0.4 +6.2354756 +1.3044508 +9 +0.4 +0.47 +4.6585327 +1.0079979 +10 +0.47 +0.55 +3.2991999 +0.83059724 +11 +0.55 +0.65 +1.622353 +0.61160322 +12 +0.65 +0.75 +0.77701967 +0.4547977 +13 +0.75 +0.9 +0.81329274 +0.25751919 + +14 +Unfolded Covariance Matrix δpT, 0 < cosθp < 0.5 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +1.25849 +1.49144 +1.04078 0.671577 0.708981 0.882038 +1.1124 +1.11745 0.738884 0.462693 0.274328 0.203402 0.138068 +Bin 2 +1.49144 +2.22714 +2.15676 +1.69512 +1.32673 +1.13649 +1.31806 +1.39276 +1.00873 0.665747 0.350695 0.213802 0.152358 +Bin 3 +1.04078 +2.15676 +2.93157 +2.77236 +1.93294 +1.05875 0.860335 0.932194 0.872113 0.730974 0.414849 0.203926 0.117906 +Bin 4 +0.671577 1.69512 +2.77236 +3.01751 +2.22762 +1.08274 0.561148 0.451988 0.550159 0.623748 0.453764 0.250835 0.118357 +Bin 5 +0.708981 1.32673 +1.93294 +2.22762 +1.96732 +1.23845 0.775358 0.482658 0.396212 0.450687 0.401102 0.301158 0.165105 +Bin 6 +0.882038 1.13649 +1.05875 +1.08274 +1.23845 +1.19483 +1.12039 0.878495 0.502895 0.346658 0.295198 +0.2806 +0.188664 +Bin 7 +1.1124 +1.31806 0.860335 0.561148 0.775358 1.12039 +1.48952 +1.4519 +0.877313 0.483014 0.294071 0.269785 0.206776 +Bin 8 +1.11745 +1.39276 0.932194 0.451988 0.482658 0.878495 +1.4519 +1.70159 +1.18593 +0.7097 +0.366275 0.260273 0.198779 +Bin 9 +0.738884 1.00873 0.872113 0.550159 0.396212 0.502895 0.877313 1.18593 +1.01606 0.747493 0.41427 0.241882 0.150558 +Bin 10 +0.462693 0.665747 0.730974 0.623748 0.450687 0.346658 0.483014 +0.7097 +0.747493 0.689892 0.455099 0.266649 0.131122 +Bin 11 +0.274328 0.350695 0.414849 0.453764 0.401102 0.295198 0.294071 0.366275 0.41427 0.455099 0.374058 0.253798 0.114996 +Bin 12 +0.203402 0.213802 0.203926 0.250835 0.301158 +0.2806 +0.269785 0.260273 0.241882 0.266649 0.253798 0.206841 0.105093 +Bin 13 +0.138068 0.152358 0.117906 0.118357 0.165105 0.188664 0.206776 0.198779 0.150558 0.131122 0.114996 0.105093 0.0663161 +Additional Smearing Matrix (AC) δpT, 0 < cosθp < 0.5 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.22272 +0.13298 +0.0258331 -0.0511626 0.0175083 +0.0911805 +0.0382431 +0.0842648 +-0.00426849 -0.0378569 -0.0225086 0.0419253 -0.0660793 +Bin 2 +0.248087 +0.187963 +0.136795 +0.0383761 +0.0795285 +0.0535091 +0.0184477 +0.0969682 +-0.0116928 +-0.0448496 -0.0497996 -0.0543121 -0.111463 +Bin 3 +0.0820694 +0.128936 +0.23268 +0.200587 +0.194396 +0.000284205 -0.0540642 -0.0110562 +-0.0223113 +-0.0137092 -0.0233107 -0.100962 +-0.115477 +Bin 4 +-0.0330883 +0.0535225 +0.197734 +0.272632 +0.285944 +0.0500552 +-0.0857662 +-0.115018 +-0.0606717 +-0.0248567 +0.0183819 -0.0213366 -0.0781114 +Bin 5 -0.00330227 +0.0321967 +0.0940884 +0.177914 +0.270792 +0.159937 +-0.0420121 -0.0926075 +-0.0894852 +-0.0730277 -0.0013067 0.0979368 -0.0285114 +Bin 6 +0.0838231 +0.0532535 +0.0132937 +0.0310214 +0.13695 +0.186134 +0.0300136 +0.0394403 +-0.0579387 +-0.106641 +-0.0433923 +0.135201 +-0.0188921 +Bin 7 +0.142679 +0.0859871 +-0.0026296 -0.0553061 0.0230832 +0.104732 +0.0900673 +0.1769 +0.0145648 +-0.0920292 -0.0773074 +0.108033 +-0.0486829 +Bin 8 +0.13154 +0.0864283 +0.0122237 -0.0823987 -0.0555617 +0.0003115 +0.0691684 +0.232366 +0.0931243 +-0.00222349 -0.0589634 0.0585327 +-0.074212 +Bin 9 +0.0630609 +0.0556105 +0.0269361 -0.0668863 -0.0690139 +-0.0722497 +0.013045 +0.151219 +0.153838 +0.141974 +0.0262699 +0.074088 +-0.0951456 +Bin 10 +-0.020076 +0.0087176 +0.0107051 -0.0319744 -0.0274829 +-0.063286 +-0.0467496 0.00419004 +0.105329 +0.219809 +0.1355 +0.2002 +-0.0515421 +Bin 11 -0.0520906 +-0.0149048 -0.0263349 -0.0175878 0.00924523 -0.00168051 -0.0594272 -0.0696059 +0.0271226 +0.138459 +0.175656 +0.40931 +0.0402768 +Bin 12 -0.0295369 -0.00861117 -0.0372217 -0.0252444 0.0201673 +0.0388275 +-0.0317564 -0.0457306 +-0.0182958 +0.0156394 +0.0890009 +0.416867 +0.108393 +Bin 13 0.00272443 +0.00559783 -0.0264635 -0.0316164 +0.014594 +0.0371156 +-0.0120505 -0.00265788 -0.0225745 +-0.0243768 +0.0170281 +0.29889 +0.113396 + +15 +Cross Section δpT, 0.5 < cosθp < 0.75 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +22.281447 +3.5209459 +2 +0.05 +0.1 +46.301135 +4.9714452 +3 +0.1 +0.15 +55.640411 +5.3155374 +4 +0.15 +0.2 +48.829592 +4.5628073 +5 +0.2 +0.25 +33.184361 +3.7312039 +6 +0.25 +0.3 +17.156771 +2.5262305 +7 +0.3 +0.35 +13.181282 +2.1836251 +8 +0.35 +0.4 +11.996287 +2.1205056 +9 +0.4 +0.47 +9.3326477 +1.7640475 +10 +0.47 +0.55 +6.2480096 +1.4376714 +11 +0.55 +0.65 +3.6493884 +0.93507907 +12 +0.65 +0.75 +2.4729482 +0.72887204 +13 +0.75 +0.9 +1.9507086 +0.43556789 +Unfolded Covariance Matrix δpT, 0.5 < cosθp < 0.75 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +12.3971 15.0891 11.5869 8.90569 7.32341 4.82125 +3.78546 +3.16132 +2.65089 +2.44325 +1.84957 +1.42184 +0.91079 +Bin 2 +15.0891 24.7153 23.5188 16.8075 10.4055 5.70924 +5.24939 +5.72964 +4.80248 +3.39335 +2.11424 +1.56541 +1.11514 +Bin 3 +11.5869 23.5188 28.2549 21.9149 11.4771 5.48075 +5.90641 +6.69548 +5.2301 +3.27912 +1.91573 +1.50614 +1.15871 +Bin 4 +8.90569 16.8075 21.9149 20.8192 13.4817 6.93739 +5.78293 +5.33078 +3.97349 +2.73153 +1.66546 +1.27175 0.999624 +Bin 5 +7.32341 10.4055 11.4771 13.4817 13.9219 8.48732 +5.33456 +3.42046 +2.33755 +1.86416 +1.34978 +1.05411 +0.79947 +Bin 6 +4.82125 5.70924 5.48075 6.93739 8.48732 6.38184 +4.55184 +2.73384 +1.45442 +1.12182 0.942322 0.896186 0.650815 +Bin 7 +3.78546 5.24939 5.90641 5.78293 5.33456 4.55184 +4.76822 +3.91388 +2.10612 +1.18628 +0.99696 +1.03137 0.677065 +Bin 8 +3.16132 5.72964 6.69548 5.33078 3.42046 2.73384 +3.91388 +4.49654 +3.16564 +1.84054 +1.18779 0.952744 0.551066 +Bin 9 +2.65089 4.80248 5.2301 +3.97349 2.33755 1.45442 +2.10612 +3.16564 +3.11186 +2.26779 +1.21667 0.607484 0.292381 +Bin 10 +2.44325 3.39335 3.27912 2.73153 1.86416 1.12182 +1.18628 +1.84054 +2.26779 +2.0669 +1.16694 0.483964 0.189204 +Bin 11 +1.84957 2.11424 1.91573 1.66546 1.34978 0.942322 0.99696 +1.18779 +1.21667 +1.16694 0.874373 0.53914 0.229182 +Bin 12 +1.42184 1.56541 1.50614 1.27175 1.05411 0.896186 1.03137 0.952744 0.607484 0.483964 0.53914 0.531254 0.286004 +Bin 13 +0.91079 1.11514 1.15871 0.999624 0.79947 0.650815 0.677065 0.551066 0.292381 0.189204 0.229182 0.286004 0.189719 + +16 +Additional Smearing Matrix (AC) δpT, 0.5 < cosθp < 0.75 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 13 +Bin 1 +0.361094 +0.157097 +0.00159787 +-0.0525027 +-0.0164129 +0.0293346 +-0.0897963 -0.0598804 -0.000507088 -0.0307189 +0.168133 +0.109396 +-0.0156839 +Bin 2 +0.394091 +0.258055 +0.135352 +-0.021201 +-0.0694085 +-0.0556852 +-0.063316 +0.0372207 +0.122425 +-0.128843 +0.0961489 +0.0377908 -0.0820481 +Bin 3 +0.202522 +0.155812 +0.234224 +0.121415 +-0.0741173 +-0.10587 +0.0703882 +0.118004 +0.136712 +-0.208631 +0.00145837 +-0.0687179 0.0302973 +Bin 4 +0.110807 +0.0205145 +0.119774 +0.193605 +0.0712887 +0.0766614 +0.113022 +0.0631669 +0.0295695 +-0.155374 0.000725702 +-0.118102 +0.0146551 +Bin 5 0.0875124 -0.0179576 +-0.020064 +0.0638735 +0.236955 +0.332884 +0.0947425 +-0.026776 +-0.0572733 +-0.0770127 +0.0541294 +-0.0782122 -0.0137013 +Bin 6 0.0528896 -0.0161488 +-0.0373754 +0.0054937 +0.132424 +0.298916 +0.137771 +0.0173448 +-0.0588866 +-0.036807 +0.0589307 +0.0103269 +0.0590854 +Bin 7 +0.032813 -0.00793252 -0.000843608 +0.0167281 +0.0324233 +0.128871 +0.195332 +0.150645 +0.0259818 +-0.0639448 +0.0768914 +0.0897107 +0.0620261 +Bin 8 0.0382811 +0.0178934 +0.0199347 +0.0217039 +-0.0160172 +-0.0166826 +0.121423 +0.213459 +0.172406 +-0.0135159 +0.103972 +0.0876201 -0.0198279 +Bin 9 0.0827459 +0.0414029 +0.0138085 +0.0122386 +-0.0232598 +-0.0558349 0.00834035 +0.162193 +0.303171 +0.147948 +0.190063 +0.0163241 +-0.119869 +Bin 10 +0.10977 +0.035141 +-0.0108148 +-0.000979296 -0.0163625 +-0.011146 +-0.0383925 +0.0509316 +0.220518 +0.237412 +0.285712 +0.00913533 -0.142502 +Bin 11 0.0946367 +0.0224776 +-0.0129716 +-0.0102307 +-0.0111713 0.00651664 -0.00291654 0.0403758 +0.104238 +0.116976 +0.316761 +0.156866 +-0.0324417 +Bin 12 0.0548384 0.00940786 +0.000978491 +-0.00663006 +-0.0112153 +0.0158145 +0.0306005 +0.0468757 +0.0320912 +-0.0243067 +0.170293 +0.19999 +0.155025 +Bin 13 0.0427095 0.00662653 +0.01137 +0.000525337 -0.00510326 0.0295829 +0.0305672 +0.0288398 +0.0153738 +-0.0492721 +0.0611095 +0.124714 +0.214593 +Cross Section δpT, 0.75 < cosθp < 1 +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +0 +0.05 +26.552912 +3.2405522 +2 +0.05 +0.1 +53.995563 +5.1475835 +3 +0.1 +0.15 +61.37349 +6.059249 +4 +0.15 +0.2 +53.806436 +5.4454423 +5 +0.2 +0.25 +39.185643 +4.0471095 +6 +0.25 +0.3 +23.753144 +3.117621 +7 +0.3 +0.35 +17.783847 +2.7236628 +8 +0.35 +0.4 +15.150194 +2.369364 +9 +0.4 +0.47 +11.095088 +1.8271559 +10 +0.47 +0.55 +8.0520338 +1.5023261 +11 +0.55 +0.65 +4.4671502 +1.1345937 +12 +0.65 +0.9 +1.3417707 +0.47866723 + +17 +Unfolded Covariance Matrix δpT, 0.75 < cosθp < 1 +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 1 +10.5012 14.3341 13.0823 +11.7797 +10.538 +7.67284 +6.5112 +5.47564 +3.72612 +3.22818 +2.70583 +1.11011 +Bin 2 +14.3341 26.4976 28.2801 +23.9494 +17.4712 10.5756 7.96809 +7.07707 +5.72545 +4.96573 +3.47859 +1.21469 +Bin 3 +13.0823 28.2801 36.7145 +31.7993 +18.9252 8.88255 6.92477 +7.90019 +7.0933 +5.44623 +3.04013 0.870473 +Bin 4 +11.7797 23.9494 31.7993 +29.6528 +18.5301 9.01118 7.24016 +8.08449 +6.77767 +4.8243 +2.53208 0.738074 +Bin 5 +10.538 17.4712 18.9252 +18.5301 +16.3791 11.1444 8.79887 +7.17077 +4.67457 +3.37689 +2.1855 +0.762508 +Bin 6 +7.67284 10.5756 8.88255 +9.01118 +11.1444 9.71956 8.03417 +5.60884 +2.87046 +2.05542 +1.66701 +0.67659 +Bin 7 +6.5112 7.96809 6.92477 +7.24016 +8.79887 8.03417 7.41834 +5.71933 +2.98684 +1.98609 +1.5819 +0.669121 +Bin 8 +5.47564 7.07707 7.90019 +8.08449 +7.17077 5.60884 5.71933 +5.61389 +3.76726 +2.51939 +1.63471 0.607925 +Bin 9 +3.72612 5.72545 +7.0933 +6.77767 +4.67457 2.87046 2.98684 +3.76726 +3.3385 +2.51902 +1.47074 0.456624 +Bin 10 +3.22818 4.96573 5.44623 +4.8243 +3.37689 2.05542 1.98609 +2.51939 +2.51902 +2.25698 +1.49992 0.494619 +Bin 11 +2.70583 3.47859 3.04013 +2.53208 +2.1855 +1.66701 +1.5819 +1.63471 +1.47074 +1.49992 +1.2873 +0.509975 +Bin 12 +1.11011 1.21469 0.870473 0.738074 0.762508 0.67659 0.669121 0.607925 0.456624 0.494619 0.509975 0.229122 +Additional Smearing Matrix (AC) δpT, 0.75 < cosθp < 1 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 1 +0.168635 +0.173589 +0.0233497 +-0.043763 +0.0108795 +-0.010602 +0.0405129 +0.0297928 +-0.0501185 +0.024681 +0.0881269 +0.106928 +Bin 2 +0.146722 +0.341233 +0.16914 +0.0151518 +-0.00291148 -0.0184496 -0.00602801 -0.0407253 +-0.0326476 +0.100236 +0.126551 +0.0595994 +Bin 3 -0.0391953 +0.244843 +0.32395 +0.180923 +-0.000730359 -0.101702 +-0.0503848 +-0.0152166 +0.0765204 +0.119403 +0.0462604 +-0.0427378 +Bin 4 -0.0367595 +0.122334 +0.232006 +0.247073 +0.0914112 +-0.0512559 -0.0069041 +0.045256 +0.0911993 +0.0886454 -0.0722632 -0.0655028 +Bin 5 +0.0783508 +0.0864368 +0.0564209 +0.0811769 +0.191763 +0.14799 +0.116726 +0.0650163 +0.00371581 0.00277014 -0.0578205 +-0.054776 +Bin 6 +0.0944449 +0.0611226 +-0.0281652 +-0.0324663 +0.139802 +0.202734 +0.183965 +0.0906654 +-0.027448 +-0.0507775 -0.0327151 -0.0140822 +Bin 7 +0.0708966 +0.0256786 +-0.0318613 +-0.0393797 +0.0788641 +0.127198 +0.193988 +0.151102 +0.0170168 +-0.0422631 -0.0482703 +0.012632 +Bin 8 +0.0351208 0.00338878 -0.0049528 -0.00956504 +0.0241403 +0.0125591 +0.112063 +0.162545 +0.115029 +0.0385198 +-0.034845 -0.00741034 +Bin 9 +0.0121434 +0.0255679 +0.0241856 +0.00100373 +-0.0141261 +-0.0653597 0.00264147 +0.094818 +0.186969 +0.174473 +0.0641824 +-0.0498915 +Bin 10 +0.02942 +0.0693101 +0.0272491 +-0.0247956 +-0.0313353 +-0.0784008 -0.0459289 -0.00189596 +0.101992 +0.216443 +0.211549 +0.00245751 +Bin 11 0.0553797 +0.0905255 +0.0192978 +-0.0545427 +-0.0382673 +-0.0609924 -0.0300345 +-0.0318398 +-0.0113708 +0.112162 +0.304671 +0.14091 +Bin 12 +0.069786 +0.090838 +0.00764978 -0.0582913 +-0.042026 +-0.0567283 -0.00702852 -0.0201669 +-0.0690734 +0.0313565 +0.287963 +0.257871 +Cross Section δαT, All events +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.047295737 +0.0069864868 +2 +22 +44 +0.044962891 +0.0058597861 +3 +44 +66 +0.044549083 +0.005667204 +4 +66 +88 +0.05044147 +0.007040037 +5 +88 +110 +0.066811541 +0.0084041599 +6 +110 +145 +0.078273769 +0.008722771 +7 +145 +180 +0.090827836 +0.010113554 + +18 +Unfolded Covariance Matrix δαT, All events +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +4.8811e-05 3.84145e-05 2.38915e-05 1.66442e-05 2.57368e-05 3.65219e-05 4.61539e-05 +Bin 2 +3.84145e-05 3.43371e-05 2.64859e-05 2.28022e-05 3.19373e-05 3.56387e-05 4.45764e-05 +Bin 3 +2.38915e-05 2.64859e-05 3.21172e-05 3.70358e-05 3.96159e-05 4.05784e-05 4.26981e-05 +Bin 4 +1.66442e-05 2.28022e-05 3.70358e-05 4.95621e-05 5.15219e-05 5.02396e-05 4.86069e-05 +Bin 5 +2.57368e-05 3.19373e-05 3.96159e-05 5.15219e-05 7.06299e-05 6.28347e-05 7.15501e-05 +Bin 6 +3.65219e-05 3.56387e-05 4.05784e-05 5.02396e-05 6.28347e-05 7.60867e-05 8.42981e-05 +Bin 7 +4.61539e-05 4.45764e-05 4.26981e-05 4.86069e-05 7.15501e-05 8.42981e-05 0.000102284 +Additional Smearing Matrix (AC) δαT, All events +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.54324 +0.413147 +0.171535 +-0.18549 +-0.0143658 +0.143843 +-0.109214 +Bin 2 +0.322934 +0.38776 +0.26295 +-0.0440761 +0.10087 +0.0292975 +-0.070621 +Bin 3 0.0898424 +0.223852 +0.310898 +0.289284 +0.177512 +-0.0261531 -0.0601582 +Bin 4 -0.0385126 0.0430092 0.219993 +0.492733 +0.317566 +-0.00458755 -0.0436167 +Bin 5 -0.0833197 0.0616928 0.0429507 +0.215413 +0.591386 +0.0607463 +0.0501594 +Bin 6 -0.0182562 0.0894267 -0.149783 -0.00911983 +0.397323 +0.617785 +0.126175 +Bin 7 -0.0773432 +0.25616 +-0.229384 +-0.425517 +0.483276 +0.676898 +0.299715 +Cross Section δαT, δpT < 0.2 GeV/c +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +22 +0.12696689 +0.022681076 +2 +22 +44 +0.14524449 +0.018478608 +3 +44 +66 +0.14734428 +0.017921719 +4 +66 +88 +0.14897292 +0.01917038 +5 +88 +110 +0.18659596 +0.020385736 +6 +110 +145 +0.1727567 +0.018089571 +7 +145 +180 +0.13435265 +0.020784081 + +19 +Unfolded Covariance Matrix δαT, δpT < 0.2 GeV/c +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.000514431 0.00033728 0.000161703 0.000116762 0.000208306 0.000253654 0.000229816 +Bin 2 +0.00033728 0.000341459 0.000252421 0.000163471 0.000207861 0.000228156 0.000253011 +Bin 3 +0.000161703 0.000252421 0.000321188 0.000289019 0.000217812 0.00019247 0.000259027 +Bin 4 +0.000116762 0.000163471 0.000289019 0.000367503 0.00029098 0.000197517 0.000232392 +Bin 5 +0.000208306 0.000207861 0.000217812 0.00029098 0.000415578 0.000310562 0.000208751 +Bin 6 +0.000253654 0.000228156 0.00019247 0.000197517 0.000310562 0.000327233 0.00025965 +Bin 7 +0.000229816 0.000253011 0.000259027 0.000232392 0.000208751 0.00025965 0.000431978 +Additional Smearing Matrix (AC) δαT, δpT < 0.2 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.559612 +0.248703 +0.0373883 +-0.167615 0.00392543 +0.083823 +0.0605731 +Bin 2 +0.264683 +0.297304 +0.213337 +-0.0484846 0.0376626 +0.0495284 +0.0116345 +Bin 3 0.0135524 +0.182539 +0.355916 +0.214361 +0.10802 +0.00104184 +-0.04539 +Bin 4 -0.0548622 0.0288009 +0.281731 +0.37798 +0.258737 +-0.00659288 -0.0536045 +Bin 5 -0.0204662 0.0143231 +0.121577 +0.215942 +0.36559 +0.144132 +-0.0283038 +Bin 6 0.0687986 0.0551935 0.0281568 -0.0033156 +0.250874 +0.378035 +0.120432 +Bin 7 +0.127228 +0.079356 -0.0493917 -0.054984 +0.0834252 +0.123854 +0.38951 +Cross Section δαT, 0.2 < δpT < 0.4 GeV/c +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +22 +0.056762524 +0.012483565 +2 +22 +44 +0.046754186 +0.011513164 +3 +44 +66 +0.052380082 +0.011718981 +4 +66 +88 +0.064783823 +0.012796541 +5 +88 +110 +0.091853641 +0.015914447 +6 +110 +145 +0.136195 +0.017895037 +7 +145 +180 +0.11214728 +0.01744216 + +20 +Unfolded Covariance Matrix δαT, 0.2 < δpT < 0.4 GeV/c +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.000155839 0.000108052 6.43911e-05 8.32881e-05 0.000105363 9.27143e-05 9.73866e-05 +Bin 2 +0.000108052 0.000132553 0.000107214 6.51669e-05 7.29807e-05 +0.00011982 0.000117103 +Bin 3 +6.43911e-05 0.000107214 0.000137335 8.80781e-05 7.19338e-05 0.000130423 0.000132043 +Bin 4 +8.32881e-05 6.51669e-05 8.80781e-05 0.000163751 0.00015084 0.000110929 0.000117216 +Bin 5 +0.000105363 7.29807e-05 7.19338e-05 +0.00015084 +0.00025327 0.000181215 0.000143795 +Bin 6 +9.27143e-05 +0.00011982 0.000130423 0.000110929 0.000181215 0.000320232 0.000267749 +Bin 7 +9.73866e-05 0.000117103 0.000132043 0.000117216 0.000143795 0.000267749 0.000304229 +Additional Smearing Matrix (AC) δαT, 0.2 < δpT < 0.4 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.442678 +0.229535 +0.0698266 -0.0294075 0.0779904 +-0.039554 +-0.0728318 +Bin 2 +0.194213 +0.382143 +0.34196 +-0.0621195 -0.048223 -0.00398776 -0.0263515 +Bin 3 -0.0800074 +0.180925 +0.582621 +0.072973 +-0.0772721 -0.00153366 0.0147672 +Bin 4 -0.0466268 -0.0680143 +0.236749 +0.343846 +0.198355 +-0.0251352 -0.0319901 +Bin 5 -0.0407601 -0.126877 -0.0267461 +0.117259 +0.565995 +0.101553 +-0.0496206 +Bin 6 -0.233464 -0.0752689 +0.12766 +-0.117169 +0.233077 +0.553286 +0.226571 +Bin 7 -0.188836 -0.0742107 +0.25044 +-0.101112 -0.0125217 +0.276288 +0.417135 +Cross Section δαT, δpT > 0.4 GeV/c +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +22 +0.0022692936 +0.0014399406 +2 +22 +44 +0.00095139453 +0.0016241049 +3 +44 +66 +0.0026372474 +0.0021132661 +4 +66 +88 +0.006840357 +0.0027823209 +5 +88 +110 +0.0068013283 +0.003786716 +6 +110 +145 +0.022077111 +0.0052624245 +7 +145 +180 +0.045353622 +0.0087474574 + +21 +Unfolded Covariance Matrix δαT, δpT > 0.4 GeV/c +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +2.07343e-06 1.32538e-06 9.86279e-07 1.87269e-06 3.02102e-06 4.62624e-06 7.90893e-06 +Bin 2 +1.32538e-06 2.63772e-06 2.41612e-06 2.26963e-06 3.67524e-06 5.80935e-06 6.00665e-06 +Bin 3 +9.86279e-07 2.41612e-06 4.46589e-06 4.12072e-06 5.34756e-06 6.75775e-06 7.52209e-06 +Bin 4 +1.87269e-06 2.26963e-06 4.12072e-06 7.74131e-06 8.43535e-06 9.73762e-06 1.43579e-05 +Bin 5 +3.02102e-06 3.67524e-06 5.34756e-06 8.43535e-06 1.43392e-05 1.64628e-05 1.86402e-05 +Bin 6 +4.62624e-06 5.80935e-06 6.75775e-06 9.73762e-06 1.64628e-05 2.76931e-05 3.7214e-05 +Bin 7 +7.90893e-06 6.00665e-06 7.52209e-06 1.43579e-05 1.86402e-05 3.7214e-05 +7.6518e-05 +Additional Smearing Matrix (AC) δαT, δpT > 0.4 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.657651 +0.127235 +-0.0763675 0.00246771 0.0195514 -9.31444e-05 0.000206455 +Bin 2 +0.324516 +0.393488 +0.150774 +-0.011284 +0.0111971 +0.014866 +-0.0120529 +Bin 3 -0.0171345 +0.216754 +0.46534 +0.168085 +0.0197781 +0.00380935 +-0.019274 +Bin 4 0.00355798 0.0241932 +0.071658 +0.747657 +0.0753558 +-0.0180982 +-0.020241 +Bin 5 +0.282211 +0.00441528 -0.00451394 +0.354826 +0.436976 +0.0990078 +-0.0553995 +Bin 6 +0.728718 +0.303365 +-0.0346546 +0.0681549 +0.289845 +0.476659 +0.0407148 +Bin 7 +1.41801 +0.240485 +-0.326576 +0.505835 +-0.475475 +0.307137 +0.52029 +Cross Section δαT, –1 < cosθμ < 0 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.0070114389 +0.0015306301 +2 +22 +44 +0.0087157755 +0.0013189318 +3 +44 +66 +0.0092671614 +0.0014031435 +4 +66 +88 +0.0085535342 +0.0012989735 +5 +88 +110 +0.010332378 +0.0014024775 +6 +110 +145 +0.012607021 +0.0013251007 +7 +145 +180 +0.013567421 +0.0015270408 + +22 +Unfolded Covariance Matrix δαT, –1 < cosθμ < 0 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +2.34283e-06 1.59166e-06 9.59126e-07 7.83249e-07 9.26983e-07 1.00775e-06 1.50872e-06 +Bin 2 +1.59166e-06 1.73958e-06 1.51974e-06 1.01556e-06 9.80254e-07 1.30153e-06 1.60824e-06 +Bin 3 +9.59126e-07 1.51974e-06 1.96881e-06 1.51658e-06 1.08321e-06 1.4886e-06 1.45628e-06 +Bin 4 +7.83249e-07 1.01556e-06 1.51658e-06 1.68733e-06 1.49869e-06 1.23385e-06 1.01977e-06 +Bin 5 +9.26983e-07 9.80254e-07 1.08321e-06 1.49869e-06 1.96694e-06 1.23942e-06 1.05445e-06 +Bin 6 +1.00775e-06 1.30153e-06 1.4886e-06 1.23385e-06 1.23942e-06 1.75589e-06 1.81435e-06 +Bin 7 +1.50872e-06 1.60824e-06 1.45628e-06 1.01977e-06 1.05445e-06 1.81435e-06 2.33185e-06 +Additional Smearing Matrix (AC) δpT, –1 < cosθμ < 0 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.330799 +0.307756 +0.024128 +-0.033425 -0.0437487 0.0387658 0.0978323 +Bin 2 +0.17292 +0.301276 +0.131722 0.0169992 -0.0324116 0.0725589 +0.090314 +Bin 3 +0.0431483 +0.171525 +0.244845 +0.16495 +0.0265722 +0.105136 +0.0731724 +Bin 4 0.00118167 0.0471798 +0.16677 +0.284999 +0.227845 +0.0855244 0.0117181 +Bin 5 -0.00662461 0.0504072 0.0595483 0.227219 +0.358279 +0.127223 -0.0358269 +Bin 6 0.00362009 +0.127122 +0.119235 +0.131012 +0.0432659 +0.338056 +0.165895 +Bin 7 +0.0827384 +0.20772 +0.0539231 0.0120919 -0.204875 +0.208733 +0.375975 +Cross Section δαT, 0 < cosθμ < 0.5 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.018649992 +0.0028683018 +2 +22 +44 +0.016519118 +0.0024557039 +3 +44 +66 +0.017917435 +0.0032169574 +4 +66 +88 +0.020440461 +0.0027421746 +5 +88 +110 +0.026117712 +0.0036023637 +6 +110 +145 +0.035285343 +0.0036686218 +7 +145 +180 +0.04006913 +0.0044840765 + +23 +Unfolded Covariance Matrix δαT, 0 < cosθμ < 0.5 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +8.22716e-06 4.51184e-06 1.69233e-06 4.77169e-06 6.81416e-06 5.22029e-06 6.10765e-06 +Bin 2 +4.51184e-06 6.03048e-06 6.28396e-06 4.54581e-06 4.47987e-06 5.33931e-06 6.79248e-06 +Bin 3 +1.69233e-06 6.28396e-06 1.03488e-05 5.57506e-06 2.54292e-06 6.83561e-06 8.83866e-06 +Bin 4 +4.77169e-06 4.54581e-06 5.57506e-06 7.51952e-06 7.07922e-06 6.42015e-06 7.19672e-06 +Bin 5 +6.81416e-06 4.47987e-06 2.54292e-06 7.07922e-06 1.2977e-05 8.68761e-06 7.3761e-06 +Bin 6 +5.22029e-06 5.33931e-06 6.83561e-06 6.42015e-06 8.68761e-06 1.34588e-05 1.38999e-05 +Bin 7 +6.10765e-06 6.79248e-06 8.83866e-06 7.19672e-06 7.3761e-06 1.38999e-05 2.01069e-05 +Additional Smearing Matrix (AC) δpT, 0 < cosθμ < 0.5 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.366512 +0.149991 +-0.019586 0.111463 +0.0965606 +-0.00385937 -0.0586131 +Bin 2 +0.135439 +0.276075 +0.231797 +0.097857 +0.0470108 +0.000220748 -0.0543718 +Bin 3 -0.0934833 +0.2038 +0.436329 +0.171209 -0.00612483 +0.0287628 +-0.0186071 +Bin 4 0.0523487 +0.0769526 +0.183949 +0.34921 +0.162031 +0.0144383 +-0.0266592 +Bin 5 +0.103209 +0.0217093 -0.0169467 0.233547 +0.376123 +0.0857796 +-0.0561585 +Bin 6 -0.126351 -0.0457043 +0.113053 +0.132987 +0.169706 +0.374586 +0.038569 +Bin 7 -0.121574 +-0.058707 +0.151933 +0.101777 0.00603485 +0.220846 +0.226775 +Cross Section δαT, 0.5 < cosθμ < 0.75 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.039046444 +0.0057487733 +2 +22 +44 +0.033051397 +0.005611716 +3 +44 +66 +0.036932838 +0.0070931129 +4 +66 +88 +0.043848093 +0.0078297063 +5 +88 +110 +0.063839561 +0.0092899599 +6 +110 +145 +0.078182522 +0.0098765961 +7 +145 +180 +0.081395804 +0.010941566 + +24 +Unfolded Covariance Matrix δαT, 0.5 < cosθμ < 0.75 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +3.30484e-05 2.66353e-05 1.94967e-05 1.98935e-05 2.64478e-05 3.24208e-05 +4.755e-05 +Bin 2 +2.66353e-05 3.14914e-05 3.34037e-05 2.90644e-05 3.14246e-05 3.80814e-05 4.51267e-05 +Bin 3 +1.94967e-05 3.34037e-05 5.03123e-05 4.6838e-05 4.71851e-05 5.33051e-05 4.98435e-05 +Bin 4 +1.98935e-05 2.90644e-05 4.6838e-05 6.13043e-05 6.77563e-05 5.8509e-05 +5.36044e-05 +Bin 5 +2.64478e-05 3.14246e-05 4.71851e-05 6.77563e-05 8.63034e-05 7.92745e-05 7.04889e-05 +Bin 6 +3.24208e-05 3.80814e-05 5.33051e-05 5.8509e-05 7.92745e-05 9.75472e-05 8.99005e-05 +Bin 7 +4.755e-05 +4.51267e-05 4.98435e-05 5.36044e-05 7.04889e-05 8.99005e-05 0.000119718 +Additional Smearing Matrix (AC) δpT, 0.5 < cosθμ < 0.75 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.409038 +0.163963 0.00992446 -0.0155609 0.0433038 -0.0230725 0.00840302 +Bin 2 +0.260699 +0.23826 +0.17837 +0.0446944 0.0322216 -0.00868523 -0.0196311 +Bin 3 0.00979316 0.176353 +0.329928 +0.161007 +0.0815721 +0.015818 +-0.0413037 +Bin 4 -0.0118926 +0.065293 +0.217609 +0.286308 +0.267722 +0.0143675 +-0.0495148 +Bin 5 -0.0040938 +0.012179 +0.0754747 +0.217616 +0.393315 +0.0887314 +-0.040917 +Bin 6 -0.0760158 0.0342302 +0.09638 +0.0501902 +0.362221 +0.273206 +0.0004465 +Bin 7 +0.19708 +0.058274 0.00893371 -0.0504265 0.201722 +0.0693701 +0.192261 +Cross Section δαT, 0.75 < cosθμ < 1 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.075870972 +0.011265281 +2 +22 +44 +0.066290677 +0.0098234579 +3 +44 +66 +0.06491299 +0.0095190088 +4 +66 +88 +0.075017427 +0.011947928 +5 +88 +110 +0.086159107 +0.014835648 +6 +110 +145 +0.084639493 +0.016739056 +7 +145 +180 +0.094523602 +0.019711266 + +25 +Unfolded Covariance Matrix δαT, 0.75 < cosθμ < 1 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.000126907 8.8084e-05 5.97118e-05 6.51985e-05 7.89206e-05 9.26714e-05 0.000126308 +Bin 2 +8.8084e-05 +9.65003e-05 7.68089e-05 5.33299e-05 7.29866e-05 8.28571e-05 7.77742e-05 +Bin 3 +5.97118e-05 7.68089e-05 9.06115e-05 8.93952e-05 9.85362e-05 9.36401e-05 8.37594e-05 +Bin 4 +6.51985e-05 5.33299e-05 8.93952e-05 0.000142753 0.000152375 0.000133901 0.000153868 +Bin 5 +7.89206e-05 7.29866e-05 9.85362e-05 0.000152375 0.000220096 0.000225933 0.000244789 +Bin 6 +9.26714e-05 8.28571e-05 9.36401e-05 0.000133901 0.000225933 0.000280196 0.000305343 +Bin 7 +0.000126308 7.77742e-05 8.37594e-05 0.000153868 0.000244789 0.000305343 0.000388534 +Additional Smearing Matrix (AC) δpT, 0.75 < cosθμ < 1 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 0.425947 0.304025 0.00262488 +-0.0225025 +-0.0451313 +0.0014715 +0.0358877 +Bin 2 0.289935 0.396661 +0.180242 +-0.000187595 +0.0298272 +0.00638319 -0.0335174 +Bin 3 0.135507 0.273836 +0.329648 +0.202551 +0.0787246 +-0.0441135 -0.0517179 +Bin 4 0.10764 0.121265 +0.279203 +0.372299 +0.125789 +-0.100968 +0.0120798 +Bin 5 0.110021 0.167548 +0.17237 +0.159153 +0.18863 +-0.0227962 0.0940123 +Bin 6 0.227898 0.29279 +0.0326139 +-0.0676499 +-0.00810853 +0.194023 +0.305873 +Bin 7 0.378786 0.232676 +-0.176513 +-0.0992115 +-0.150365 +0.10993 +0.566057 +Cross Section δαT, –1 < cosθp < 0 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.0033470124 +0.0014792221 +2 +22 +44 +0.0032858289 +0.0011873971 +3 +44 +66 +0.0034797314 +0.0012390371 +4 +66 +88 +0.0039534109 +0.0012219203 +5 +88 +110 +0.0059104064 +0.0012232133 +6 +110 +145 +0.0081354051 +0.0014539827 +7 +145 +180 +0.0087585646 +0.0013637679 + +26 +Unfolded Covariance Matrix δαT, –1 < cosθp < 0 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +2.1881e-06 1.57807e-06 1.06945e-06 9.21849e-07 1.20172e-06 1.25221e-06 8.79356e-07 +Bin 2 +1.57807e-06 1.40991e-06 1.28098e-06 1.10909e-06 +1.046e-06 +9.00749e-07 6.40266e-07 +Bin 3 +1.06945e-06 1.28098e-06 1.53521e-06 1.41945e-06 1.04956e-06 5.96465e-07 4.65152e-07 +Bin 4 +9.21849e-07 1.10909e-06 1.41945e-06 1.49309e-06 1.2193e-06 6.41929e-07 5.34355e-07 +Bin 5 +1.20172e-06 +1.046e-06 +1.04956e-06 1.2193e-06 1.49625e-06 1.37198e-06 1.15677e-06 +Bin 6 +1.25221e-06 9.00749e-07 5.96465e-07 6.41929e-07 1.37198e-06 2.11407e-06 1.81519e-06 +Bin 7 +8.79356e-07 6.40266e-07 4.65152e-07 5.34355e-07 1.15677e-06 1.81519e-06 1.85986e-06 +Additional Smearing Matrix (AC) δαT, –1 < cosθp < 0 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 0.353665 +0.197248 +0.143426 +-0.0247056 0.096808 +0.0888466 -0.0418628 +Bin 2 0.186275 +0.200818 +0.25582 +0.0327222 0.0797883 0.0417735 -0.0359414 +Bin 3 0.0398379 0.182496 +0.35627 +0.117344 +0.113199 -0.0204464 -0.0299969 +Bin 4 0.0154123 0.118764 +0.288282 +0.159889 +0.200098 -0.0380683 -0.0330582 +Bin 5 0.0859166 0.0694509 +0.128086 +0.062132 +0.250404 +0.0624778 -0.0297557 +Bin 6 0.174453 0.0933979 0.00524077 -0.158033 +0.18521 +0.34685 +0.00390055 +Bin 7 0.078135 0.0345203 +-0.040221 +-0.184097 +0.129048 +0.25827 +0.0376223 +Cross Section δαT, 0 < cosθp < 0.5 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.026104529 +0.0040353837 +2 +22 +44 +0.024505852 +0.0037360674 +3 +44 +66 +0.024197958 +0.0038524767 +4 +66 +88 +0.027064782 +0.0042700252 +5 +88 +110 +0.034754005 +0.0048521009 +6 +110 +145 +0.031651427 +0.0042703155 +7 +145 +180 +0.021955772 +0.0030895824 + +27 +Unfolded Covariance Matrix δαT, 0 < cosθp < 0.5 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +1.62843e-05 1.3511e-05 9.63143e-06 8.01442e-06 9.87581e-06 8.70088e-06 6.29121e-06 +Bin 2 +1.3511e-05 1.39582e-05 1.18765e-05 1.03869e-05 1.11568e-05 9.67926e-06 7.03057e-06 +Bin 3 +9.63143e-06 1.18765e-05 1.48416e-05 1.48149e-05 1.41359e-05 1.08329e-05 7.29988e-06 +Bin 4 +8.01442e-06 1.03869e-05 1.48149e-05 1.82331e-05 1.89368e-05 1.30213e-05 8.63293e-06 +Bin 5 +9.87581e-06 1.11568e-05 1.41359e-05 1.89368e-05 2.35429e-05 1.83041e-05 1.20837e-05 +Bin 6 +8.70088e-06 9.67926e-06 1.08329e-05 1.30213e-05 1.83041e-05 1.82356e-05 1.23339e-05 +Bin 7 +6.29121e-06 7.03057e-06 7.29988e-06 8.63293e-06 1.20837e-05 1.23339e-05 9.54552e-06 +Additional Smearing Matrix (AC) δαT, 0 < cosθp < 0.5 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.380503 +0.299028 0.159307 -0.0340054 +0.0057237 +0.0378798 -0.050901 +Bin 2 +0.269668 +0.30467 +0.239139 +0.0493354 -0.00927125 0.0453116 -0.0194677 +Bin 3 0.0659561 +0.17779 +0.365896 +0.186955 +0.0140686 +0.0455855 -0.0415884 +Bin 4 -0.0597793 0.105096 0.293758 +0.261671 +0.127807 +0.0551629 -0.0236081 +Bin 5 -0.0626226 0.104576 0.166897 +0.172745 +0.19565 +0.179574 +0.0254134 +Bin 6 -0.0436541 0.159144 0.143877 +0.0241893 +0.111338 +0.438064 +0.120016 +Bin 7 -0.0296371 0.119989 0.0687627 -0.007604 +0.041625 +0.245946 +0.133478 +Cross Section δαT, 0.5 < cosθp < 0.75 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.062216152 +0.0089630236 +2 +22 +44 +0.059951055 +0.0078957292 +3 +44 +66 +0.059177472 +0.0085398267 +4 +66 +88 +0.065201941 +0.0093877473 +5 +88 +110 +0.082828711 +0.010863918 +6 +110 +145 +0.078871598 +0.010172356 +7 +145 +180 +0.065773388 +0.0086853806 + +28 +Unfolded Covariance Matrix δαT, 0.5 < cosθp < 0.75 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +8.03358e-05 5.7748e-05 3.76609e-05 4.39106e-05 5.98196e-05 5.11445e-05 3.96994e-05 +Bin 2 +5.7748e-05 6.23425e-05 5.59059e-05 5.21495e-05 5.81918e-05 5.09319e-05 3.99492e-05 +Bin 3 +3.76609e-05 5.59059e-05 7.29286e-05 6.94112e-05 5.97778e-05 4.56153e-05 3.77974e-05 +Bin 4 +4.39106e-05 5.21495e-05 6.94112e-05 8.81298e-05 8.49071e-05 5.74431e-05 4.56942e-05 +Bin 5 +5.98196e-05 5.81918e-05 5.97778e-05 8.49071e-05 0.000118025 9.81096e-05 6.96303e-05 +Bin 6 +5.11445e-05 5.09319e-05 4.56153e-05 5.74431e-05 9.81096e-05 0.000103477 7.8964e-05 +Bin 7 +3.96994e-05 3.99492e-05 3.77974e-05 4.56942e-05 6.96303e-05 +7.8964e-05 +7.54358e-05 +Additional Smearing Matrix (AC) δαT, 0.5 < cosθp < 0.75 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.334472 +0.249471 0.0270459 -0.00610521 0.0292026 +0.0089557 +-0.0484225 +Bin 2 +0.174902 +0.343491 +0.215734 +0.0607089 +0.0372224 0.00220286 -0.0208021 +Bin 3 0.00125669 0.277549 +0.374603 +0.192053 +0.0634971 -0.0211967 -0.0392409 +Bin 4 0.0260908 0.188651 +0.245967 +0.287528 +0.189263 -0.00217654 -0.0597624 +Bin 5 +0.101162 +0.169468 0.0608076 +0.126747 +0.294648 +0.143136 +-0.0101688 +Bin 6 +0.129998 +0.211603 -0.0179664 -0.0793443 +0.289785 +0.343654 +0.148302 +Bin 7 0.0454118 0.124145 -0.0316148 +-0.068097 +0.136798 +0.196956 +0.232458 +Cross Section δαT, 0.75 < cosθp < 1 +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +22 +0.04591813 +0.0069123709 +2 +22 +44 +0.037772034 +0.0062443702 +3 +44 +66 +0.037785815 +0.007421952 +4 +66 +88 +0.047121348 +0.0079437151 +5 +88 +110 +0.077744533 +0.0098333753 +6 +110 +145 +0.11889674 +0.011720873 +7 +145 +180 +0.18746467 +0.019375934 + +29 +Unfolded Covariance Matrix δαT, 0.75 < cosθp < 1 +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +4.77809e-05 3.35626e-05 2.08944e-05 2.22106e-05 +3.8981e-05 +4.88853e-05 6.95466e-05 +Bin 2 +3.35626e-05 3.89922e-05 3.72422e-05 3.27322e-05 3.29349e-05 3.92358e-05 +5.8936e-05 +Bin 3 +2.08944e-05 3.72422e-05 5.50854e-05 5.16498e-05 3.38577e-05 3.36478e-05 +6.3498e-05 +Bin 4 +2.22106e-05 3.27322e-05 5.16498e-05 6.31026e-05 +5.5915e-05 +4.90748e-05 7.33681e-05 +Bin 5 +3.8981e-05 3.29349e-05 3.38577e-05 5.5915e-05 +9.66953e-05 9.75465e-05 0.000106435 +Bin 6 +4.88853e-05 3.92358e-05 3.36478e-05 4.90748e-05 9.75465e-05 0.000137379 0.00019473 +Bin 7 +6.95466e-05 5.8936e-05 +6.3498e-05 7.33681e-05 0.000106435 0.00019473 0.000375427 +Additional Smearing Matrix (AC) δαT, 0.75 < cosθp < 1 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 1 +0.53353 +0.274614 0.00706012 -0.0991495 0.0393246 0.0100931 -0.0223494 +Bin 2 0.385527 +0.339548 +0.175278 +0.0538596 0.0175478 -0.0109439 -0.0297243 +Bin 3 0.137504 +0.222149 +0.339707 +0.281916 +0.0333939 -0.0334954 -0.0312188 +Bin 4 0.0152392 0.0803645 +0.258554 +0.382452 +0.189684 +0.0140591 -0.0483661 +Bin 5 0.0883442 0.0254829 0.00783223 +0.167434 +0.419872 +0.164006 +-0.0479162 +Bin 6 0.141884 0.0461857 +-0.203894 +-0.0753264 0.385821 +0.411758 +0.133568 +Bin 7 0.105821 0.0145124 +-0.225604 +-0.17243 +-0.032952 +0.430741 +0.522392 + +30 +Cross Section δφT, All events +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg 40Ar] Uncertainty [10–38 +cm2 +deg 40Ar] +1 +0 +12.5 +0.31441652 +0.032284057 +2 +12.5 +25 +0.20358551 +0.01870319 +3 +25 +37.5 +0.10729928 +0.010816997 +4 +37.5 +50 +0.062616734 +0.0076583118 +5 +50 +60 +0.049715971 +0.0079764994 +6 +60 +75 +0.030944993 +0.0054822582 +7 +75 +90 +0.024593059 +0.0046791168 +8 +90 +106 +0.020754743 +0.0043203551 +9 +106 +126 +0.016671524 +0.0034825657 +10 +126 +145 +0.013700322 +0.0032524063 +11 +145 +162 +0.011648936 +0.003485876 +12 +162 +180 +0.010095428 +0.0036017253 +Unfolded Covariance Matrix δφT, All events +Units in [10–38 +cm2 +deg 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 1 +0.00104226 0.000565695 0.000212993 0.000129284 0.00014732 0.000103356 7.71478e-05 7.39125e-05 7.19796e-05 6.6085e-05 5.66317e-05 4.66297e-05 +Bin 2 +0.000565695 0.000349809 0.000166989 9.79334e-05 8.85369e-05 6.08468e-05 5.17649e-05 4.8598e-05 4.32292e-05 3.83852e-05 3.3036e-05 2.71163e-05 +Bin 3 +0.000212993 0.000166989 0.000117007 7.29332e-05 5.00803e-05 3.12038e-05 3.14489e-05 2.83589e-05 2.12085e-05 1.76343e-05 1.64419e-05 1.47933e-05 +Bin 4 +0.000129284 9.79334e-05 7.29332e-05 5.86497e-05 5.19122e-05 3.13999e-05 +2.3882e-05 1.89824e-05 1.49976e-05 1.42339e-05 1.53562e-05 1.52523e-05 +Bin 5 +0.00014732 +8.85369e-05 5.00803e-05 5.19122e-05 6.36245e-05 4.11897e-05 2.61136e-05 1.8979e-05 1.71083e-05 +1.759e-05 +1.85968e-05 1.81645e-05 +Bin 6 +0.000103356 6.08468e-05 3.12038e-05 3.13999e-05 4.11897e-05 3.00552e-05 2.19022e-05 1.69321e-05 1.42942e-05 1.32223e-05 1.24432e-05 1.11306e-05 +Bin 7 +7.71478e-05 5.17649e-05 3.14489e-05 +2.3882e-05 +2.61136e-05 2.19022e-05 2.18941e-05 1.92413e-05 1.43235e-05 1.10321e-05 8.66788e-06 6.74309e-06 +Bin 8 +7.39125e-05 +4.8598e-05 +2.83589e-05 1.89824e-05 +1.8979e-05 +1.69321e-05 1.92413e-05 1.86655e-05 1.41362e-05 1.04727e-05 7.91366e-06 5.99661e-06 +Bin 9 +7.19796e-05 4.32292e-05 2.12085e-05 1.49976e-05 1.71083e-05 1.42942e-05 1.43235e-05 1.41362e-05 1.21283e-05 1.03374e-05 8.82126e-06 7.31607e-06 +Bin 10 +6.6085e-05 +3.83852e-05 1.76343e-05 1.42339e-05 +1.759e-05 +1.32223e-05 1.10321e-05 1.04727e-05 1.03374e-05 1.05781e-05 1.05216e-05 9.59436e-06 +Bin 11 +5.66317e-05 +3.3036e-05 +1.64419e-05 1.53562e-05 1.85968e-05 1.24432e-05 8.66788e-06 7.91366e-06 8.82126e-06 1.05216e-05 1.21513e-05 1.21733e-05 +Bin 12 +4.66297e-05 2.71163e-05 1.47933e-05 1.52523e-05 1.81645e-05 1.11306e-05 6.74309e-06 5.99661e-06 7.31607e-06 9.59436e-06 1.21733e-05 1.29724e-05 + +31 +Additional Smearing Matrix (AC) δφT, All events +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 12 +Bin 1 +0.381677 +0.71643 +0.309001 +-0.287563 +0.124765 +0.0761652 +-0.516932 +-0.083428 +0.532529 +0.55729 +-0.295725 +-0.539467 +Bin 2 +0.0826612 +0.542361 +0.537582 +-0.0483494 +-0.042471 +0.0152274 +-0.0905658 -0.143624 +0.173102 +0.336114 -0.0942919 -0.305344 +Bin 3 -0.0504963 0.194702 +0.627858 +0.22029 +-0.0313114 -0.00398141 +0.154973 +-0.139891 -0.0581769 0.103333 -0.0115146 -0.0463378 +Bin 4 -0.0540718 0.0498613 +0.313394 +0.3257 +0.25885 +0.191202 +0.0818221 +-0.200978 -0.0610753 0.0523683 -0.0323176 0.0591564 +Bin 5 -0.0306164 0.0220316 +0.040342 +0.194026 +0.377733 +0.338733 +0.0711508 +-0.174145 +0.0103402 0.0720419 -0.0814814 0.0441278 +Bin 6 -0.0330035 0.024731 0.00310732 0.0624084 +0.270565 +0.422856 +0.262603 +-0.0418201 0.0733859 +0.112627 +-0.125328 -0.0194687 +Bin 7 -0.0331649 0.011074 +0.0893456 +-0.0572414 0.00101423 +0.237031 +0.432296 +0.192196 +0.113787 +0.141398 +-0.109989 -0.0665918 +Bin 8 -0.0212715 0.0107062 0.0967939 +-0.101428 +-0.103897 +0.0987013 +0.346076 +0.299594 +0.222047 +0.187744 -0.0623878 -0.0473283 +Bin 9 -0.0208628 0.0328734 0.0557834 +-0.104579 +-0.0790618 +0.0421831 +0.175664 +0.208283 +0.323647 +0.333451 +0.0158844 +0.0294013 +Bin 10 -0.0329926 0.0455895 0.0294585 +-0.0540332 +-0.01069 +-0.0136866 +0.024888 +0.0430887 +0.257975 +0.383854 +0.136702 +0.162919 +Bin 11 -0.0445607 0.0381367 0.0350268 0.00237989 0.0375327 +-0.0502259 -0.0781496 -0.0281611 +0.170051 +0.305406 +0.224414 +0.330539 +Bin 12 -0.0507084 0.0270306 0.0451747 +0.0362038 +0.0720245 +-0.0706123 +-0.138258 -0.0566988 +0.134707 +0.241407 +0.259188 +0.477808 +Cross Section δφT, δpT < 0.2 GeV/c +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +12.5 +1.364992 +0.1215575 +2 +12.5 +25 +0.73123664 +0.061852255 +3 +25 +37.5 +0.21435948 +0.029315593 +4 +37.5 +180 +0.0081386225 +0.0018999008 +Unfolded Covariance Matrix δφT, δpT < 0.2 GeV/c +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 1 +0.0147762 +0.00563554 +0.00153079 0.000119854 +Bin 2 +0.00563554 +0.0038257 +0.00154756 +6.30612e-05 +Bin 3 +0.00153079 +0.00154756 0.000859404 3.97385e-05 +Bin 4 +0.000119854 6.30612e-05 3.97385e-05 3.60962e-06 + +32 +Additional Smearing Matrix (AC) δφT, δpT < 0.2 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 1 +0.651056 +0.421486 +0.122573 +0.0277306 +Bin 2 +0.110053 +0.517767 +0.601269 +0.00261283 +Bin 3 -0.0336902 +0.165396 +0.461469 +0.438494 +Bin 4 -0.00884042 -0.0308033 0.24402 +0.656027 +Cross Section δφT, 0.2 < δpT < 0.4 GeV/c +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +20 +0.15286824 +0.026137739 +2 +20 +40 +0.25116958 +0.033748704 +3 +40 +60 +0.15996139 +0.020901006 +4 +60 +80 +0.059045083 +0.012094918 +5 +80 +100 +0.019126701 +0.0088509973 +6 +100 +120 +0.010656056 +0.0075925506 +7 +120 +140 +0.010600178 +0.0071329061 +8 +140 +160 +0.0077681367 +0.0058117466 +9 +160 +180 +0.0044826701 +0.0051335927 +Unfolded Covariance Matrix δφT, 0.2 < δpT < 0.4 GeV/c +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.000683181 0.000782497 0.000308231 8.80183e-05 6.53958e-05 8.2675e-05 8.27199e-05 6.63707e-05 6.20541e-05 +Bin 2 +0.000782497 0.00113898 0.000567719 0.000148218 5.50718e-05 7.72663e-05 9.96908e-05 9.12437e-05 8.26303e-05 +Bin 3 +0.000308231 0.000567719 0.000436852 0.000179468 5.41443e-05 2.4909e-05 3.38677e-05 4.4793e-05 5.42253e-05 +Bin 4 +8.80183e-05 0.000148218 0.000179468 0.000146287 8.34225e-05 3.37221e-05 9.43335e-06 9.13399e-06 2.35642e-05 +Bin 5 +6.53958e-05 5.50718e-05 5.41443e-05 8.34225e-05 7.83402e-05 5.33834e-05 2.75344e-05 1.31263e-05 1.24748e-05 +Bin 6 +8.2675e-05 +7.72663e-05 +2.4909e-05 +3.37221e-05 5.33834e-05 5.76468e-05 4.66638e-05 2.81806e-05 1.35196e-05 +Bin 7 +8.27199e-05 9.96908e-05 3.38677e-05 9.43335e-06 2.75344e-05 4.66638e-05 5.08783e-05 3.75999e-05 1.89923e-05 +Bin 8 +6.63707e-05 9.12437e-05 +4.4793e-05 +9.13399e-06 1.31263e-05 2.81806e-05 3.75999e-05 3.37764e-05 2.32405e-05 +Bin 9 +6.20541e-05 8.26303e-05 5.42253e-05 2.35642e-05 1.24748e-05 1.35196e-05 1.89923e-05 2.32405e-05 2.63538e-05 + +33 +Additional Smearing Matrix (AC) δφT, 0.2 < δpT < 0.4 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.150203 +0.303586 +0.11801 +-0.153051 +-0.176382 +0.157878 +0.0899184 -0.0201824 0.160691 +Bin 2 +0.0773194 +0.506929 +0.496119 +-0.0920515 +-0.301065 +-0.0363322 +0.243829 +0.188924 +0.340157 +Bin 3 +-0.108102 +0.167158 +0.629034 +0.285196 +0.00523665 -0.112789 -0.0267549 +0.100544 +0.241081 +Bin 4 -0.0944427 +-0.0351664 +0.25333 +0.450718 +0.349804 +0.136766 +-0.13861 +-0.113241 0.0225582 +Bin 5 -0.0436973 +-0.0483563 0.0358167 +0.284406 +0.355228 +0.325699 +0.0596138 -0.0622666 0.0360514 +Bin 6 -0.0141004 +-0.010273 +-0.0115866 0.0759759 +0.205177 +0.353802 +0.27358 +0.0945595 +0.158547 +Bin 7 -0.00644838 0.0143535 +0.0163168 -0.0477205 0.0499331 +0.264343 +0.35991 +0.222794 +0.276615 +Bin 8 -0.00919507 0.00899754 0.0448134 -0.0640158 -0.0359387 +0.133664 +0.277892 +0.232028 +0.303309 +Bin 9 +-0.012742 +-0.0103688 0.0497679 -0.0202434 -0.0538902 0.0356291 +0.126572 +0.149203 +0.257174 +Cross Section δφT, δpT > 0.4 GeV/c +Bin # Low edge [deg] High edge [deg] Cross Section [10–38 +cm2 +deg (GeV/c) 40Ar] Uncertainty [10–38 +cm2 +deg (GeV/c) 40Ar] +1 +0 +25 +0.0019354696 +0.002538475 +2 +25 +50 +0.0099343841 +0.0044054612 +3 +50 +75 +0.023950775 +0.0063413422 +4 +75 +105 +0.023479868 +0.0051466413 +5 +105 +145 +0.016668485 +0.0037441977 +6 +145 +180 +0.01542767 +0.0041453489 +Unfolded Covariance Matrix δφT, δpT > 0.4 GeV/c +Units in [10–38 +cm2 +deg (GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 1 +6.44386e-06 9.82677e-06 9.77984e-06 7.09684e-06 6.21115e-06 6.19111e-06 +Bin 2 +9.82677e-06 1.94081e-05 2.48625e-05 1.67106e-05 1.05301e-05 1.02424e-05 +Bin 3 +9.77984e-06 2.48625e-05 4.02126e-05 2.9252e-05 1.45126e-05 1.14699e-05 +Bin 4 +7.09684e-06 1.67106e-05 2.9252e-05 2.64879e-05 1.54525e-05 1.05853e-05 +Bin 5 +6.21115e-06 1.05301e-05 1.45126e-05 1.54525e-05 1.4019e-05 1.32789e-05 +Bin 6 +6.19111e-06 1.02424e-05 1.14699e-05 1.05853e-05 1.32789e-05 1.71839e-05 + +34 +Additional Smearing Matrix (AC) δφT, δpT > 0.4 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 1 0.562728 +0.0395993 +0.0625026 -0.0464433 0.0414228 0.000360551 +Bin 2 +0.60219 +0.0636863 +0.324516 +0.0566229 -0.0563008 +0.0257986 +Bin 3 0.138634 -0.0506159 +0.632292 +0.378733 +-0.124571 +-0.00498109 +Bin 4 -0.014064 -0.206313 +0.415088 +0.560776 +0.165248 +-0.0430022 +Bin 5 0.213661 +-0.134225 +0.0393532 +0.235741 +0.513988 +0.219026 +Bin 6 +0.16613 +0.018933 +-0.0475621 -0.0964813 +0.45158 +0.497684 +Cross Section δpT,x, All events +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV/c) 40Ar] +1 +-0.55 +-0.45 +1.3018458 +0.46828784 +2 +-0.45 +-0.35 +2.563478 +0.59039688 +3 +-0.35 +-0.25 +5.5731346 +0.89649529 +4 +-0.25 +-0.15 +11.707396 +1.3957857 +5 +-0.15 +-0.05 +20.510914 +2.3000463 +6 +-0.05 +0.05 +26.019769 +3.0236457 +7 +0.05 +0.15 +19.707973 +2.2296908 +8 +0.15 +0.25 +11.103249 +1.4077904 +9 +0.25 +0.35 +5.5394189 +0.91807296 +10 +0.35 +0.45 +2.5183878 +0.62427657 +11 +0.45 +0.55 +1.2015817 +0.45888133 + +35 +Unfolded Covariance Matrix δpT,x, All events +Units in [10–38 +cm2 +(GeV/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.219294 0.232594 0.232466 0.277171 0.38592 0.492393 0.423579 0.306975 0.245541 0.216313 0.183504 +Bin 2 +0.232594 0.348568 0.465303 0.532551 0.638571 0.797177 0.722886 0.58234 0.469885 0.332926 0.212854 +Bin 3 +0.232466 0.465303 0.803704 1.08054 +1.35252 +1.64411 +1.44492 +1.08867 0.778097 0.461675 0.239296 +Bin 4 +0.277171 0.532551 1.08054 +1.94822 +2.99076 +3.66436 +2.88731 +1.76956 +1.0176 +0.543675 0.291185 +Bin 5 +0.38592 0.638571 1.35252 +2.99076 +5.29021 +6.72269 +4.92385 +2.57229 +1.27913 +0.66189 0.372158 +Bin 6 +0.492393 0.797177 1.64411 +3.66436 +6.72269 +9.14243 +6.49303 +3.04629 +1.58698 +0.88808 0.417559 +Bin 7 +0.423579 0.722886 1.44492 +2.88731 +4.92385 +6.49303 +4.97152 +2.71033 +1.45241 0.756429 0.366246 +Bin 8 +0.306975 0.58234 +1.08867 +1.76956 +2.57229 +3.04629 +2.71033 +1.98187 +1.15239 0.562354 0.308449 +Bin 9 +0.245541 0.469885 0.778097 +1.0176 +1.27913 +1.58698 +1.45241 +1.15239 0.842858 0.504153 0.266441 +Bin 10 +0.216313 0.332926 0.461675 0.543675 0.66189 +0.88808 0.756429 0.562354 0.504153 0.389721 0.241085 +Bin 11 +0.183504 0.212854 0.239296 0.291185 0.372158 0.417559 0.366246 0.308449 0.266441 0.241085 0.210572 +Additional Smearing Matrix (AC) δpT,x, All events +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.608825 +0.108505 +0.0835085 -0.0150919 0.00881438 -0.0236513 0.0252189 -0.00903844 -0.106234 +0.0637926 +0.202277 +Bin 2 +0.336808 +0.298309 +0.233055 +0.0132687 +0.0263009 -0.0438141 0.0153022 +0.00167781 -0.0731335 0.0992754 +0.131753 +Bin 3 -0.0298876 +0.310249 +0.405831 +0.192639 +0.0787247 -0.0789994 0.0158152 +0.0294692 +-0.0372995 0.0640728 +0.06749 +Bin 4 +-0.151514 +0.08982 +0.271785 +0.519261 +0.252748 +-0.107838 +0.0726825 +0.0512266 +-0.0890651 -0.101554 +0.149703 +Bin 5 +-0.116121 +-0.0407329 +-0.0962132 +0.748375 +0.483298 +-0.0724205 +0.1907 +0.0517504 +-0.1443 +-0.239649 +0.1736 +Bin 6 +0.0341671 +-0.116137 +-0.265799 +0.758626 +0.411838 +0.0712349 +0.410269 +-0.122196 +-0.0459439 0.0519191 +-0.21337 +Bin 7 +0.0515604 +-0.0572882 +-0.0835953 +0.464375 +0.286917 +-0.0794126 +0.412473 +0.147668 +0.0244409 -0.0225528 -0.151986 +Bin 8 -0.0727135 +0.0790082 +0.17944 +0.161495 +0.197438 +-0.170782 +0.18109 +0.354514 +0.111235 +-0.135537 +0.145258 +Bin 9 -0.0500579 +0.132511 +0.258026 +0.0306147 +0.0413035 -0.0693188 0.0680742 +0.133181 +0.107136 +0.208208 +0.19282 +Bin 10 +0.113542 +0.0714151 +0.180497 +0.00151348 -0.0140447 +-0.012518 +0.0307881 +-0.0113679 -0.0272015 +0.329209 +0.385128 +Bin 11 +0.212937 +-0.000824422 0.0940523 0.00290975 0.00586203 -0.017935 0.00115087 0.00945362 -0.0983132 +0.127357 +0.602292 + +36 +Cross Section δpT,x, δpT,y < –0.15 GeV/c +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c)2 40Ar] Uncertainty [10–38 +cm2 +(GeV/c)2 40Ar] +1 +-0.55 +-0.45 +1.6218493 +0.40791533 +2 +-0.45 +-0.35 +2.8287626 +0.53220357 +3 +-0.35 +-0.25 +4.9296487 +0.828875 +4 +-0.25 +-0.15 +6.1584596 +0.9657805 +5 +-0.15 +-0.05 +7.9144932 +1.0811938 +6 +-0.05 +0.05 +8.7493754 +1.1882755 +7 +0.05 +0.15 +8.5097752 +1.2183604 +8 +0.15 +0.25 +7.2111143 +1.0521717 +9 +0.25 +0.35 +5.6686658 +0.86446124 +10 +0.35 +0.45 +2.9004347 +0.56376489 +11 +0.45 +0.55 +1.2983579 +0.41069205 +Unfolded Covariance Matrix δpT,x, δpT,y < –0.15 GeV/c +Units in [10–38 +cm2 +(GeV/c)2 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.166395 0.201863 0.258106 0.269995 0.312093 0.338651 0.32072 0.265329 0.207234 0.119101 0.0822108 +Bin 2 +0.201863 0.283241 0.408043 0.433027 0.471449 0.471055 0.423703 0.351969 0.307575 0.207386 0.142093 +Bin 3 +0.258106 0.408043 0.687034 0.775398 0.807246 0.729945 0.613065 0.546349 0.53454 0.379706 0.253616 +Bin 4 +0.269995 0.433027 0.775398 0.932732 0.997874 0.902642 0.774481 0.711383 0.679866 0.458514 0.293436 +Bin 5 +0.312093 0.471449 0.807246 0.997874 1.16898 +1.18836 +1.0983 +0.967973 0.837022 0.507984 0.306419 +Bin 6 +0.338651 0.471055 0.729945 0.902642 1.18836 +1.412 +1.40756 +1.18307 0.924202 0.490718 +0.26714 +Bin 7 +0.32072 +0.423703 0.613065 0.774481 +1.0983 +1.40756 +1.4844 +1.24681 0.920607 0.440942 0.213245 +Bin 8 +0.265329 0.351969 0.546349 0.711383 0.967973 1.18307 +1.24681 +1.10707 0.856729 0.417615 0.200895 +Bin 9 +0.207234 0.307575 0.53454 0.679866 0.837022 0.924202 0.920607 0.856729 0.747293 0.429783 0.238263 +Bin 10 +0.119101 0.207386 0.379706 0.458514 0.507984 0.490718 0.440942 0.417615 0.429783 0.317831 +0.21315 +Bin 11 +0.0822108 0.142093 0.253616 0.293436 0.306419 0.26714 0.213245 0.200895 0.238263 0.21315 +0.168668 + +37 +Additional Smearing Matrix (AC) δpT,x, δpT,y < –0.15 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.447534 +0.14004 +0.0824181 +0.013213 +0.0134962 +0.0141533 +0.0309477 +0.0147085 +-0.0395867 -0.0417552 -0.121439 +Bin 2 +0.381085 +0.227344 +0.179102 +0.0389878 +0.022696 +0.0110257 +0.0307066 +-0.00252424 +-0.0623969 +0.0150945 0.0218938 +Bin 3 +0.275047 +0.241499 +0.348206 +0.111194 +0.0541287 +0.00683236 -0.0033379 0.000325739 -0.0595093 +0.103088 +0.271062 +Bin 4 +0.198536 +0.146091 +0.33603 +0.138329 +0.0944051 +-0.00290425 0.00955882 +0.0551334 +-0.0146238 +0.151595 +0.36073 +Bin 5 +0.244691 +0.10116 +0.251439 +0.0551704 +0.112665 +0.0252269 +0.118278 +0.138008 +0.0637798 +0.154138 +0.347274 +Bin 6 +0.314214 +0.0892305 +0.126163 +-0.0852992 +0.0489167 +0.0711353 +0.287957 +0.223998 +0.163602 +0.149672 +0.209465 +Bin 7 +0.31175 +0.0688128 +0.0109573 +-0.151359 -0.00941576 +0.0479762 +0.367957 +0.300857 +0.237652 +0.176638 +0.0657609 +Bin 8 +0.23284 +-0.0286886 0.00820142 -0.122808 +-0.0213499 +0.0160578 +0.256609 +0.310144 +0.285849 +0.227142 +0.087401 +Bin 9 +0.0640037 -0.0681636 0.0717852 +-0.076997 +-0.0222277 +0.0134912 +0.132323 +0.206629 +0.253756 +0.313632 +0.343409 +Bin 10 -0.0767339 -0.0327306 0.0914475 +-0.02496 +-0.0120183 +0.0121447 +0.0416083 +0.0535827 +0.0833627 +0.262042 +0.510595 +Bin 11 -0.0594818 -0.0273933 0.0730349 -0.0112767 -0.00528466 +0.0131894 +0.00910853 -0.00415322 -0.00755429 +0.128803 +0.461581 +Cross Section δpT,x, |δpT,y| < 0.15 GeV/c +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c)2 40Ar] Uncertainty [10–38 +cm2 +(GeV/c)2 40Ar] +1 +-0.55 +-0.45 +1.2115661 +0.61126513 +2 +-0.45 +-0.35 +2.6765554 +0.82125417 +3 +-0.35 +-0.25 +7.37408 +1.0556462 +4 +-0.25 +-0.15 +22.199696 +2.2045935 +5 +-0.15 +-0.05 +46.575073 +4.1450698 +6 +-0.05 +0.05 +64.119322 +5.923925 +7 +0.05 +0.15 +46.97485 +4.3491347 +8 +0.15 +0.25 +21.92865 +2.390739 +9 +0.25 +0.35 +7.1639576 +1.1869639 +10 +0.35 +0.45 +2.4893963 +0.95587785 +11 +0.45 +0.55 +0.65304935 +0.72558594 + +38 +Unfolded Covariance Matrix δpT,x, |δpT,y| < 0.15 GeV/c +Units in [10–38 +cm2 +(GeV/c)2 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.373645 0.445282 0.423656 0.403378 0.461031 0.835512 0.837471 0.577603 0.397967 0.374982 0.295969 +Bin 2 +0.445282 0.674458 0.751372 0.73561 0.882513 1.64898 +1.44795 0.774405 0.525903 0.549017 0.403667 +Bin 3 +0.423656 0.751372 1.11439 +1.77711 +2.64896 +3.67804 +2.8346 +1.44581 0.780921 0.65691 0.432323 +Bin 4 +0.403378 0.73561 +1.77711 +4.86023 +8.44834 +10.349 +7.40225 +3.89238 +1.54195 0.725906 0.398222 +Bin 5 +0.461031 0.882513 2.64896 +8.44834 +17.1816 +22.9511 +16.0817 +7.05194 +2.25342 0.966815 0.564219 +Bin 6 +0.835512 1.64898 +3.67804 +10.349 +22.9511 +35.0929 +24.6529 +9.46626 +2.77758 +1.7282 +1.23794 +Bin 7 +0.837471 1.44795 +2.8346 +7.40225 +16.0817 +24.6529 +18.915 +8.4283 +2.86791 +1.5491 +0.931342 +Bin 8 +0.577603 0.774405 1.44581 +3.89238 +7.05194 +9.46626 +8.4283 +5.71563 +2.44987 0.959545 0.435987 +Bin 9 +0.397967 0.525903 0.780921 1.54195 +2.25342 +2.77758 +2.86791 +2.44987 +1.40888 0.838764 0.477882 +Bin 10 +0.374982 0.549017 0.65691 0.725906 0.966815 +1.7282 +1.5491 +0.959545 0.838764 0.913702 0.652589 +Bin 11 +0.295969 0.403667 0.432323 0.398222 0.564219 1.23794 0.931342 0.435987 0.477882 0.652589 0.526475 +Additional Smearing Matrix (AC) δpT,x, |δpT,y| < 0.15 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 10 +Bin 11 +Bin 1 +0.238492 +0.119914 +0.111824 +0.014165 +-0.00438664 -0.0148515 0.000493344 +0.0208499 +-0.0381854 -0.0308014 +0.0726088 +Bin 2 +0.253155 +0.30288 +0.239134 +0.0327858 +-0.00911913 -0.0132125 +0.00397062 +0.0187421 +-0.0552825 +0.0371577 +0.0757129 +Bin 3 +0.0945286 +0.32685 +0.335038 +0.144346 +0.0273997 +-0.0198814 +0.00502543 +0.0236643 +-0.0379872 +0.0607432 +0.035615 +Bin 4 +-0.175762 +0.160394 +0.274843 +0.449634 +0.162518 +-0.0123893 +0.0090526 +0.0890296 +0.0311237 +-0.127903 +-0.139488 +Bin 5 +-0.638782 +0.0165204 +0.0552459 +0.474797 +0.343918 +0.114644 +0.161547 +0.0720988 +-0.0271003 +-0.276907 +-0.242082 +Bin 6 +-0.885233 +0.190623 +-0.103157 +0.214902 +0.274942 +0.346465 +0.39788 +0.0844324 +-0.255099 +-0.166916 +0.298163 +Bin 7 +-0.467011 +0.13461 +-0.101665 +0.0459403 +0.163284 +0.150109 +0.388642 +0.300894 +0.0285275 -0.00885317 -0.0377221 +Bin 8 +-0.163271 -0.00240758 -0.0713958 +0.0979268 +0.0744572 +-0.0301267 +0.0996823 +0.466168 +0.378591 +-0.0239726 +-0.239044 +Bin 9 +-0.100751 +-0.0380883 +0.052072 +0.042795 +0.0164917 +-0.0470341 +0.0146872 +0.217556 +0.23444 +0.237981 +0.0951803 +Bin 10 -0.117178 +-0.0254209 +0.11418 +-0.00697291 -0.0179875 +-0.0147786 +0.0211936 +0.040311 +0.0400128 +0.386113 +0.420585 +Bin 11 -0.0730312 -0.0391132 +0.0727811 -0.00989024 -0.0190981 0.00178702 0.00973777 -0.00435176 -0.031091 +0.225363 +0.409018 +Cross Section δpT,x, δpT,y > 0.15 GeV/c +Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 +cm2 +(GeV/c)2 40Ar] Uncertainty [10–38 +cm2 +(GeV/c)2 40Ar] +1 +-0.55 +-0.35 +0.2342011 +0.22671042 +2 +-0.35 +-0.25 +1.006492 +0.55670098 +3 +-0.25 +-0.15 +2.7324655 +0.88391345 +4 +-0.15 +-0.05 +5.7874941 +1.3844421 +5 +-0.05 +0.05 +7.4255073 +1.5313866 +6 +0.05 +0.15 +5.8817046 +1.2031302 +7 +0.15 +0.25 +3.1660393 +0.75858938 +8 +0.25 +0.35 +1.4156509 +0.51675371 +9 +0.35 +0.55 +0.46780295 +0.27308941 + +39 +Unfolded Covariance Matrix δpT,x, δpT,y > 0.15 GeV/c +Units in [10–38 +cm2 +(GeV/c)2 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.0513976 0.108815 0.125996 0.158666 0.177259 0.156161 0.114426 0.0781145 0.0332451 +Bin 2 +0.108815 +0.309916 0.440532 0.564084 0.550037 0.400661 0.265051 0.181625 0.0796011 +Bin 3 +0.125996 +0.440532 0.781303 +1.1477 +1.10821 0.681115 0.366268 0.242835 +0.11692 +Bin 4 +0.158666 +0.564084 +1.1477 +1.91668 +1.99182 +1.2354 +0.596827 0.337503 +0.155594 +Bin 5 +0.177259 +0.550037 +1.10821 +1.99182 +2.34514 +1.67829 0.850309 0.395843 +0.139324 +Bin 6 +0.156161 +0.400661 0.681115 +1.2354 +1.67829 +1.44752 0.834993 0.373839 +0.103365 +Bin 7 +0.114426 +0.265051 0.366268 0.596827 0.850309 0.834993 0.575458 0.323607 +0.111993 +Bin 8 +0.0781145 0.181625 0.242835 0.337503 0.395843 0.373839 0.323607 0.267034 +0.128279 +Bin 9 +0.0332451 0.0796011 0.11692 0.155594 0.139324 0.103365 0.111993 0.128279 0.0745778 +Additional Smearing Matrix (AC) δpT,x, δpT,y > 0.15 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.283981 +0.142652 +0.0578745 +-0.00406982 -0.00596516 +0.021526 +0.0219778 +0.0247479 -0.0248516 +Bin 2 +0.325935 +0.248141 +0.188942 +0.0533708 +0.00616095 0.00869173 -0.0138248 0.0123266 +0.0148468 +Bin 3 +0.229278 +0.284816 +0.364996 +0.185996 +0.0506093 +0.00616198 -0.0665842 -0.034873 +0.11406 +Bin 4 0.0929379 +0.184043 +0.447192 +0.344133 +0.131196 +0.089738 +-0.0468907 -0.0740597 +0.162968 +Bin 5 0.0474994 +0.0439422 +0.288426 +0.276995 +0.17011 +0.294853 +0.149114 +-0.0128844 +0.052895 +Bin 6 +0.125303 +-0.0173728 +0.0492296 +0.0722596 +0.0911906 +0.368141 +0.331983 +0.16347 +-0.0126541 +Bin 7 +0.118282 +-0.0288781 -0.0296457 +-0.0111587 +0.0193112 +0.205607 +0.285423 +0.30351 +0.16427 +Bin 8 +0.019745 +-0.0309126 -0.00663667 -0.00622196 -0.00269483 0.0412556 +0.144161 +0.303342 +0.366314 +Bin 9 -0.0682454 -0.036675 +0.0235291 +0.00866636 +-0.0040182 +-0.0297024 0.0696102 +0.261339 +0.529614 + +40 +Cross Section ECal, All events +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +GeV 40Ar] Uncertainty [10–38 +cm2 +GeV 40Ar] +1 +0.2 +0.35 +2.4280363 +0.79460421 +2 +0.35 +0.5 +8.514759 +1.2297474 +3 +0.5 +0.65 +13.117445 +1.3413158 +4 +0.65 +0.8 +14.6252 +1.4420716 +5 +0.8 +0.95 +13.51041 +1.3446012 +6 +0.95 +1.1 +11.08195 +1.2200177 +7 +1.1 +1.25 +7.766869 +1.1386025 +8 +1.25 +1.4 +4.8493563 +1.0879941 +9 +1.4 +1.6 +1.8229971 +0.55094844 +Unfolded Covariance Matrix ECal, All events +Units in [10–38 +cm2 +GeV 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.631396 0.745221 0.690303 0.711899 0.750373 0.760828 0.793249 0.723599 0.340987 +Bin 2 +0.745221 1.51228 +1.4763 +1.25687 +1.29711 +1.36377 +1.31737 +1.09756 0.493087 +Bin 3 +0.690303 +1.4763 +1.79913 +1.50347 +1.57962 +1.50756 +1.32938 +1.05655 0.460105 +Bin 4 +0.711899 1.25687 +1.50347 +2.07957 +1.73883 +1.49125 +1.33977 +1.10003 0.502543 +Bin 5 +0.750373 1.29711 +1.57962 +1.73883 +1.80795 +1.58111 +1.24597 0.877393 0.358545 +Bin 6 +0.760828 1.36377 +1.50756 +1.49125 +1.58111 +1.48844 +1.25231 0.938906 0.39754 +Bin 7 +0.793249 1.31737 +1.32938 +1.33977 +1.24597 +1.25231 +1.29642 +1.15885 0.542827 +Bin 8 +0.723599 1.09756 +1.05655 +1.10003 0.877393 0.938906 1.15885 +1.18373 0.585281 +Bin 9 +0.340987 0.493087 0.460105 0.502543 0.358545 0.39754 0.542827 0.585281 0.303544 + +41 +Additional Smearing Matrix (AC) ECal, All events +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.468578 +-0.007816 0.0266806 -0.00958129 0.0604614 -0.0932427 0.180416 +0.11656 +0.0343321 +Bin 2 -0.240475 +0.548541 +0.14521 +-0.0981671 +0.152397 +-0.10336 +0.307903 +0.165633 +0.0374963 +Bin 3 -0.317461 0.0529613 0.660139 +-0.0492356 +0.309486 +-0.135296 +0.251195 +0.125541 +0.0119831 +Bin 4 -0.0710508 -0.133718 +0.162421 +0.701283 +0.223554 +-0.102423 +0.223106 -0.0536489 0.0178971 +Bin 5 -0.0407812 -0.110339 +0.26655 +0.25144 +0.538018 +-0.0148715 0.168779 -0.114466 -0.0575493 +Bin 6 -0.0598932 0.129188 +0.210026 +0.087116 +0.35386 +0.00673019 0.217555 -0.0193128 -0.0275789 +Bin 7 +0.111012 +0.194644 +0.186548 +0.1013 +0.0465323 +-0.181841 +0.340983 +0.17554 +0.0578717 +Bin 8 +0.253029 +0.133768 +0.197589 +0.117109 +-0.163744 +-0.26775 +0.288197 +0.343627 +0.126662 +Bin 9 +0.204053 +0.0719783 0.120343 +0.0989725 +-0.16922 +-0.195653 +0.18008 +0.23869 +0.135617 +Cross Section ECal, δpT < 0.2 GeV/c +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +(GeV2/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV2/c) 40Ar] +1 +0.2 +0.34 +7.4778516 +1.663791 +2 +0.34 +0.48 +23.721989 +2.6923951 +3 +0.48 +0.62 +37.375296 +2.99564 +4 +0.62 +0.76 +42.367813 +3.0983986 +5 +0.76 +0.9 +39.159892 +3.1222198 +6 +0.9 +1.04 +35.757642 +3.2385658 +7 +1.04 +1.18 +30.738633 +3.3618913 +8 +1.18 +1.32 +23.333565 +3.1753137 +9 +1.32 +1.6 +8.3184135 +1.333598 + +42 +Unfolded Covariance Matrix ECal, δpT < 0.2 GeV/c +Units in [10–38 +cm2 +(GeV2/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +2.7682 4.01429 3.30596 2.93741 3.24048 +3.259 +3.05339 2.96903 1.40275 +Bin 2 +4.01429 7.24899 6.76053 6.07509 5.98125 5.97772 5.64916 5.37548 2.40711 +Bin 3 +3.30596 6.76053 8.97386 8.14642 7.1032 6.60378 6.24297 5.68398 2.34706 +Bin 4 +2.93741 6.07509 8.14642 9.60007 8.67603 6.79065 5.50217 4.89311 2.21397 +Bin 5 +3.24048 5.98125 7.1032 8.67603 9.74826 8.30445 6.89028 5.88424 2.40028 +Bin 6 +3.259 +5.97772 6.60378 6.79065 8.30445 10.4883 9.41828 7.99101 3.2708 +Bin 7 +3.05339 5.64916 6.24297 5.50217 6.89028 9.41828 11.3023 9.98777 3.33803 +Bin 8 +2.96903 5.37548 5.68398 4.89311 5.88424 7.99101 9.98777 10.0826 3.61221 +Bin 9 +1.40275 2.40711 2.34706 2.21397 2.40028 3.2708 3.33803 3.61221 1.77848 +Additional Smearing Matrix (AC) ECal, δpT < 0.2 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.332012 +0.246473 +-0.0882899 0.0495375 0.00236788 -0.0420704 +0.0122056 +0.0659321 0.00125015 +Bin 2 0.0967835 +0.557742 +-0.0101648 0.160388 +-0.0699365 -0.0336817 +0.0330535 +0.109563 +0.027758 +Bin 3 -0.126033 +0.122804 +0.45176 +0.325083 +-0.093336 +0.00256258 -0.00608461 0.124206 +0.0298445 +Bin 4 -0.131224 +-0.0253861 +0.121591 +0.69184 +0.0797448 +0.0158361 +-0.038969 +0.0884846 0.0190246 +Bin 5 -0.0344541 0.00293491 -0.0872603 +0.4646 +0.315479 +0.145182 +0.0447158 +0.114306 +-0.041489 +Bin 6 -0.125819 +0.172002 +-0.0928248 0.181306 +0.137621 +0.401902 +0.16364 +0.154655 +-0.0110214 +Bin 7 -0.122548 +0.27646 +-0.092208 0.0377869 0.0706819 +0.242878 +0.296523 +0.30989 +0.00395701 +Bin 8 -0.0212106 +0.259486 +-0.105298 0.0579449 0.0254825 +0.0962355 +0.188686 +0.40643 +0.0771504 +Bin 9 0.0502941 +0.203684 +-0.127859 +0.141478 +-0.0231762 +0.0469033 +0.0698983 +0.217821 +0.177127 + +43 +Cross Section ECal, 0.2 < δpT < 0.4 GeV/c +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +(GeV2/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV2/c) 40Ar] +1 +0.2 +0.34 +8.5177464 +1.377228 +2 +0.34 +0.48 +17.368425 +2.2344492 +3 +0.48 +0.62 +21.658076 +2.2270906 +4 +0.62 +0.76 +23.0641 +2.1795824 +5 +0.76 +0.9 +20.629847 +2.3712022 +6 +0.9 +1.04 +15.884387 +2.6178185 +7 +1.04 +1.18 +12.990481 +2.6000633 +8 +1.18 +1.32 +9.9747439 +2.5419014 +9 +1.32 +1.6 +3.7079479 +0.97366846 +Unfolded Covariance Matrix ECal, 0.2 < δpT < 0.4 GeV/c +Units in [10–38 +cm2 +(GeV2/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +1.89676 2.63506 2.38123 1.77538 1.35356 1.57752 1.43207 1.3801 0.619098 +Bin 2 +2.63506 4.99276 4.47115 3.31143 2.63655 2.75769 3.11913 +3.007 +1.19924 +Bin 3 +2.38123 4.47115 4.95993 4.11754 3.66967 3.69833 3.58682 3.37203 1.32496 +Bin 4 +1.77538 3.31143 4.11754 4.75058 4.67059 4.34432 3.96921 3.42904 1.25923 +Bin 5 +1.35356 2.63655 3.66967 4.67059 5.6226 5.44488 5.09807 4.19559 1.43705 +Bin 6 +1.57752 2.75769 3.69833 4.34432 5.44488 6.85297 6.15936 4.97345 1.83354 +Bin 7 +1.43207 3.11913 3.58682 3.96921 5.09807 6.15936 6.76033 6.18423 2.11273 +Bin 8 +1.3801 +3.007 +3.37203 3.42904 4.19559 4.97345 6.18423 6.46126 2.21694 +Bin 9 +0.619098 1.19924 1.32496 1.25923 1.43705 1.83354 2.11273 2.21694 0.94803 + +44 +Additional Smearing Matrix (AC) ECal, 0.2 < δpT < 0.4 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.109164 +0.0419037 +0.0836673 -0.00930175 -0.104853 +0.0065885 +0.00186561 +0.0118106 +-0.0052122 +Bin 2 -0.0220518 +0.278052 +0.187143 +-0.0504702 +-0.162508 -9.86369e-05 0.0446662 +0.0259156 +0.0292819 +Bin 3 -0.094654 +0.0567177 +0.345528 +-0.0226151 -0.0715755 +0.0586704 +0.00837126 +0.0187044 +0.0227379 +Bin 4 -0.159663 -0.0877275 0.0882331 +0.245747 +0.0401781 +0.0994074 +0.0291937 +0.00222292 -0.0287235 +Bin 5 -0.206936 +-0.15431 +0.0158151 +0.159918 +0.17807 +0.172444 +0.0870625 -0.00340846 -0.0510671 +Bin 6 -0.170821 -0.0811259 -0.0239962 +0.0665726 +0.04262 +0.350127 +0.116693 +-0.0289014 -0.0274388 +Bin 7 -0.187403 +0.0758085 -0.0671069 +-0.0145557 +0.038634 +0.186595 +0.170184 +0.0744945 +0.0103675 +Bin 8 -0.150738 +0.113654 +-0.0379305 +-0.0510391 +0.0203275 +0.0382183 +0.0940384 +0.167877 +0.0873667 +Bin 9 -0.100333 +0.0766541 0.00370471 -0.0403667 -0.0341803 +0.00518112 +0.0394386 +0.0455563 +0.228015 +Cross Section ECal, δpT > 0.4 GeV/c +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +(GeV2/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV2/c) 40Ar] +1 +0.2 +0.34 +1.1821006 +0.29856414 +2 +0.34 +0.48 +3.3696642 +0.63185827 +3 +0.48 +0.62 +4.9647507 +0.68253959 +4 +0.62 +0.76 +5.347174 +0.72095291 +5 +0.76 +0.9 +4.604007 +0.73427148 +6 +0.9 +1.04 +3.872309 +0.72258138 +7 +1.04 +1.18 +2.8535645 +0.64178201 +8 +1.18 +1.32 +2.1345734 +0.64319549 +9 +1.32 +1.6 +0.59841284 +0.23714856 +Unfolded Covariance Matrix ECal, δpT > 0.4 GeV/c +Units in [10–38 +cm2 +(GeV2/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.0891405 0.175269 0.154321 0.126164 +0.13086 0.151896 0.140726 0.133609 0.0499376 +Bin 2 +0.175269 0.399245 0.381907 0.341759 0.327208 0.34601 0.322685 0.312683 0.112147 +Bin 3 +0.154321 0.381907 0.46586 +0.453613 0.401772 0.388672 0.352275 0.329322 0.111957 +Bin 4 +0.126164 0.341759 0.453613 0.519773 0.492251 0.449318 0.353818 0.288997 0.0922748 +Bin 5 +0.13086 +0.327208 0.401772 0.492251 0.539155 0.49638 0.380723 0.287547 0.085787 +Bin 6 +0.151896 +0.34601 0.388672 0.449318 +0.49638 0.522124 0.412197 0.331062 0.108951 +Bin 7 +0.140726 0.322685 0.352275 0.353818 0.380723 0.412197 0.411884 0.38548 +0.123694 +Bin 8 +0.133609 0.312683 0.329322 0.288997 0.287547 0.331062 0.38548 +0.4137 +0.143313 +Bin 9 +0.0499376 0.112147 0.111957 0.0922748 0.085787 0.108951 0.123694 0.143313 0.0562394 + +45 +Additional Smearing Matrix (AC) ECal, δpT > 0.4 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.0931121 +0.0983705 0.115428 -0.0559073 -0.0250682 -0.00763155 0.0848454 0.0285786 +0.0211121 +Bin 2 -0.00853425 +0.2951 +0.311508 -0.00449115 -0.040433 +-0.0367123 +0.173613 +0.0649463 +0.0362442 +Bin 3 -0.0725362 +0.0397894 0.535149 +0.194827 +-0.019212 -0.00703728 +0.19008 +0.0303788 +-0.020959 +Bin 4 +-0.143708 +-0.099961 0.418759 +0.370076 +0.134729 +0.0760182 +0.195705 -0.0648952 +-0.076964 +Bin 5 -0.0768097 +-0.123574 0.225145 +0.281358 +0.294061 +0.131329 +0.242453 -0.0855012 +-0.102749 +Bin 6 -0.0452569 +-0.108786 0.178781 +0.162803 +0.163218 +0.216386 +0.280406 -0.0290967 0.00340337 +Bin 7 -0.0859781 -0.0158235 0.195278 +0.0166949 +0.0422117 +0.0759005 +0.321599 +0.156297 +0.0757575 +Bin 8 +-0.101102 +0.0765364 0.210047 -0.0540062 -0.0696197 -0.00934385 0.282294 +0.2638 +0.208489 +Bin 9 -0.0241305 +0.0801451 0.141856 -0.0465726 +-0.103011 -0.00132606 +0.15648 +0.18027 +0.241303 +Cross Section ECal, δαT < 45o +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +deg GeV 40Ar] Uncertainty [10–38 +cm2 +deg GeV 40Ar] +1 +0.2 +0.38 +0.022911113 +0.0049556794 +2 +0.38 +0.55 +0.046506764 +0.0058656604 +3 +0.55 +0.73 +0.05371068 +0.0053429329 +4 +0.73 +0.9 +0.057122311 +0.0051425099 +5 +0.9 +1.08 +0.044246553 +0.0050250762 +6 +1.08 +1.23 +0.031770412 +0.0052311462 +7 +1.23 +1.4 +0.019595659 +0.0049797225 +8 +1.4 +1.6 +0.0086228162 +0.00271678 +Unfolded Covariance Matrix ECal, δαT < 45o +Units in [10–38 +cm2 +deg GeV 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 1 +2.45588e-05 2.3115e-05 1.72193e-05 9.56483e-06 1.37585e-05 1.36684e-05 1.41928e-05 9.04982e-06 +Bin 2 +2.3115e-05 +3.4406e-05 2.60071e-05 1.92979e-05 1.74194e-05 1.87576e-05 1.70236e-05 9.58294e-06 +Bin 3 +1.72193e-05 2.60071e-05 2.85469e-05 2.3178e-05 1.93935e-05 1.80045e-05 1.55008e-05 8.48334e-06 +Bin 4 +9.56483e-06 1.92979e-05 2.3178e-05 2.64454e-05 2.16351e-05 1.74016e-05 1.13098e-05 5.15249e-06 +Bin 5 +1.37585e-05 1.74194e-05 1.93935e-05 2.16351e-05 2.52514e-05 2.19635e-05 1.55291e-05 7.43748e-06 +Bin 6 +1.36684e-05 1.87576e-05 1.80045e-05 1.74016e-05 2.19635e-05 2.73649e-05 2.31378e-05 1.11024e-05 +Bin 7 +1.41928e-05 1.70236e-05 1.55008e-05 1.13098e-05 1.55291e-05 2.31378e-05 2.47976e-05 1.28476e-05 +Bin 8 +9.04982e-06 9.58294e-06 8.48334e-06 5.15249e-06 7.43748e-06 1.11024e-05 1.28476e-05 7.38089e-06 + +46 +Additional Smearing Matrix (AC) ECal, δαT < 45o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 1 +0.214015 +0.14431 +0.0935853 -0.0436162 +0.0413904 +-0.00736974 0.0722259 +0.0121885 +Bin 2 0.0350815 +0.347665 +0.16401 +0.0652129 -0.00761471 +0.0455451 +0.1054 +0.0108441 +Bin 3 -0.0274569 0.0626681 +0.41405 +0.217889 +0.0349829 +0.0451783 +0.0651346 +0.00383067 +Bin 4 +-0.070977 +-0.0115522 0.204198 +0.440867 +0.150709 +0.0575478 +0.00200467 -0.000697258 +Bin 5 0.0186721 -0.0536033 0.111678 +0.291431 +0.318587 +0.108481 +0.00846444 +0.00171664 +Bin 6 0.00347665 0.0245411 0.0686577 +0.160001 +0.108357 +0.201719 +0.119724 +0.0043447 +Bin 7 0.0275845 +0.0468213 0.0809657 0.0890458 +-0.0217563 +0.133628 +0.227601 +0.0217404 +Bin 8 0.0339496 +0.0284494 0.0602311 0.0294924 +-0.0396266 +0.039581 +0.12042 +0.0146859 +Cross Section ECal, 45o < δαT < 90o +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +deg GeV 40Ar] Uncertainty [10–38 +cm2 +deg GeV 40Ar] +1 +0.2 +0.34 +0.015223769 +0.0040708036 +2 +0.34 +0.48 +0.042871964 +0.0060792512 +3 +0.48 +0.62 +0.061405294 +0.0058409485 +4 +0.62 +0.76 +0.067888001 +0.0062125014 +5 +0.76 +0.9 +0.06182615 +0.006244721 +6 +0.9 +1.04 +0.04785571 +0.008022795 +7 +1.04 +1.18 +0.02950621 +0.0085592201 +8 +1.18 +1.39 +0.011741284 +0.0064936311 +9 +1.39 +1.6 +0.0050906485 +0.0035720766 +Unfolded Covariance Matrix ECal, 45o < δαT < 90o +Units in [10–38 +cm2 +deg GeV 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +1.65714e-05 2.17096e-05 1.61165e-05 1.34064e-05 1.61927e-05 2.45648e-05 2.62167e-05 1.75668e-05 1.08752e-05 +Bin 2 +2.17096e-05 3.69573e-05 3.01948e-05 2.14091e-05 2.50269e-05 3.33579e-05 3.64985e-05 2.57546e-05 1.49574e-05 +Bin 3 +1.61165e-05 3.01948e-05 3.41167e-05 2.95355e-05 2.49843e-05 2.57874e-05 2.66386e-05 1.99403e-05 1.25383e-05 +Bin 4 +1.34064e-05 2.14091e-05 2.95355e-05 3.85952e-05 3.04728e-05 2.36242e-05 2.07416e-05 1.66306e-05 1.15262e-05 +Bin 5 +1.61927e-05 2.50269e-05 2.49843e-05 3.04728e-05 3.89965e-05 4.10664e-05 3.61737e-05 2.20696e-05 1.22399e-05 +Bin 6 +2.45648e-05 3.33579e-05 2.57874e-05 2.36242e-05 4.10664e-05 6.43652e-05 6.17283e-05 3.41484e-05 1.80472e-05 +Bin 7 +2.62167e-05 3.64985e-05 2.66386e-05 2.07416e-05 3.61737e-05 6.17283e-05 7.32602e-05 4.94426e-05 2.44258e-05 +Bin 8 +1.75668e-05 2.57546e-05 1.99403e-05 1.66306e-05 2.20696e-05 3.41484e-05 4.94426e-05 4.21672e-05 2.12141e-05 +Bin 9 +1.08752e-05 1.49574e-05 1.25383e-05 1.15262e-05 1.22399e-05 1.80472e-05 2.44258e-05 2.12141e-05 1.27597e-05 + +47 +Additional Smearing Matrix (AC) ECal, 45o < δαT < 90o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.144511 +0.132867 +0.00188425 -0.00411617 +0.00806955 +-0.0159754 +0.0290823 +0.0221901 +0.059924 +Bin 2 +0.0564224 +0.326603 +0.110783 +-0.0144956 -0.000544054 -0.0322722 +0.0383533 +0.0315212 0.0634079 +Bin 3 +0.0100221 +0.0883864 +0.30553 +0.135166 +-0.0509968 +-0.0452151 +0.0114401 +0.0356079 0.0321712 +Bin 4 +0.065177 +-0.0777837 +0.110321 +0.366968 +0.0148477 +-0.0485207 -0.00457134 0.0179172 0.0115288 +Bin 5 -0.00387751 -0.00143439 -0.0134921 +0.103056 +0.202124 +0.0425935 +0.0880865 +0.00201993 0.031266 +Bin 6 +-0.04297 +0.0464894 +-0.00102566 +-0.115312 +0.129873 +0.158581 +0.220214 +0.0249775 0.0412134 +Bin 7 -0.0428489 +0.0525865 +0.025311 +-0.175886 +0.0607949 +0.064551 +0.270719 +0.174932 +0.0606612 +Bin 8 0.00232508 +0.024175 +0.0303037 +-0.11509 +0.0234896 +-0.0543038 +0.148606 +0.359526 +0.118717 +Bin 9 +0.0529726 +-0.00217489 0.00802375 -0.00110526 +-0.0124215 +-0.042141 +0.0302965 +0.140164 +0.109106 +Cross Section ECal, 90o < δαT < 135o +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +deg GeV 40Ar] Uncertainty [10–38 +cm2 +deg GeV 40Ar] +1 +0.2 +0.34 +0.014030087 +0.0051703105 +2 +0.34 +0.48 +0.054311971 +0.0076341954 +3 +0.48 +0.62 +0.090180705 +0.0077883177 +4 +0.62 +0.76 +0.098905179 +0.0080743704 +5 +0.76 +0.9 +0.087012541 +0.0097326103 +6 +0.9 +1.04 +0.069321234 +0.011079443 +7 +1.04 +1.18 +0.056595104 +0.010273923 +8 +1.18 +1.39 +0.033671172 +0.0070344961 +9 +1.39 +1.6 +0.013865221 +0.0031854942 +Unfolded Covariance Matrix ECal, 90o < δαT < 135o +Units in [10–38 +cm2 +deg GeV 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +2.67321e-05 3.31333e-05 2.11468e-05 2.11514e-05 2.69741e-05 +3.4113e-05 +2.92199e-05 1.96907e-05 1.15361e-05 +Bin 2 +3.31333e-05 5.82809e-05 4.9377e-05 3.62148e-05 3.59862e-05 4.27673e-05 4.20475e-05 3.04206e-05 1.69097e-05 +Bin 3 +2.11468e-05 4.9377e-05 6.06579e-05 4.61376e-05 3.65051e-05 3.72283e-05 3.58315e-05 2.58819e-05 1.42286e-05 +Bin 4 +2.11514e-05 3.62148e-05 4.61376e-05 6.51955e-05 6.57262e-05 +5.4932e-05 +4.37034e-05 2.82061e-05 1.41509e-05 +Bin 5 +2.69741e-05 3.59862e-05 3.65051e-05 6.57262e-05 9.47237e-05 +9.3255e-05 +7.35524e-05 4.16414e-05 1.77432e-05 +Bin 6 +3.4113e-05 4.27673e-05 3.72283e-05 5.4932e-05 +9.3255e-05 0.000122754 0.000102112 5.28708e-05 2.18916e-05 +Bin 7 +2.92199e-05 4.20475e-05 3.58315e-05 4.37034e-05 7.35524e-05 0.000102112 0.000105554 6.53115e-05 2.46442e-05 +Bin 8 +1.96907e-05 3.04206e-05 2.58819e-05 2.82061e-05 4.16414e-05 5.28708e-05 6.53115e-05 4.94841e-05 1.98204e-05 +Bin 9 +1.15361e-05 1.69097e-05 1.42286e-05 1.41509e-05 1.77432e-05 2.18916e-05 2.46442e-05 1.98204e-05 1.01474e-05 + +48 +Additional Smearing Matrix (AC) ECal, 90o < δαT < 135o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.193155 +0.174788 +0.0468135 -0.0145914 +0.00807388 +0.00751103 -0.0211459 -0.00381463 0.0802165 +Bin 2 0.0559721 +0.405876 +0.259704 -0.0226756 +-0.024953 +0.00286068 0.0118828 +0.00843255 +0.0706839 +Bin 3 +-0.12228 +0.14142 +0.549514 +0.0972058 +-0.0316195 +0.0299529 +0.0235975 -0.00458832 0.0206822 +Bin 4 -0.0890518 -0.0502697 0.270657 +0.393017 +0.151964 +-0.0133933 0.0320262 +-0.0113839 +0.0218976 +Bin 5 -0.121637 -0.0734024 0.124125 +0.227054 +0.379764 +0.0589708 +0.082431 +-0.00108853 0.00595713 +Bin 6 -0.191335 -0.0225609 0.161979 +0.032355 +0.220928 +0.233661 +0.183945 +0.0067701 +0.0216689 +Bin 7 -0.259258 +0.0984878 +0.148876 +-0.039466 +0.107229 +0.152703 +0.235997 +0.126265 +0.0104258 +Bin 8 -0.194727 +0.14524 +0.146635 +-0.037525 +0.0614335 +0.0474292 +0.170983 +0.217023 +0.0981407 +Bin 9 -0.0230308 +0.056371 +0.0772496 0.0016455 -0.000934953 0.0198736 +0.0301298 +0.0482617 +0.127642 +Cross Section ECal, 135o < δαT < 180o +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +deg GeV 40Ar] Uncertainty [10–38 +cm2 +deg GeV 40Ar] +1 +0.2 +0.38 +0.027377992 +0.004790503 +2 +0.38 +0.55 +0.084359465 +0.0095187811 +3 +0.55 +0.73 +0.1123024 +0.011355117 +4 +0.73 +0.9 +0.10788839 +0.010553428 +5 +0.9 +1.08 +0.080871908 +0.010456613 +6 +1.08 +1.25 +0.056206713 +0.012229283 +7 +1.25 +1.43 +0.029605819 +0.010305779 +8 +1.43 +1.6 +0.011512073 +0.0052540905 +Unfolded Covariance Matrix ECal, 135o < δαT < 180o +Units in [10–38 +cm2 +deg GeV 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 1 +2.29489e-05 3.86213e-05 4.23844e-05 3.54022e-05 3.49212e-05 3.47533e-05 2.28861e-05 9.91373e-06 +Bin 2 +3.86213e-05 9.06072e-05 9.89917e-05 8.19396e-05 5.81525e-05 +3.9288e-05 +2.03703e-05 7.70905e-06 +Bin 3 +4.23844e-05 9.89917e-05 0.000128939 0.000101578 7.74726e-05 5.68293e-05 2.58845e-05 8.42033e-06 +Bin 4 +3.54022e-05 8.19396e-05 0.000101578 0.000111375 8.91661e-05 +6.5271e-05 +3.86204e-05 1.60688e-05 +Bin 5 +3.49212e-05 5.81525e-05 7.74726e-05 8.91661e-05 0.000109341 0.000111267 7.70787e-05 3.45722e-05 +Bin 6 +3.47533e-05 3.9288e-05 +5.68293e-05 +6.5271e-05 +0.000111267 0.000149555 0.000116661 5.49348e-05 +Bin 7 +2.28861e-05 2.03703e-05 2.58845e-05 3.86204e-05 7.70787e-05 0.000116661 0.000106209 5.34096e-05 +Bin 8 +9.91373e-06 7.70905e-06 8.42033e-06 1.60688e-05 3.45722e-05 5.49348e-05 5.34096e-05 2.76055e-05 + +49 +Additional Smearing Matrix (AC) ECal, 135o < δαT < 180o +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 1 0.344614 0.0994744 +0.0134076 -0.00019442 0.00242534 +0.0149544 +0.0130152 +-0.0138546 +Bin 2 0.252109 +0.432082 +0.080048 +0.0402081 +-0.00473542 -0.00423299 +0.015471 +-0.00174856 +Bin 3 0.191022 +0.23672 +0.31122 +0.123228 +0.0326861 +0.00579486 -0.0141573 +-0.0106008 +Bin 4 0.105969 0.0940126 +0.0540609 +0.387431 +0.118332 +-0.00866266 0.00485879 -0.00601292 +Bin 5 0.205879 -0.0810976 -0.0288859 +0.28526 +0.233881 +0.0803451 +0.0698116 +-0.0140106 +Bin 6 0.26836 +-0.162649 -0.0394283 +0.165016 +0.107319 +0.197615 +0.20912 +-0.0159508 +Bin 7 0.192951 -0.118214 -0.0596226 +0.106598 +0.0252027 +0.107841 +0.339042 +0.00785766 +Bin 8 0.081445 -0.0469623 -0.033978 +0.0436873 +0.000148688 +0.0355065 +0.183864 +0.0120461 +Cross Section ECal, δpT,y < –0.15 GeV/c +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +(GeV2/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV2/c) 40Ar] +1 +0.2 +0.34 +1.9750005 +0.44493023 +2 +0.34 +0.48 +5.6431053 +0.81690274 +3 +0.48 +0.62 +8.1495146 +0.90801221 +4 +0.62 +0.76 +8.7619175 +0.93199333 +5 +0.76 +0.9 +7.70625 +0.99277051 +6 +0.9 +1.04 +6.5646567 +1.0696243 +7 +1.04 +1.18 +5.0179828 +1.0467697 +8 +1.18 +1.32 +3.2678194 +0.94332837 +9 +1.32 +1.6 +0.84086961 +0.27510692 +Unfolded Covariance Matrix ECal, δpT,y < –0.15 GeV/c +Units in [10–38 +cm2 +(GeV2/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.197963 0.336851 0.304578 0.245693 0.218786 0.205849 0.192956 0.203003 0.0719656 +Bin 2 +0.336851 +0.66733 +0.66017 0.551068 0.477125 0.459131 0.449094 0.415296 0.125247 +Bin 3 +0.304578 +0.66017 0.824486 0.744252 0.665787 0.639772 0.519784 0.405001 +0.11917 +Bin 4 +0.245693 0.551068 0.744252 0.868612 0.857476 0.76633 0.631517 0.509084 +0.14212 +Bin 5 +0.218786 0.477125 0.665787 0.857476 0.985593 0.977139 0.848584 0.695622 0.190083 +Bin 6 +0.205849 0.459131 0.639772 0.76633 0.977139 +1.1441 +1.04791 0.836266 0.218672 +Bin 7 +0.192956 0.449094 0.519784 0.631517 0.848584 1.04791 +1.09573 0.935545 0.241009 +Bin 8 +0.203003 0.415296 0.405001 0.509084 0.695622 0.836266 0.935545 0.889868 0.245135 +Bin 9 +0.0719656 0.125247 0.11917 +0.14212 0.190083 0.218672 0.241009 0.245135 0.0756838 + +50 +Additional Smearing Matrix (AC) ECal, δpT,y < –0.15 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 0.258982 +0.290163 +0.0598335 +-0.0237732 +0.00654955 -0.00767649 +-0.0154331 +-0.0100231 0.0546274 +Bin 2 0.224771 +0.599993 +0.197485 +-0.000855549 0.0450191 +-0.0333677 +0.00943202 +0.00550107 0.0755554 +Bin 3 0.0596168 +0.336637 +0.42362 +0.1307 +0.137331 +0.0234496 +-0.0091154 +-0.0343488 0.0407716 +Bin 4 +-0.05069 +0.111528 +0.214557 +0.39926 +0.337856 +-0.00136191 -0.00700122 -0.0222311 0.0177868 +Bin 5 -0.108289 -0.0122851 +0.111822 +0.289434 +0.469815 +0.0926546 +0.0358627 +-0.00577917 0.0288427 +Bin 6 -0.158342 -0.0274417 +0.119077 +0.06809 +0.393725 +0.200294 +0.136086 +0.031297 +0.0317284 +Bin 7 -0.221689 0.0947182 +0.0227254 +-0.0300817 +0.292491 +0.0939459 +0.212888 +0.123889 +0.104562 +Bin 8 -0.126722 +0.105206 +-0.0338211 +-0.0583892 +0.217202 +-0.000494976 +0.144867 +0.17211 +0.161664 +Bin 9 0.0204245 0.0152219 +-0.013884 +-0.0438123 +0.101108 +-0.0142269 +0.0421754 +0.0727728 +0.14113 +Cross Section ECal, |δpT,y| < 0.15 GeV/c +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +(GeV2/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV2/c) 40Ar] +1 +0.2 +0.34 +5.7256169 +1.2122548 +2 +0.34 +0.48 +19.727215 +1.9508107 +3 +0.48 +0.62 +27.381216 +2.2053728 +4 +0.62 +0.76 +32.181157 +2.3524262 +5 +0.76 +0.9 +29.561199 +2.4085133 +6 +0.9 +1.04 +25.427838 +2.2917764 +7 +1.04 +1.18 +20.137455 +2.1749865 +8 +1.18 +1.32 +14.830218 +2.1226077 +9 +1.32 +1.6 +4.9781531 +0.90011043 +Unfolded Covariance Matrix ECal, |δpT,y| < 0.15 GeV/c +Units in [10–38 +cm2 +(GeV2/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +1.46956 +1.82924 +1.81576 +1.50746 +1.75206 1.70651 1.61039 1.43728 0.578361 +Bin 2 +1.82924 +3.80566 +3.60408 +3.40082 +2.86004 2.76553 2.60945 2.27019 0.873838 +Bin 3 +1.81576 +3.60408 +4.86367 +4.31341 +3.89315 3.13913 2.64071 2.24748 0.853925 +Bin 4 +1.50746 +3.40082 +4.31341 +5.53391 +4.77603 +3.6633 2.77124 2.20546 0.864764 +Bin 5 +1.75206 +2.86004 +3.89315 +4.77603 +5.80094 4.81499 3.65324 2.61433 0.885788 +Bin 6 +1.70651 +2.76553 +3.13913 +3.6633 +4.81499 5.25224 4.51429 3.23939 1.04951 +Bin 7 +1.61039 +2.60945 +2.64071 +2.77124 +3.65324 4.51429 4.73057 4.16847 1.50947 +Bin 8 +1.43728 +2.27019 +2.24748 +2.20546 +2.61433 3.23939 4.16847 4.50546 1.82207 +Bin 9 +0.578361 0.873838 0.853925 0.864764 0.885788 1.04951 1.50947 1.82207 0.810199 + +51 +Additional Smearing Matrix (AC) ECal, |δpT,y| < 0.15 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.322497 +0.0905649 +0.101626 +0.0300083 +-0.0356452 -0.0588684 0.0707156 0.0231266 -0.0124691 +Bin 2 -0.0135338 +0.501382 +0.216449 +0.183919 +-0.170235 +-0.0297659 0.111656 0.0484389 +0.0195058 +Bin 3 0.0130477 +0.0470315 +0.666263 +0.243777 +-0.0736494 -0.0235973 0.0678795 0.0727558 +0.0272982 +Bin 4 -0.092557 +0.039126 +0.167226 +0.705128 +0.0567686 +0.028069 +0.0486624 0.0415374 -0.00851025 +Bin 5 -0.0777826 -0.0716627 0.0940598 +0.344559 +0.397144 +0.13688 +0.136813 0.0228841 -0.0567614 +Bin 6 -0.148728 +0.0903508 0.0229516 +0.0968807 +0.249563 +0.295088 +0.273221 0.0715431 -0.0421295 +Bin 7 -0.114256 +0.148935 +0.0241555 +-0.0109053 +0.0973428 +0.178685 +0.324859 +0.233572 +0.0742204 +Bin 8 -0.027769 +0.104303 +0.0448909 -0.000806135 0.00407994 -0.0202481 0.257128 +0.359773 +0.183541 +Bin 9 0.0141602 +0.0494249 0.0315971 +0.0431423 +-0.043055 +-0.107492 +0.15496 +0.30638 +0.193736 +Cross Section ECal, δpT,y > 0.15 GeV/c +Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 +cm2 +(GeV2/c) 40Ar] Uncertainty [10–38 +cm2 +(GeV2/c) 40Ar] +1 +0.2 +0.34 +2.3812738 +0.53728629 +2 +0.34 +0.48 +3.8519267 +0.7584091 +3 +0.48 +0.62 +4.8841718 +0.70560257 +4 +0.62 +0.76 +5.1290618 +0.62295676 +5 +0.76 +0.9 +4.2405823 +0.62866421 +6 +0.9 +1.04 +3.211598 +0.73798187 +7 +1.04 +1.18 +1.9052577 +0.87651596 +8 +1.18 +1.32 +0.91661655 +0.95293063 +9 +1.32 +1.6 +0.36476708 +0.49297537 +Unfolded Covariance Matrix ECal, δpT,y > 0.15 GeV/c +Units in [10–38 +cm2 +(GeV2/c) 40Ar]2 +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 +0.288677 0.328153 0.239634 0.141798 0.116787 0.135921 0.161568 0.151674 0.0778962 +Bin 2 +0.328153 0.575184 0.473237 0.297274 0.203548 0.201145 0.234967 0.252571 0.120788 +Bin 3 +0.239634 0.473237 0.497875 0.379256 +0.23748 0.191696 0.206691 0.212745 0.103976 +Bin 4 +0.141798 0.297274 0.379256 0.388075 0.315144 0.248043 0.202497 0.172576 0.0777672 +Bin 5 +0.116787 0.203548 0.23748 +0.315144 0.395219 0.404325 0.345754 0.266076 0.111748 +Bin 6 +0.135921 0.201145 0.191696 0.248043 0.404325 0.544617 0.585026 0.524348 0.243676 +Bin 7 +0.161568 0.234967 0.206691 0.202497 0.345754 0.585026 0.76828 +0.78999 +0.391573 +Bin 8 +0.151674 0.252571 0.212745 0.172576 0.266076 0.524348 0.78999 0.908077 0.465227 +Bin 9 +0.0778962 0.120788 0.103976 0.0777672 0.111748 0.243676 0.391573 0.465227 0.243025 + +52 +Additional Smearing Matrix (AC) ECal, δpT,y > 0.15 GeV/c +Bin 1 +Bin 2 +Bin 3 +Bin 4 +Bin 5 +Bin 6 +Bin 7 +Bin 8 +Bin 9 +Bin 1 0.0574863 +0.119834 +0.20923 +-0.0525759 +0.0109354 +-0.026514 +-0.00386619 0.0460792 -0.0233909 +Bin 2 -0.0525597 +0.290424 +0.500753 +0.0115298 -0.00980539 -0.0415917 -0.00717367 0.104371 -0.0721812 +Bin 3 -0.0604904 +0.132088 +0.619888 +0.223617 +0.0470369 +-0.0407812 +0.00482642 0.0836055 -0.080892 +Bin 4 -0.051691 +-0.0608881 +0.438255 +0.353263 +0.28528 +0.0263162 +0.0151576 +0.0518864 -0.094828 +Bin 5 -0.0550683 -0.0903235 +0.148248 +0.224764 +0.523968 +0.126263 +0.100147 +0.0507039 -0.106761 +Bin 6 -0.0931912 -0.0394578 0.00797704 +0.0831667 +0.488213 +0.154978 +0.211066 +0.159143 +-0.054027 +Bin 7 -0.132024 +-0.0004211 -0.00944179 0.0139328 +0.290612 +0.082325 +0.255604 +0.343521 +0.0374567 +Bin 8 -0.160573 +0.0183179 -0.00291147 0.0200926 +0.106735 +-0.00380633 +0.20166 +0.490899 +0.11169 +Bin 9 -0.148341 0.00295388 0.00980972 +0.0333003 +0.019359 +-0.0418381 +0.17709 +0.518324 +0.160304 +PARTICLE IDENTIFICATION AND EVENT SELECTION +We used the log-likelihood ratio particle identification (LLR PID) score method [? ] to obtain our +muon and proton candidates. As illustrated in Fig. 6 of [? ], muons tend to have higher LLR score +values than protons. Thus, the candidate track with the greater LLR PID score is assigned the label +of the candidate muon, while the track with the smaller score is our candidate proton. +We studied the effect of cutting on different values of the candidate proton LLR PID score, which +has a strong discrimination power rejecting MC non-CC1p0π background, out-of-cryostat (Dirt) and +cosmic events. Maximizing the purity × efficiency product yielded an optimal cut on the proton +candidate LLR PID score < 0.05, which is indicated by the dashed line in Fig. 1. +MicroBooNE Data +Cosmic +π +MC CC1p0 +π +MC Non-CC1p0 +Dirt +Proton Candidate LLR PID Score +0 +200 +400 +600 +800 +1000 +# Events / 6.79e+20 + = 41.7 % +π +CC1p0 +Cosmic = 21.5 % +1 +− +0.5 +− +0 +0.5 +1 +Proton Candidate LLR PID Score +0.5 +1 +1.5 +Simulation +Data +FIG. 1. +The proton candidate LLR PID score distribution, illustrating the fitness of a cut at LLR PID < 0.05 to reject cosmic +and non-CC1p0π background events. +To minimize the contribution of misreconstructed tracks, we took advantage of the fact that we had +two muon momentum reconstruction methods available for contained tracks, namely the momentum +from range [? ] and the momentum from Multiple Coulomb Scattering (MCS) [? ]. We applied a +quality cut on the contained muons by requiring the range and MCS momenta to be in agreement + +53 +within 25%. The effect of the quality cut is shown in Fig. 2. +0 +100 +200 +300 +400 +500 +600 +700 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +True Muon Momentum [GeV/c] +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 + Range Reco Muon Momentum [GeV/c] +0 +100 +200 +300 +400 +500 +600 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +True Muon Momentum [GeV/c] +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 + Range Reco Muon Momentum [GeV/c] +FIG. 2. Muon momentum reconstruction (left) before and (right) after the application of the muon momentum quality cut +using contained muon tracks. +In order to avoid mis-reconstructed track directions, we further required that the distance between +the track start points and the vertex is smaller than the corresponding distance between the track +end points and the vertex. We also demanded that the distance between the start points of the two +candidate tracks is smaller than the distance between the two end points. +FINAL STATE INTERACTION SMEARING +Figures 3-5 show the effect of final state interactions (FSI) on the CC1p0π selection using the G18 +configuration of GENIE. The addition of FSI allows for more non-QE events to satisfy the CC1p0π +signal definition. Furthermore, FSI effects smear the δpT distribution to higher values (Fig. 3), +introduce an asymmetric behavior in δαT (Fig. 4), and lead to larger δφT values (Fig. 5). +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +σ +d +QE +MEC +RES +DIS +MicroBooNE Data + Shape) +⊕ +(Stat +6.79e+20 POT +Norm +(a) G18, All events +0 +0.2 +0.4 +0.6 +0.8 + [GeV/c] +T +p +δ +0 +10 +20 +30 +40 +50 +GeV/c Ar +2 +cm + +-38 +10 + +T +p +δ +d +σ +d +QE +MEC +RES +DIS +MicroBooNE Data + Shape) +⊕ +(Stat +6.79e+20 POT +Norm +(b) G18 No FSI, All events +FIG. 3. Cross section interaction breakdown for the selected events for the G18 configuration (left) with FSI effects, and (right) +without FSI effects as a function of δpT. + +54 +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +deg Ar +2 +cm + +-38 +10 + +T +α +δ +d +σ +d +QE +MEC +RES +DIS +MicroBooNE Data + Shape) +⊕ +(Stat +6.79e+20 POT +Norm +(a) G18, All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +α +δ +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +deg Ar +2 +cm + +-38 +10 + +T +α +δ +d +σ +d +QE +MEC +RES +DIS +MicroBooNE Data + Shape) +⊕ +(Stat +6.79e+20 POT +Norm +(b) G18 No FSI, All events +FIG. 4. Cross section interaction breakdown for the selected events for the G18 configuration (left) with FSI effects, and (right) +without FSI effects as a function of δαT. +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +deg Ar +2 +cm + +-38 +10 + +T +φ +δ +d +σ +d +QE +MEC +RES +DIS +MicroBooNE Data + Shape) +⊕ +(Stat +6.79e+20 POT +Norm +(a) G18, All events +0 +20 +40 +60 +80 +100 120 140 160 180 + [deg] +T +φ +δ +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +deg Ar +2 +cm + +-38 +10 + +T +φ +δ +d +σ +d +QE +MEC +RES +DIS +MicroBooNE Data + Shape) +⊕ +(Stat +6.79e+20 POT +Norm +(b) G18 No FSI, All events +FIG. 5. Cross section interaction breakdown for the selected events for the G18 configuration (left) with FSI effects, and (right) +without FSI effects as a function of δφT. + diff --git a/pNE2T4oBgHgl3EQfKQZI/content/tmp_files/load_file.txt b/pNE2T4oBgHgl3EQfKQZI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..69af0a3f256d6853ec9b0d4a372f35aef3787023 --- /dev/null +++ b/pNE2T4oBgHgl3EQfKQZI/content/tmp_files/load_file.txt @@ -0,0 +1,12576 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf,len=12575 +page_content='Multi-Differential Cross Section Measurements of νµ-Argon Quasielastic-like Reactions with the MicroBooNE Detector P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko,35 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Alterkait,35 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Andrade Aldana,15 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Anthony,5 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Arellano,20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Asaadi,34 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ashkenazi,32 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Balasubramanian,12 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Baller,12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Barr,25 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Barrow,21, 32 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Basque,12 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Benevides Rodrigues,31 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Berkman,12 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bhanderi,20 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bhattacharya,12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bishai,3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Blake,17 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bogart,22 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bolton,16 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Book,14 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Camilleri,10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Caratelli,4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Caro Terrazas,9 F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cavanna,12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cerati,12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Chen,28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Conrad,21 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Convery,28 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cooper-Troendle,39 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Crespo-Anad´on,6 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Del Tutto,12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dennis,5 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Detje,5 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Devitt,17 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Diurba,2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Djurcic,1 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dorrill,15 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Duffy,25 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dytman,26 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Eberly,30 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ereditato,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Evans,20 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fine,18 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Finnerud,20 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Foreman,15 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fleming,39 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Foppiani,14 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Franco,39 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furmanski,23 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Garcia-Gamez,13 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gardiner,12 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ge,10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gollapinni,33, 18 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Goodwin,20 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gramellini,12 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Green,20, 25 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Greenlee,12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gu,3 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Guenette,20 P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Guzowski,20 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Hagaman,39 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Hen,21 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Hicks,18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Hilgenberg,23 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Horton-Smith,16 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Irwin,23 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Itay,28 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' James,12 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ji,3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Jiang,37 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Jo,3, 39 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Johnson,8 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Jwa,10 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kalra,10 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kamp,21 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Karagiorgi,10 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ketchum,12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kirby,12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kobilarcik,12 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kreslo,2 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Leibovitch,4 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lepetic,27 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Li,11 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Li,39 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Li,3 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lin,27 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Littlejohn,15 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Louis,18 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Luo,4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mariani,37 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Marsden,20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Marshall,38 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Martinez,16 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Martinez Caicedo,29 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mason,35 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mastbaum,27 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' McConkey,20, 36 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Meddage,16 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Miller,7 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mills,35 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mogan,9 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mohayai,12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mooney,9 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Moor,5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Moore,12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mora Lepin,20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mousseau,22 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mulleriababu,2 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Naples,26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Navrer-Agasson,20 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nayak,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nebot-Guinot,11 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nowak,17 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Oza,10, 18 O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Palamara,12 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pallat,23 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Paolone,26 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Papadopoulou,1, 21 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Papavassiliou,24 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Parkinson,11 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pate,24 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Patel,17 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pavlovic,12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Piasetzky,32 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ponce-Pinto,39 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pophale,17 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Prince,14 X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Qian,3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Raaf,12 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Radeka,3 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rafique,1 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Reggiani-Guzzo,20 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ren,24 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rochester,28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rodriguez Rondon,29 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rosenberg,35 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ross-Lonergan,18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rudolf von Rohr,2 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Scanavini,39 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Schmitz,7 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Schukraft,12 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Seligman,10 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Shaevitz,10 R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sharankova,12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Shi,5 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Snider,12 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Soderberg,31 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' S¨oldner-Rembold,20 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Spitz,22 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Stancari,12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' St.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' John,12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Strauss,12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sword-Fehlberg,24 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Szelc,11 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tang,33 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Taniuchi,5 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Terao,28 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Thorpe,17 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Torbunov,3 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Totani,4 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Toups,12 Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tsai,28 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tyler,16 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Uchida,5 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Usher,28 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Viren,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Weber,2 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wei,19 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' White,39 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Williams,34 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wolbers,12 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wongjirad,35 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wospakrik,12 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wresilo,5 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wright,21 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wu,12 E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yandel,4 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yang,12 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yates,12 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yu,3 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Zeller,12 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Zennamo,12 and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Zhang3 (The MicroBooNE Collaboration)∗ 1Argonne National Laboratory (ANL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lemont,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 60439,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 2Universit¨at Bern,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bern CH-3012,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Switzerland 3Brookhaven National Laboratory (BNL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 4University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Santa Barbara,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 93106,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 5University of Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cambridge CB3 0HE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 6Centro de Investigaciones Energ´eticas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Medioambientales y Tecnol´ogicas (CIEMAT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Madrid E-28040,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Spain 7University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 60637,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 8University of Cincinnati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cincinnati,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' OH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 45221,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 9Colorado State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fort Collins,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' CO,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 80523,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 10Columbia University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' New York,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 10027,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 11University of Edinburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Edinburgh EH9 3FD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 12Fermi National Accelerator Laboratory (FNAL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Batavia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IL 60510,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 13Universidad de Granada,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Granada E-18071,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Spain 14Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MA 02138,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 15Illinois Institute of Technology (IIT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IL 60616,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 16Kansas State University (KSU),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Manhattan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' KS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 66506,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 17Lancaster University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lancaster LA1 4YW,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 18Los Alamos National Laboratory (LANL),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Los Alamos,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 87545,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 19Louisiana State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Baton Rouge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' LA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 70803,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 20The University of Manchester,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Manchester M13 9PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 21Massachusetts Institute of Technology (MIT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 02139,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 22University of Michigan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ann Arbor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 48109,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 23University of Minnesota,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Minneapolis,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 55455,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 24New Mexico State University (NMSU),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Las Cruces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 88003,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 25University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Oxford OX1 3RH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 26University of Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pittsburgh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' PA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 15260,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 27Rutgers University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Piscataway,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NJ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 08854,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 28SLAC National Accelerator Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Menlo Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' CA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 94025,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 29South Dakota School of Mines and Technology (SDSMT),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rapid City,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' SD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 57701,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03700v1 [hep-ex] 9 Jan 2023 2 30University of Southern Maine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Portland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ME,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 04104,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 31Syracuse University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Syracuse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NY,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 13244,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 32Tel Aviv University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tel Aviv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Israel,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 69978 33University of Tennessee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Knoxville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' TN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 37996,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 34University of Texas,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Arlington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' TX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 76019,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 35Tufts University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Medford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 02155,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 36University College London,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' London WC1E 6BT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 37Center for Neutrino Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Virginia Tech,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Blacksburg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 24061,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA 38University of Warwick,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Coventry CV4 7AL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' United Kingdom 39Wright Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yale University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' New Haven,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 06520,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' USA (Dated: January 11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 2023) We report on a flux-integrated multi-differential measurement of charged-current muon neutrino scattering on argon with one muon and one proton in the final state using the Booster Neutrino Beam and MicroBooNE detector at Fermi National Accelerator Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The data are studied as a function of various kinematic imbalance variables and of a neutrino energy estimator, and are compared to a number of event generator predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We find that the measured cross sections in different phase-space regions are sensitive to nuclear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Our results provide precision data to test and improve the neutrino-nucleus interaction models needed to perform high-accuracy oscillation analyses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Specific regions of phase-space are identified where further model refinements are most needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' INTRODUCTION High-precision measurements of the neutrino mix- ing angles, mass differences, and charge-parity violat- ing phase, and the search for physics beyond the Stan- dard Model are the primary physics goals of many cur- rently operating as well as next-generation neutrino ex- periments [1–6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' These measurements require reliable comparisons of measured and theoretically-expected neu- trino interaction rates in the corresponding detectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Thus, understanding the neutrino-nucleus scattering pro- cesses in detail is a prerequisite for these experiments to reach their discovery potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A number of neutrino oscillation experiments employ liquid argon time projec- tion chambers (LArTPCs) [3–5, 7–9] to detect the par- ticles produced in neutrino interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The ultimate goal of these efforts is both to reconstruct the energy of the neutrino based on the kinematics of the outgo- ing particles and to enable few-percent-level modeling of neutrino-argon interaction rates [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Therefore, high- accuracy modeling of neutrino-argon interactions is of the utmost importance [11–13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This work presents the first measurement of flux- integrated single- and double-differential cross sections for muon-neutrino-argon (νµ-Ar) charged-current (CC) quasielastic (QE)-like scattering reactions as a func- tion of kinematic imbalance variables [14–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Double- differential measurements as a function of a neutrino en- ergy estimator are further reported for the first time in kinematic imbalance bins on argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Motivated by a pre- vious analysis with a similar signal event topology [19], we focus on reactions where a single muon-proton pair is reconstructed with no additional detected particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The results reported here use the MicroBooNE detector [20] ∗ microboone info@fnal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='gov with an exposure of 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 × 1020 protons on target from the Booster Neutrino Beam (BNB) [21] at Fermi National Accelerator Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The experimental setup is presented in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' II, followed by the signal definition and event selection in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The observables of interest are defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sec- tion V describes the cross section extraction and system- atics procedure and Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VI outlines the modeling config- urations used for comparison to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The results are reported in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VII and the conclusions are discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' EXPERIMENTAL SETUP The MicroBooNE LArTPC has an active volume that contains 85 tonnes of argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' It is exposed to BNB neutri- nos, with an energy spectrum that peaks around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 GeV and extends to 2 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Charged particles are produced after the primary neu- trino interaction with the argon nuclei in the LArTPC active volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Scintillation light and electron ionization trails are produced while these charged particles travel through the liquid argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the presence of an electric field of 273 V/cm, the ionization electrons drift towards a system of three anode wire planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Photomultiplier tubes (PMTs) are used to measure the scintillation light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' If the PMT signals are in time coincidence with the beam arrival time, then events are recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Trigger hardware and software selection criteria are designed to minimize the contribution from background events, which are primarily cosmic muons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' After these are ap- plied, enriched data samples are obtained in which a neutrino interaction occurs in ≈ 15% of selected beam spills [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Individual particle tracks are reconstructed with Pandora pattern recognition algorithms based on the measured ionization signals in the enriched data sam- 3 ples [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Particles are identified based on the measured track energy deposition profile, while the particle mo- menta are obtained based on the track length [24, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' SIGNAL DEFINITION & EVENT SELECTION The QE-like signal definition used in this analysis in- cludes all νµ-Ar scattering events with a final-state muon with momentum 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 < pµ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c, and exactly one proton with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 < pp < 1 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Events with final-state neutral pions at any momentum are excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Signal events may contain any number of protons below 300 MeV/c or above 1 GeV/c, neutrons at any momentum, and charged pions with momentum lower than 70 MeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We refer to the events passing this definition as CC1p0π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This signal consists predominantly of QE events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' More complex interactions, namely meson exchange currents (MEC), resonance interactions (RES) and deep inelastic scattering events (DIS), can mimic the experimental sig- nature of true QE events due to final-state interactions (FSI) or particles not satisfying the signal definition as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Candidate muon-proton pairs are isolated by requiring the existence of precisely two track-like and no shower- like objects, as classified by Pandora using a track-score variable [26, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The log-likelihood ratio (LLR) parti- cle identification (PID) score [28] is used to identify the muon and proton candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Muons tend to have higher LLR PID score values than protons, thus the track with the highest score is tagged as the candidate muon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mean- while, the track with the lower score is treated as the candidate proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cosmic muon and non-CC1p0π contamination back- grounds were significantly reduced by applying a require- ment on the candidate proton LLR PID score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We stud- ied the effect of cutting on different values of this quan- tity, which has a strong discrimination power for rejecting MC non-CC1p0π background, out-of-cryostat and cosmic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' That yielded an optimal cut on the proton candi- date LLR score of < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' To further minimize the con- tribution of mis-reconstructed track directions, we took advantage of two muon momentum reconstruction meth- ods available for contained tracks, namely the momen- tum from range [29] and the momentum from Multiple Coulomb Scattering (MCS) [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The range and MCS muon momenta needed to be in agreement within 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We required that the distance between the track start points and the vertex is smaller than the corresponding distance between the track end points and the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We also demanded that the distance between the start points of the two candidate tracks is smaller than the distance between the two end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' More details are provided in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Further reduction of the cosmic tracks and minimiza- tion of bin-migration effects is achieved by considering only fully contained candidate muon-proton pairs within a fiducial volume of 10 cm inside the edge of the detector active volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We retain 9051 data events that satisfy all event selection criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In order to provide an accurate description of the dominant cosmic backgrounds pertinent to surface de- tectors, the full Monte Carlo (MC) simulation consists of a combination of simulated neutrino interactions over- laid on top of beam-off background data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This approach has been extensively used by MicroBooNE [19, 31–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 event generator is used to simulate neutrino interactions with the G18 10a 02 11a configu- ration [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The CCQE and CCMEC predictions have been additionally tuned to T2K νµ-carbon CC0π data [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We refer to the corresponding prediction as G18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' All the final state particles following the primary neutrino interaction are generated by GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' They are further propagated in GENIE through the nucleus to ac- count for FSI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The propagation of the particles outside the nucleus is simulated using GEANT4 [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The Micro- BooNE detector response is modeled using the LArSoft framework [39, 40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Based on this MC prediction, we obtain a purity of ≈ 70% and an efficiency for selecting CC1p0π events of ≈ 10%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' OBSERVABLES In neutrino-nucleus scattering events, there is an im- balance between the true initial neutrino momentum and the true sum of final-state lepton and hadron momenta as a result of nuclear effects [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A schematic represen- tation of the kinematic imbalance variables of interest in this work is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Schematic representation of the kinematic imbalance variables on the plane transverse to the beam direction using CC1p0π events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Using the CC1p0π candidate muon-proton pair kine- matics, the missing momentum in the plane transverse to the beam direction is defined as δpT = |⃗pT µ + ⃗pT p|, (1) where ⃗pT µ and ⃗pT p are the projections of the momenta of the outgoing lepton and proton on the transverse plane, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the absence of nuclear effects, purely QE interactions would yield δpT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the presence of the SPTX o0 T SPTy ph SPT SPT Sα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 dense nuclear medium, this variable encapsulates infor- mation related to the Fermi motion, but it is smeared due to FSI and non-QE interactions, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Further discussion on the FSI smearing effects can be found in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' BNB Data Cosmic (8%) MC QE (58%) MC MEC (19%) MC RES (13%) MC DIS (2%) 0 500 1000 1500 Number of events / bin 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ Reconstructed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Prediction Data Prediction Uncertainty POT 20 10 × MicroBooNE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Distribution of the selected CC1p0π events as a function of the transverse missing momentum δpT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Only sta- tistical uncertainties are shown on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The interac- tion contributions are obtained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The bottom panel shows the ratio of data to prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The direction of the transverse momentum imbalance δpT is described by the angle δαT = arccos � − ⃗pT µ · δ⃗pT pT µ δpT � , (2) which is uniformly distributed in the absence of FSI due to the isotropic nature of the Fermi motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the pres- ence of FSI, the proton momentum is generally reduced and the δαT distribution becomes weighted towards 180◦, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The opening angle δφT between the correlated candi- date muon-proton pair on the transverse plane is given by δφT = arccos � − ⃗pT µ · ⃗pT p pT µ pT p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (3) In the absence of nuclear effects, QE events would be con- centrated at δφT = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' When nuclear effects are present, QE events can occupy wider angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' At the same time, non-QE events are dominant in the high δφT part of the tail and their contribution is fairly flat across all angles, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The muon-proton momentum imbalances transverse and longitudinal to the transverse lepton momentum [17] are defined as δpT,x = (ˆpν × ˆpµ T ) · δ⃗pT δpT,y = −ˆpµ T · δ⃗pT , (4) BNB Data Cosmic (8%) MC QE (58%) MC MEC (19%) MC RES (13%) MC DIS (2%) 0 500 1000 1500 2000 Number of events / bin 0 20 40 60 80 100 120 140 160 180 [deg] T α δ Reconstructed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Prediction Data Prediction Uncertainty POT 20 10 × MicroBooNE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Distribution of the selected CC1p0π events as a function of the transverse missing momentum direction δαT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Only statistical uncertainties are shown on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The interaction contributions are obtained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The bottom panel shows the ratio of data to prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' BNB Data Cosmic (8%) MC QE (58%) MC MEC (19%) MC RES (13%) MC DIS (2%) 0 1000 2000 3000 Number of events / bin 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ Reconstructed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Prediction Data Prediction Uncertainty POT 20 10 × MicroBooNE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Distribution of the selected CC1p0π events as a function of the muon-proton transverse opening angle δφT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Only statistical uncertainties are shown on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The interaction contributions are obtained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The bottom panel shows the ratio of data to prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' and can also be written as δpT,x = δpT · sin δαT δpT,y = δpT · cos δαT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (5) These distributions can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 5 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 6, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The δpT,x distribution is symmetric around 0 GeV/c due to the presence of the sin δαT factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 5 and the fact that δαT ranges from 0o to 180o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The width of the distribution is driven by the Fermi motion that affects the δpT magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Unlike δpT,x, the δpT,y dis- tribution is asymmetric with an enhanced contribution from negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The asymmetry is driven by the 5 presence of the cos δαT factor in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 5 and the fact that δαT is mainly peaked around 180o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Given that the for- ward δαT peak is driven by FSI, the size of the δpT,y asymmetry is also sensitive to the FSI strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' BNB Data Cosmic (8%) MC QE (58%) MC MEC (19%) MC RES (13%) MC DIS (2%) 0 500 1000 1500 2000 2500 Number of events / bin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ Reconstructed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Prediction Data Prediction Uncertainty POT 20 10 × MicroBooNE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Distribution of the selected CC1p0π events as a function of the perpendicular component of the transverse missing momentum δpT,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Only statistical uncertainties are shown on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The interaction contributions are ob- tained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The bottom panel shows the ratio of data to prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' BNB Data Cosmic (8%) MC QE (58%) MC MEC (19%) MC RES (13%) MC DIS (2%) 0 500 1000 1500 2000 Number of events / bin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,y p δ Reconstructed 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Prediction Data Prediction Uncertainty POT 20 10 × MicroBooNE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Distribution of the selected CC1p0π events as a func- tion of the longitudinal component of the transverse missing momentum δpT,y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Only statistical uncertainties are shown on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The interaction contributions are obtained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The bottom panel shows the ratio of data to pre- diction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Finally, the calorimetric energy reconstruction ECal = Eµ + Tp + BE (6) is investigated, where Eµ is the muon energy, Tp is the proton kinetic energy and BE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 GeV/c is the aver- age binding energy for argon [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This energy estimator, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 7, is an approximation for the true energy of the incoming neutrino and is used in oscillation searches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' BNB Data Cosmic (8%) MC QE (58%) MC MEC (19%) MC RES (13%) MC DIS (2%) 0 1000 2000 3000 Number of events / bin 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal Reconstructed E 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Prediction Data Prediction Uncertainty POT 20 10 × MicroBooNE 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Distribution of the selected CC1p0π events as a function of the calorimetric energy reconstruction ECal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Only statistical uncertainties are shown on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The interac- tion contributions are obtained from simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The bottom panel shows the ratio of data to prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' CROSS SECTION EXTRACTION & SYSTEMATICS The flux-averaged differential event rate as a function of a given variable x in bin i is obtained by dR dxi = Ni − Bi T · Φν · ∆i (7) where Ni and Bi are the number of measured events and the expected background events, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' T is the number of target argon nuclei in the fiducial volume of in- terest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Φν corresponds to the integrated BNB flux and ∆i corresponds to the i-th bin width or area for the single- and double-differential results, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We report the extracted cross sections for CC1p0π in- teractions using the Wiener singular value decomposition (Wiener-SVD) unfolding technique as a function of un- folded kinematic variables [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This unfolding procedure corrects a measured event rate for inefficiency and resolu- tion effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This is achieved by performing a minimiza- tion of a χ2 score that compares data to a prediction and allows for a regularization term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A Wiener filter deter- mines the level of regularization that is required to mini- mize the mean square error between the variance and bias of the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In addition to the measured event rate, the method uses a covariance matrix calculated from simu- lated events accounting for the statistical and systematic uncertainties on the measurement as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' It also re- quires the construction of a response matrix describing 6 the expected detector smearing and reconstruction effi- ciency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The output of the method is an unfolded differential cross section, a covariance matrix describing the total uncertainty on the unfolded result, and an additional smearing matrix that we refer to as AC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The latter con- tains information about the regularization and bias of the measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The corresponding AC matrices have been applied to all the cross section predictions included in this work when a comparison to the unfolded data is performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The AC matrix should be applied to any in- dependent theoretical prediction when a comparison is performed to the data reported in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The data release, the unfolded covariance matrices, and the ad- ditional matrices AC can be found in the Supplemental Material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The total covariance matrix Eij = Estat ij + Esyst ij in- cludes the statistical and systematic uncertainties on the differential event rate associated with our measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Estat ij is a diagonal covariance matrix with the statisti- cal uncertainties and Esyst ij is a covariance matrix that incorporates the total systematic uncertainties detailed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The neutrino flux is predicted using the flux simula- tion of the MiniBooNE collaboration that used the same beam line [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Neutrino cross section modeling uncer- tainties were estimated using the GENIE framework of event reweighting [34, 35, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The rescattering uncer- tainties were obtained using GEANT4 and the relevant reweighting package [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' For each of these sources of uncertainty, we use a multisim technique [45], which con- sists of generating a large number of MC replicas, each one called a “universe”, where model parameters are var- ied within their uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The simultaneous varying of many model parameters provides a correct treatment of their correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A total of n such universes are used to construct a covariance matrix corresponding to each source of uncertainty, Eij = 1 n k=n � k=1 � Rk i − RCV i � � Rk j − RCV j � (8) where RCV i (RCV j ) and Rk i (Rk j ) are the flux-averaged event rates for the central value and systematic universe k in a measured bin i (j), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The resulting covariance matrices are summed together to estimate the relevant uncertainty from each source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' An additional cross section uncertainty using the NuWro v19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 event generator prediction [46] as an alternative universe has been added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The relevant mod- eling is detailed in section VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated NuWro cross sections are obtained using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 7 and the corre- sponding covariance matrices are constructed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8 and a single universe (n = 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' For detector model systematic uncertainties, one de- tector parameter is varied each time by 1σ and is re- ferred to as a “unisim”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' These include variations in the light yield, the ionization electron recombination model, space-charge effects, and waveform deconvolution [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We then examine the impact of each parameter varia- tion on the MC event rates by obtaining the differences with respect to the central value on a bin-by-bin basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We define the total detector 1σ systematic uncertainty by summing in quadrature the effect of m detector vari- ations using the formalism introduced in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8, Eij = k=m � k=1 � Rk i − RCV i � � Rk j − RCV j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (9) The full fractional uncertainty on the integrated to- tal cross section is 11% and includes contributions from the neutrino flux prediction (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3%), neutrino interaction cross section modeling (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3%), detector response mod- eling (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9%), beam exposure (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3%), statistics (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5%), number-of-scattering-targets (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2%), reinteractions (1%), and out-of-cryostat interaction modeling (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the results presented below, the inner error bars on the reported cross sections correspond to the statisti- cal uncertainties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The systematic uncertainties were de- composed into shape- and normalization-related sources following the procedure outlined in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The cross- term uncertainties were incorporated in the normaliza- tion part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The outer error bars on the reported cross sections correspond to statistical and shape uncertain- ties added in quadrature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The normalization uncertain- ties are presented with the gray band at the bottom of each plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Overflow (underflow) values are included in the last (first) bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MODELING CONFIGURATIONS The nominal MC neutrino interaction prediction (G18) uses the local Fermi gas (LFG) model [49], the Nieves CCQE scattering prescription [50] which includes Coulomb corrections for the outgoing muon [51] and ran- dom phase approximation (RPA) corrections [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Addi- tionally, it uses the Nieves MEC model [53], the KLN- BS RES [54–57] and Berger-Sehgal coherent (COH) [58] scattering models, the hA2018 FSI model [59], and MicroBooNE-specific tuning of model parameters [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Our results are also compared to a number of alterna- tive event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' GiBUU 2021 (GiBUU) uses sim- ilar models, but they are implemented in a coherent way by solving the Boltzmann-Uehling-Uhlenbeck trans- port equation [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The modeling includes the LFG model [49], a standard CCQE expression [61], an em- pirical MEC model and a dedicated spin dependent res- onance amplitude calculation following the MAID anal- ysis [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The DIS model is from PYTHIA [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' GiBUU’s FSI treatment propagates the hadrons through the resid- ual nucleus in a nuclear potential which is consistent with the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NuWro v19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 (NuWro) uses the LFG model [49], the Llewellyn Smith model for QE events [63], the Nieves model for MEC events [64], the Adler-Rarita-Schwinger formalism to calculate the 7 ∆ resonance explicitly [57], the BS COH [58] scat- tering model and an intranuclear cascade model for FSI [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0 (NEUT) uses the LFG model [49], the Nieves CCQE scattering prescription [50], the Nieves MEC model [53], the BS RES [54–57] and BS COH [58] scattering models, and FSI with Oset medium corrections for pions [34, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In addition to the alternative event generators, our results are compared to a number of differ- ent GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' These include an older version, GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10 (Gv2) [34, 35], which uses the Bodek-Ritchie Fermi Gas model, the Llewellyn Smith CCQE scattering prescription [63], the em- pirical MEC model [65], a Rein-Sehgal RES and COH scattering model [66], and a data driven FSI model denoted as “hA” [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Another model, “Untuned”, uses the GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 G18 10a 02 11a configuration without additional MicroBooNE-specific tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Finally, the newly added theory-driven GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0 G21 11b 00 000 configuration (G21) is shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This includes the SuSAv2 prediction for the QE and MEC scattering parts [68] and the hN2018 FSI model [69].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The modeling options for RES, DIS, and COH interactions are the same as for G18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The χ2/bins data comparison for each generator shown on all the figures takes into account the total covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Theoretical uncertainties on the models them- selves are not included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' RESULTS Along with the aforementioned kinematic imbalance and energy estimator results, the data are also pre- sented as a function of the lepton angular orientation (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Previous MicroBooNE measurements using dif- ferent signal definitions [19, 70, 71] showed discrepan- cies in that quantity, primarily in the forward direc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' These analyses used an older simulation prediction, namely GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2, to account for the efficiency cor- rections and beam-induced backgrounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This work illus- trates that all generator (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8a) and GENIE configura- tion (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8b) predictions are in good agreement with the data when reported as a function of cosθµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 9 and 10 show the measured single-differential cross sections as a function of δpT using all the events (panel a), as well as the double-differential results as a function of the same kinematic variable in δαT bins (pan- els b-e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the presence of FSI, the proton can rescat- ter or be absorbed, yielding larger kinematic imbalances on the transverse plane and δpT values that extend be- yond the Fermi momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furthermore, the same ex- tended tail can be obtained when pions produced due to multi-nucleon effects (MEC or RES) are either absorbed or below the detection threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The single-differential result shows such a high-momentum tail that extends above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This picture is consistent with the results reported by the T2K and MINERvA collabora- tions [15, 16, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Unlike the single-differential result, the double differential results with low δαT extend only slightly above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' That indicates that this region contains minimal FSI and multi-nucleon effects and the δpT distribution is driven by the nucleon Fermi motion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' On the other hand, the higher δαT values correspond to δpT distributions that extend beyond 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 GeV/c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This behavior is indicative of the presence of FSI and multi- nucleon effects that smear the δpT distribution to higher values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Future multi-differential results can help further disentangle the contributions from these effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 9 shows the comparisons to a number of available neutrino event generators with NuWro and G18 showing the best agreement over all events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 10 shows the same results compared to a number of GENIE configurations illustrating that Gv2 is disfavored, an observation that is driven by the Gv2 low δpT behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furthermore, Untuned shows a good χ2/bins performance across all slices but predicts lower values than data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 11 shows the double-differential results as a function of δpT in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In a factorized nuclear model such as the LFG, the Fermi motion part of δpT should stay constant in terms of the shape as a function of the outgoing lepton kinematics, since in such models the initial state nucleon momentum is a property of the nucleus that cannot be affected by the interaction mo- mentum or energy transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' That is indeed the observed behavior in the reported results across all event genera- tors and configurations, where no evidence of the inad- equacy of the factorization approach is observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fig- ure 11 shows the comparisons to a number of available neutrino event generators, where the G18 prediction is favored based on the χ2/ndf results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Apart from the factorization, a better separation between QE and non- QE can be gained depending on the cosθµ region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12 for G18, MEC events play a more pronounced role for forward muon scattering and in the high δpT tail, as opposed to backward scattering angles, which are much more strongly populated by QE events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furthermore, the G18 cross section prediction falls be- low the data in the -1 < cosθµ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 region, as seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12a and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' That could indicate that addi- tional contribution from the QE part of the G18 predic- tion is needed beyond the MicroBooNE tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 13 shows the same interaction breakdown for GiBUU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Unlike G18, GiBUU illustrates a peak shift to the right, which be- comes more pronounced in the backward direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This shift is driven by the enhanced MEC contribution in higher δpT values and the reduced QE contribution at smaller values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the backward direction, GiBUU further shows a cross section excess driven by the MEC contri- bution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 14 shows the same results compared to a number of GENIE configurations illustrating that Gv2 is disfavored due to the low δpT bin behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 15 and 16 show the double-differential cross section as a function of δpT in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The factoriza- tion of the nuclear motion is mostly preserved in cosθp bins, analogously to the previous result in cosθµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fig- 8 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 µ θ cos 0 5 10 15 20 Ar 2 cm 38 10 µ θ dcos σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/18) GiBUU (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/18) NEUT (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/18) NuWro (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/18) (a) 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 µ θ cos 0 5 10 15 20 Ar 2 cm 38 10 µ θ dcos σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/18) Untuned (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/18) G21 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/18) Gv2 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/18) (b) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated single-differential cross sections as a function of cosθµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (a) Generator and (b) GENIE configuration predictions are compared to data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ure 15 shows the comparisons to a number of available neutrino event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The GiBUU prediction is sig- nificantly lower than the data in the backward proton angle for low δpT values, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 15a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fig- ure 16 shows the same results compared to a number of GENIE configurations illustrating that Gv2 is disfavored across all cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 17, this particularly poor performance is driven by the QE con- tribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' For backward scattering events (panel a), the QE contribution predicted by Llewellyn Smith is signifi- cantly overestimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' For intermediate angles (0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5), the same QE model results in an unphysical dou- ble peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' For forward scattering (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 1), the Gv2 QE prediction yields a pronounced contribution at lower values of δpT compared to the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 18 and 19 show the single-differential cross sec- tions as a function of δαT using all the events (panel a), as well as the double-differential results in the same kinematic variable in δpT bins (panels b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The single- differential results shown in panel a yield some inter- esting observations when compared to the relevant T2K and MINERvA results [15, 16, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Our distribution il- lustrates a slightly asymmetric behavior, similar to the one reported by the T2K collaboration at a compara- ble energy with MicroBooNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Within the precision of the data sets, the mass-number dependence of the nu- clear effects seems to be reasonably well-modeled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Un- like our result, the measurement by MINERvA reports a more pronounced asymmetry on hydrocarbon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The breakdown plots in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 18 in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [72] show that this be- havior is driven by enhanced pion-production rates due to the higher average beam energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Low δpT values re- sult in a fairly uniform δαT distribution indicative of the absence of FSI effects in that part of the phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' On the other hand, higher δpT values result in a highly asym- metric δαT distribution, which is driven by the strength of the FSI interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 18 shows the compar- isons to a number of available neutrino event generators, where NuWro is the generator with the most conservative FSI strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 19 shows the same results com- pared to a number of GENIE configurations, where Gv2 yields the highest χ2/bins result, especially in the lowest δpT region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 20, this is driven by the Gv2 QE performance, which results in peaks at the edges of the distribution, unlike the data result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 21 and 22 show the double-differential results as a function of δαT in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' All the bins illustrate an asymmetric δαT distribution, with the exception of the region where cosθµ ≈ 1, with the latter implying that this part of phase-space includes events with minimal FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 21 shows the comparisons to a number of available neutrino event generators with GiBUU giving the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 22 shows the same results com- pared to a number of GENIE configurations, illustrating that Gv2 is disfavored in the region where cosθµ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 23 and 24 show the double-differential cross sections as a function of δαT in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The results in the region with 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 show a fairly flat distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The cross section distributions correspond- ing to forward and backward proton scattering exhibit an FSI-driven asymmetric behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 23 shows the comparisons to a number of available neutrino event gen- erators, where NuWro yields a prediction that is disfavored for forward scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 24 shows the same results compared to a number of GENIE configurations, illustrat- ing that Gv2 is disfavored across all cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the -1 < cosθp < 0 region shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 24a, all the predictions illustrate a peak close to 180◦ with the exception of Gv2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The driving force for this difference is the Gv2 QE con- tribution, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This is indicative of potential modeling issues in the Llewellyn Smith QE cross section and of the hA FSI performance used in the 9 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 GeV/c Ar 2 cm 38 10 T p δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) GiBUU (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) NEUT (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) NuWro (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) (a) All events 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/11) GiBUU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/11) NEUT (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/11) NuWro (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/11) o < 45 T α δ (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) GiBUU (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/12) NEUT (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/12) NuWro (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) o < 90 T α δ < o (c) 45 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) GiBUU (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) NEUT (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) NuWro (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) o < 135 T α δ < o (d) 90 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) GiBUU (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/13) NEUT (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/13) NuWro (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) o < 180 T α δ < o (e) 135 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-e) double- (in δαT bins) differential cross sections as a function of δpT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gv2 prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Unlike Gv2, the theory-driven GENIE v3 family of predictions (G18, Untuned, and G21) closely fol- low the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 26 and 27 show the single-differential cross sec- tions as a function of δφT using all the events (panel a), as well as the double-differential results as a func- 10 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 GeV/c Ar 2 cm 38 10 T p δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) Untuned (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) G21 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) Gv2 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/13) (a) All events 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/11) Untuned (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/11) G21 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/11) Gv2 (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/11) o < 45 T α δ (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) Untuned (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) G21 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/12) Gv2 (31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/12) o < 90 T α δ < o (c) 45 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) Untuned (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) G21 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) Gv2 (54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/13) o < 135 T α δ < o (d) 90 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) Untuned (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) G21 (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) Gv2 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/13) o < 180 T α δ < o (e) 135 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-e) double- (in δαT bins) differential cross sections as a function of δpT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' tion of the same kinematic variable in δpT bins (panels b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 26 shows the comparisons to a number of available neutrino event generators, with all the genera- tors illustrating a fairly good performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This result is consistent with the one reported by the T2K collabora- tion [15, 72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the lowest δpT region shown in panel b, 11 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 1 2 3 4 5 6 7 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) GiBUU (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/13) NEUT (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) NuWro (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/13) < 0 µ θ (a) -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 5 10 15 20 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) GiBUU (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) NEUT (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) NuWro (32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 µ θ (b) 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) GiBUU (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/13) NEUT (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) NuWro (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 µ θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) GiBUU (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) NuWro (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) < 1 µ θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δpT in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NuWro is the generator with the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fig- ure 27 shows the same results compared to a number of GENIE configurations, where Gv2 is disfavored in all regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' At small δpT values the cross section is decreas- ing and zero above ≈ 40◦ which indicates the absence of multi-nucleon and FSI effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Higher δpT values lead to δφT cross sections that extend up to 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This behavior is primarily driven by multi-body effects with hadrons below the detection threshold that introduce large kine- matic imbalances, as can be seen in panels c-d of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 29 and 30 show the single-differential cross sec- tions as a function of δpT,x using all the events (panel a), as well as the double-differential results in the same kinematic variable in δpT,y slices (panels b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fig- ure 29 shows the comparisons to a number of avail- able neutrino event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The central region with |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c is dominated by QE interactions, while the broader distributions with |δpT,y| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c are mainly driven by MEC events, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In the MEC dominated region of δpT,y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c, all the generators, apart from GiBUU, seem to be lacking in terms of the peak strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' GiBUU seems to be overestimating that MEC contribution in the δpT,y < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' With the exception of NEUT, all the event generators illustrate a good performance in the |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 30 shows the same results compared to a number of GENIE configurations, where Gv2 shows the worst performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The aforementioned results in kinematic imbalance variables illustrate significant differences across the event generators and configurations used for comparison, espe- cially in the case of the double-differential studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yet, the quantity that enters the oscillation probability is the true neutrino energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Neutrino energy estimators, such as the calorimetric energy ECal defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 6, are used as a proxy for the true quantity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The studies reported next present the results as a function of ECal in bins of the kinematic imbalance variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 1 2 3 4 5 6 7 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0 µ θ (a) G18, -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 5 10 15 20 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 µ θ (b) G18, 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 µ θ (c) G18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 1 µ θ (d) G18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the flux-integrated double-differential cross sections as a function of δpT for data and the G18 GENIE prediction in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the G18 GENIE prediction for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 32 and 33 show the single-differential cross sec- tions as a function of ECal using all the events (panel a), as well as the double-differential results in the same kine- matic variable in δpT bins (panels b-d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 32 shows the comparisons to a number of available neutrino event generators, where the ECal distribution covers the same energy spectrum across all bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' All the event generators illustrate an equally good performance in the lowest δpT bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NEUT and NuWro show a deficit relative to the data in the highest δpT bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 33 shows the same results compared to a number of GENIE configurations, where G18 illustrates the best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Interestingly, all the alternative GENIE configurations illustrate a plateau in the highest δpT bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 34 and 35 show the double-differential results as a function of ECal in δαT bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 34 shows the comparisons to a number of available neutrino event gen- erators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Once again, the ECal distribution covers the same energy spectrum across all of our results and all the event generators show fairly good behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 35 shows the same results compared to a number of GENIE configurations, where all the GENIE configurations except for G18 illustrate shape and strength differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figures 36 and 37 show the double-differential results as a function of ECal in δpT,y bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 36 shows the comparisons to a number of available neutrino event gen- erators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' All event generators predict very similar cross sections for -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 < δpT,y < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c (panel a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Unlike this central region, the |δpT,y| > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c results yield a wide spread across the generator predictions (panels b-c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furthermore, apart from GiBUU, all the predictions lack strength in the δpT,y < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c bin (panel b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Additionally, NEUT illustrates the same deficit in the δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c bin (panel c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Figure 37 shows the same results compared to a number of GENIE configurations, where all the GENIE configurations but G18 illustrate a poor performance due to shape and strength issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 13 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 1 2 3 4 5 6 7 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0 µ θ (a) GiBUU, -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 5 10 15 20 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 µ θ (b) GiBUU, 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 µ θ (c) GiBUU, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 1 µ θ (d) GiBUU, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the flux-integrated double-differential cross sections as a function of δpT for data and the GiBUU prediction in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the GiBUU prediction for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' CONCLUSIONS This work reports on measurements of flux-integrated differential cross sections for event topologies with a sin- gle muon and a single proton detected in the final state using the Booster Neutrino Beam at Fermi National Ac- celerator Laboratory and the MicroBooNE detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The data were studied for the first time in the form of single- differential cross sections in kinematic imbalance vari- ables on argon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furthermore, the first double-differential cross sections in these variables were reported on the same nucleus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Additionally, novel double-differential cross section measurements of a neutrino energy esti- mator in bins of these variables were presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The results were compared to a number of event genera- tors and model configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The predictions as a function of the energy estimator across all generators and model configurations remain mostly unchanged re- gardless of the kinematic variable used for the double- differential measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The good agreement observed across the calorimetric energy distributions suggests that the energy dependence is largely well-modeled across all predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Unlike the energy estimator results, we found that the measured kinematic imbalance cross sec- tions in different phase-space regions are sensitive to nu- clear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The performance of the event generators and configurations varies depending on the observable of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Overall, the GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 G18 10a 02 11a cross section predictions with the MicroBooNE-specific tuning (G18) fit the data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' On the other hand, the GENIE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10 (Gv2) cross section predictions are systematically a poor fit to data with significant shape differences across all variables of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 G18 10a 02 11a configuration without additional tuning (Untuned) shows a systematic deficit of ∼ 20% which necessitated the development of the afore- mentioned tune.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The GENIE v3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0 G21 11b 00 000 configuration (G21) serves as an example of a theory- 14 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 1 2 3 4 5 6 7 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) Untuned (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) G21 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) Gv2 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) < 0 µ θ (a) -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 5 10 15 20 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) Untuned (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/13) G21 (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/13) Gv2 (65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 µ θ (b) 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) Untuned (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/13) G21 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) Gv2 (34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 µ θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 µ θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) Untuned (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/13) G21 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/13) Gv2 (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) < 1 µ θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δpT in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' driven GENIE configuration that shows good agreement with data in most variables without the need for addi- tional tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' GiBUU 2021 (GiBUU) shows good agree- ment with data in most kinematic variables, with the exception of δpT , where a systematic shift to higher val- ues of δpT has been identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A potential source of this shift is due to the GiBUU MEC modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The NuWro v19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 (NuWro) prediction falls bellow the data due to poor FSI modeling and shows significant shape differences in FSI-dominated parts of the phase-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' NEUT v5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0 (NEUT) also results in predictions mostly falling below the data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' This mismodeling remains largely unnoticed when combined into the calorimetric energy estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yet, future neutrino oscillation mea- surements will rely on accurate cross section predictions and a precise mapping between measured and true neu- trino energies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Therefore, such mismodeling effects might impact their experimental sensitivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The reported re- sults both provide precision data to benchmark neutrino- nucleus interaction models and establish phase-space re- gions where precise reaction modeling is still needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' IX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ACKNOWLEDGMENTS This document was prepared by the MicroBooNE col- laboration using the resources of the Fermi National Ac- celerator Laboratory (Fermilab), a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Department of Energy, Office of Science, HEP User Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fermilab is managed by Fermi Research Alliance, LLC (FRA), act- ing under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' DE-AC02-07CH11359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Micro- BooNE is supported by the following: the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Depart- ment of Energy, Office of Science, Offices of High En- ergy Physics and Nuclear Physics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' National Sci- ence Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' the Swiss National Science Foundation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' the Science and Technology Facilities Council (STFC), part of the United Kingdom Research and Innovation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' the Royal Society (United Kingdom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' the UK Research 15 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/8) GiBUU (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/8) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/8) NuWro (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/8) < 0 p θ (a) -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 2 4 6 8 10 12 14 16 18 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) GiBUU (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) NEUT (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) NuWro (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 p θ (b) 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/13) GiBUU (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) NEUT (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) NuWro (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 p θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 70 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/12) GiBUU (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/12) NEUT (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/12) NuWro (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/12) < 1 p θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δpT in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' and Innovation (UKRI) Future Leaders Fellowship;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' and The European Union’s Horizon 2020 Marie Sklodowska- Curie Actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Additional support for the laser calibra- tion system and cosmic ray tagger was provided by the Albert Einstein Center for Fundamental Physics, Bern, Switzerland.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We also acknowledge the contributions of technical and scientific staff to the design, construction, and operation of the MicroBooNE detector as well as the contributions of past collaborators to the development of MicroBooNE analyses, without whom this work would not have been possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' For the purpose of open access, the authors have applied a Creative Commons Attribu- tion (CC BY) license to any Author Accepted Manuscript version arising from this submission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 16 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/8) Untuned (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/8) G21 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/8) Gv2 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/8) < 0 p θ (a) -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 2 4 6 8 10 12 14 16 18 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) Untuned (21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/13) G21 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/13) Gv2 (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 p θ (b) 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/13) Untuned (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/13) G21 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/13) Gv2 (70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/13) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 p θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 70 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/12) Untuned (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/12) G21 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) Gv2 (39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) < 1 p θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δpT in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 17 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 3 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0 p θ (a) Gv2 , -1 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 2 4 6 8 10 12 14 16 18 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 p θ (b) Gv2 , 0 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 p θ (c) Gv2 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 60 70 GeV/c Ar 2 cm 38 10 p θ dcos T p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 1 p θ (d) Gv2 , 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the flux-integrated double-differential cross sections as a function of δpT for data and the Gv2 GENIE prediction in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the Gv2 GENIE prediction for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 18 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12 deg Ar 2 cm 38 10 T α δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) GiBUU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) NEUT (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/7) NuWro (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) (a) All events 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) GiBUU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) NEUT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) NuWro (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) NEUT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) NuWro (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) GiBUU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) NEUT (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) NuWro (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δαT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 19 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12 deg Ar 2 cm 38 10 T α δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) Untuned (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) G21 hN (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) Gv2 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) (a) All events 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) Untuned (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) G21 hN (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) Gv2 (84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) Untuned (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) G21 hN (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) Gv2 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) Untuned (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) G21 hN (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) Gv2 (17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δαT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 20 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (a) G18, 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) Gv2, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the data flux-integrated double-differential cross section as a function of δαT for events in the region δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c region against the G18 and Gv2 GENIE predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the (a) G18 and (b) Gv2 GENIE predictions for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 21 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) GiBUU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) NuWro (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) < 0 µ θ (a) -1 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) NEUT (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) NuWro (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 µ θ (b) 0 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) NEUT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) NuWro (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 µ θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/7) GiBUU (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) NEUT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) NuWro (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) < 1 µ θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δαT in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 22 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) Untuned (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) G21 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) Gv2 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) < 0 µ θ (a) -1 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) Untuned (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) G21 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) Gv2 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 µ θ (b) 0 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) Untuned (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) G21 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/7) Gv2 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 µ θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 deg Ar 2 cm 38 10 µ θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/7) Untuned (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) G21 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) Gv2 (26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) < 1 µ θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δαT in cosθµ bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 23 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) GiBUU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) NEUT (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) NuWro (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) < 0 p θ (a) -1 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) GiBUU (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) NuWro (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 p θ (b) 0 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) GiBUU (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) NEUT (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) NuWro (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 p θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) GiBUU (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) NEUT (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) NuWro (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) < 1 p θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δαT in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 24 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) Untuned (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) G21 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) Gv2 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) < 0 p θ (a) -1 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) Untuned (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/7) G21 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) Gv2 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 p θ (b) 0 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) Untuned (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/7) G21 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/7) Gv2 (27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/7) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 p θ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/7) Untuned (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/7) G21 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/7) Gv2 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/7) < 1 p θ (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of δαT in cosθp bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 25 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0 p θ (a) G18, -1 < cos 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012 deg Ar 2 cm 38 10 p θ dcos T α δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0 p θ (b) Gv2, -1 < cos FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the data flux-integrated double-differential cross section as a function of δαT for events in the region -1 < cosθp < 0 region against the G18 and Gv2 GENIE predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the (a) G18 and (b) Gv2 GENIE predictions for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 26 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg Ar 2 cm 38 10 T φ δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/12) GiBUU (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/12) NEUT (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) NuWro (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/12) (a) All events 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/4) GiBUU (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/4) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/4) NuWro (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/4) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) GiBUU (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/9) NEUT (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) NuWro (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/6) GiBUU (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/6) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/6) NuWro (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/6) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δφT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 27 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg Ar 2 cm 38 10 T φ δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/12) Untuned (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/12) G21 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/12) Gv2 (41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/12) (a) All events 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/4) Untuned (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/4) G21 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/4) Gv2 (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/4) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) Untuned (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) G21 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) Gv2 (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/6) Untuned (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/6) G21 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/6) Gv2 (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/6) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of δφT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 28 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 deg Ar 2 cm 38 10 T φ δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS (a) G18, All events 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) G18, 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) G18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 deg GeV/c Ar 2 cm 38 10 T p δ d T φ δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) G18, FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the flux-integrated double- (in δpT bins) differential cross sections as a function of δφT for data and the G18 GENIE prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the G18 prediction for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 5 10 15 20 25 30 35 GeV/c Ar 2 cm 38 10 T,x p δ d σ d (a) All events MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/11) GiBUU (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/11) NEUT (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/11) NuWro (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 2 4 6 8 10 12 14 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (b) MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/11) GiBUU (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/11) NEUT (25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/11) NuWro (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 20 40 60 80 100 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (c) | MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/11) GiBUU (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/11) NEUT (24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/11) NuWro (19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/11) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 2 4 6 8 10 12 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (d) MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) GiBUU (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/9) NEUT (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) NuWro (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/9) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT,y bins) differential cross sections as a function of δpT,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 5 10 15 20 25 30 35 GeV/c Ar 2 cm 38 10 T,x p δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/11) Untuned (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/11) G21 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/11) Gv2 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/11) (a) All events 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 2 4 6 8 10 12 14 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/11) Untuned (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/11) G21 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/11) Gv2 (79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/11) < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 20 40 60 80 100 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/11) Untuned (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/11) G21 (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/11) Gv2 (107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/11) | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (c) | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 2 4 6 8 10 12 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) Untuned (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) G21 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) Gv2 (74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT,y bins) differential cross sections as a function of δpT,x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 31 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 5 10 15 20 25 30 35 GeV/c Ar 2 cm 38 10 T,x p δ d σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS (a) All events 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 2 4 6 8 10 12 14 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 20 40 60 80 100 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (c) | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 [GeV/c] T,x p δ 0 2 4 6 8 10 12 Ar 2 /c 2 GeV 2 cm 38 10 T,y p δ d T,x p δ d σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm QE MEC RES DIS > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Comparison between the flux-integrated double- (in δpT,y bins) differential cross sections as a function of δpT,x for data and the G18 GENIE prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored stacked histograms show the results of theoretical cross section calculations using the G18 GENIE prediction for QE (blue), MEC (orange), RES (green), and DIS (red) interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 5 10 15 20 GeV Ar 2 cm 38 10 Cal dE σ d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/9) GiBUU (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) NEUT (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) NuWro (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/9) (a) All events 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 20 40 60 /c Ar 2 GeV 2 cm 38 10 T p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) GiBUU (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/9) NEUT (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/9) NuWro (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 10 20 30 /c Ar 2 GeV 2 cm 38 10 T p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) NEUT (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) NuWro (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 2 4 6 8 /c Ar 2 GeV 2 cm 38 10 T p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/9) NEUT (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) NuWro (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- (in δpT bins) differential cross sections as a function of ECal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 5 10 15 20 GeV Ar 2 cm 38 10 Cal dE σ d MicroBooNE Data 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Shape ⊕ Stat Norm G18 (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/9) Untuned (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) G21 (51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) Gv2 (63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) (a) All events 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 20 40 60 /c Ar 2 GeV 2 cm 38 10 T p δ d Cal dE σ 2 d MicroBooNE Data 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Shape ⊕ Stat Norm G18 (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) Untuned (71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) G21 (76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) Gv2 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c T p δ (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 10 20 30 /c Ar 2 GeV 2 cm 38 10 T p δ d Cal dE σ 2 d MicroBooNE Data 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Shape ⊕ Stat Norm G18 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) Untuned (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) G21 (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/9) Gv2 (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/9) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (c) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 2 4 6 8 /c Ar 2 GeV 2 cm 38 10 T p δ d Cal dE σ 2 d MicroBooNE Data 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Shape ⊕ Stat Norm G18 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) Untuned (18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) G21 (23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/9) Gv2 (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c T p δ (d) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated (a) single- and (b-d) double- in δpT bins differential cross sections as a function of ECal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/8) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/8) NEUT (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/8) NuWro (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/8) o < 45 T α δ (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) GiBUU (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) NEUT (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/9) NuWro (9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) o < 90 T α δ < o (b) 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) GiBUU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) NEUT (13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) NuWro (14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) o < 135 T α δ < o (c) 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/8) GiBUU (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/8) NEUT (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/8) NuWro (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/8) o < 180 T α δ < o (d) 135 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of ECal in δαT bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/8) Untuned (40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/8) G21 (29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/8) Gv2 (36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/8) o < 45 T α δ (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) Untuned (45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) G21 (37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) Gv2 (73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) o < 90 T α δ < o (b) 45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) Untuned (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) G21 (49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/9) Gv2 (88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) o < 135 T α δ < o (c) 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 deg GeV Ar 2 cm 38 10 T α δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/8) Untuned (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9/8) G21 (33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/8) Gv2 (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/8) o < 180 T α δ < o (d) 135 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of ECal in δαT bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 10 20 30 40 /c Ar 2 GeV 2 cm 38 10 T,y p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) GiBUU (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) NEUT (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) NuWro (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/9) | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (a) | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 2 4 6 8 10 12 /c Ar 2 GeV 2 cm 38 10 T,y p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) GiBUU (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) NEUT (16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) NuWro (28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5/9) < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 2 4 6 8 /c Ar 2 GeV 2 cm 38 10 T,y p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) GiBUU (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) NEUT (20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/9) NuWro (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of ECal in δpT,y bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 GENIE (blue), GiBUU (green), NEUT (pink), and NuWro (red) event generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 37 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 10 20 30 40 /c Ar 2 GeV 2 cm 38 10 T,y p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) Untuned (67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) G21 (58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3/9) Gv2 (93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6/9) | < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (a) | 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 2 4 6 8 10 12 /c Ar 2 GeV 2 cm 38 10 T,y p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) Untuned (42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) G21 (59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1/9) Gv2 (69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2/9) < -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 [GeV] Cal E 0 2 4 6 8 /c Ar 2 GeV 2 cm 38 10 T,y p δ d Cal dE σ 2 d MicroBooNE Data POT 20 10 × 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79 Shape ⊕ Stat Norm G18 (12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8/9) Untuned (22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4/9) G21 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7/9) Gv2 (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0/9) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c T,y p δ (c) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The flux-integrated double-differential cross sections as a function of ECal in δpT,y bins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Inner and outer error bars show the statistical and total (statistical and shape systematic) uncertainty at the 1σ, or 68%, confidence level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The gray band shows the normalization systematic uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Colored lines show the results of theoretical cross section calculations using the G18 (light blue), Untuned (magenta), G21 (orange), and Gv2 (dark blue) GENIE configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The numbers in parentheses show the χ2/bins calculation for each one of the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 38 [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tanabashi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (Particle Data Group), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 98, 030001 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [2] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (T2K Collaboration), Nature 580, 339 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (DUNE Collaboration), arXiv (2018), 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10334 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [4] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (DUNE Collaboration), arXiv (2018), 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10327 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [5] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (DUNE Collaboration), arXiv (2018), 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10340 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [6] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (Hyper-Kamiokande Collaboration), arXiv (2018), 1805.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04163 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Antonello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE, LAr1-ND, ICARUS- WA104), arXiv (2015), 1503.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01520 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [8] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tortorici, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bellini, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sutera (ICARUS Collab- oration), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1056, 012057 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [9] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (DUNE Collaboration), arXiv (2018), 1807.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10334 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='ins-det].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [10] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (DUNE Collaboration), arXiv (2020), 2002.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03005 [hep-ex].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [11] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dolan, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mosel, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gallmeister, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pickering, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bolognesi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 98, 045502 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [12] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rocco, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lovato, and O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Benhar, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 116, 192501 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [13] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rocco, Frontiers in Physics 8, 116 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [14] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pickering, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dolan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Barr, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Coplowe, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Uchida, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wascko, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Weber, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Yuan, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 94, 015503 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [15] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (T2K Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 98, 032003 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [16] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='-G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MINERvA Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 121, 022504 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MINERvA Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 101, 092001 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [18] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bathe-Peters, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gardiner, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Guenette, arXiv (2022), 2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04664 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [19] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 125, 201803 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [20] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Acciarri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In- strum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12, P02017 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [21] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Aguilar-Arevalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MiniBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 79, 072002 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [22] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kaleko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 8, C09009 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [23] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Acciarri et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 78, 82 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [24] “Table 289: Muons in liquid argon (Ar),” http://pdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='gov/2012/AtomicNuclearProperties/ MUON_ELOSS_TABLES/muonloss_289.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='pdf (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [25] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bichsel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Groom, “Passage of particles through matter,” PDG Chapter 27, Figure 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 http: //pdg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='lbl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='gov/2005/reviews/passagerpp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='pdf (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [26] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Van De Pontseele, Search for Electron Neutrino Anomalies with the MicroBooNE Detector, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' thesis, Oxford U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [27] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 105, 112004 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [28] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' High Energ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 2021, 153 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [29] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Berger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Coursey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Zucker, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Chang, “Stop- ping powers and ranges for protons and alpha particles,” NIST Standard Reference Database 124 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [30] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In- strum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12, P10010 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [31] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Adams et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 79, 673 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [32] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 128, 151801 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [33] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 105, L051102 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [34] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Andreopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=', Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A 614, 87 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [35] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Andreopoulos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=', arXiv (2015), 1510.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05494 [hep- ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [36] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (T2K Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 93, 112012 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [37] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 105, 072001 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [38] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Agostinelli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (GEANT4), Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A 506, 250 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [39] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pordes and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Snider, PoS ICHEP2016, 182 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [40] E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Snider and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Petrillo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 898, 042057 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [41] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bodek and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cai, Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 79, 293 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [42] W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Li, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Qian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Wei, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 12, P10002–P10002 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [43] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Aguilar-Arevalo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MiniBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 88 (2013), 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1103/PhysRevD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [44] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Calcutt, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Thorpe, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mahn, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Fields, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In- strum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 16, P08042 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [45] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Roe, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Instrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Meth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A 570, 159 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [46] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Golan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sobczyk, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ˙Zmuda, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' B Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 229-232, 499 (2012), Neutrino 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [47] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Eur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 82, 454 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [48] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mahn, A search for muon neutrino and antineutrino disappearance in the Booster Neutrino Beam, Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' the- sis, Columbia University (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [49] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Carrasco and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Oset, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A 536, 445 (1992).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [50] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nieves, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sanchez, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ruiz Simo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Vicente Va- cas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 85, 113008 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [51] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Engel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 57, 2004 (1998).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [52] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nieves, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Amaro, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Valverde, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 70, 055503 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [53] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Schwehr, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cherdack, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gran, arXiv (2016), 1601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02038 [hep-ph].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [54] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nowak (MiniBooNE Collaboration), AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1189, 243 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [55] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Kuzmin, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lyubushkin, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Naumov, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 35, S133 (2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [56] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Berger and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sehgal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 76, 113004 (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [57] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Graczyk and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sobczyk, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 77, 053001 (2008), [Erratum: Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='D 79, 079903 (2009)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [58] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Berger and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sehgal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 79, 053003 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [59] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ashery, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Navon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Azuelos, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Walter, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Pfeiffer, and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Schleputz, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 23, 2173 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [60] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mosel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' G 46, 113001 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [61] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Leitner, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Alvarez-Ruso, and U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mosel, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 73, 065502 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [62] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sjostrand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mrenna, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Skands, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' High En- erg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 05, 026 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [63] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Llewellyn Smith, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 3, 261 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 39 [64] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Nieves, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Simo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Vacas, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' C 83, 045501 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [65] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Katori, AIP Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1663, 030001 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [66] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rein and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sehgal, Annals Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 133, 79 (1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [67] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Mashnik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sierk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Gudima, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Baznat, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Ser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 41, 340 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [68] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dolan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Megias, and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Bolognesi, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 101, 033003 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [69] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Dytman, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Hayato, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Raboanary, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Sobczyk, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Tena-Vidal, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Vololoniaina, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 104, 053006 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [70] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 123, 131801 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [71] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Abratenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' (MicroBooNE Collaboration), Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 102, 112013 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' [72] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Avanzini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=', Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' D 105, 092004 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1 Multi-Differential Cross-Section Measurements in νμ-Argon Quasielastic-like Reactions with the MicroBooNE Detector (Dated: January 11, 2023) DATA RELEASE Overflow (underflow) values are included in the last (first) bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The additional smearing matrix AC should be applied to an independent theoretical prediction when a comparison is performed to the data reported herein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The AC matrices are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cross Section δpT, All events Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='619713 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5914076 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='267059 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4668375 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='993846 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8932224 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='591946 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6680086 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='613702 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9077564 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='443932 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2848366 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='568607 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2445447 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='864394 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1026524 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5845512 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6582198 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7027679 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3718027 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6962171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97287168 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5539428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77973266 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1167688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50570624 Unfolded Covariance Matrix δpT, All events Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71539 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25146 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17662 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39751 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01335 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85967 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31444 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44092 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05538 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37278 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='897909 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='568438 Bin 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25146 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3112 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0706 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9652 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14157 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26953 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68817 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33499 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44135 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61673 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='621567 Bin 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17662 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3112 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1572 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9752 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81664 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72466 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60095 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58379 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05767 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19748 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9267 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='451845 Bin 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39751 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0706 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9752 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4543 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72979 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64056 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50451 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55976 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1161 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32919 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0333 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='487379 Bin 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01335 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9652 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81664 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72979 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45505 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92495 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1107 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68871 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82418 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90884 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='703471 Bin 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85967 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14157 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72466 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64056 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92495 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22048 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9238 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23195 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82268 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06617 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54814 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='766047 Bin 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31444 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26953 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60095 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50451 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1107 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9238 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03798 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4844 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92434 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07059 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='832317 Bin 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44092 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68817 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58379 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55976 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68871 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23195 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4844 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42115 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19164 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35671 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67218 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='771476 Bin 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05538 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33499 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05767 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1161 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82268 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92434 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19164 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74969 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19561 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='521087 Bin 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37278 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44135 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19748 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32919 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82418 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06617 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07059 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35671 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19561 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88184 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='811564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='413841 Bin 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03097 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61673 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9267 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0333 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90884 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54814 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67218 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45701 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='946479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='383166 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='897909 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12272 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04027 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10892 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29809 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26196 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33828 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='811564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='607983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='382896 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='568438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='621567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='451845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='487379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='703471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='766047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='832317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='771476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='521087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='413841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='383166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='382896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='255739 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03700v1 [hep-ex] 9 Jan 2023 2 Additional Smearing Matrix (AC) δpT, All events Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='608391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='218619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00823167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00556978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0206322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0337794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0201499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158207 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0523749 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00223226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='146027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0347278 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='518315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='317154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0423483 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0342697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0715514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0997334 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0152325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0625475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127988 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='199797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='338918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0663399 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0799824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0656408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0537008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0733254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='230515 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0797768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0737304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='254057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200673 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00453236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0487452 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0285734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0869409 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0752719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205579 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0370381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0532791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='300893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00465611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0522884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='047862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0611183 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0375769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='272525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='130546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='152064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116554 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0210045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0405033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0473373 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00896947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0535465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0131631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='256089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0521243 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0106936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0032654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0925498 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0653087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0112558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0217705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00246232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0457396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0988458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0808979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0204205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0809313 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00863876 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00860406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0381422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0412628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0152354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0405864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0271734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0944415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='327082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0283769 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0382239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0104248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0388503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0401655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0314258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0221658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0990147 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0341562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='389518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='333004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00375582 Bin 11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0192523 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00391206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00749073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00898804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0399121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='043164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0655587 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0432441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0350014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='247309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='402844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101832 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00557305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0253563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0254045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0338288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0575915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00973714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0123258 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0433998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0470516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='383929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='215191 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0297581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0108119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0397229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0414738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='062826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0117404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0414886 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0638342 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0383003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269655 Cross Section δpT, δαT < 45o Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='092457234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='017843801 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1564988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020582334 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18951511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021746966 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17474694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022869393 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12335155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='018393756 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='054899234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01141765 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='028157619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010923284 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013531823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0082138729 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0076007037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0046983808 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0070376678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0029152099 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0013784316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00037879095 Unfolded Covariance Matrix δpT, δαT < 45o Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000318401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000325794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000165729 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51192e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76839e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70784e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46573e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35268e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42137e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56596e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68585e-07 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000325794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000423632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000341819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00021417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000140789 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94733e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000106398 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25507e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59273e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00488e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='659e-06 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000165729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000341819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000472931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000443965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000292367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00012278 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46405e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18356e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32803e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87653e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4326e-06 Bin 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51192e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00021417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000443965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000523009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000384776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000151883 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24282e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12025e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72727e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94327e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19496e-06 Bin 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76839e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000140789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000292367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000384776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00033833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000171181 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62622e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76139e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21197e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96973e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02607e-06 Bin 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70784e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94733e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00012278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000151883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000171181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000130363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109843 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10706e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24314e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43608e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71258e-06 Bin 7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46573e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000106398 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46405e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24282e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62622e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119318 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47709e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45946e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03844e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10519e-07 Bin 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35268e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25507e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18356e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12025e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76139e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10706e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47709e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74677e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23546e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05961e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99402e-07 Bin 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42137e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59273e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32803e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72727e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21197e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24314e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45946e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23546e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20748e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05918e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46038e-07 Bin 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56596e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00488e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87653e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94327e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96973e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43608e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03844e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05961e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05918e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49845e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80001e-07 Bin 11 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68585e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='659e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4326e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19496e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02607e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71258e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10519e-07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99402e-07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46038e-07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80001e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43483e-07 3 Additional Smearing Matrix (AC) δpT, δαT < 45o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='334149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='234164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0223846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0620358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0284375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0494776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0593212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0774406 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='270181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='292076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='102139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='055425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0788581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0293701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0602708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='476025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239922 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0625865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='199135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0651717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179045 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0530697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='219075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='589391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297954 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0430749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0400979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='280637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0705921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='220829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='453966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='209747 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0288676 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0010926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0377225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='178237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='267568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0362133 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0608711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00947498 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0194727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00610772 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00391155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0344214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0993643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0834889 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0419699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126522 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0412064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0233472 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00687408 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00733942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='260493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='248838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101021 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0811857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153201 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0396195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0282355 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00404456 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0240852 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0165453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='097616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='166702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='210901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0161929 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0703993 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='035856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00465383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00251156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00149926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0282931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0383342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0516738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='332516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162915 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0485199 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0176183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='025551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0038694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0136261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0126307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00168902 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0313676 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129974 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0123295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138957 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00243578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00920375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00908874 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000293633 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0268065 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0291088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='056371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0689814 Cross Section δpT, 45o < δαT < 90o Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='080836614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016092518 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16504577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='023783968 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21304334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='023374406 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19022336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021716322 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11467602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01723157 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='053371488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01148308 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040890294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010114624 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='038336569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0094371064 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='025850733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0062116622 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016096871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0044190856 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0093470892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0027906058 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014248176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00037616895 Unfolded Covariance Matrix δpT, 45o < δαT < 90o Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000258969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000344307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000215312 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22798e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44531e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94356e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1977e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87235e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07598e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73143e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61582e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09881e-07 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000344307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000565677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0004613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000238555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00010029 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58054e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000126494 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24862e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38332e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52389e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77052e-07 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000215312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0004613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000546363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000424386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000210141 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61126e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38352e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87849e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9264e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3406e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27006e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48542e-06 Bin 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22798e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000238555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000424386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000471599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000315713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000132261 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91192e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39685e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84039e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26593e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46019e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10894e-06 Bin 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44531e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00010029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000210141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000315713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000296927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000166211 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13575e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16139e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74452e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72486e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35031e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19913e-06 Bin 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94356e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58054e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61126e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000132261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000166211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000131861 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28267e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89367e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31562e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09889e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20397e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70845e-06 Bin 7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1977e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105136 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38352e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91192e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13575e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28267e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000102306 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57727e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91643e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39093e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42752e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45229e-07 Bin 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87235e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000126494 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87849e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39685e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16139e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89367e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57727e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9059e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79009e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58739e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07243e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95662e-07 Bin 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07598e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24862e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9264e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84039e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74452e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31562e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91643e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79009e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85847e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01415e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36753e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27451e-07 Bin 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73143e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38332e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3406e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26593e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72486e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09889e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39093e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58739e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01415e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95283e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08504e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26302e-06 Bin 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61582e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52389e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27006e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46019e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35031e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20397e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42752e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07243e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36753e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08504e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78748e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93848e-07 Bin 12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09881e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77052e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48542e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10894e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19913e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70845e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45229e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95662e-07 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27451e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26302e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93848e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41503e-07 4 Additional Smearing Matrix (AC) δpT, 45o < δαT < 90o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='074978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0829661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0977091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0384619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0783802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='196463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0805772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='363502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197379 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00315365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0762631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0791543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='311593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='358342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0809632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='523708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='214217 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0918317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000146056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0507077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='230081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='219773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='199698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129839 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0251096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0207458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='279568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='196315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0758035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0472131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0142602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0122951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0499588 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0741814 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0160966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0442857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0268192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='270119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='335156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0721694 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0675044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0446691 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0836594 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0157009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0162483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0581193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0182767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184832 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0559395 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0238669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0328982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0415853 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0846101 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0257014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0343673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0183917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22408 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='230904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0662662 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00497163 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10248 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00979932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0439613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='041259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0715393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0802027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104415 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0145792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0388795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0202758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0384119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0151164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0295031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32618 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0518135 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0787813 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0055833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0130169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0109926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0039168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00709286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000215394 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0273113 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0330379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0781422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='448427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00179196 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00931318 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00507384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00942823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00328942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00886275 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0291231 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0615924 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0182208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='260284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='307959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0560094 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00765898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00148581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000489381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00190886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00282892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00532749 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0094215 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0212541 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0629965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0958337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0234679 Cross Section δpT, 90o < δαT < 135o Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09521135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01745665 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18478331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021506156 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22497344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021801294 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20256678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022339224 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1579132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019450883 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11380471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015790683 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='099324443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014449218 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='093102719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01471589 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='070194885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010694 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='044756146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0077854456 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01945489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0048989672 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0082820388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0031612306 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0034413958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0013179012 Unfolded Covariance Matrix δpT, 90o < δαT < 135o Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000304735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000327676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000209902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000125501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000115852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000122321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000137878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000139792 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96145e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62393e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50149e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8124e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85779e-06 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000327676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000462515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000393721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000257562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000216126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000199646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000191164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000172847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000135942 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83516e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97423e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23294e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0287e-05 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000209902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000393721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000475296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000413123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000314666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000227026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000194574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000174989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000131558 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6443e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21309e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19954e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06881e-06 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000125501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000257562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000413123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000499041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00038842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000237391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000196556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000183499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00010499 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28512e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06394e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36578e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93751e-06 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000115852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000216126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000314666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00038842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000378337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000276951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000209686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000157816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000101404 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59041e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99723e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9417e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12336e-05 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000122321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000199646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000227026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000237391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000276951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000249346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000203314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000148675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000103954 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34626e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30704e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83436e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16739e-05 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000137878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000191164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000194574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000196556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000209686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000203314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00020878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000191563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119474 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54265e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52392e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57002e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78634e-06 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000139792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000172847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000174989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000183499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000157816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000148675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000191563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000216557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000134291 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22813e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82534e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16058e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46107e-06 Bin 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96145e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000135942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000131558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00010499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000101404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000103954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000134291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000114362 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78858e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65907e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67683e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71225e-06 Bin 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62393e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83516e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6443e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28512e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59041e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34626e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54265e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22813e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78858e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06132e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20014e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53492e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19316e-06 Bin 11 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50149e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97423e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21309e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06394e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99723e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30704e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52392e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82534e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65907e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20014e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39999e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33174e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0908e-06 Bin 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8124e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23294e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19954e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36578e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9417e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83436e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57002e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16058e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67683e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53492e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33174e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99338e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79228e-06 Bin 13 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85779e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0287e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06881e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93751e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12336e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16739e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78634e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46107e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71225e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19316e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0908e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79228e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73686e-06 5 Additional Smearing Matrix (AC) δpT, 90o < δαT < 135o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0472538 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294652 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00355256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0380359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0579156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239484 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='230434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0142559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='030142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0542944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0720635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0824348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0305376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='271931 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0250846 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='060762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='166201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0704369 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0611954 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='049026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0652899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0338124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='164223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0668586 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00206288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0155551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='272921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='102726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0401011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0469898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0498202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0760499 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0298625 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0154717 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0935439 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0210431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0613301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0633331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0319431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0520368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0111069 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0231149 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0129635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='176306 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00943557 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0015439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0216698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0559392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0911468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00640754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0245296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0296632 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0548393 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0197375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26659 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0119003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0103708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00503933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0229602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0980477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0344193 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0620902 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0111392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0917068 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0258947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0118838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0004686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0179612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0119837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0132319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0646848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='268814 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0718173 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0452721 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019704 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0979195 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00982194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0126429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0268634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0209681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00252312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0302897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0445011 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0544066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0904394 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00745678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0157421 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000976458 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0337072 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00361503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00472199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00500756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0532215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='210571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0942449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0500002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0759872 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00700856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0175687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00748022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0127108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00241449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0172558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0235768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0257031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='236822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='045016 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0253061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00841863 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00909259 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00441502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0150895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00102935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0085354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120066 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00858064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='023581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0924414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24741 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0210545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00218508 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00561258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000436053 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00365227 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0207961 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00377806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00778932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0226378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0974349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='247006 Cross Section δpT, 135o < δαT < 180o Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='066581268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014820438 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16062795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021214869 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22961992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='025079325 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23266351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024402602 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17853598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022262933 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11627078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019363331 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10123569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019007703 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1049638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='018703819 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09874567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016174564 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='091015041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013985124 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='067935328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010618221 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='049079696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0090362775 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027273377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006218243 Unfolded Covariance Matrix δpT, 135o < δαT < 180o Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000219645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000265891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000226961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000210609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000202423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000160416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000138227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00011654 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90464e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46404e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13283e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92613e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26661e-05 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000265891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000450071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000478877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000420906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000320775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000214812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000195546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000200779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000180606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000168301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000134319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00010283 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95601e-05 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000226961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000478877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000628973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000575656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000391684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000237633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000228029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00026615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000257532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000225661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000155179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105935 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79933e-05 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000210609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000420906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000575656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000595487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000472681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000319052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000295682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000312616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000279274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000231367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00015546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109746 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18038e-05 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000202423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000320775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000391684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000472681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000495638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000403406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000366668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000328443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000246268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000194413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000141749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109205 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48968e-05 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000160416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000214812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000237633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000319052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000403406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000374939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000354332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000301474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000202591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000152555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000116983 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5053e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77128e-05 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000138227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000195546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000228029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000295682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000366668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000354332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000361293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000331578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000233655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000172412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000124508 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74835e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81115e-05 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00011654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000200779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00026615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000312616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000328443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000301474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000331578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000349833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000281757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000214005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000141829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000101832 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84934e-05 Bin 9 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90464e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000180606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000257532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000279274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000246268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000202591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000233655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000281757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000261617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000214167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000138578 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22713e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01577e-05 Bin 10 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46404e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000168301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000225661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000231367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000194413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000152555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000172412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000214005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000214167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000195584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000137531 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15925e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80177e-05 Bin 11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13283e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000134319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000155179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00015546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000141749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000116983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000124508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000141829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000138578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000137531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000112747 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5784e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88272e-05 Bin 12 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92613e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00010283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109746 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109205 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5053e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74835e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000101832 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22713e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15925e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5784e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16543e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37207e-05 Bin 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26661e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95601e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79933e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18038e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48968e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77128e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81115e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84934e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01577e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80177e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88272e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37207e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86665e-05 6 Additional Smearing Matrix (AC) δpT, 135o < δαT < 180o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00752731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0361286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0224066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312236 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0461844 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0663142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0110903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0784301 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0593238 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241906 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='247664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='169545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0748076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0519574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00292192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='044334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0445275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='053145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000647254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0514641 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0577072 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0222758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0794506 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02916e-05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0349125 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0165781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0101567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0276959 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0559446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0139131 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0742549 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00791078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0839733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00815828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0177592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00371927 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0937358 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0460174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0177146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0377159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='246198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0846931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0577698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00290768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0112792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0264449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00507141 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0990717 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0465907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0106864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0261667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0508792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='095473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00442573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0229454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0168466 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0814339 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0133032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0170859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0211366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='034229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='207841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0389739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00508968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00841402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0697206 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0272604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0142441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0103163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0467388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0977147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0449523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00611837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0315543 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0517233 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0477319 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00224557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0455374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0703165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0643685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0361518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0508241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='115317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='057052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0575197 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0463255 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00509726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0326052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0471358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0475981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00784017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00872553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0278992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0353218 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0386023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0774963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00033048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0119997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0364747 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0311501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00835365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='282291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0196984 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0543867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0738844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000826599 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0205318 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00770406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0270939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0421725 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0440621 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0180303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0589739 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='268149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10793 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='057112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0658198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00132882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0282947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='048399 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0420272 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0155515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0216731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='281048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='173459 Cross Section δpT, –1 < cosθμ < 0 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8969562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47028307 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8791802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59171167 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2153662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69443198 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2967538 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68637835 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8681037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60571957 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8610855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48818439 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0740716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39655185 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0155115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35083279 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5788478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28480735 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2206318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22960154 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7124747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15544671 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40178047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10230988 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21327361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='049330314 7 Unfolded Covariance Matrix δpT, –1 < cosθμ < 0 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='221166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='164153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0731202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0462467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0449851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='041953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0328209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0159533 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='350123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='350368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0872878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0723317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='052663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0356552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0181032 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='350368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='482236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='441593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='145446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0859852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0469973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0137991 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='441593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='471115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='359526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0818173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0437763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0248323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141402 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='164153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='359526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='195187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='147844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0891233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0640818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0447763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0162131 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='238324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0591081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0467065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0401737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0287793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140729 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='195187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0653398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0499785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0379558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0257813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0121777 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0731202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='145446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='147844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0908358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='064964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0389044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0105873 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0462467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0872878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117437 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0891233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0591081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0653398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0908358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0811152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0606037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0336237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0169986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00690663 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0449851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0723317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0859852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0818173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0640818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0467065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0499785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='064964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0606037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0527169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0322509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0159149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00577918 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='041953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='052663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0469973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0437763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0447763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0401737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0379558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0389044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0336237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0322509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0137844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00528131 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0328209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0356552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0248323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0287793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0257813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0169986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0159149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0137844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0104673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00464599 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0159533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0181032 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0137991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0162131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0121777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0105873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00690663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00577918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00528131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00464599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00243348 Additional Smearing Matrix (AC) δpT, –1 < cosθμ < 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='413364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='274925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0374968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0434767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0265977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0272224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00289372 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0801879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0658809 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00357411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0784738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='280037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0311748 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='362092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='173483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0426272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0332632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0219173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0162116 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0474379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0093844 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00420526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0756319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119043 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0607942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0533021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0416542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0445716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0367766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0785281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00949505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0423003 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00943888 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241983 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0628411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0873893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='291781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='073805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00738086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0497339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0841751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0258297 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0481042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0252116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229496 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0380659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0139799 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0219465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0112417 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00497227 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0277828 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0379502 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0907513 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0240416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0311056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0273816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0853475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='226957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='292168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='262563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0242655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0460384 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00998375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0424374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='182585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0402536 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0529068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0165569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00735209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0453947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='252875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0946865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0319989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0307269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0795483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='198964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0553282 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0847243 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000492521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0295183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0455743 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0674227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0654574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='134996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0979346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0716237 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0877119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0200718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0525439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0391908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0310312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0336011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='280695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='250589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135464 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0348986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0418936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0032057 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='035941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00988257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='102974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='236121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144808 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0258231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0429484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0108961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0245312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0150959 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00208762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0316845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0825304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='393631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='304011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0625754 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0375132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0355029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0190569 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00308078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00734749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0247791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0195876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0522745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154564 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='284765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0260269 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0260829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00885009 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00920575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00306917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00713788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0116325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00869197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00432296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0111207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0484439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0546125 8 Cross Section δpT, 0 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4072537 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0364815 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='534754 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='488025 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='328263 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8050866 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='583625 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7108738 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='715974 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4129169 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2660878 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='059323 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1185826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89294996 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6069672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85032139 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6332932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74466871 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2549493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5943019 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5113146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4710171 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4982949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42198144 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94577651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25023832 Unfolded Covariance Matrix δpT, 0 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07429 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='994964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='834872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='854299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='699379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='570867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='397501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='256108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='255389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165957 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28147 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21422 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33875 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95335 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='796529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='688155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='688667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='555966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='419068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='215381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156787 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='994964 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33875 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25834 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91462 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='746269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='614595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='821877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='792989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='573593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='130297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101644 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='834872 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95335 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91462 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92709 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03556 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='735775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='780957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='717699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='573808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143515 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='854299 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36991 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72745 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03556 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99633 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='960586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='672993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='459362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='436117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='408744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='226405 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='699379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='796529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='746269 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00675 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32676 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='867808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='283356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='314284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='313813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193063 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='570867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='688155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='614595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='735775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='960586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='867808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='631608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='396269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='224494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137193 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='397501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='688667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='821877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='780957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='672993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='631608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='723046 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='580104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='404737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='067166 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='555966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='792989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='717699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='459362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='396269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='580104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='554531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='406636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0864203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0365171 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='419068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='573593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='573808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='436117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='283356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='404737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='406636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0656834 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='256108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='408744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='314284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='221857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0946957 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='255389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='215381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='130297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='313813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='224494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0864203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='178068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100845 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='226405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='067166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0365171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0656834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0946957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0626192 9 Additional Smearing Matrix (AC) δpT, 0 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='357572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0104737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='066634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0650186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0392803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0133809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0532685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0771741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00408395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0309646 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='357927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='298237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0128014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0583549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='091999 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0106305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0505337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000757665 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0701626 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0316451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0118552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='048686 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0842722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0268115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0936062 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0890018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0427286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='066392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0172387 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0268197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0919518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0846838 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0995551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0389345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0623978 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0277434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00645468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0142712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00602731 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00858007 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0964352 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0886072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0165199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0204467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='130513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='265468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0317295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0765981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0602305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0189137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0518218 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00397283 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0240503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0542346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0520921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0629133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00653457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0376774 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00176019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0127556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0560215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0265764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0629988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='268896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0481723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0204946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00366693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0695446 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0509405 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0319358 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00840165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0194119 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00206434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0628278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='182657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0667404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0166855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0150684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0697172 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0706499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0139773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00430313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00333253 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0340493 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0149317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0830574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='325345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0697084 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00941066 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0847108 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0520724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00707522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00745826 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198208 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0128821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0220114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='251086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0497093 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00588436 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00557116 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0368843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0075801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0427189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0305049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0389183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00164327 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0496212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='044596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0190006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0756961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468722 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0286461 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00793796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='130229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='298471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0425192 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0695293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0302512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0309085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0308317 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0151455 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0672554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0524307 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0316419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0426197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0121328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0731626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='224195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0637115 Cross Section δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='379123 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='670063 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='849612 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='460739 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='658288 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9965989 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='430432 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8611275 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='702258 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2700023 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='658558 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32953 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73851 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2388642 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='081407 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2761151 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2445367 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='779744 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4507555 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4475703 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2613597 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1132975 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0109188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84604385 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1824419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49126105 10 Unfolded Covariance Matrix δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12924 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58474 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48054 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27361 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0579 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26568 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01388 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80377 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22939 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88815 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47779 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='759175 Bin 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58474 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9767 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2655 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0005 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97839 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5272 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31907 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32399 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30219 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51253 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='917322 Bin 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48054 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2655 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9728 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2992 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67621 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51433 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66386 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41488 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34696 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54665 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86364 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='917689 Bin 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27361 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0005 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2992 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9083 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4896 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64198 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80614 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87122 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90659 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42538 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88893 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='877689 Bin 5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0579 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97839 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67621 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4896 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6929 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99471 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0572 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74327 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65522 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20813 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73062 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='831861 Bin 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26568 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5272 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51433 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64198 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99471 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42671 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66446 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79722 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49175 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78439 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29141 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='643628 Bin 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01388 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31907 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66386 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80614 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0572 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66446 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01251 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76323 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14527 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99638 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='579584 Bin 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80377 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32399 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41488 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87122 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74327 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79722 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76323 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1807 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75465 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43922 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='909718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='524239 Bin 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22939 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30219 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34696 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90659 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65522 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49175 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14527 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75465 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16749 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36357 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44847 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='865411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='438761 Bin 10 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88815 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51253 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54665 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42538 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20813 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78439 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99638 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43922 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36357 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='939208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='446597 Bin 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47779 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86364 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88893 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73062 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29141 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26072 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42866 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44847 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4778 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='879134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='429904 Bin 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19179 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41025 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43893 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41695 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33865 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='909718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='865411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='939208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='879134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38796 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='759175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='917322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='917689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='877689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='831861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='643628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='579584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='524239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='438761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='446597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='429904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241337 Additional Smearing Matrix (AC) δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='338247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0557819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0444002 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0182889 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='054567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0395433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00988355 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0266796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0177639 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0350631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00037824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='042754 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='325456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='285417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0430611 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0305097 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0562853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294053 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0187443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0337263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0355299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0512976 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='291007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168938 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0517172 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0878593 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0438666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0295244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0386872 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0419223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0106127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0632565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0382287 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0314852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0705993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0789847 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0243366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0310157 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0551673 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0260121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0181269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0558271 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0516305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0411916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0699132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='263477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00288842 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0337281 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0105199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000285898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0645719 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='057608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00662162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0675663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185605 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='102479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0161758 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00474702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0544455 Bin 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75665e-05 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0374968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00259733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='058263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0769511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0181891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0199415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0538951 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0233856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0132565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00890178 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0222703 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0169091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0185733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='290818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='086702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0464809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0477687 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0489283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00537801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00622217 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0370408 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0441174 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0359678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0788245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='218017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0327569 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0487309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00293825 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0052965 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0305208 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0345482 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='049295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00253759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='086631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0954233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='282867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='329908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0365455 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00938233 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0390696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00198104 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00948851 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0260064 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0244436 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0443374 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0323653 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00459105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0220972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='403501 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0624616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0037073 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0301862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00337277 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0054649 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0243847 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0171204 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281379 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0235188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0255029 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0142924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='260613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0989811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0193207 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0320992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00155677 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00628957 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0246112 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0149941 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0239619 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0127776 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0143898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0735423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0707074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0209479 11 Cross Section δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθμ < 1 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024279 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2181418 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='708221 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4406778 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='810979 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8134699 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='485573 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7402503 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='507298 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2066832 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='374607 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6765154 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07327 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6889473 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='475107 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3116438 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='281648 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4681369 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5689957 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8988171 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2056831 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2528841 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0276347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90117192 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3485493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50604703 Unfolded Covariance Matrix δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθμ < 1 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3564 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9318 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9987 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86061 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37093 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84204 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39126 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60981 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20114 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49122 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10149 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92183 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13277 Bin 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9318 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7196 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2693 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7595 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='662 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86436 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86699 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24425 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91836 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21191 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17774 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41868 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26071 Bin 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9987 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2693 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1695 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3263 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25441 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59276 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60727 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9277 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8794 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89052 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35565 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0187 Bin 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86061 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7595 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3263 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9645 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7067 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0128 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8524 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39735 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56404 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99395 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='939984 Bin 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37093 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='662 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9645 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6962 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0497 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0273 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6709 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='368 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80895 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39959 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='962426 Bin 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84204 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86436 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25441 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7067 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0497 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5168 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2687 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2135 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15019 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43723 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57686 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='881016 Bin 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39126 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86699 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59276 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0128 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0273 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2687 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6083 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='758 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43027 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5269 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='900989 Bin 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60981 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24425 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60727 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8524 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6709 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2135 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='758 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='967 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59913 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97607 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84478 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84793 Bin 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20114 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91836 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9277 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39735 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='368 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15019 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43027 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59913 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0917 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43575 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60782 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65104 Bin 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49122 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21191 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8794 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56404 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80895 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43723 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5269 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97607 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43575 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60551 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25839 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55603 Bin 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10149 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17774 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89052 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99395 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39959 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57686 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59107 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84478 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60782 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25839 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56972 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='478498 Bin 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92183 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41868 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35565 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26321 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07478 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72074 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74543 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77212 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5091 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3149 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='812111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='435865 Bin 13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13277 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26071 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='939984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='962426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='881016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='900989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='478498 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='435865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='256084 12 Additional Smearing Matrix (AC) δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθμ < 1 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='384657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0553495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0296859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0793514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0403141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0829512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0150641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194276 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='451664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='302344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0518724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0551758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00303118 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0504308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0243592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0759207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='207352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='065638 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='264871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0887745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0460066 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0268092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0587237 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0189124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0766786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='096092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139785 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='262679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0206973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0249262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='046453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0519168 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000114072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158453 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0723883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0627335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0897787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='178182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0341738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0394512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='068061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0575902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0267137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00932699 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0582858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='025271 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00138134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0451777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='363188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='198199 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0702425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0244084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00948488 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100322 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000996697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113323 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='037793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00892294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0265868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0130587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341985 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='210836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0314291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0131758 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0927936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0524476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150021 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00487896 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00922399 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0110112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0298237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0880319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='234799 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0649594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0674215 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0214187 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117802 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0471862 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0253277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0262304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0663415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='070356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='146863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0467986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0495879 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0946817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0931032 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0489527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0101504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0425763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0628849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0395834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0546652 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0229053 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00655114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0726846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188842 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='196471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104566 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00936115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00929629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0286803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00912331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='023517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0343762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0280602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0157977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185503 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0648266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0219428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00328263 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0108632 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00564771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230018 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115905 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0226026 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0231344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0450974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='358325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='242847 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0807166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0237072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0182799 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0260993 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00891949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0286955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00153088 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0172138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='034022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='300882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='263973 Cross Section δpT, –1 < cosθp < 0 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3785276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33040672 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0322901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42979616 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9818081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35210571 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6100049 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30139374 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22411673 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80073041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20208985 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54891848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15741415 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36990176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10005425 Unfolded Covariance Matrix δpT, –1 < cosθp < 0 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='098696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0434857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141377 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0729882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0267644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0253857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='028319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0197072 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='098696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0884386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0369666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0174466 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0434857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0729882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0884386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0908382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0618047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0266923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138663 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0267644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0618047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0502283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0411323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107688 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0253857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0369666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0411323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0408403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0278607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141436 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='028319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0266923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0278607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0247792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146779 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141377 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0197072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0174466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100109 13 Additional Smearing Matrix (AC) δpT, –1 < cosθp < 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0565382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0850707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0160541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108007 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0416427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174316 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0518873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='182727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0370454 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0887635 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='145365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192806 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00628258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0484866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='146595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0675518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0374136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0443475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0400332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0739565 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='035414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00642723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='060506 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0816017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0437941 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468351 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0630991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0299789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00880477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0245643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='328881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0359311 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0938149 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0326063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0123868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00587622 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0306397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0589522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='262663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0106106 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0165222 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000979674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00380812 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0443815 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0151857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0988266 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00986243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='001694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00970362 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0327057 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0362059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0577859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142234 Cross Section δpT, 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2035597 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1218257 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='611311 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4923618 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='317929 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7121838 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='722065 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='737097 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='970913 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4026135 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='623598 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0930818 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6757667 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2204584 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2354756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3044508 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6585327 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0079979 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2991999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83059724 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='622353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61160322 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77701967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4547977 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81329274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25751919 14 Unfolded Covariance Matrix δpT, 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25849 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49144 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='671577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='708981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='882038 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1124 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='738884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='462693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='274328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138068 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49144 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22714 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15676 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69512 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32673 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13649 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31806 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39276 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='665747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='350695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='152358 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04078 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15676 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93157 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77236 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93294 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='860335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='932194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='872113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='730974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='414849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117906 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='671577 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69512 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77236 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01751 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22762 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='561148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='451988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='550159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='623748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='453764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='250835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118357 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='708981 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32673 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93294 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22762 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96732 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='775358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='482658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='396212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='450687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='401102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='301158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165105 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='882038 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13649 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05875 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08274 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23845 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19483 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='878495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='502895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='346658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188664 Bin 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1124 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='860335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='561148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='775358 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48952 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4519 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='877313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='483014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='294071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206776 Bin 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11745 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='932194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='451988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='482658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='878495 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4519 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70159 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='260273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='198779 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='738884 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='872113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='550159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='396212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='502895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='877313 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18593 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='747493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150558 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='462693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='665747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='730974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='623748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='450687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='346658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='483014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='747493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='689892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='455099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131122 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='274328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='350695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='414849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='453764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='401102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='294071 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='455099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='374058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114996 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='250835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='301158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='260273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105093 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='152358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='198779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0663161 Additional Smearing Matrix (AC) δpT, 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0258331 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0511626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0175083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0911805 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0382431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0842648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00426849 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0378569 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0419253 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0660793 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='248087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='187963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0383761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0795285 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0535091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0184477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0969682 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0116928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0448496 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0497996 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0543121 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111463 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0820694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000284205 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0540642 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0110562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0223113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0137092 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0233107 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='115477 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0330883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0535225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='272632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='285944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0500552 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0857662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='115018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0606717 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0248567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0183819 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0213366 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0781114 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00330227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0321967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0940884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='270792 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0420121 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0926075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0894852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0730277 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0013067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0979368 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0285114 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0838231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0532535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0132937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0310214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0300136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0394403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0579387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0433923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0188921 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0859871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0026296 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0553061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230832 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0900673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0145648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0920292 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0773074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0486829 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0864283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0122237 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0823987 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0555617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0003115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0691684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0931243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00222349 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0589634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0585327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='074212 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0630609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0556105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0269361 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0668863 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0690139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0722497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013045 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0262699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='074088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0951456 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0087176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107051 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0319744 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0274829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='063286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0467496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00419004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='219809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0515421 Bin 11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0520906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0149048 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0263349 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0175878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00924523 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00168051 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0594272 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0696059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0271226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40931 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0402768 Bin 12 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0295369 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00861117 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0372217 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0252444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0201673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0388275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0317564 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0457306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0182958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0156394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0890009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='416867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108393 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00272443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00559783 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0264635 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0316164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0371156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120505 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00265788 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0243768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0170281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113396 15 Cross Section δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='281447 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5209459 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='301135 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9714452 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='640411 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3155374 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='829592 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5628073 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184361 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7312039 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156771 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5262305 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181282 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1836251 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='996287 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1205056 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3326477 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7640475 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2480096 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4376714 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6493884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93507907 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4729482 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72887204 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9507086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43556789 Unfolded Covariance Matrix δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3971 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0891 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5869 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90569 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32341 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82125 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78546 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16132 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65089 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44325 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84957 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91079 Bin 2 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0891 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7153 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5188 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8075 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4055 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70924 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24939 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72964 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80248 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39335 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11424 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11514 Bin 3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5869 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5188 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2549 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9149 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4771 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48075 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90641 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69548 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2301 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27912 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91573 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50614 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15871 Bin 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90569 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8075 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9149 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8192 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4817 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93739 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78293 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33078 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97349 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73153 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='999624 Bin 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32341 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4055 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4771 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4817 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9219 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48732 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33456 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42046 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33755 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86416 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34978 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79947 Bin 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82125 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70924 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48075 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93739 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48732 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38184 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55184 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73384 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45442 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='942322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='896186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='650815 Bin 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78546 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24939 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90641 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78293 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33456 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55184 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76822 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91388 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10612 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99696 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='677065 Bin 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16132 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72964 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69548 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33078 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42046 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73384 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91388 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49654 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16564 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84054 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='952744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='551066 Bin 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65089 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80248 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2301 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97349 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33755 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45442 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10612 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16564 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11186 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26779 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='607484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='292381 Bin 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44325 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39335 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27912 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73153 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86416 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12182 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18628 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84054 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26779 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0669 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='483964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='189204 Bin 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84957 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11424 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91573 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='942322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99696 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18779 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21667 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='874373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229182 Bin 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42184 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56541 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50614 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27175 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='896186 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='952744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='607484 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='483964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='531254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='286004 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91079 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11514 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='999624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='650815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='677065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='551066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='292381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='189204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='286004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='189719 16 Additional Smearing Matrix (AC) δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 13 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='361094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00159787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0525027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0164129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0293346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0897963 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0598804 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000507088 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0307189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0156839 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='394091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135352 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0694085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0556852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='063316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0372207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0961489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0377908 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0820481 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='234224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121415 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0741173 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0703882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208631 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00145837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0687179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0302973 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='110807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0205145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193605 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0712887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0766614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0631669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0295695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000725702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146551 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0875124 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0179576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0638735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='236955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='332884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0947425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0572733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0770127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0541294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0782122 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0137013 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0528896 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0161488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0373754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0054937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='298916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0173448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0588866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='036807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0589307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0103269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0590854 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032813 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00793252 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000843608 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0167281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0324233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='195332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0259818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0639448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0768914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0897107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0620261 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0382811 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0178934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0199347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0217039 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0160172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0166826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0135159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0876201 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198279 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0827459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0414029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0122386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0232598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0558349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00834035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162193 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='303171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='147948 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0163241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119869 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='035141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0108148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000979296 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0163625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0383925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0509316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='220518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='285712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00913533 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142502 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0946367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0224776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0129716 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0102307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0111713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00651664 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00291654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0403758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='316761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0324417 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0548384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00940786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000978491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00663006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0112153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0158145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0306005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0320912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0243067 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155025 Bin 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0427095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00662653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000525337 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00510326 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0295829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0305672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0288398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0153738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0492721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0611095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='124714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='214593 Cross Section δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθp < 1 Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='552912 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2405522 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='995563 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1475835 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37349 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='059249 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='806436 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4454423 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185643 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0471095 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='753144 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117621 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='783847 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7236628 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150194 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='369364 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='095088 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8271559 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0520338 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5023261 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4671502 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1345937 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3417707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47866723 17 Unfolded Covariance Matrix δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθp < 1 Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5012 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3341 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0823 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7797 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='538 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67284 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5112 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47564 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72612 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22818 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70583 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11011 Bin 2 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3341 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4976 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2801 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9494 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4712 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5756 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96809 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07707 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72545 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96573 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47859 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21469 Bin 3 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0823 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2801 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7145 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7993 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9252 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88255 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92477 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90019 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0933 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44623 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='870473 Bin 4 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7797 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9494 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7993 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6528 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5301 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01118 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24016 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08449 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77767 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8243 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='738074 Bin 5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='538 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4712 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9252 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5301 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3791 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1444 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79887 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17077 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67457 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37689 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1855 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='762508 Bin 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67284 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5756 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88255 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01118 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1444 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71956 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03417 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60884 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87046 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67659 Bin 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5112 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96809 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92477 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24016 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79887 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03417 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41834 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71933 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98684 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98609 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='669121 Bin 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47564 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07707 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90019 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08449 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17077 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60884 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71933 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61389 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76726 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51939 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='607925 Bin 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72612 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72545 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0933 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77767 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67457 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87046 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98684 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76726 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3385 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51902 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='456624 Bin 10 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22818 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96573 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44623 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8243 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37689 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05542 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98609 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51939 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51902 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25698 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='494619 Bin 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70583 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47859 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04013 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53208 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1855 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66701 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5819 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63471 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47074 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49992 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='509975 Bin 12 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11011 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='870473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='738074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='762508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='669121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='607925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='456624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='494619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='509975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229122 Additional Smearing Matrix (AC) δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθp < 1 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='173589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0233497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='043763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0108795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0405129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0297928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0501185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0881269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106928 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='146722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0151518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00291148 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0184496 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00602801 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407253 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0326476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100236 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0595994 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0391953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000730359 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0503848 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0152166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0765204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0462604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0427378 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0367595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='247073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0914112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0512559 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0069041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='045256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0911993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0886454 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0722632 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0655028 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0783508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0864368 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0564209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0811769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191763 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14799 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0650163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00371581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00277014 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0578205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='054776 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0944449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0611226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0324663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0906654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0507775 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0327151 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140822 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0708966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0256786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0318613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0393797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0788641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0170168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0422631 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0482703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012632 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0351208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00338878 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0049528 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00956504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0125591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='115029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0385198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='034845 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00741034 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0121434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0255679 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00100373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0653597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00264147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='094818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0641824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0498915 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0693101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0272491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0247956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0313353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0784008 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0459289 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00189596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00245751 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0553797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0905255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0192978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0545427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0382673 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0609924 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0300345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0318398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0113708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='304671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14091 Bin 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='069786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='090838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00764978 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0582913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='042026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0567283 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00702852 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0201669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0690734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0313565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='287963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='257871 Cross Section δαT, All events Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='047295737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0069864868 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='044962891 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0058597861 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='044549083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='005667204 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05044147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='007040037 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='066811541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0084041599 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='078273769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='008722771 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='090827836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010113554 18 Unfolded Covariance Matrix δαT, All events Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8811e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84145e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38915e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66442e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57368e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65219e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61539e-05 Bin 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84145e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43371e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64859e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28022e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19373e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56387e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45764e-05 Bin 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38915e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64859e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21172e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70358e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96159e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05784e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26981e-05 Bin 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66442e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28022e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70358e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95621e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15219e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02396e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86069e-05 Bin 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57368e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19373e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96159e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15219e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06299e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28347e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15501e-05 Bin 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65219e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56387e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05784e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02396e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28347e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60867e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42981e-05 Bin 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61539e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45764e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26981e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86069e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15501e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42981e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000102284 Additional Smearing Matrix (AC) δαT, All events Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='413147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0143658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109214 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='322934 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0440761 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0292975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='070621 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0898424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='223852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='310898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0261531 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0601582 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0385126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0430092 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='219993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='492733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='317566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00458755 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0436167 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0833197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0616928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0429507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='215413 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='591386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0607463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0501594 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0182562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0894267 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149783 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00911983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='397323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='617785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126175 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0773432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='425517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='483276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='676898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='299715 Cross Section δαT, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12696689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022681076 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14524449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='018478608 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14734428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='017921719 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14897292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01917038 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18659596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020385736 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1727567 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='018089571 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13435265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020784081 19 Unfolded Covariance Matrix δαT, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000514431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00033728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000161703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000116762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000208306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000253654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000229816 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00033728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000341459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000252421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000163471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000207861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000228156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000253011 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000161703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000252421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000321188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000289019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000217812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00019247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000259027 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000116762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000163471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000289019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000367503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00029098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000197517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000232392 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000208306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000207861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000217812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00029098 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000415578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000310562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000208751 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000253654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000228156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00019247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000197517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000310562 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000327233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00025965 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000229816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000253011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000259027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000232392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000208751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00025965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000431978 Additional Smearing Matrix (AC) δαT, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='559612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='248703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0373883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00392543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='083823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0605731 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='264683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297304 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213337 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0484846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0376626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0495284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0116345 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0135524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='182539 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='355916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='214361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00104184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04539 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0548622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0288009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='281731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258737 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00659288 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0536045 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0204662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0143231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='215942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0283038 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0687986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0551935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281568 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0033156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='250874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='378035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120432 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='079356 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0493917 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='054984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0834252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38951 Cross Section δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='056762524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012483565 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='046754186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011513164 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='052380082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011718981 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='064783823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012796541 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='091853641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015914447 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='017895037 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11214728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01744216 20 Unfolded Covariance Matrix δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000155839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000108052 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43911e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32881e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105363 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27143e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73866e-05 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000108052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000132553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000107214 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51669e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29807e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00011982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000117103 Bin 3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43911e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000107214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000137335 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80781e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19338e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000130423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000132043 Bin 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32881e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51669e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80781e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000163751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00015084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000110929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000117216 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105363 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29807e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19338e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00015084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00025327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000181215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000143795 Bin 6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27143e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00011982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000130423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000110929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000181215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000320232 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000267749 Bin 7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73866e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000117103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000132043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000117216 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000143795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000267749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000304229 Additional Smearing Matrix (AC) δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='442678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0698266 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0779904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='039554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0728318 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='382143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0621195 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='048223 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00398776 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0263515 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0800074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='582621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='072973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0772721 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00153366 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0147672 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0466268 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0680143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='236749 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='198355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0251352 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0319901 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407601 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126877 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0267461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='565995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0496206 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233464 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0752689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117169 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='553286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='226571 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188836 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0742107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101112 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0125217 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='417135 Cross Section δαT, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0022692936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014399406 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00095139453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0016241049 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0026372474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0021132661 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006840357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0027823209 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0068013283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='003786716 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022077111 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0052624245 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='045353622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0087474574 21 Unfolded Covariance Matrix δαT, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07343e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32538e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86279e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87269e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02102e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62624e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90893e-06 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32538e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63772e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41612e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26963e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67524e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80935e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00665e-06 Bin 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86279e-07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41612e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46589e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12072e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34756e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75775e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52209e-06 Bin 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87269e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26963e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12072e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74131e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43535e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73762e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43579e-05 Bin 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02102e-06 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67524e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34756e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43535e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43392e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64628e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86402e-05 Bin 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62624e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80935e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75775e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73762e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64628e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76931e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7214e-05 Bin 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90893e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00665e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52209e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43579e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86402e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7214e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6518e-05 Additional Smearing Matrix (AC) δαT, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='657651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0763675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00246771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0195514 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31444e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000206455 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='324516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='393488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0111971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120529 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0171345 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0197781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00380935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019274 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00355798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='071658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='747657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0753558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0180982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020241 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='282211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00441528 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00451394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='354826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='436976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0990078 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0553995 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='728718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='303365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0346546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0681549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='476659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407148 Bin 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='240485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='326576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='505835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='475475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='307137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52029 Cross Section δαT, –1 < cosθμ < 0 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0070114389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0015306301 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0087157755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0013189318 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0092671614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014031435 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0085535342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0012989735 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010332378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014024775 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012607021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0013251007 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013567421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0015270408 22 Unfolded Covariance Matrix δαT, –1 < cosθμ < 0 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34283e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59166e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59126e-07 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83249e-07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26983e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00775e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50872e-06 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59166e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73958e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51974e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01556e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80254e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30153e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60824e-06 Bin 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59126e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51974e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96881e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51658e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08321e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4886e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45628e-06 Bin 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83249e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01556e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51658e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68733e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49869e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23385e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01977e-06 Bin 5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26983e-07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80254e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08321e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49869e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96694e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23942e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05445e-06 Bin 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00775e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30153e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4886e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23385e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23942e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75589e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81435e-06 Bin 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50872e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60824e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45628e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01977e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05445e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81435e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33185e-06 Additional Smearing Matrix (AC) δpT, –1 < cosθμ < 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='330799 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='307756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033425 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0437487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0387658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0978323 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='301276 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0169992 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0324116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0725589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='090314 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0431483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0265722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0731724 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00118167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0471798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='284999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0855244 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0117181 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00662461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0504072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0595483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='358279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127223 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0358269 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00362009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0432659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='338056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165895 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0827384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0539231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120919 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='375975 Cross Section δαT, 0 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='018649992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0028683018 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016519118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0024557039 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='017917435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0032169574 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020440461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0027421746 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026117712 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0036023637 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='035285343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0036686218 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04006913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0044840765 23 Unfolded Covariance Matrix δαT, 0 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22716e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51184e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69233e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77169e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81416e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22029e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10765e-06 Bin 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51184e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03048e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28396e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54581e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47987e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33931e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79248e-06 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69233e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28396e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03488e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57506e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54292e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83561e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83866e-06 Bin 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77169e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54581e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57506e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51952e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07922e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42015e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19672e-06 Bin 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81416e-06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47987e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54292e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07922e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2977e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68761e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3761e-06 Bin 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22029e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33931e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83561e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42015e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68761e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34588e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38999e-05 Bin 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10765e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79248e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83866e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19672e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3761e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38999e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01069e-05 Additional Smearing Matrix (AC) δpT, 0 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0965606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00385937 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0586131 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='231797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='097857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0470108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000220748 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0543718 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0934833 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='436329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='171209 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00612483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0287628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0186071 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0523487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0769526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0144383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0266592 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0217093 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0169467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='376123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0857796 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0561585 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126351 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0457043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='169706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='374586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='038569 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='058707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00603485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='220846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='226775 Cross Section δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='039046444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0057487733 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033051397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='005611716 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='036932838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0070931129 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='043848093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0078297063 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='063839561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0092899599 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='078182522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0098765961 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='081395804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010941566 24 Unfolded Covariance Matrix δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30484e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66353e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94967e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98935e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64478e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24208e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='755e-05 Bin 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66353e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14914e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34037e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90644e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14246e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80814e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51267e-05 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94967e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34037e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03123e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6838e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71851e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33051e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98435e-05 Bin 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98935e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90644e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6838e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13043e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77563e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8509e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36044e-05 Bin 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64478e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14246e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71851e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77563e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63034e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92745e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04889e-05 Bin 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24208e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80814e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33051e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8509e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92745e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75472e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99005e-05 Bin 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='755e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51267e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98435e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36044e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04889e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99005e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119718 Additional Smearing Matrix (AC) δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθμ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='409038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00992446 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0155609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0433038 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00840302 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='260699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23826 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0446944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0322216 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00868523 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0196311 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00979316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='176353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='329928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0815721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0413037 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0118926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='065293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='286308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='267722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0143675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0495148 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0040938 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0754747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='393315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0887314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040917 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0760158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0342302 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0501902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='362221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='273206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0004465 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='058274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00893371 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0504265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='201722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0693701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192261 Cross Section δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθμ < 1 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='075870972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011265281 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='066290677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0098234579 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06491299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0095190088 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='075017427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011947928 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='086159107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014835648 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='084639493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016739056 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='094523602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019711266 25 Unfolded Covariance Matrix δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθμ < 1 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000126907 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8084e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97118e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51985e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89206e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26714e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000126308 Bin 2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8084e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65003e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68089e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33299e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29866e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28571e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77742e-05 Bin 3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97118e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68089e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06115e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93952e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85362e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36401e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37594e-05 Bin 4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51985e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33299e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93952e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000142753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000152375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000133901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000153868 Bin 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89206e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29866e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85362e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000152375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000220096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000225933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000244789 Bin 6 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26714e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28571e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36401e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000133901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000225933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000280196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000305343 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000126308 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77742e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37594e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000153868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000244789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000305343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000388534 Additional Smearing Matrix (AC) δpT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθμ < 1 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='425947 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='304025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00262488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0451313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0358877 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='396661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180242 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000187595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0298272 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00638319 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0335174 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='273836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='329648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0787246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0441135 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0517179 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='279203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='372299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125789 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120798 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='110021 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0227962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0940123 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0326139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0676499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00810853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305873 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='378786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='176513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0992115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='566057 Cross Section δαT, –1 < cosθp < 0 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0033470124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014792221 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0032858289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0011873971 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0034797314 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0012390371 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0039534109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0012219203 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0059104064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0012232133 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0081354051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0014539827 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0087585646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0013637679 26 Unfolded Covariance Matrix δαT, –1 < cosθp < 0 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1881e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57807e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06945e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21849e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20172e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25221e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79356e-07 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57807e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40991e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28098e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10909e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='046e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00749e-07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40266e-07 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06945e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28098e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53521e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41945e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04956e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96465e-07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65152e-07 Bin 4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21849e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10909e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41945e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49309e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2193e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41929e-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34355e-07 Bin 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20172e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='046e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04956e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2193e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49625e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37198e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15677e-06 Bin 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25221e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00749e-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96465e-07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41929e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37198e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11407e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81519e-06 Bin 7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79356e-07 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40266e-07 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65152e-07 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34355e-07 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15677e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81519e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85986e-06 Additional Smearing Matrix (AC) δαT, –1 < cosθp < 0 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0247056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='096808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0888466 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0418628 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0327222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0797883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0417735 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0359414 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0398379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='182496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113199 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0204464 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0299969 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0154123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200098 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0380683 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0330582 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0859166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0694509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='062132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='250404 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0624778 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0297557 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174453 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0933979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00524077 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00390055 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='078135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0345203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0376223 Cross Section δαT, 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026104529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0040353837 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024505852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0037360674 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024197958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0038524767 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027064782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0042700252 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='034754005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0048521009 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='031651427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0042703155 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021955772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0030895824 27 Unfolded Covariance Matrix δαT, 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62843e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3511e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63143e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01442e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87581e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70088e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29121e-06 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3511e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39582e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18765e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03869e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11568e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67926e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03057e-06 Bin 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63143e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18765e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48416e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48149e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41359e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08329e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29988e-06 Bin 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01442e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03869e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48149e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82331e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89368e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30213e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63293e-06 Bin 5 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87581e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11568e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41359e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89368e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35429e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83041e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20837e-05 Bin 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70088e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67926e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08329e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30213e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83041e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82356e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23339e-05 Bin 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29121e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03057e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29988e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63293e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20837e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23339e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54552e-06 Additional Smearing Matrix (AC) δαT, 0 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='380503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='299028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159307 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0340054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0057237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0378798 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='050901 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0493354 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00927125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0453116 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0194677 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0659561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17779 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='365896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0455855 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0415884 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0597793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105096 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='293758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='261671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0551629 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236081 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0626226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='166897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0254134 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0436541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='438064 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120016 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0296371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0687627 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='007604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='041625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245946 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133478 Cross Section δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='062216152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0089630236 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='059951055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0078957292 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='059177472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0085398267 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='065201941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0093877473 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='082828711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010863918 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='078871598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010172356 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='065773388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0086853806 28 Unfolded Covariance Matrix δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03358e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7748e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76609e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39106e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98196e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11445e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96994e-05 Bin 2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7748e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23425e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59059e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21495e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81918e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09319e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99492e-05 Bin 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76609e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59059e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29286e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94112e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97778e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56153e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77974e-05 Bin 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39106e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21495e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94112e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81298e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49071e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74431e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56942e-05 Bin 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98196e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81918e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97778e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49071e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000118025 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81096e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96303e-05 Bin 6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11445e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09319e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56153e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74431e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81096e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000103477 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8964e-05 Bin 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96994e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99492e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77974e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56942e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96303e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8964e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54358e-05 Additional Smearing Matrix (AC) δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 < cosθp < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='334472 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='249471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0270459 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00610521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0292026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0089557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0484225 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='215734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0607089 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0372224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00220286 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0208021 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00125669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='374603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0634971 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0211967 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0392409 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0260908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='287528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='189263 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00217654 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0597624 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='169468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0608076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='294648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0101688 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211603 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0179664 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0793443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148302 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0454118 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='124145 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0316148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='068097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='196956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232458 Cross Section δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθp < 1 Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04591813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0069123709 2 22 44 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='037772034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0062443702 3 44 66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='037785815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='007421952 4 66 88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='047121348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0079437151 5 88 110 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='077744533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0098333753 6 110 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11889674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011720873 7 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18746467 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019375934 29 Unfolded Covariance Matrix δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθp < 1 Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77809e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35626e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08944e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22106e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8981e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88853e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95466e-05 Bin 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35626e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89922e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72422e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27322e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29349e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92358e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8936e-05 Bin 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08944e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72422e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50854e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16498e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38577e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36478e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3498e-05 Bin 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22106e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27322e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16498e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31026e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5915e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90748e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33681e-05 Bin 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8981e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29349e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38577e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5915e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66953e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75465e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000106435 Bin 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88853e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92358e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36478e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90748e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75465e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000137379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00019473 Bin 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95466e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8936e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3498e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33681e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000106435 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00019473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000375427 Additional Smearing Matrix (AC) δαT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75 < cosθp < 1 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='274614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00706012 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0991495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0393246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100931 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0223494 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='385527 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='339548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0538596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0175478 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0109439 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0297243 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='339707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='281916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0333939 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0334954 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0312188 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0152392 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0803645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='382452 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='189684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140591 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0483661 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0883442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0254829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00783223 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='419872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='164006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0479162 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0461857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0753264 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='385821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='411758 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133568 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0145124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='225604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17243 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='430741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='522392 30 Cross Section δφT, All events Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg 40Ar] Uncertainty [10–38 cm2 deg 40Ar] 1 0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31441652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032284057 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20358551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01870319 3 25 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10729928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010816997 4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='062616734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0076583118 5 50 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='049715971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0079764994 6 60 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='030944993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0054822582 7 75 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024593059 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0046791168 8 90 106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020754743 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0043203551 9 106 126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016671524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0034825657 10 126 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013700322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0032524063 11 145 162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011648936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='003485876 12 162 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010095428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0036017253 Unfolded Covariance Matrix δφT, All events Units in [10–38 cm2 deg 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00104226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000565695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000212993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000129284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00014732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000103356 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71478e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39125e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19796e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6085e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66317e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66297e-05 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000565695 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000349809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000166989 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79334e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85369e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08468e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17649e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8598e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32292e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83852e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3036e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71163e-05 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000212993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000166989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000117007 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29332e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00803e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12038e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14489e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83589e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12085e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76343e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64419e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47933e-05 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000129284 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79334e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29332e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86497e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19122e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13999e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3882e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89824e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49976e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42339e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53562e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52523e-05 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00014732 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85369e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00803e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19122e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36245e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11897e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61136e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8979e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71083e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='759e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85968e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81645e-05 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000103356 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08468e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12038e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13999e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11897e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00552e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19022e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69321e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42942e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32223e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24432e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11306e-05 Bin 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71478e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17649e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14489e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3882e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61136e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19022e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18941e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92413e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43235e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10321e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66788e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74309e-06 Bin 8 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39125e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8598e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83589e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89824e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8979e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69321e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92413e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86655e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41362e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04727e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91366e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99661e-06 Bin 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19796e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32292e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12085e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49976e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71083e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42942e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43235e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41362e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21283e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03374e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82126e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31607e-06 Bin 10 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6085e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83852e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76343e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42339e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='759e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32223e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10321e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04727e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03374e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05781e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05216e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59436e-06 Bin 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66317e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3036e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64419e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53562e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85968e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24432e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66788e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91366e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82126e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05216e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21513e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21733e-05 Bin 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66297e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71163e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47933e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52523e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81645e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11306e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74309e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99661e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31607e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59436e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21733e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29724e-05 31 Additional Smearing Matrix (AC) δφT, All events Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 12 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='381677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='309001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='287563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='124765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0761652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='516932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='083428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='532529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='539467 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0826612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='542361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='537582 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0483494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='042471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0152274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0905658 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='173102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='336114 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0942919 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305344 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0504963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='627858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0313114 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00398141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139891 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0581769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103333 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115146 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0463378 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0540718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0498613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='313394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0818221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200978 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0610753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0523683 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0323176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0591564 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0306164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0220316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='377733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='338733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0711508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0103402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0720419 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0814814 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0441278 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0330035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024731 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00310732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0624084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='270565 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='422856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='262603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0418201 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0733859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125328 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0194687 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0331649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0893456 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0572414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00101423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='432296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109989 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0665918 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0212715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0107062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0967939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103897 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0987013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='346076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='299594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='187744 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0623878 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0473283 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0208628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0328734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0557834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0790618 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0421831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='323647 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='333451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0158844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294013 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0329926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0455895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294585 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0540332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0136866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0430887 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='257975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='383854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162919 Bin 11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0445607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0381367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0350268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00237989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0375327 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0502259 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0781496 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0281611 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='305406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='224414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='330539 Bin 12 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0507084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0270306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0451747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0362038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0720245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0706123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138258 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0566988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='134707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241407 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='477808 Cross Section δφT, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='364992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1215575 2 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73123664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='061852255 3 25 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21435948 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='029315593 4 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0081386225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0018999008 Unfolded Covariance Matrix δφT, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0147762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00563554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00153079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119854 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00563554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0038257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00154756 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30612e-05 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00153079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00154756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000859404 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97385e-05 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000119854 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30612e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97385e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60962e-06 32 Additional Smearing Matrix (AC) δφT, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='651056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='421486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0277306 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='110053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='517767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='601269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00261283 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0336902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='461469 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='438494 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00884042 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0308033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24402 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='656027 Cross Section δφT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15286824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026137739 2 20 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25116958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033748704 3 40 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15996139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020901006 4 60 80 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='059045083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012094918 5 80 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019126701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0088509973 6 100 120 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010656056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0075925506 7 120 140 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010600178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0071329061 8 140 160 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0077681367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0058117466 9 160 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0044826701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0051335927 Unfolded Covariance Matrix δφT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000683181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000782497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000308231 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80183e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53958e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2675e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27199e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63707e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20541e-05 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000782497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00113898 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000567719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000148218 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50718e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72663e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96908e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12437e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26303e-05 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000308231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000567719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000436852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000179468 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41443e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4909e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38677e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4793e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42253e-05 Bin 4 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80183e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000148218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000179468 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000146287 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34225e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37221e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43335e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13399e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35642e-05 Bin 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53958e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50718e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41443e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34225e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83402e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33834e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75344e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31263e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24748e-05 Bin 6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2675e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72663e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4909e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37221e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33834e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76468e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66638e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81806e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35196e-05 Bin 7 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27199e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96908e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38677e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43335e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75344e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66638e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08783e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75999e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89923e-05 Bin 8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63707e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12437e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4793e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13399e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31263e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81806e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75999e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37764e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32405e-05 Bin 9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20541e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26303e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42253e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35642e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24748e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35196e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89923e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32405e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63538e-05 33 Additional Smearing Matrix (AC) δφT, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='303586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='153051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='176382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='157878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0899184 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0201824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160691 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0773194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='506929 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='496119 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0920515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='301065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0363322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='340157 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='629034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='285196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00523665 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112789 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0267549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241081 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0944427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0351664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='450718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='349804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225582 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0436973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0483563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0358167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='284406 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='355228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='325699 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0596138 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0622666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0360514 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010273 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0759759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353802 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0945595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158547 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00644838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0143535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0163168 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0477205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0499331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='264343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35991 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='222794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276615 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00919507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00899754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0448134 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0640158 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0359387 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133664 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='303309 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0103688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0497679 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0202434 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0538902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0356291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149203 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='257174 Cross Section δφT, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin # Low edge [deg] High edge [deg] Cross Section [10–38 cm2 deg (GeV/c) 40Ar] Uncertainty [10–38 cm2 deg (GeV/c) 40Ar] 1 0 25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0019354696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='002538475 2 25 50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0099343841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0044054612 3 50 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='023950775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0063413422 4 75 105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='023479868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0051466413 5 105 145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='016668485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0037441977 6 145 180 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01542767 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0041453489 Unfolded Covariance Matrix δφT, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Units in [10–38 cm2 deg (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44386e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82677e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77984e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09684e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21115e-06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19111e-06 Bin 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82677e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94081e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48625e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67106e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05301e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02424e-05 Bin 3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77984e-06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48625e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02126e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9252e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45126e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14699e-05 Bin 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09684e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67106e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9252e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64879e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54525e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05853e-05 Bin 5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21115e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05301e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45126e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54525e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4019e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32789e-05 Bin 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19111e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02424e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14699e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05853e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32789e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71839e-05 34 Additional Smearing Matrix (AC) δφT, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='562728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0395993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0625026 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0464433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0414228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000360551 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0636863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='324516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0566229 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0563008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0257986 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138634 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0506159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='632292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='378733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='124571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00498109 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014064 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='415088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='560776 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0430022 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='134225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0393532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='513988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='219026 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='018933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0475621 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0964813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='497684 Cross Section δpT,x, All events Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c) 40Ar] Uncertainty [10–38 cm2 (GeV/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3018458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46828784 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='563478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59039688 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5731346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89649529 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='707396 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3957857 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='510914 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3000463 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019769 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236457 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='707973 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2296908 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103249 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4077904 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5394189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91807296 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5183878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62427657 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2015817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45888133 35 Unfolded Covariance Matrix δpT,x, All events Units in [10–38 cm2 (GeV/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='219294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='492393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='423579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='306975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183504 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='348568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='465303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='532551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='638571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='797177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='722886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='469885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='332926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212854 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='232466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='465303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='803704 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08054 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35252 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64411 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44492 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='778097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='461675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239296 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='277171 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='532551 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08054 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94822 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99076 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66436 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88731 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76956 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='543675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='291185 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38592 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='638571 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35252 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99076 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29021 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72269 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92385 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57229 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='372158 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='492393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='797177 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64411 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66436 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72269 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14243 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49303 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04629 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='417559 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='423579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='722886 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44492 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88731 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92385 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49303 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97152 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71033 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='756429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366246 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='306975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58234 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08867 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76956 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57229 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04629 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71033 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98187 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='562354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='308449 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='469885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='778097 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0176 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27913 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58698 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45241 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15239 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='842858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='504153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266441 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='332926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='461675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='543675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66189 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='756429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='562354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='504153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='389721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241085 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212854 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='291185 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='372158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='417559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='308449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266441 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='210572 Additional Smearing Matrix (AC) δpT,x, All events Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='608825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108505 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0835085 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0150919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00881438 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236513 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0252189 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00903844 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0637926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202277 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='336808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='298309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0132687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0263009 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0438141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0153022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00167781 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0731335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0992754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131753 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0298876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='310249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='405831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192639 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0787247 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0789994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0158152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0372995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0640728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06749 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='271785 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='519261 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='252748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0726825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0512266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0890651 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149703 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0962132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='748375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='483298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0724205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0517504 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1736 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0341671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='265799 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='758626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='411838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0712349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='410269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0459439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0519191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21337 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0515604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0572882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0835953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='464375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='286917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0794126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='412473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='147668 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0244409 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225528 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151986 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0727135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0790082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17944 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161495 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='354514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135537 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='145258 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0500579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0306147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0413035 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0693188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0680742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107136 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19282 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113542 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0714151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00151348 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0307881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0113679 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0272015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='329209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='385128 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000824422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0940523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00290975 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00586203 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='017935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00115087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00945362 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0983132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='602292 36 Cross Section δpT,x, δpT,y < –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c)2 40Ar] Uncertainty [10–38 cm2 (GeV/c)2 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6218493 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40791533 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8287626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53220357 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9296487 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='828875 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1584596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9657805 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9144932 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0811938 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7493754 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1882755 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5097752 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2183604 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2111143 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0521717 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6686658 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86446124 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9004347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56376489 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2983579 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41069205 Unfolded Covariance Matrix δpT,x, δpT,y < –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Units in [10–38 cm2 (GeV/c)2 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='166395 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='201863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='312093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='338651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='265329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='207234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0822108 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='201863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='283241 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='408043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='433027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='471449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='471055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='423703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='351969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='307575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='207386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142093 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='408043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='687034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='775398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='807246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='729945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='613065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='546349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='379706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253616 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='269995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='433027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='775398 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='932732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='997874 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='902642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='774481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='711383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='679866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='458514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='293436 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='312093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='471449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='807246 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='997874 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16898 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18836 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='967973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='837022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='507984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='306419 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='338651 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='471055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='729945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='902642 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18836 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='412 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='924202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='490718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26714 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='423703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='613065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='774481 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0983 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4844 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='920607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='440942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213245 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='265329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='351969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='546349 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='711383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='967973 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18307 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24681 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10707 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='856729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='417615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200895 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='207234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='307575 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='679866 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='837022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='924202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='920607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='856729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='747293 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='429783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='238263 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='207386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='379706 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='458514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='507984 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='490718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='440942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='417615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='429783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='317831 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21315 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0822108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253616 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='293436 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='306419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26714 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='213245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200895 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='238263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168668 37 Additional Smearing Matrix (AC) δpT,x, δpT,y < –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='447534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0824181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0309477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0147085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0395867 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0417552 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121439 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='381085 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='179102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0389878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0110257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0307066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00252424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0623969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0150945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0218938 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='275047 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='348206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0541287 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00683236 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0033379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000325739 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0595093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='271062 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='198536 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='146091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0944051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00290425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00955882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0551334 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146238 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36073 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='244691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='251439 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0551704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0252269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='138008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0637798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='347274 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='314214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0892305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0852992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0489167 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0711353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='287957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='223998 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='209465 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0688128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0109573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151359 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00941576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0479762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='367957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='300857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='176638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0657609 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0286886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00820142 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122808 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0213499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0160578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='256609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='310144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='285849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='087401 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0640037 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0681636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0717852 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='076997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0222277 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206629 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='313632 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343409 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0767339 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0327306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0914475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0121447 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0416083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0535827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0833627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='262042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='510595 Bin 11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0594818 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0273933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0730349 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0112767 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00528466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0131894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00910853 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00415322 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00755429 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='461581 Cross Section δpT,x, |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c)2 40Ar] Uncertainty [10–38 cm2 (GeV/c)2 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2115661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61126513 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6765554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82125417 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37408 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0556462 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='199696 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2045935 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='575073 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1450698 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119322 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='923925 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97485 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3491347 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92865 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='390739 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1639576 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1869639 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4893963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95587785 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65304935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72558594 38 Unfolded Covariance Matrix δpT,x, |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Units in [10–38 cm2 (GeV/c)2 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='373645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='445282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='423656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='403378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='461031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='835512 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='837471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='577603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='397967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='374982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295969 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='445282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='674458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='751372 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='882513 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64898 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='774405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='525903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='549017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='403667 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='423656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='751372 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11439 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77711 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64896 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67804 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8346 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='780921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='432323 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='403378 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73561 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77711 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86023 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44834 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='349 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40225 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89238 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='725906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='398222 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='461031 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='882513 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64896 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44834 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1816 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9511 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0817 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05194 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='966815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='564219 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='835512 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64898 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67804 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='349 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9511 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0929 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6529 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46626 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77758 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7282 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23794 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='837471 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44795 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8346 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40225 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0817 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6529 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='915 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4283 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86791 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='931342 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='577603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='774405 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44581 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89238 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05194 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46626 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4283 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='71563 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='959545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='435987 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='397967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='525903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='780921 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54195 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25342 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77758 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86791 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44987 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='838764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='477882 Bin 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='374982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='549017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='725906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='966815 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7282 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='959545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='838764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='913702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='652589 Bin 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='403667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='432323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='398222 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='564219 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23794 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='931342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='435987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='477882 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='652589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='526475 Additional Smearing Matrix (AC) δpT,x, |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 10 Bin 11 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='238492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111824 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00438664 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0148515 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000493344 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0208499 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0381854 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0308014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0726088 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30288 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0327858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00911913 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0132125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00397062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0187421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0552825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0371577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0757129 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0945286 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='335038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0273997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198814 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00502543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0236643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0379872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0607432 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='035615 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160394 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='274843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='449634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0123893 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0090526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0890296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0311237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139488 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='638782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0165204 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0552459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='474797 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0720988 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0271003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='242082 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='885233 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190623 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103157 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='214902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='274942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='346465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0844324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='255099 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='166916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='298163 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='467011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0459403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='388642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='300894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0285275 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00885317 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0377221 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163271 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00240758 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0713958 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0979268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0744572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0301267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0996823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='466168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='378591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0239726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239044 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100751 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0380883 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='052072 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='042795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0164917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0470341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146872 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='237981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0951803 Bin 10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117178 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0254209 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11418 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00697291 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0179875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0147786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0211936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0400128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='386113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='420585 Bin 11 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0730312 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0391132 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0727811 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00989024 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0190981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00178702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00973777 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00435176 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='031091 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='225363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='409018 Cross Section δpT,x, δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin # Low edge [GeV/c] High edge [GeV/c] Cross Section [10–38 cm2 (GeV/c)2 40Ar] Uncertainty [10–38 cm2 (GeV/c)2 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2342011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22671042 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55670098 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7324655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88391345 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7874941 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3844421 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4255073 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5313866 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8817046 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2031302 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1660393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75858938 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4156509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51675371 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46780295 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27308941 39 Unfolded Covariance Matrix δpT,x, δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Units in [10–38 cm2 (GeV/c)2 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0513976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0781145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0332451 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='309916 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='440532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='564084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='550037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='400661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='265051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0796011 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='440532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='781303 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1477 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='681115 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='242835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11692 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158666 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='564084 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1477 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91668 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99182 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='596827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='337503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155594 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='550037 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10821 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99182 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34514 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='850309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='395843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139324 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='400661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='681115 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2354 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67829 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='834993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='373839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103365 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='265051 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366268 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='596827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='850309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='834993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='575458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='323607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111993 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0781145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='242835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='337503 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='395843 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='373839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='323607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='267034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128279 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0332451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0796011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11692 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='155594 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='139324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103365 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='128279 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0745778 Additional Smearing Matrix (AC) δpT,x, δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='283981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='142652 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0578745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00406982 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00596516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='021526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0219778 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0247479 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0248516 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='325935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='248141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188942 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0533708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00616095 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00869173 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0123266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0148468 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='229278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='284816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='364996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='185996 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0506093 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00616198 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0665842 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='034873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11406 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0929379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='184043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='447192 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='344133 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131196 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='089738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468907 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0740597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162968 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0474994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0439422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288426 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='276995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='294853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='149114 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0128844 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='052895 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0173728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0492296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0722596 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0911906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='368141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='331983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0126541 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118282 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0288781 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0296457 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0111587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0193112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205607 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='285423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30351 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16427 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0309126 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00663667 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00622196 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00269483 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0412556 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144161 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='303342 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366314 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0682454 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='036675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0235291 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00866636 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0040182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0297024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0696102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='261339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='529614 40 Cross Section ECal, All events Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 GeV 40Ar] Uncertainty [10–38 cm2 GeV 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4280363 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79460421 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='514759 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2297474 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117445 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3413158 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6252 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4420716 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51041 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3446012 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08195 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2200177 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='766869 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1386025 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8493563 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0879941 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8229971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55094844 Unfolded Covariance Matrix ECal, All events Units in [10–38 cm2 GeV 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='631396 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='745221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='690303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='711899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='750373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='760828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='793249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='723599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='340987 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='745221 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51228 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4763 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25687 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29711 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36377 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31737 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='493087 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='690303 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4763 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79913 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50347 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57962 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32938 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='460105 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='711899 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25687 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50347 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07957 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73883 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33977 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='502543 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='750373 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29711 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57962 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73883 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80795 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='877393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='358545 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='760828 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36377 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49125 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58111 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48844 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='938906 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39754 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='793249 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31737 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32938 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33977 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24597 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25231 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29642 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='542827 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='723599 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09756 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05655 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='877393 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='938906 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15885 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18373 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='585281 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='340987 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='493087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='460105 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='502543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='358545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='542827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='585281 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='303544 41 Additional Smearing Matrix (AC) ECal, All events Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='468578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='007816 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0266806 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00958129 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0604614 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0932427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='180416 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0343321 Bin 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='240475 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='548541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0981671 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='152397 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='307903 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0374963 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='317461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0529613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='660139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0492356 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='309486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='251195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0119831 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0710508 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162421 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='701283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='223554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='102423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='223106 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0536489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0178971 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407812 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='110339 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='538018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0148715 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='168779 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114466 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0575493 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0598932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129188 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='210026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='087116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00673019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217555 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0193128 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0275789 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194644 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1013 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0465323 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='340983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17554 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0578717 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='253029 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197589 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='117109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126662 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0719783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120343 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0989725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16922 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='195653 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135617 Cross Section ECal, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 (GeV2/c) 40Ar] Uncertainty [10–38 cm2 (GeV2/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4778516 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='663791 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='721989 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6923951 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='375296 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99564 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='367813 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0983986 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159892 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1222198 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='757642 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2385658 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='738633 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3618913 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='333565 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1753137 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3184135 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='333598 42 Unfolded Covariance Matrix ECal, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Units in [10–38 cm2 (GeV2/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7682 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01429 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30596 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93741 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24048 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05339 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96903 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40275 Bin 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01429 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24899 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76053 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07509 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98125 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97772 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64916 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37548 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40711 Bin 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30596 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76053 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97386 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14642 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1032 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60378 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24297 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68398 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34706 Bin 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93741 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07509 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14642 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60007 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67603 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79065 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50217 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89311 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21397 Bin 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24048 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98125 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1032 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67603 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74826 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30445 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89028 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88424 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40028 Bin 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97772 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60378 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79065 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30445 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4883 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41828 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99101 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2708 Bin 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05339 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64916 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24297 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50217 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89028 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41828 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3023 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98777 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33803 Bin 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96903 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37548 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68398 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89311 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='88424 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99101 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98777 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0826 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61221 Bin 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40275 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40711 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34706 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21397 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40028 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2708 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33803 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61221 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77848 Additional Smearing Matrix (AC) ECal, δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='332012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='246473 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0882899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0495375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00236788 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0420704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0122056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0659321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00125015 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0967835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='557742 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0101648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0699365 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0336817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0330535 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027758 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122804 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45176 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='325083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='093336 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00256258 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00608461 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='124206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0298445 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0253861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121591 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0797448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0158361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='038969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0884846 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0190246 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0344541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00293491 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0872603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='315479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='145182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0447158 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='041489 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0928248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181306 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='401902 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0110214 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='122548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27646 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='092208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0377869 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0706819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='242878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='296523 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00395701 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0212106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0579449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0254825 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0962355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='188686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40643 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0771504 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0502941 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0231762 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0469033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0698983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217821 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='177127 43 Cross Section ECal, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 (GeV2/c) 40Ar] Uncertainty [10–38 cm2 (GeV2/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5177464 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='377228 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='368425 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2344492 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='658076 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2270906 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0641 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1795824 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='629847 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3712022 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='884387 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6178185 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='990481 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6000633 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9747439 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5419014 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7079479 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97366846 Unfolded Covariance Matrix ECal, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Units in [10–38 cm2 (GeV2/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89676 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63506 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38123 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77538 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35356 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57752 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43207 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3801 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='619098 Bin 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63506 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99276 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47115 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31143 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63655 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75769 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11913 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='007 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19924 Bin 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38123 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47115 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95993 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11754 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66967 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69833 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58682 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37203 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32496 Bin 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77538 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31143 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11754 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75058 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67059 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34432 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96921 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42904 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25923 Bin 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35356 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63655 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66967 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67059 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6226 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44488 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09807 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19559 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43705 Bin 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57752 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75769 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69833 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34432 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44488 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85297 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15936 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97345 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83354 Bin 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43207 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11913 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58682 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96921 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09807 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15936 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76033 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18423 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11273 Bin 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3801 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='007 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37203 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42904 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19559 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='97345 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18423 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46126 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21694 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='619098 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19924 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32496 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25923 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43705 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='83354 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11273 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94803 44 Additional Smearing Matrix (AC) ECal, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 < δpT < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0419037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0836673 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00930175 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0065885 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00186561 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0118106 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0052122 Bin 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0220518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='278052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='187143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0504702 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162508 -9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86369e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0446662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0259156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0292819 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='094654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0567177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='345528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0226151 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0715755 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0586704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00837126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0187044 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0227379 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159663 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0877275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0882331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245747 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0401781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0994074 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0291937 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00222292 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0287235 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206936 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0158151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159918 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0870625 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00340846 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0510671 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170821 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0811259 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0239962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0665726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='350127 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0289014 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0274388 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='187403 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0758085 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0671069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0145557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='038634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='186595 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0744945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0103675 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='113654 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0379305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0510391 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0203275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0382183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0940384 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167877 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0873667 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100333 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0766541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00370471 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0403667 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0341803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00518112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0394386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0455563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='228015 Cross Section ECal, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 (GeV2/c) 40Ar] Uncertainty [10–38 cm2 (GeV2/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1821006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29856414 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3696642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='63185827 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9647507 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68253959 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='347174 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72095291 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='604007 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73427148 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='872309 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72258138 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8535645 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64178201 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1345734 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64319549 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59841284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23714856 Unfolded Covariance Matrix ECal, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Units in [10–38 cm2 (GeV2/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0891405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0499376 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='399245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='381907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='327208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='322685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='312683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112147 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='381907 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='453613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='401772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='388672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='352275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='329322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111957 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='341759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='453613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='519773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='492251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='449318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0922748 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='327208 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='401772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='492251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='539155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='380723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='287547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='085787 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='388672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='449318 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='522124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='412197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='331062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108951 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='322685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='352275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353818 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='380723 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='412197 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='411884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123694 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='312683 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='329322 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='287547 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='331062 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4137 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143313 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0499376 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='112147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0922748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='085787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143313 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0562394 45 Additional Smearing Matrix (AC) ECal, δpT > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0931121 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0983705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='115428 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0559073 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0250682 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00763155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0848454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0285786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0211121 Bin 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00853425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2951 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='311508 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00449115 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='040433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0367123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='173613 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0649463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0362442 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0725362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0397894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='535149 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194827 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019212 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00703728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0303788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='020959 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='143708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='099961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='418759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='370076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='134729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0760182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='195705 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0648952 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='076964 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0768097 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123574 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='225145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='281358 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='294061 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='131329 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='242453 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0855012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='102749 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0452569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='178781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162803 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='163218 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216386 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='280406 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0290967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00340337 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0859781 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0158235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='195278 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0166949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0422117 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0759005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='321599 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='156297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0757575 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101102 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0765364 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='210047 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0540062 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0696197 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00934385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='282294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='208489 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241305 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0801451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141856 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0465726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103011 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00132606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15648 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241303 Cross Section ECal, δαT < 45o Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 deg GeV 40Ar] Uncertainty [10–38 cm2 deg GeV 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='022911113 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0049556794 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='046506764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0058656604 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05371068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0053429329 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='057122311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0051425099 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='044246553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0050250762 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='031770412 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0052311462 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019595659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0049797225 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0086228162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00271678 Unfolded Covariance Matrix ECal, δαT < 45o Units in [10–38 cm2 deg GeV 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45588e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3115e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72193e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56483e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37585e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36684e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41928e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04982e-06 Bin 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3115e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4406e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60071e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92979e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74194e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87576e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70236e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58294e-06 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72193e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60071e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85469e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3178e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93935e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80045e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55008e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48334e-06 Bin 4 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='56483e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92979e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3178e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64454e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16351e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74016e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13098e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15249e-06 Bin 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37585e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74194e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93935e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16351e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='52514e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19635e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55291e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43748e-06 Bin 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36684e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87576e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80045e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74016e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19635e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73649e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31378e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11024e-05 Bin 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41928e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70236e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55008e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13098e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55291e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31378e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47976e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28476e-05 Bin 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04982e-06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58294e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48334e-06 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15249e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43748e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11024e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28476e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38089e-06 46 Additional Smearing Matrix (AC) ECal, δαT < 45o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='214015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0935853 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0436162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0413904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00736974 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0722259 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0121885 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0350815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='347665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0652129 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00761471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0455451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0108441 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0274569 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0626681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41405 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0349829 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0451783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0651346 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00383067 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='070977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115522 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='204198 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='440867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='150709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0575478 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00200467 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000697258 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0186721 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0536033 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111678 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='291431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='318587 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00846444 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00171664 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00347665 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0245411 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0686577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108357 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='201719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119724 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0043447 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0275845 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0809657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0890458 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0217563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='133628 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0217404 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0339496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0284494 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0602311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0294924 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0396266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='039581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0146859 Cross Section ECal, 45o < δαT < 90o Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 deg GeV 40Ar] Uncertainty [10–38 cm2 deg GeV 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015223769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0040708036 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='042871964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0060792512 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='061405294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0058409485 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='067888001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0062125014 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06182615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='006244721 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04785571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='008022795 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02950621 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0085592201 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011741284 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0064936311 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0050906485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0035720766 Unfolded Covariance Matrix ECal, 45o < δαT < 90o Units in [10–38 cm2 deg GeV 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65714e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17096e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61165e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34064e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61927e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45648e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62167e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75668e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08752e-05 Bin 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17096e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69573e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01948e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14091e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50269e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33579e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64985e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57546e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49574e-05 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61165e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01948e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41167e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95355e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49843e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57874e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66386e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99403e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25383e-05 Bin 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34064e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14091e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95355e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='85952e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04728e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36242e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07416e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66306e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15262e-05 Bin 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61927e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50269e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49843e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04728e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89965e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10664e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61737e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20696e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22399e-05 Bin 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45648e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='33579e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57874e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36242e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10664e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43652e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17283e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41484e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80472e-05 Bin 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62167e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64985e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66386e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='07416e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61737e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17283e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32602e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94426e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44258e-05 Bin 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75668e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57546e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99403e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66306e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20696e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41484e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94426e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='21672e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12141e-05 Bin 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08752e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49574e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25383e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15262e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='22399e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80472e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44258e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12141e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27597e-05 47 Additional Smearing Matrix (AC) ECal, 45o < δαT < 90o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144511 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00188425 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00411617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00806955 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0159754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0290823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0221901 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='059924 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0564224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='326603 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='110783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0144956 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000544054 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0322722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0383533 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0315212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0634079 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0883864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30553 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0509968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0452151 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0114401 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0356079 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0321712 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='065177 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0777837 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='110321 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='366968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0148477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0485207 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00457134 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0179172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115288 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00387751 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00143439 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103056 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202124 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0425935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0880865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00201993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='031266 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0464894 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00102566 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='115312 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='129873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158581 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='220214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0249775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0412134 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0428489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0525865 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='025311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='175886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0607949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='064551 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='270719 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174932 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0606612 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00232508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0303037 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11509 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0234896 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0543038 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148606 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='359526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118717 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0529726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00217489 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00802375 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00110526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0124215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='042141 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0302965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='140164 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='109106 Cross Section ECal, 90o < δαT < 135o Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 deg GeV 40Ar] Uncertainty [10–38 cm2 deg GeV 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='014030087 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0051703105 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='054311971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0076341954 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='090180705 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0077883177 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='098905179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0080743704 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='087012541 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0097326103 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='069321234 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011079443 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='056595104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010273923 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033671172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0070344961 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013865221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0031854942 Unfolded Covariance Matrix ECal, 90o < δαT < 135o Units in [10–38 cm2 deg GeV 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='67321e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31333e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11468e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11514e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69741e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4113e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92199e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96907e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15361e-05 Bin 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31333e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82809e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9377e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62148e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59862e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27673e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20475e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04206e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69097e-05 Bin 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11468e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9377e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06579e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61376e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65051e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72283e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58315e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58819e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42286e-05 Bin 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11514e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62148e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61376e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51955e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57262e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4932e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37034e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82061e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41509e-05 Bin 5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69741e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='59862e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65051e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='57262e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47237e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3255e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35524e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16414e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77432e-05 Bin 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4113e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27673e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='72283e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4932e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3255e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000122754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000102112 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28708e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18916e-05 Bin 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='92199e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20475e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58315e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='37034e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35524e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000102112 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000105554 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53115e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46442e-05 Bin 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='96907e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04206e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58819e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82061e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16414e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28708e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53115e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94841e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98204e-05 Bin 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15361e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='69097e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42286e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='41509e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77432e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18916e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46442e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='98204e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='01474e-05 48 Additional Smearing Matrix (AC) ECal, 90o < δαT < 135o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='174788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0468135 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0145914 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00807388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00751103 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0211459 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00381463 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0802165 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0559721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='405876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259704 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0226756 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='024953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00286068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0118828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00843255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0706839 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14142 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='549514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0972058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0316195 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0299529 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0235975 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00458832 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0206822 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0890518 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0502697 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='270657 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='393017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151964 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0133933 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0320262 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0113839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0218976 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='121637 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0734024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='124125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='227054 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='379764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0589708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='082431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00108853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00595713 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191335 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0225609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161979 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='032355 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='220928 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183945 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0067701 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0216689 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='259258 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0984878 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148876 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='039466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107229 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='152703 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='235997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126265 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0104258 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='194727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14524 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='146635 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='037525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0614335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0474292 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170983 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0981407 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0230308 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='056371 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0772496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0016455 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000934953 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0198736 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0301298 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0482617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='127642 Cross Section ECal, 135o < δαT < 180o Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 deg GeV 40Ar] Uncertainty [10–38 cm2 deg GeV 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027377992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='004790503 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='084359465 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0095187811 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1123024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011355117 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='10788839 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010553428 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='080871908 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010456613 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='056206713 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='012229283 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='029605819 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='010305779 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='011512073 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0052540905 Unfolded Covariance Matrix ECal, 135o < δαT < 180o Units in [10–38 cm2 deg GeV 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='29489e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86213e-05 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23844e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54022e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49212e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47533e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28861e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91373e-06 Bin 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86213e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06072e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89917e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19396e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81525e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9288e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03703e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70905e-06 Bin 3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23844e-05 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89917e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000128939 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000101578 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74726e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68293e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58845e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42033e-06 Bin 4 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='54022e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='19396e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000101578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000111375 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91661e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5271e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86204e-05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60688e-05 Bin 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49212e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81525e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='74726e-05 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91661e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000109341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000111267 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70787e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45722e-05 Bin 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='47533e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9288e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='68293e-05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5271e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000111267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000149555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000116661 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49348e-05 Bin 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28861e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='03703e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='58845e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86204e-05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70787e-05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000116661 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000106209 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34096e-05 Bin 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91373e-06 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70905e-06 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42033e-06 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60688e-05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='45722e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49348e-05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34096e-05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76055e-05 49 Additional Smearing Matrix (AC) ECal, 135o < δαT < 180o Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='344614 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0994744 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0134076 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00019442 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00242534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0149544 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0130152 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0138546 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='252109 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='432082 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='080048 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0402081 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00473542 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00423299 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='015471 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00174856 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191022 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31122 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123228 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0326861 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00579486 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0106008 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0940126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0540609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='387431 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118332 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00866266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00485879 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00601292 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205879 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0810976 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0288859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28526 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0803451 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0698116 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0140106 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='26836 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='162649 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0394283 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='165016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107319 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197615 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20912 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0159508 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192951 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='118214 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0596226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0252027 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='339042 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00785766 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='081445 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0469623 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='033978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0436873 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000148688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0355065 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183864 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0120461 Cross Section ECal, δpT,y < –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 (GeV2/c) 40Ar] Uncertainty [10–38 cm2 (GeV2/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9750005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='44493023 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6431053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81690274 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1495146 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90801221 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7619175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='93199333 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70625 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='99277051 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5646567 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0696243 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0179828 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0467697 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2678194 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='94332837 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='84086961 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27510692 Unfolded Covariance Matrix ECal, δpT,y < –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Units in [10–38 cm2 (GeV2/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='336851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='304578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='218786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0719656 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='336851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66733 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='551068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='477125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='459131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='449094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='415296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125247 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='304578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='66017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='824486 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='744252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='665787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='639772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='519784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='405001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11917 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245693 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='551068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='744252 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='868612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='857476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='631517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='509084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14212 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='218786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='477125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='665787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='857476 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='985593 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='977139 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='848584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='695622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190083 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='205849 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='459131 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='639772 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='977139 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1441 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04791 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='836266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='218672 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='192956 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='449094 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='519784 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='631517 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='848584 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04791 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='09573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='935545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241009 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='415296 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='405001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='509084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='695622 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='836266 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='935545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='889868 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245135 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0719656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='125247 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='190083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='218672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='241009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='245135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0756838 50 Additional Smearing Matrix (AC) ECal, δpT,y < –0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='258982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='290163 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0598335 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0237732 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00654955 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00767649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0154331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0100231 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0546274 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='224771 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='599993 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='197485 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000855549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0450191 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0333677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00943202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00550107 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0755554 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0596168 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='336637 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='42362 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1307 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137331 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0234496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0091154 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0343488 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407716 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='214557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='39926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='337856 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00136191 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00700122 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0222311 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0177868 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='108289 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0122851 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111822 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='289434 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='469815 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0926546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0358627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00577917 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0288427 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='158342 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0274417 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06809 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='393725 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='200294 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='031297 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0317284 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='221689 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0947182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0227254 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0300817 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='292491 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0939459 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='123889 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104562 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126722 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='105206 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0338211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0583892 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='217202 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000494976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='144867 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17211 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161664 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0204245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0152219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='013884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0438123 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0142269 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0421754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0727728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='14113 Cross Section ECal, |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 (GeV2/c) 40Ar] Uncertainty [10–38 cm2 (GeV2/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7256169 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2122548 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='727215 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9508107 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='381216 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2053728 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='181157 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3524262 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='561199 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4085133 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='427838 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2917764 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='137455 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1749865 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='830218 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1226077 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9781531 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='90011043 Unfolded Covariance Matrix ECal, |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Units in [10–38 cm2 (GeV2/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='46956 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82924 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81576 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50746 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75206 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70651 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61039 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='578361 Bin 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82924 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80566 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60408 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40082 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86004 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76553 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='873838 Bin 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81576 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60408 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86367 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31341 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89315 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13913 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64071 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='853925 Bin 4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50746 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='40082 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='31341 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53391 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77603 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6633 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77124 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20546 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='864764 Bin 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='75206 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='86004 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='89315 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77603 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='80094 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81499 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65324 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61433 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='885788 Bin 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70651 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76553 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13913 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6633 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='81499 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25224 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51429 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23939 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04951 Bin 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61039 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='60945 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='64071 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='77124 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='65324 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='51429 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73057 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16847 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50947 Bin 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='43728 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='27019 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='24748 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20546 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='61433 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23939 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='16847 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50546 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82207 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='578361 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='873838 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='853925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='864764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='885788 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04951 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='50947 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='82207 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='810199 51 Additional Smearing Matrix (AC) ECal, |δpT,y| < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='322497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0905649 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='101626 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0300083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0356452 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0588684 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0707156 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0231266 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0124691 Bin 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0135338 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='501382 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='216449 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183919 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='170235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0297659 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111656 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0484389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0195058 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0130477 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0470315 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='666263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243777 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0736494 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0235973 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0678795 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0727558 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0272982 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='092557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='039126 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='167226 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='705128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0567686 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='028069 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0486624 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0415374 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00851025 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0777826 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0716627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0940598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='344559 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='397144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='13688 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='136813 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0228841 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0567614 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0903508 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0229516 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0968807 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='249563 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='295088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='273221 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0715431 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0421295 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='114256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0241555 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0109053 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0973428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='178685 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='324859 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='233572 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0742204 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='027769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104303 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0448909 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='000806135 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00407994 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0202481 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='257128 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='359773 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='183541 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0141602 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0494249 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0315971 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0431423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='043055 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='107492 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15496 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='30638 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='193736 Cross Section ECal, δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin # Low edge [GeV] High edge [GeV] Cross Section [10–38 cm2 (GeV2/c) 40Ar] Uncertainty [10–38 cm2 (GeV2/c) 40Ar] 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3812738 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='53728629 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8519267 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7584091 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='48 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8841718 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='70560257 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1290618 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62295676 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2405823 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='62866421 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='73798187 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='9052577 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='87651596 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='18 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='91661655 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='95293063 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='36476708 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='49297537 Unfolded Covariance Matrix ECal, δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Units in [10–38 cm2 (GeV2/c) 40Ar]2 Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='288677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='328153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0778962 Bin 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='328153 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='575184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='473237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='201145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='234967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='252571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120788 Bin 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='239634 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='473237 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='497875 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='379256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103976 Bin 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='141798 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='297274 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='379256 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='388075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='315144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='248043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0777672 Bin 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='116787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='203548 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='23748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='315144 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='395219 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='404325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='345754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111748 Bin 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='135921 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='201145 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='191696 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='248043 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='404325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='544617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='585026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='524348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243676 Bin 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='161568 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='234967 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='206691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='202497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='345754 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='585026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='76828 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='391573 Bin 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='151674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='252571 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='212745 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='172576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='266076 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='524348 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='78999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='908077 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='465227 Bin 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0778962 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='120788 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='103976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0777672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='111748 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243676 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='391573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='465227 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='243025 52 Additional Smearing Matrix (AC) ECal, δpT,y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 GeV/c Bin 1 Bin 2 Bin 3 Bin 4 Bin 5 Bin 6 Bin 7 Bin 8 Bin 9 Bin 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0574863 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='119834 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20923 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0525759 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0109354 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='026514 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00386619 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0460792 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0233909 Bin 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0525597 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='290424 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='500753 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0115298 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00980539 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0415917 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00717367 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='104371 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0721812 Bin 3 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0604904 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132088 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='619888 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='223617 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0470369 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0407812 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00482642 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0836055 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='080892 Bin 4 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='051691 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0608881 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='438255 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='353263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='28528 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0263162 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0151576 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0518864 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='094828 Bin 5 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0550683 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0903235 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148248 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='224764 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='523968 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='126263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='100147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0507039 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106761 Bin 6 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0931912 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0394578 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00797704 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0831667 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='488213 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='154978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='211066 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='159143 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='054027 Bin 7 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='132024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0004211 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00944179 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0139328 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='290612 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='082325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='255604 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='343521 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0374567 Bin 8 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0183179 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00291147 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0200926 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='106735 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00380633 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='20166 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='490899 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='11169 Bin 9 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='148341 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00295388 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='00980972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0333003 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='019359 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='0418381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='17709 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='518324 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='160304 PARTICLE IDENTIFICATION AND EVENT SELECTION We used the log-likelihood ratio particle identification (LLR PID) score method [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ] to obtain our muon and proton candidates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 6 of [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ], muons tend to have higher LLR score values than protons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Thus, the candidate track with the greater LLR PID score is assigned the label of the candidate muon, while the track with the smaller score is our candidate proton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We studied the effect of cutting on different values of the candidate proton LLR PID score, which has a strong discrimination power rejecting MC non-CC1p0π background, out-of-cryostat (Dirt) and cosmic events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Maximizing the purity × efficiency product yielded an optimal cut on the proton candidate LLR PID score < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05, which is indicated by the dashed line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' MicroBooNE Data Cosmic π MC CC1p0 π MC Non-CC1p0 Dirt Proton Candidate LLR PID Score 0 200 400 600 800 1000 # Events / 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='7 % π CC1p0 Cosmic = 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 % 1 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 − 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 Proton Candidate LLR PID Score 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='5 Simulation Data FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The proton candidate LLR PID score distribution, illustrating the fitness of a cut at LLR PID < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 to reject cosmic and non-CC1p0π background events.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' To minimize the contribution of misreconstructed tracks, we took advantage of the fact that we had two muon momentum reconstruction methods available for contained tracks, namely the momentum from range [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ] and the momentum from Multiple Coulomb Scattering (MCS) [?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We applied a quality cut on the contained muons by requiring the range and MCS momenta to be in agreement 53 within 25%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The effect of the quality cut is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 0 100 200 300 400 500 600 700 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 True Muon Momentum [GeV/c] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Range Reco Muon Momentum [GeV/c] 0 100 200 300 400 500 600 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 True Muon Momentum [GeV/c] 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 Range Reco Muon Momentum [GeV/c] FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Muon momentum reconstruction (left) before and (right) after the application of the muon momentum quality cut using contained muon tracks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' In order to avoid mis-reconstructed track directions, we further required that the distance between the track start points and the vertex is smaller than the corresponding distance between the track end points and the vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' We also demanded that the distance between the start points of the two candidate tracks is smaller than the distance between the two end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' FINAL STATE INTERACTION SMEARING Figures 3-5 show the effect of final state interactions (FSI) on the CC1p0π selection using the G18 configuration of GENIE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' The addition of FSI allows for more non-QE events to satisfy the CC1p0π signal definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Furthermore, FSI effects smear the δpT distribution to higher values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 3), introduce an asymmetric behavior in δαT (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 4), and lead to larger δφT values (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 GeV/c Ar 2 cm 38 10 T p δ d σ d QE MEC RES DIS MicroBooNE Data Shape) ⊕ (Stat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Norm (a) G18, All events 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='8 [GeV/c] T p δ 0 10 20 30 40 50 GeV/c Ar 2 cm 38 10 T p δ d σ d QE MEC RES DIS MicroBooNE Data Shape) ⊕ (Stat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Norm (b) G18 No FSI, All events FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cross section interaction breakdown for the selected events for the G18 configuration (left) with FSI effects, and (right) without FSI effects as a function of δpT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 54 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12 deg Ar 2 cm 38 10 T α δ d σ d QE MEC RES DIS MicroBooNE Data Shape) ⊕ (Stat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Norm (a) G18, All events 0 20 40 60 80 100 120 140 160 180 [deg] T α δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='12 deg Ar 2 cm 38 10 T α δ d σ d QE MEC RES DIS MicroBooNE Data Shape) ⊕ (Stat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Norm (b) G18 No FSI, All events FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cross section interaction breakdown for the selected events for the G18 configuration (left) with FSI effects, and (right) without FSI effects as a function of δαT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 deg Ar 2 cm 38 10 T φ δ d σ d QE MEC RES DIS MicroBooNE Data Shape) ⊕ (Stat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Norm (a) G18, All events 0 20 40 60 80 100 120 140 160 180 [deg] T φ δ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='35 deg Ar 2 cm 38 10 T φ δ d σ d QE MEC RES DIS MicroBooNE Data Shape) ⊕ (Stat 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content='79e+20 POT Norm (b) G18 No FSI, All events FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} +page_content=' Cross section interaction breakdown for the selected events for the G18 configuration (left) with FSI effects, and (right) without FSI effects as a function of δφT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/pNE2T4oBgHgl3EQfKQZI/content/2301.03700v1.pdf'} diff --git a/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf b/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..43dbed55bfa4560508372677810c96c91229f9c7 --- /dev/null +++ b/ptFRT4oBgHgl3EQfdjek/content/2301.13568v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:8c41b5c11e3833b5090b53ced3d945bb7cd2cc8df04b1667a0546d264cf89346 +size 30910131 diff --git a/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf b/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fe94b4539c95dac86875c2d65efb60d0b9fb100e --- /dev/null +++ b/qdE0T4oBgHgl3EQfaQC1/content/2301.02333v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:52e111e915c3372ac3d8689f0a7dac9c872b21cd30d4fe1e3696a327dc301ba0 +size 28151281 diff --git a/qtFPT4oBgHgl3EQf9DUp/content/2301.13210v1.pdf b/qtFPT4oBgHgl3EQf9DUp/content/2301.13210v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..4e2717764c90ddd253f163b1c189f64210fc9392 --- /dev/null +++ b/qtFPT4oBgHgl3EQf9DUp/content/2301.13210v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5ea2864c537a6b978f84ebab7e169c7d90a092b2a80e0efb6df85d59b5ca9351 +size 731157 diff --git a/qtFPT4oBgHgl3EQf9DUp/vector_store/index.faiss b/qtFPT4oBgHgl3EQf9DUp/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..35f1d65e5e4ceac735a02f28855f410b73fb4d24 --- /dev/null +++ b/qtFPT4oBgHgl3EQf9DUp/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ad2554038cc1264e8fea1123bce7936983ac2b6579144f58544e862b03275324 +size 2621485 diff --git a/qtFPT4oBgHgl3EQf9DUp/vector_store/index.pkl b/qtFPT4oBgHgl3EQf9DUp/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..83030fdbdc5e8d297ae1715c638d085c34dbb22f --- /dev/null +++ b/qtFPT4oBgHgl3EQf9DUp/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ab7a379cfa630eb1a556f126e7a05364b2ed571bbc62dca1d44853a41b582b50 +size 110966 diff --git a/rNFRT4oBgHgl3EQffDdP/vector_store/index.faiss b/rNFRT4oBgHgl3EQffDdP/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..5a4172c6487758a4f8493f249f4d5924fb363ce1 --- /dev/null +++ b/rNFRT4oBgHgl3EQffDdP/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:d38d7275e9e4f74e8e9c6054f21477b2f50a434a7029e51a9b6e212031b90b74 +size 5046317 diff --git a/rdAzT4oBgHgl3EQfrf1b/content/2301.01644v1.pdf b/rdAzT4oBgHgl3EQfrf1b/content/2301.01644v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..41b336c1aec6cf2b1e5774874f6277fac089e431 --- /dev/null +++ b/rdAzT4oBgHgl3EQfrf1b/content/2301.01644v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:a20f34ee6afa493e74713cd6acab2478d20f7b20eaccfc07138789c3b4891c97 +size 10224931 diff --git a/rdAzT4oBgHgl3EQfrf1b/vector_store/index.faiss b/rdAzT4oBgHgl3EQfrf1b/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..b78580b633a70aa621912d727f1fc7826918fb95 --- /dev/null +++ b/rdAzT4oBgHgl3EQfrf1b/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f03390030a1bf16b797dec17f848a0f48703eba072e22b65fec09a3743e9022a +size 3670061 diff --git a/rdAzT4oBgHgl3EQfrf1b/vector_store/index.pkl b/rdAzT4oBgHgl3EQfrf1b/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..f5f264449a74e81168315fca9ce18b6d7fef816b --- /dev/null +++ b/rdAzT4oBgHgl3EQfrf1b/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:94056af63e65cead1603f4f4cd99f6b453bbe71c5e39a3543dbb2d64524048ed +size 144861 diff --git a/sNFJT4oBgHgl3EQfcCyS/content/tmp_files/2301.11542v1.pdf.txt b/sNFJT4oBgHgl3EQfcCyS/content/tmp_files/2301.11542v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..0882eb69bdb53ed2c8a891ba339e4df33870f316 --- /dev/null +++ b/sNFJT4oBgHgl3EQfcCyS/content/tmp_files/2301.11542v1.pdf.txt @@ -0,0 +1,1988 @@ +Feasibility and Transferability of Transfer Learning: A +Mathematical Framework +Haoyang Cao ∗ +Haotian Gu † +Xin Guo ‡ +Mathieu Rosenbaum § +January 26, 2023 +Abstract +Transfer learning is an emerging and popular paradigm for utilizing existing knowledge from +previous learning tasks to improve the performance of new ones. Despite its numerous empirical +successes, theoretical analysis for transfer learning is limited. In this paper we build for the +first time, to the best of our knowledge, a mathematical framework for the general procedure +of transfer learning. Our unique reformulation of transfer learning as an optimization problem +allows for the first time, analysis of its feasibility. Additionally, we propose a novel concept of +transfer risk to evaluate transferability of transfer learning. Our numerical studies using the +Office-31 dataset demonstrate the potential and benefits of incorporating transfer risk in the +evaluation of transfer learning performance. +1 +Introduction +The basic idea of transfer learning is simple: it is to leverage knowledge from a well-studied learning +problem, known as the source task, to improve the performance of a new learning problem with +similar features, known as the target task. Transfer learning has seen success in a variety of field, +including natural language processing (Ruder et al., 2019; Devlin et al., 2019; Sung et al., 2022), +sentiment analysis (Jiang and Zhai, 2007; Deng et al., 2013; Liu et al., 2019), computer vision (Deng +et al., 2009; Long et al., 2015; Ganin et al., 2016; Wang and Deng, 2018), activity recognition (Cook +et al., 2013; Wang et al., 2018), medical data analysis (Zeng et al., 2019; Wang et al., 2022; Kim +et al., 2022), bio-informatics (Hwang and Kuang, 2010), finance (Leal et al., 2020; Rosenbaum and +Zhang, 2021), recommendation system (Pan et al., 2010; Yuan et al., 2019), and fraud detection +(Lebichot et al., 2020). See also review papers (Pan and Yang, 2010; Tan et al., 2018; Zhuang et al., +2020). Transfer learning is a versatile and enduring paradigm in the rapidly changing AI landscape +where new machine learning techniques and tools mushroom with a breakneck speed. +Despite its empirical successes, studies on transfer learning are primarily based on trial-and-error +heuristics. Virtually there are neither basic theoretical frameworks for the general procedure of +transfer learning, nor studies on the fundamental issue of it feasibility. +Existing theoretical works of transfer learning. +Earlier theoretical works for transfer learning +tend to focus on specific learning problems, such as classification, and derive upper bounds of +∗Centre de Mathématiques Appliquées, Ecole Polytechnique. Email: haoyang.cao@polytechnique.edu +†Department of Mathematics, UC Berkeley. Email: haotian_gu@berkeley.edu +‡Department of Industrial Engineering & Operations Research, UC Berkeley. Email: xinguo@berkeley.edu +§Centre de Mathématiques Appliquées, Ecole Polytechnique. Email: mathieu.rosenbaum@polytechnique.edu +1 +arXiv:2301.11542v1 [cs.LG] 27 Jan 2023 + +generalization error under different measurements. There are the VC-dimension of the hypothesis +space adopted in (Blitzer et al., 2007), total variation distance in (Ben-David et al., 2010), f-divergence +in (Harremoës and Vajda, 2011), Jensen-Shannon divergence in (Zhao et al., 2019), H-score in (Bao +et al., 2019), mutual information in (Bu et al., 2020), and more recently X 2-divergence in (Tong +et al., 2021), and variations of optimal transport cost in (Tan et al., 2021). +Another line of theoretical studies interprets transferability for transfer learning as a measurement +of similarity between the source and the target data using various divergences, such as low-rank +common information in (Saenko et al., 2010), KL-divergence in (Ganin and Lempitsky, 2015; Ganin +et al., 2016; Tzeng et al., 2017), l2-distance in (Long et al., 2014), and the optimal transport cost in +(Courty et al., 2017). +Our work. +In this paper, we address the issues of feasibility and transferability for transfer learning +through rigorous and comprehensive mathematical analysis. +• We build, for the first time to the best of our knowledge, a mathematical framework for the +general procedure of transfer learning, identifying its three key steps and components. +• We reformulate this three-step transfer learning procedure as an optimization problem, enabling +us to analyze, for the first time, its feasibility. This is accomplished via analyzing the well- +definedness of the corresponding optimization problem. +• Additionally, we propose a novel concept of transfer risk to evaluate the transferability of +transfer learning. Our form of transfer risk accounts for both the compatibility between the +output and the input data and the compatibility between the models in the source and the +target tasks, allowing for the study of the trade-off between the two. This novel notion of +transfer risk generalizes earlier works on transferability, including the H-score proposed in a +particular classification setting in (Bao et al., 2019) and (Saenko et al., 2010; Ganin et al., +2016; Long et al., 2014) on the relation between source and target inputs. +• In the special case of linear regression with Gaussian data, we show that the regret in the +learning problem can be lower bounded by Wasserstein-based transfer risk, which in turn is +useful for prescreening unsuitable candidate pretrained models or source tasks. +• Our numerical studies using the Office-31 dataset show the consistency of the transfer risk with +existing statistical metrics in evaluating the performance of transfer learning; and demonstrate +the potential and benefit of adopting transfer risk to improve computational efficiency. +2 +Mathematical Framework and Feasibility of Transfer Learning +In this section, we will establish necessary concepts and a mathematical framework for the entire +procedure of transfer learning. We will then reformulate transfer learning as an optimization problem, +the well-definedness of which yields the feasibility of transfer learning. +For ease of exposition and without loss of generality, we will focus on a supervised setting, with +a source task S and a target task T on a probability space (Ω, F, P). +2.1 +Mathematical Framework for Transfer Learning +Target task T. +In the target task T, denote XT and YT as its input and output spaces, respectively, +and (XT , YT ) as a pair of XT × YT -valued random variables. Here, (XT , ∥ · ∥XT ) and (YT , ∥ · ∥YT ) are +2 + +Banach spaces with norms ∥ · ∥XT and ∥ · ∥YT , respectively. Let LT : YT × YT → R be a real-valued +function, and assume that the learning objective for the target task is +min +f∈AT LT (fT ) = min +fT ∈AT E[LT (YT , fT (XT ))], +(1) +where LT (fT ) is a loss function that measures a model fT : XT → YT for the target task T, and AT +denotes the set of target models such that +AT ⊂ {fT |fT : XT → YT }. +(2) +Take the image classification task as an example, XT is a space containing images as high +dimensional vectors, YT is a space containing image labels, (XT , YT ) is a pair of random variables +satisfying the empirical distribution of target images and their corresponding labels, and LT is the +cross-entropy loss function between the actual label YT and the predicted label fT (XT ). For the +image classification task using neural networks, AT will depend on the neural network architecture +as well as the constraints applied to the network parameters. +Let f∗ +T denote the optimizer for the optimization problem (1), and PT = Law(f∗ +T (XT )) for the +probability distribution of its output. Then the model distribution PT depends on three factors: LT , +the conditional distribution Law(YT |XT ), and the marginal distribution Law(XT ). Note that in +direct learning, this optimizer f∗ +T ∈ AT is solved directly by analyzing the optimization problem (1), +whereas in transfer learning, one leverages knowledge from the source task to facilitate the search of +f∗ +T . +Source task S. +In the source task S, denote XS and YS as the input and output spaces of +the source task, respectively, and (XS, YS) as a pair of XS × YS-valued random variables. Here, +(XS, ∥ · ∥XS) and (YS, ∥ · ∥YS) are Banach spaces with norms ∥ · ∥XS and ∥ · ∥YS, respectively. Let +LS : YS × YS → R be a real-valued function and let us assume that the learning objective for the +source task is +min +fS∈AS LS(fS) = min +f∈AS E[LS(YS, fS(XS))], +(3) +where LS(fS) is the loss function for a model fS : XS → YS for the source task S. Here AS denotes +the set of source task models such that +AS ⊂ {fS|fS : XS → YS}. +(4) +Moreover, denote the optimal solution for this optimization problem (3) as f∗ +S, and the probability +distribution of the output of f∗ +S by PS = Law(f∗ +S(XS)). Meanwhile, similar as the target model, the +model distribution PS will depend on the function LS, the conditional distribution Law(YS|XS), +and the marginal distribution Law(XS). +Back to the image classification example, the target task may only contain images of items in +an office environment, the source task may have more image samples from a richer dataset, e.g., +ImageNet. Meanwhile, XS and YS may have different dimensions compared with XT and YT , since +the image resolution and the class number vary from task to task. Similar to the admissible set AT +in the target task, AS depends on the task description, and f∗ +S is usually a deep neural network with +parameters pretrained using the source data. +In transfer learning, the optimal model f∗ +S for the source task is also referred to as a pretrained +model. The essence of transfer learning is to utilize this pretrained model f∗ +S in the source task to +accomplish the optimization objective (1). We now define this procedure in three steps. +3 + +Step 1. Input transport. +Since XT is not necessarily contained by the source input space XS, +the first step is therefore to make an appropriate adaptation to the target input XT ∈ XT . In the +example of image classification, popular choices for input transport may include resizing, cropping, +rotation, and grayscale. We define this adaptation as an input transport mapping. +Definition 2.1 (Input transport mapping). A function +T X ∈ {finput|finput : XT → XS} +(5) +is called an input transport mapping with respect to the source and target task pair (S, T) if it takes +any data point in the target input space XT and maps it into the source input space XS. +With an input transport mapping T X, the first step of transfer learning can be represented as +follows. +XT ∋ XT +Step 1. Input transport by T X +�−−−−−−−−−−−−−−−−−−−→ T X(XT ) ∈ XS. +In a class of transfer learning called domain adaption, it is assumed that the difference between +the source input distribution Law(XS) and target input distribution Law(XT ) is the only factor to +motivate the transfer, while the labeling function of the source and target tasks stays the same. (See +also Section 2.3 for more details on domain adaptation). Therefore, once a proper input transport +mapping T X is found, transfer learning is accomplished. Definition 2.1 is thus consistent with +(Courty et al., 2017), in which domain adaption is formulated as an optimal transport from the +target input to the source input. +For most transfer learning problems, however, one needs both a transport mapping for the input +and a transport mapping for the output. For instance, the labeling function for different classes of +computer vision tasks, such as object detection, instance segmentation, and image classification, can +vary greatly and depend on the specific task. Hence, the following two more steps are required. +Step 2. Applying pretrained model. +After applying an input transport mapping T X to the +target input XT , the pretrained model f∗ +S will take the transported data T X(XT ) ∈ XS as an input. +That is, +XS ∋ T X(XT ) +Step 2. Apply f∗ +S +�−−−−−−−−−−→ (f∗ +S ◦ T X)(XT ) ∈ YS, +where (f∗ +S ◦ T X)(XT ) denotes the corresponding output of the pretrained model f∗ +S. Note here the +composed function f∗ +S ◦ T X ∈ {fint|fint : XT → YS}. +Step 3. Output transport. +After utilizing the pretrained model f∗ +S, the resulting model f∗ +S ◦ T X +may, however, still be inadequate for the target model: one may need to map the YS-valued output +into the target output space YT . Hence, it is necessary to define an output transport mapping. +Definition 2.2 (Output transport mapping). A function +T Y ∈ {foutput|foutput : XT × YS → YT } +(6) +is called an output transport mapping with respect to the source and target task pair (S, T) if, for an +optimal source model f∗ +S : XS → YS, the composed function T Y (·, f∗ +S(·)) ∈ AT . +Now, this third and the final step in transfer learning can be expressed as +XT × YS ∋ (XT , (f∗ +S ◦ T X)(XT )) +Step 3. Output transport by T Y +�−−−−−−−−−−−−−−−−−−−−→ T Y � +XT , (f∗ +S ◦ T X)(XT ) +� +∈ YT . +4 + +For the image classification task with transfer learning, the optimal source model usually consists +of the first few layers of the neural network for feature extraction, and the output transport mapping +is the subsequent prediction layers that map the features from the optimal source model to the +target output labels. See Section 2.3 for more details. +An output transport mapping can also be viewed as an operation to tailor the optimal source +model into a suitable target model. For instance, in (Xia et al., 2022), a large language model +is a collection of optimal pretrained transformer models and each model consists of a multi-head +self-attention layer and feed-forward layer. Thus, the output transport mapping is the structure +pruning with distillation operation applied to each optimal transformer model, where pruning reduces +the original transformer model to a simplified sub-model which is more suitable for the corresponding +down-stream tasks, and where distillation ensures the proper knowledge is passed from the source +model down to the target model. +Combining these three steps, transfer learning can be presented by the following diagram, +XS ∋ XS +Pretrained model f∗ +S from (3) +====================⇒ +f∗ +S(XS) ∈ YS +T X��� +���T Y +XT ∋ XT +Direct learning (1) +− − − − − − − → +f∗ +T ∈arg min +f∈AT +LT (fT ) +f∗ +T (XT ) ∈ YS +(7) +2.2 +Optimization Formulation and Feasibility of Transfer Learning +In summary, transfer learning aims to find an appropriate pair of input and output transport +mappings T X and T Y , where the input transport mapping T X translates the target input XT +back to the source input space XS in order to utilize the optimal source model f∗ +S, and the output +transport mapping T Y transforms a YS-valued model to a YT -valued model. This is in contrast to +the direct learning, where the optimal model f∗ +T is derived by solving the optimization problem in +the target task (1). In other words, transfer learning is the following optimization problem. +Definition 2.3 (Transfer learning). The three-step transfer learning procedure presented in (7) is to +solve the optimization problem +min +T X∈TX,T Y ∈TY LT +� +T Y (·, (f∗ +S ◦ T X)(·)) +� += E +� +LT +� +YT , T Y (XT , (f∗ +S ◦ T X)(XT )) +�� +. +(8) +Here, TX and TY are proper sets of transport mappings such that +� +T Y (·, (f∗ +S ◦ T X)(·))|T X ∈ TX, T Y ∈ TY � +⊂ AT . +In particular, when XS = XT (resp. YS = YT ), the identity mapping idX(x) = x (resp. idY (x, y) = y) +is included in TX (resp. TY ). +This optimization reformulation of the three-step transfer learning procedure provides potentially +a unified framework to analyze the impact and implications of various transfer learning techniques, +including resizing, cropping, pruning, and distillation. Moreover, it enables us to analyze the +feasibility of transfer learning, which we establish in terms of the following well-definedness of the +corresponding optimization problem (8). +Theorem 2.1. Under suitable choices of loss functions for LT and appropriate compactness as- +sumptions, there exists optimal solutions for optimization problem (8). +5 + +Detailed assumptions and proof for Theorem 2.1 is deferred to Appendix A.1. +The procedure of solving this optimization problem is often referred to as fine-tuning in the +literature of transfer learning. It is to choose some initial transport mappings T X +0 ∈ TX +0 ⊂ TX and +T Y +0 ∈ TY +0 ⊂ TY to derive an intermediate model fST ∈ AT with +fST (x) = T Y +0 (x, (f∗ +S ◦ T X +0 )(x)), +∀x ∈ XT , +(9) +with the set of possible intermediate models denoted as +I = +� +T Y +0 (·, (f∗ +S ◦ T X +0 )(·)) +��T X +0 ∈ TX +0 , T Y +0 ∈ TY +0 +� +. +(10) +This fine-tuning procedure allows for computationally efficient evaluation of transferability in +terms of transfer risk, to be introduced in Section 3.1. +2.3 +Examples. +Image classification. +Consider a transfer learning task in image classification using the Office-31 +(Saenko et al., 2010) benchmark dataset, which consists of images from three domains: Amazon (A), +Webcam (W) and DSLR (D). In total, the dataset contains 4110 images of 31 categories of objects +typically found in an office environment. Samples from the Office-31 dataset are shown in Figure 1. +Figure 1: Samples from Office-31. +The neural network architecture for the image classification task is shown in Figure 2. +It +sequentially consists of: 1) a data-preprocessing module which resizes a input image to 3 × 244 × 244 +dimension; 2) ResNet50 as a feature extractor whose output is a 2048-dimensional feature vector; +and 3) a two-layer neural network which maps a 2048-dimensional feature vector to a 31-dimensional +probability vector. +6 + +Amazon +DSLR +WebcamFigure 2: Neural network architecture for Office-31. +In this example, the source task can be chosen from any of three domains (A, D, or W), with +XS = R3×244×244 being the space of resized image samples from the source domain, and +YS = ∆31 := {p ∈ R31 : +31 +� +1 +pi = 1, pi ≥ 0, ∀1 ≤ i ≤ 31} +being the space of image class labels. Similarly, for any target task (A, D, or W), +XT = XS = R3×244×244 +is the space of resized image samples from the target domain, and YT = YS = ∆31. For both the +source and the target tasks, the loss function LS = LT is chosen to be the cross entropy between the +actual label and the predicted label. +As introduced in Figure 2, the set of source models are given by +AS = {fNN ◦ fRes : XS → YS|fNN ∈ NN31 +2048, fRes ∈ Res2048 +3×244×244}. +Here Res2048 +3×244×244 denotes all ResNet50 architectures with 3×244×244-dimensional input and 2048- +dimensional output, and NN31 +2048 denotes all two-layer neural networks which map a 2048-dimensional +feature vector to a 31-dimensional probability vector in YS. The source model f∗ +Res,S and f∗ +NN,S is +obtained by solving the source task optimization (3). +To transfer the source model to the target task, the pretrained ResNet50 model f∗ +Res,S will be +fixed, while the last two-layer classifier fNN ∈ NN31 +2048 will be fine-tuned using part of the data from +the target domain (XT , YT ). The input transport set TX in this example is a singleton set whose +element is the identity mapping on R3×244×244. Meanwhile, the set of output transport mappings is +given by +TY = {fNN ◦ f∗ +Res,S : XT → YT |fNN ∈ NN31 +2048}. +(11) +The transfer learning task is formulated as +min +T Y ∈TY E +� +LT +� +YT , T Y (XT ) +�� +. +7 + +Resize: +Feature Extractor: +Source +Fully Connected +Source +3x244x244 +ResNet50 +Feature +Layer +Label +Source Task +Finetuning +Target Task +Target Feature +Target LabelNote the formulation is slightly simpler than (8) because in this particular example, the output +transport in TY takes inputs from XT instead of XT × YS. Furthermore, in this example, there is no +additional constraint on intermediate models defined in (9). Therefore, the set I defined in (10) is +equivalent to TY in (11). +Domain adaption. +This class of transfer learning problem considers the case where the output +variable for the source and target tasks coincides, i.e., YS = YT = Y ∈ Y, and there exists some +one-to-one input transport T X such that T X(XT ) = XS almost surely (Courty et al., 2017). Here +we define the family of admissible (initial) output transport mappings as TY +0 = TY = {idY}, +where idY denotes the identity mapping on Y; and define the family of admissible (initial) input +transport mappings as TX +0 = TX = {T X : XT → XS | T is one-to-one}. Then I = {f∗ +S ◦ T|T ∈ TX +0 }. +When the loss functions for the source and the target tasks are also in the same form such that +LS = LT = L : Y × Y → R, it can be shown that the optimal source model and optimal target +model satisfy the relation f∗ +T = f∗ +S ◦ T X, where +f∗ +· := arg min +f:X·→Y +E[L(Y, f(X·))]. +From the transfer learning perspective, T X is also the optimal solution to the optimization problem +(8). In particular, the transfer learning model f∗ +T = f∗ +S ◦ T X is equivalent to the optimal model from +the direct learning, while solving the transfer learning problem (8) may require much less data. +3 +Transfer Risk and Transferability of Transfer Learning +Given the mathematical framework and after the feasibility analysis of transfer learning, we will +now propose a novel notion of transfer risk, to analyze the effectiveness and the appropriateness of +transfer learning over the set of all intermediate models I given by (10). +3.1 +Transfer Risk +The idea is to re-interpret the transfer learning framework (7) in a sequential manner: the mapping +T X first transports Law(XT ) to some probability distribution Law(T X(XT )); then, applying the +pretrained model f∗ +S for the optimization problem (3) yields the distribution ˜PS = Law(f∗ +S(T X(XT ))). +Finally, an output transport mapping T Y , together with the target input XT , transports the +distribution ˜PS to PT . That is, the transfer learning scheme can be viewed as the composition of the +following two steps. +1. (Psuedo) Domain adaption, which can also be seen as optimal transport from Law(XT ) to +Law(XS). +2. Optimal transport from ˜PS = Law(f∗ +S(T X(XT ))) to PT over T Y . +In other words, in parallel to the three-step procedure in transfer learning, there are two major +sources of transfer risk for a fixed intermediate model fST ∈ I: the risk that measures the mismatch +between the output distributions of the intermediate model fST and the optimal target model f∗ +T , +and the risk reflecting the difference between the transported target input and the source input. +Let us first define the risk associated the output transport mapping. +Definition 3.1 (Output transport risk). Let EO : AT → R be a real-valued function on the set of +target models. For any fST ∈ I ⊂ AT , EO(fST ) is called an output transport risk of intermediate +model fST if it satisfies +8 + +1. EO(fST ) ≥ 0, i.e., transfer learning always incurs a non-negative effort; +2. EO(fST ) = 0 if and only if PT = PST , where PT := Law(fT (XT )) and PST := Law(fST (XT )). +That is, the output transport risk vanishes when the intermediate model fST completely recovers +the distribution of the optimal target task. +Clearly, the smaller this output risk, the more effective the transfer scheme with the intermediate +model fST . +We next define the risk associated with the input transfer. +Definition 3.2 (Input transfer risk). Let EI : TX → R be a real-valued function on the set of input +transport mappings. Given an import transport mapping T X +0 ∈ TX +0 ⊂ TX, EI(T X +0 ) is called an input +transport risk if it satisfies +1. EI(T X +0 ) ≥ 0, i.e., transfer learning always incurs a non-negative effort; +2. EI(T X +0 ) = 0 if and only if T X +0 #Law(XT ) = Law(XS). +The smaller this input risk, the higher the similarity between the transported target input +T X +0 (XT ) and the source input XS. +Note that these definitions of risks involve the sets of initial transport mappings TX +0 and TY +0 , +instead of the sets of all possible transport mappings TX and TY . These reduced sets allow for +efficient evaluation of transfer risk prior to starting the full-scale transfer learning. +Both the input transfer risk and the output transfer risk are functions characterizing the divergence +between probability distributions, and their exact forms can be task dependent. Nevertheless, there +is a key difference between these two forms of risks: in the output transport risk, PT , the output +distribution of the optimal target model, is unknown, and no prior knowledge about f∗ +T is assumed. +Therefore, analyzing the output transport risk is decisively more complicated. See more detailed +discussions in Section 3.3. +We are now ready to propose the notion of transfer risk by considering all intermediate models +in I, in order to measure the effectiveness of a transfer learning framework (8). +Definition 3.3 (Transfer risk). For a transfer learning procedure characterized by the 6-tuple +(S, T, TX, TX +0 , TY , TY +0 ) in (8), the transfer risk of the transfer learning framework (8) from source +task S to target task T is defined as +C(S, T) = +inf +fST ∈I C(S, T|fST ). +(12) +Here, for a given fST = T Y +0 (·, (f∗ +S ◦ T X +0 )(·)) ∈ I, C(S, T|fST ) is called model-specific transfer risk +such that C(S, T|fST ) ≥ 0 with the following properties: +1. Let C : R × R → R with C(0, 0) = 0. C(S, T|fST ) = C(EO(fST ), EI(T X +0 )) is non-decreasing in +EO(fST ) under any fixed EI(T X +0 ) and non-decreasing in EI(T X +0 ) under any fixed E(fST ); +2. C(S, T|fST ) is Lipschitz in the sense that for any other transfer problem characterized by +( ¯S, ¯T, ¯TX, ¯TX +0 , ¯TY , ¯TY +0 ) and one of its intermediate models ¯fST = ¯T Y +0 (·, ( ¯f∗ +S ◦ ¯T X +0 )(·)) ∈ ¯I, there +exists a constant L > 0 such that +|C(S, T|fST ) − C( ¯S, ¯T| ¯fST )| ≤ L(|EO(fST ) − EO( ¯fST )| ++ |EI(T X +0 ) − EI( ¯T X +0 )|). +9 + +The expression of this Lipschitz property in Definition 3.3 is to emphasize the dependence of +transfer risk on a given transfer learning problem. This Lipschitz property is satisfied when the +function C in Definition 3.3 is Lipschitz continuous. +One simple example of the model-specific transfer risk is +Cλ(S, T|fST ) = EO(fST ) + λEI(T X +0 ), +(13) +where λ > 0 is a pre-specified parameter modulating the weight of the input transport in the transfer +learning problem (7). +Transfer risk in Definition 3.3 unifies the analysis of the risk from both the input and the output +transport mappings. It allows for studying the trade-off between them. Moreover, two of its key +components, the input and the output transfer risks in Definitions 3.2 and 3.1 generalize earlier works +on transferability. For instance, the H-score proposed in (Bao et al., 2019) addresses transferability of +a particular classification setting and can be incorporated into the output transfer risk in Definition +3.1. Earlier works on the relation between source and target inputs such as (Saenko et al., 2010; +Ganin et al., 2016; Long et al., 2014) correspond to the special case in Definition 3.2 with T X +0 +being +the identity mapping. +Furthermore, one can establish the following properties of transfer risk: a) there is zero transfer +risk if the source and the target tasks are identical; and b) transfer risk is continuous in the input +distribution and robust with respect to the pretrained model. (See the exact mathematical statement +and analysis of these properties in Appendix A.2). The continuity of the transfer risk in terms of +the changes in the input and the pretrained model is useful to exclude a priori inappropriate source +tasks when compared against existing viable source tasks. +3.2 +Examples +We now revisit some examples in Section 2.3 and their associated transfer risks based on Definition +3.3. In particular, we will illustrate how the two key components of the transfer risk, namely, the +input transport risk EI and the output transport risk EO, are embedded in transfer learning for a +given intermediate model fST . +Transfer risk in domain adaption. +Recall the domain adaptation problem in Section 2.3, and +consider the case where the transfer risk is independent of the output transport risk, i.e., the input +risk EI(T X +0 ) completely determine the transfer risk: +C(S, T|fST ) = EI(T X +0 ). +In this case, there exists a one-to-one input mapping T X ∈ TX +0 such that T X(XT ) = XS almost surely, +implying T X#Law(XT ) = Law(XS) and consequently C(S, T) = C(S, T|f∗ +S ◦ T X) = EI(T X) = 0. +Therefore, vanishing input transport risk is a necessary condition for the domain adaptation framework +to hold. Thus, the input transport risk may be adopted to check the viability of domain adaptation +on certain tasks. +Transfer risk in image classification. +Recall the image classification problem introduced in +Section 2.3. Fix a source task S and a target task T. Since the input transport set TX in this problem +is a singleton set, the input transport risk EI is a constant depending on Law(XS) and Law(XT ), +with the output transport risk denoted as EO(fST ) for any fST ∈ I in (11). By Definition 3.3, the +model-specific transfer risk C(S, T|fST ) = C(EI, EO(fST )) for some appropriate function C : R2 → R +satisfying conditions stated in Definition 3.3. In particular, since the function C is non-decreasing +10 + +with respect to EO(fST ), minimizing C(S, T|fST ) over fST ∈ I is equivalent to minimizing EO(fST ) +over fST ∈ I: +arg min +fST ∈I +C(S, T|fST ) = arg min +fST ∈I +EO(fST ). +And consequently, +C(S, T) = min +fST ∈I C(S, T|fST ) = C(EI, min +fST ∈I EO(fST )). +3.3 +Transfer Risk and Choices of Divergence Functions +Clearly, different learning tasks may require different choices of divergence functions for assessment +of transfer risk. In this section, we present two forms of transfer risks based on two divergence +functions, and analyze their properties and relations. +KL-based output transport risk. +For learning tasks such as the classification problem, one +may use cross-entropy as the loss function. +Specifically, let PT = ˜PT + P0 be its unique Lebesgue decomposition, i.e., for any measurable +set B ⊂ YT , there exists some function hST : YT → R+ such that ˜PT (B) = +� +B hST dPST , with P0 +singular with respect to PST . Then the KL-based output risk can be defined as +EO +KL(fST ) := DKL(˜PT ∥PST ) + H(P0), +where H(P0) is the entropy function of P0. +Proposition 3.1. For a classification problem over K ∈ N classes with cross entropy as the training +loss, for any fST ∈ I, +K +� +i=1 +log pST (i) ≤ H(PT , PST ) − H(Law(YT ), PST ) ≤ − +K +� +i=1 +log pST (i), +where pST denotes the probability mass function for PST . +Note that H(Law(YT ), PST ) is indeed the cross-entropy loss for the classifier fST . Therefore, in +actual training, one may use H(Law(YT ), PST ) ± �K +i=1 log pST (i) to replace EO +KL(fST ). +Wasserstein-based output transport risk. +For learning problems such as GANs or supervised +learning with domain adaption, Wasserstein and related distances are popular choices to measure the +distance between the generative distribution and the target distribution. Therefore, a Wassertein- +based output risk is a natural choice related to such learning targets. +More specifically, for p ≥ 1, let Pp(YT ) be the set of probability measures over YT such that +� +RdO,T ∥x∥p +YT dµ(x) < ∞, +∀µ ∈ Pp(YT ). +The Wasserstein-based output risk is defined as +EO +W (fST ) := Wp(PST , PT )p := +inf +γ∈Π(PST ,PT ) +� +RdO,T ×RdO,T ∥x − y∥p +YT dγ(dx, dy), +(14) +for some suitable choice of p ≥ 1, where Π(PST , PT ) denotes the set of couplings of probability +measures PST and PT . +Analogy to Proposition 3.1 is the following property for EO +W (fST ), based on the triangle inequality +of the Wasserstein distance. +11 + +Proposition 3.2. The Wasserstein-based output risk EO +W in (14) is upper bounded in the following +sense: +EO +W (fST ) ≤ 2p−1[Wp(PST , Law(YT ))p + Wp(PT , Law(YT ))p]. +Now, consider any intermediate model fST , then Talagrand’s inequality (Talagrand, 1996) gives +EI +W (T X +0 ) ≤ 2EI +KL(T Y +0 ), EO +W (fST ) ≤ 2EO +KL(fST ). +In particular, the linear transfer risk defined in (13) satisfies +Cλ +W (S, T|fST ) := EO +W (fST ) + λ · EI +W (T X +0 ) ≤ 2Cλ +KL(S, T|fST ) := 2(EO +KL(fST ) + λ · EI +KL(T X +0 )) +(15) +Such a relation between KL- and Wasserstein-based linear transfer risks (15) gives the following +proposition. +Proposition 3.3. Consider transfer risk in linear form as in (15). Suppose YT is a finite-dimensional +Euclidean space and PT ≪ PST . Then for a given transfer learning problem (S, T, TX, TY , T0 +X, T0 +Y ), +CW (S, T) ≤ 2CKL(S, T). +3.4 +Transfer Risk and Regret +We will establish the connection between the transfer risk (12) and the transfer learning performance +through a linear regression example. +Consider a source task S and a target task T with the same input space XS = XT = Rd and +the same input space YS = YT = R. Both source and target data satisfy two (d + 1)-dimensional +Gaussian distributions: (X·, Y·) ∼ N(µ·, Σ·) with +µ· = +�µ·,X +µ·,Y +� +, +Σ· = +� Σ·,X +Σ·,XY +Σ·,Y X +Σ·,Y +� +, +(16) +where µ·,Y and Σ·,Y ∈ R, µ·,X and Σ·,XY ∈ Rd, Σ·,Y X = Σ⊤ +·,XY , and Σ·,X ∈ Rd×d. Define the sets of +admissible source and target models AS = AT = {f : Rd → R}. For any f ∈ AS = AT , define the +loss function as +LS(f) = E∥YS − f(XS)∥2 +2, LT (f) = E∥YT − f(XT )∥2 +2. +(17) +Under such a setting, the optimal source and target models are obtained by direct computations: +f∗ +· (x) = w⊤ +· x + b· with +w· = Σ−1 +·,XΣ·,XY , +b· = µ·,Y − Σ·,Y XΣ−1 +·,Xµ·,X. +(18) +Transfer learning. +Take the above linear regression example, and consider a simple setting where +the input (resp. output) transport set TX (resp. TY ) is a singleton set only containing the identical +mapping on Rd (resp. R). Then, the transfer learning scheme (8) is equivalent to directly applying +the optimal source model f∗ +S to the target task. Consequently, the intermediate model set I in (10) +is also a singleton set with I = {f∗ +S}. +Now, define the transfer risk in this linear regression problem as the Wassersteinn-based output +transport risk as in (14): +CW (S, T) = CW (S, T|f∗ +S) = EO +W (f∗ +S). +(19) +12 + +Regret. +Next, define the notion of regret as the gap between the transfer learning and the direct +learning: +R(S, T) := LT (f∗ +S) − LT (f∗ +T ) +(20) +Then, the following proposition shows that the transfer risk serves as a lower bound of the regret. +Proposition 3.4. For transfer learning in linear regression with Gaussian data, the regret with +respect to the chosen intermediate model R(S, T) in (20) is lower bounded by the Wasserstein-based +transfer risk in (19), +CW (S, T) ≤ R(S, T). +Proposition 3.4 suggests that in evaluating the transfer learning scheme (8), transfer risk provides a +proper initial indication of its effectiveness, especially for eliminating unsuitable candidate pretrained +models or source tasks if the transfer risk is large. The proof of Proposition 3.4, together with +detailed analysis of transfer risk and regret with Gaussian data, is in Appendix B. +4 +Numerical Experiments with Office-31 +In this section, we will demonstrate the correlation between the performance of the transfer learning +scheme (8) and the transfer risk (3.3), through numerical experimentation using the Office-31 dataset +for image classification. +4.1 +Experiment Set-up +Recall the neural network architecture for the experiment introduced in Section 2.3. For each pair of +the source and the target tasks, the source model is first trained using the source data, and then +the fully connected layer of the pretrained model is fine tuned using half of the target data. The +performance of the model is measured by the classification accuracy using the remaining of the +target data. +Transfer risk. +Now let us define the explicit form of transfer risk for this example. Fix a source- +target pair (S, T). Recall that the input transport risk EI is a constant since the input transport set +TX is a singleton set. More specifically, we define the input transport risk as +EI := W1(Law(XS), Law(XT )), +(21) +which is the Wasserstein-1 distance between the empirical distribution of (resized) source images +Law(XS) and the empirical distribution of (resized) target images Law(XT ). Meanwhile, for any +fST ∈ I (11), we define the output transport risk as EO(fST ) = W1(PST , PT ). Furthermore, as +discussed in Section 3.2, the transfer risk is given by +C(S, T) = C(EI, min +fST ∈I EO(fST )), +(22) +for some function C : R2 → R satisfying the regularity conditions in Definition 3.3. Note the the +optimal target distribution PT in the definition of EO(fST ) is unknown a priori. Thus, as suggested +by Proposition 3.2, we approximate EO(fST ) by W1(PST , Law(YT )). Denote the approximated +output transfer risk as +�EO = min +fST ∈I W1(PST , Law(YT )). +(23) +Finding �EO (23) is an optimization problem over a neural network function class fST ∈ I (11), which +is solved by gradient descent in the numerical experiment. Finally, the (approximated) transfer risk +is obtained by plugging �EO into (22). +13 + +4.2 +Numerical Result +Three different domains in Office-31 (A, D, and W) lead to 3 × 2 = 6 source-target pairs in total. +The accuracy, the input transport risk EI (21), and the output transport risk �EO (23) for each pair +of source and target tasks are reported in the first three rows of Table 1. Here the input transport +risk is rescaled by a constant factor to achieve the same scale as the other metrics. +In order to compute the transfer risk C in (22) given EI in (21) and �EO in (23), an appropriate +form of function C in (22) need to be determined. In this experiment, we search C from the class of +second order polynomials, so as to maximize the (absolute value of) correlation between the transfer +learning accuracy and the transfer risk. In particular, we define the risk in the following form: +C(S, T) = 0.31 · EI + 0.92 · +� +�EO�2 +. +(24) +Transfer risks for all source-target pair are reported in the last row of Table 1. +Metric\Task +A-W +A-D +W-A +W-D +D-A +D-W +Accuracy +80.9% +83.1% +66.9% +94.5% +66.6% +87.8% +Input Risk +0.181 +0.263 +0.181 +0.148 +0.263 +0.148 +Output Risk +0.428 +0.380 +0.545 +0.084 +0.543 +0.412 +Transfer Risk +0.224 +0.214 +0.330 +0.052 +0.353 +0.201 +Table 1: Accuracy and transfer risk. +Accuracy v.s. transfer risk. +Figure 3 demonstrates a significant negative correlation between +the transfer learning accuracy and the transfer risk: the higher the risk, the lower the transfer +learning accuracy. For example, it can be observed from Figure 3 that transfer learning between +DSLR and Webcam (D-W or W-D) results in low risk and high accuracy; while transfer learning from +those domains to Amazon (D-A or W-A) is risky and suffers from low accuracy. Those numerical +findings demonstrate the potential of transfer risk as an informative metric for the effectiveness of +transfer learning task. +14 + +Figure 3: Accuracy and transfer risk. +Computational benefit of transfer risk. +In this numerical experiment on the Office-31 dataset, +assessing transfer risk is computationally efficient and guaranteed by the early-stopping trick in +deep learning: for each source-target pair, the optimization problem (23) is solved by running the +gradient descent for a small and fixed number (∼10) of epochs, while the transfer learning problem +is solved until the accuracy converges, which may take up to 100 epochs. This early stopping trick is +essentially equivalent to shrinking the search space of the output mapping from TY in (11) to some +smaller class of neural networks TY +0 ⊂ TY . +Indeed, as emphasized in Section 3.1, computing transfer risk (12) is to solve an optimization +problem over the sets TX +0 and TY +0 , which can be much smaller than the function classes TX and TY +involved in the transfer learning problem (8). This reduction of the function classes demonstrates +the potential and benefit of adopting transfer risk for computational efficiency: one can first perform +the much easier computing task of the transfer risk, and then assess whether or not to resort to the +full-scale and more computationally intense form of transfer learning. +5 +Conclusion +This paper establishes a mathematical framework for transfer learning, and addresses issues of +feasibility and transferability through rigorous and comprehensive mathematical analysis. A novel +concept of transfer risk is introduced, which not only generalizes existing notions for transferability +but also provides a unified framework for future studies on the impact and implications of various +transfer learning techniques, including resizing, cropping, pruning, and distillation. +15 + +Tranfer Accuracy V.S. Transfer Risk +A-W +A-D +W-A +Accuracy of transfer learning +W-D +0.9 +D-A +D-W +0.8 +0.7 +0.6 +0 +0.1 +0.2 +0.3 +0.4 +0.5 +Transfer risk = 0.31 * input risk + 0.92 * (output risk)2References +Bao, Y., Li, Y., Huang, S.-L., Zhang, L., Zheng, L., Zamir, A., and Guibas, L. (2019). An information- +theoretic approach to transferability in task transfer learning. +In 2019 IEEE International +Conference on Image Processing, pages 2309–2313. IEEE. +Ben-David, S., Blitzer, J., Crammer, K., Kulesza, A., Pereira, F., and Vaughan, J. W. (2010). A +theory of learning from different domains. Machine learning, 79(1):151–175. +Blitzer, J., Crammer, K., Kulesza, A., Pereira, F., and Wortman, J. (2007). Learning bounds for +domain adaptation. In Proceedings of the 20th International Conference on Neural Information +Processing Systems, volume 20, page 129–136. Curran Associates Inc. +Bu, Y., Zou, S., and Veeravalli, V. V. (2020). Tightening mutual information-based bounds on +generalization error. IEEE Journal on Selected Areas in Information Theory, 1(1):121–130. +Cook, D., Feuz, K. D., and Krishnan, N. C. (2013). Transfer learning for activity recognition: A +survey. Knowledge and Information Systems, 36:537–556. +Courty, N., Flamary, R., Tuia, D., and Rakotomamonjy, A. (2017). Optimal transport for domain +adaptation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(9):1853–1865. +Deng, J., Dong, W., Socher, R., Li, L.-J., Li, K., and Fei-Fei, L. (2009). ImageNet: A large-scale +hierarchical image database. In Proceedings of the IEEE Conference on Computer Vision and +Pattern Recognition, pages 248–255. IEEE. +Deng, J., Zhang, Z., Marchi, E., and Schuller, B. (2013). Sparse autoencoder-based feature transfer +learning for speech emotion recognition. In Proceedings of the 2013 Humaine Association Conference +on Affective Computing and Intelligent Interaction, pages 511–516. IEEE. +Devlin, J., Chang, M.-W., Lee, K., and Toutanova, K. (2019). +BERT: Pre-training of deep +bidirectional transformers for language understanding. In Proceedings of the 2019 Conference of +the North American Chapter of the Association for Computational Linguistics: Human Language +Technologies, volume 1, pages 4171–4186. Association for Computational Linguistics. +Ganin, Y. and Lempitsky, V. (2015). Unsupervised domain adaptation by backpropagation. In +Proceedings of the 32nd International Conference on Machine Learning, volume 37, pages 1180–1189. +PMLR. +Ganin, Y., Ustinova, E., Ajakan, H., Germain, P., Larochelle, H., Laviolette, F., Marchand, M., +and Lempitsky, V. (2016). Domain-adversarial training of neural networks. Journal of Machine +Learning Research, 17(59):1–35. +Harremoës, P. and Vajda, I. (2011). +On pairs of f-divergences and their joint range. +IEEE +Transactions on Information Theory, 57(6):3230–3235. +Hwang, T. and Kuang, R. (2010). A heterogeneous label propagation algorithm for disease gene +discovery. In Proceedings of the 2010 SIAM International Conference on Data Mining, pages +583–594. SIAM. +Jiang, J. and Zhai, C. (2007). Instance weighting for domain adaptation in nlp. In Proceedings of +the 45th Annual Meeting of the Association of Computational Linguistics, pages 264–271. +16 + +Kim, H. E., Cosa-Linan, A., Santhanam, N., Jannesari, M., Maros, M. E., and Ganslandt, T. (2022). +Transfer learning for medical image classification: A literature review. BMC Medical Imaging, +22(1):69. +Leal, L., Laurière, M., and Lehalle, C.-A. (2020). Learning a functional control for high-frequency +finance. arXiv preprint arXiv:2006.09611. +Lebichot, B., Le Borgne, Y.-A., He-Guelton, L., Oblé, F., and Bontempi, G. (2020). Deep-learning +domain adaptation techniques for credit cards fraud detection. In Recent Advances in Big Data +and Deep Learning: Proceedings of the 2019 INNS Big Data and Deep Learning Conference, pages +78–88. Springer. +Liu, R., Shi, Y., Ji, C., and Jia, M. (2019). A survey of sentiment analysis based on transfer learning. +IEEE Access, 7:85401–85412. +Long, M., Cao, Y., Wang, J., and Jordan, M. (2015). Learning transferable features with deep +adaptation networks. In Proceedings of the 32nd International Conference on Machine Learning, +volume 37, pages 97–105. PMLR. +Long, M., Wang, J., Ding, G., Sun, J., and Yu, P. S. (2014). Transfer joint matching for unsupervised +domain adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern +Recognition, pages 1410–1417. IEEE. +Pan, S. J. and Yang, Q. (2010). A survey on transfer learning. IEEE Transactions on Knowledge +and Data Engineering, 22(10):1345–1359. +Pan, W., Xiang, E., Liu, N., and Yang, Q. (2010). Transfer learning in collaborative filtering +for sparsity reduction. In Proceedings of the 24th AAAI Conference on Artificial Intelligence, +volume 24, pages 230–235. AAAI Press. +Rosenbaum, M. and Zhang, J. (2021). Deep calibration of the quadratic rough heston model. arXiv +preprint arXiv:2107.01611. +Ruder, S., Peters, M. E., Swayamdipta, S., and Wolf, T. (2019). Transfer learning in natural +language processing. In Proceedings of the 2019 Conference of the North American Chapter of the +Association for Computational Linguistics: Tutorials, pages 15–18. Association for Computational +Linguistics. +Saenko, K., Kulis, B., Fritz, M., and Darrell, T. (2010). Adapting visual category models to new +domains. In Proceedings of the 11th European Conference on Computer Vision, pages 213–226. +Springer. +Sung, Y.-L., Cho, J., and Bansal, M. (2022). Vl-adapter: Parameter-efficient transfer learning +for vision-and-language tasks. In Proceedings of the IEEE Conference on Computer Vision and +Pattern Recognition, pages 5227–5237. IEEE. +Talagrand, M. (1996). Transportation cost for gaussian and other product measures. Geometric & +Functional Analysis GAFA, 6(3):587–600. +Tan, C., Sun, F., Kong, T., Zhang, W., Yang, C., and Liu, C. (2018). A survey on deep transfer +learning. In International Conference on Artificial Neural Networks, pages 270–279. Springer. +17 + +Tan, Y., Li, Y., and Huang, S.-L. (2021). OTCE: A transferability metric for cross-domain cross- +task representations. In Proceedings of the IEEE Conference on Computer Vision and Pattern +Recognition, pages 15779–15788. IEEE. +Tong, X., Xu, X., Huang, S.-L., and Zheng, L. (2021). A mathematical framework for quantifying +transferability in multi-source transfer learning. In Proceedings of the 35th International Conference +on Neural Information Processing Systems, volume 34, pages 26103–26116. Curran Associates, Inc. +Tzeng, E., Hoffman, J., Saenko, K., and Darrell, T. (2017). Adversarial discriminative domain +adaptation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, +pages 7167–7176. IEEE. +Wang, G., Kikuchi, Y., Yi, J., Zou, Q., Zhou, R., and Guo, X. (2022). Transfer learning for retinal +vascular disease detection: A pilot study with diabetic retinopathy and retinopathy of prematurity. +arXiv preprint arXiv:2201.01250. +Wang, J., Chen, Y., Hu, L., Peng, X., and Philip, S. Y. (2018). Stratified transfer learning for +cross-domain activity recognition. In Proceedings of the 2013 IEEE International Conference on +Pervasive Computing and Communications, pages 1–10. IEEE. +Wang, M. and Deng, W. (2018). Deep visual domain adaptation: A survey. Neurocomputing, +312:135–153. +Xia, M., Zhong, Z., and Chen, D. (2022). Structured pruning learns compact and accurate models. In +Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics, volume 1, +pages 1513–1528. +Yuan, F., Yao, L., and Benatallah, B. (2019). Darec: deep domain adaptation for cross-domain +recommendation via transferring rating patterns. In Proceedings of the 28th International Joint +Conference on Artificial Intelligence, pages 4227–4233. AAAI Press. +Zeng, M., Li, M., Fei, Z., Yu, Y., Pan, Y., and Wang, J. (2019). Automatic icd-9 coding via deep +transfer learning. Neurocomputing, 324:43–50. +Zhao, H., Des Combes, R. T., Zhang, K., and Gordon, G. (2019). On learning invariant representations +for domain adaptation. In Proceedings of the 36th International Conference on Machine Learning, +volume 97, pages 7523–7532. PMLR. +Zhuang, F., Qi, Z., Duan, K., Xi, D., Zhu, Y., Zhu, H., Xiong, H., and He, Q. (2020). A comprehensive +survey on transfer learning. Proceedings of the IEEE, 109(1):43–76. +18 + +Appendix +A +Mathematical Proofs +A.1 +Proof of Theorem 2.1 +We will show that the optimization problem (8) is well-defined in the sense that an optimal pair +of transport mappings (T X,∗, T Y,∗) for (8) is obtainable, under certain regularity conditions. More +specifically, we will focus on the following type of loss function LT . +Definition A.1 (Proper loss function). Let (X, Y ) be a pair of XT × YT -valued random variables +with Law(XT , YT ) ∈ P(XT × YT ). A loss functional LT over AT is said to be proper with respect to +(X, Y ) if there exist a corresponding function LT : YT × YT → R bounded from below such that for +any f ∈ AT , +LT (f) = E[LT (Y, f(X))] = E[E[LT (Y, f(X))|X]]; +moreover, the function ˜LT : YT → R given by +˜LT (y) = E[LT (Y, Y ′)|Y ′ = y], +∀y ∈ YT , +is continuous. +Examples of proper loss functions include mean squared error and KL-divergence and more +generally the Bregman divergence, assuming that the first and second moments of Y conditioned on +Y ′ = y is continuous with respect to y. +Without loss of generality, we shall in this section assume the input transport set TX contains all +functions from XT to XS. We then specify the following assumptions for the well-definedness of (8). +Assumption A.1. Assume the following regularity conditions hold. +1. LT is a proper loss functional with respect to (XT , YT ); +2. the image f∗ +S(XS) is compact in (YS, ∥ · ∥YS); +3. the set TY is such that the following set of functions +˜TY = { ˜T Y : XT → YT | ∃T Y ∈ TY s.t. ˜T Y (x) = +inf +y∈f∗ +S(XS) +˜LT (T Y (x, y)), +∀x ∈ XT } +is compact in ({f|f : XT → YT }, ∥·∥∞), where for any f : XT → YT , ∥f∥∞ := supx∈XT ∥f(x)∥YT . +The proper choice of loss functions for LT is fairly general and includes the mean squared +error, the KL-divergence, and more generally the Bregman divergence; the compactness assump- +tions can be fairly flexible as long as the target optimal model f∗ +T can be written as f∗ +T (x) = +T Y (x, f∗ +S(T X(x))), +∀x ∈ XT . This compactness condition can be implemented by choosing a +particular family of activation functions or imposing boundaries restrictions to weights and biases +when constructing machine learning models. +Now we are ready to prove Theorem 2.1 under Assumption A.1. +Proof of Theorem 2.1. Since LT is proper, there exists a function LT : YT × YT → R such that +inf +(y,y′)∈YT ×YT +LT (y, y′) > −∞, +19 + +and +LT (T Y (·, (f∗ +S ◦ T X)(·))) = E[LT (YT , T Y (XT , (f∗ +S ◦ T X)(XT )))], +∀T X ∈ TX, T X ∈ TX. +Therefore, for the function ˜LT (·) = E[LT (Y, Y ′)|Y ′ = ·], there exists m ∈ R such that ˜LT (y) ≥ m +for any y ∈ YT . +Now fix any T Y ∈ TY . The continuity of ˜LT and the continuity of T Y (x, ·) for each x ∈ XT +guarantee the continuity of ˜LT (T y(x, ·)). Together with the compactness of f∗ +S(XS), we have that +for any x ∈ XT , +Mx = arg min +y∈f∗ +S(XS) +˜LT (T Y (x, y)) ̸= ∅. +Therefore, for any T Y , one can construct ˜T X ∈ TX such that ˜T X(x) ∈ Mx +T Y for any x ∈ XT and +min +T X∈TX LT (T Y (·, (f∗ +S ◦ T X)(·))) = E[˜LT ( ˜T Y (XT ))] =: ˜LT ( ˜T Y ). +The continuity of the new loss functional ˜LT comes from the continuity of the function ˜L, and the +particular choice of the function space ({f|f : XT → YT }, ∥ · ∥∞), where {f|f : XT → YT } contains +all functions from XT to YT . Since ˜TY is compact in ({f|f : XT → YT }, ∥ · ∥∞), the minimum over +˜TY is attained at some ˜T Y,∗. According to the definition of ˜TY , there exists T Y,∗ ∈ TY such that +˜T Y,∗(·) = infy∈f∗ +S(XS T Y,∗(·, y). Let T X,∗ be the ˜T X ∈ TX corresponding to T Y,∗. For any T X ∈ TX +and T Y ∈ TY , we have +LT (T Y (·, (f∗ +S ◦ T X)(·))) ≥ LT (T Y (·, (f∗ +S ◦ ˜T X)(·))) += ˜LT ( ˜T Y (·)) ≥ ˜LT ( ˜T Y,∗(·)) += LT (T Y,∗(·, (f∗ +S ◦ T X,∗))(·)) ≥ +min +T X∈TX,T Y ∈TY LT +� +T Y (·, (f∗ +S ◦ T X)(·)) +� +. +Therefore, the transfer learning problem (8) is well-defined and it attains its minimum at (T X,∗, T Y,∗) +described above. +If one removes the compactness assumptions in Assumption (A), then a sufficiently rich family +of output transport mappings is needed, such that the target optimal model f∗ +T can be written +as f∗ +T (x) = T Y (x, f∗ +S(T X(x))), +∀x ∈ XT . However, it is often difficult to verify if the set TY is +sufficiently rich, due to the construction of neural networks as well as the choices of optimization +algorithms. The compactness conditions, on the other hand, can be implemented through choosing +a particular family of activation functions or imposing boundaries restrictions to weights and biases +when constructing machine learning models. +A.2 +Properties of Transfer risk +In this section, the mathematical properties of transfer risk (12) will be studied under mild assump- +tions. In the following discussion, we will fix a target task T and explore how transfer risk is affected +by the choice of source task S. +There are two vital pieces of information obtained from the source task S based on the optimization +problem in (8), and transfer risks in (12). One is the probability distribution of source input Law(XS), +and the other is the pretrained model f∗ +S in (3) Therefore, we can characterize source task S by +(Law(XS), f∗ +S). More specifically, given a target task T and the source input and output spaces XS +and YS, we can define a corresponding set of pretrained source tasks S ⊂ P(XS) × AS. Without +ambiguity on the target task T, we denote C(S, T) = C(S) = C(µ, f) for any S = (µ, f) ∈ S. For the +20 + +set of probability measures over XS, P(XS), we can adopt a metric function D : P(XS)×P(XS) → R. +Then for the set of functions AS, fix a sufficiently large constant M > 0 and define the following +metric: ∀f1, f2 ∈ AS, +dM(f1, f2) := min{M, sup +x∈XS +∥f1(x) − f2(x)∥YS}. +Then for any S1, S2 ∈ S such that S1 = (µ1, f1) and S2 = (µ2, f2), define +dS(S1, S2) := D(µ1, µ2) + dM(f1, f2). +(25) +It is easy to verify that dS is a metric over S. +In the following discussion on continuity, the next assumption is necessary. Assumption A.2 +ensures that the choice of input transfer risk is consistent with the metric dS in (25) defined between +source tasks. +Assumption A.2. For any input transport mapping T X +0 +∈ TX +0 , assume the input transfer risk +EI(T X +0 ) take the form EI(T X +0 ) := D(T X +0 #Law(XT ), Law(XS)), where D : P(XS) × P(XS) → R is +the distance function appearing in (25). +By definition, the following degenerate case holds immediately. +Proposition A.1 (Zero transfer risk). Suppose XT = XS, YT = YS and the target task T ∈ S. +Then C(T) = 0. +Now, we consider source tasks S1, S2 ∈ S that differ only in the input distribution, i.e., S1 +f = (µ1, f) +and S2 +f = (µ2, f). Then we have the following continuity property for C. +Proposition A.2 (Continuity in input distribution). Assume Assumption A.2. Fix f ∈ AS. C(·, f) +is continuous on (P(XS), D). +Proof of Proposition A.2. Fix an arbitrary ϵ > 0. Take any µ ∈ P(XS). Then we first establish the +lower semi-continuity: For any T X +0 ∈ TX +0 and T Y +0 ∈ TY +0 , let fI denote the corresponding intermediate +model from source model f. By Definition 3.3, we have +C(µ, f) − 1 +2ϵ < C(µ, f|fI). +By triangle inequality of D and the Lipschitz property of C(µ, f|fI), take δ = +ϵ +2L for any µ′ ∈ +Bδ(µ) ⊂ P(XS), +C(µ, f|fI) ≤ C(µ′, f|fI) + Lδ. +Notice that the choice of δ is independent of T X +0 +and T Y +0 . Therefore, +C(µ, f) − ϵ < C(µ′, f; fI) ⇒ C(µ, f) − ϵ < C(µ′, f). +Now we show the upper semi-continuity. From the infimum nature of C, there exists ¯T X +0 ∈ TX +0 +and ¯T Y +0 ∈ TY +0 , with corresponding intermediate model ¯fI, such that +C(µ, f| ¯fI) < C(µ, f) + 1 +2ϵ. +Again, by triangle inequality of D and the Lipschitz property of C(µ, f|fI), take δ = +ϵ +2L for any +µ′ ∈ Bδ(µ) ⊂ P(XS), +C(µ, f| ¯fI) ≥ C(µ′, f| ¯fI) − δ. +Then we have +C(µ′, f) ≤ C(µ′, f| ¯fI) < C(µ, f) + ϵ. +Hence, we conclude that C(·, f) is continuous on (P(XS), D). +21 + +This proposition shows that transfer risk will change continuously along with any modification +in source input. Its proof indicates that the sensitivity of transfer risk with respect to the change in +source input distribution depends on the Lipschitz constant L of C. Therefore, one can modulate this +sensitivity by carefully designing the C function in Definition 3.3. For instance, for linear transfer +risk Cλ in (13), the sensitivity can be controlled by varying the value of λ. +Next, consider source tasks S1, S2 ∈ S that differ only in the pretrained model, i.e., S1 +µ = (µ, f1) +and S2 +µ = (µ, f2). Then we have the robustness of the transferability in terms of the continuity of +C(µ, ·) in pretrained model f ∈ (AS, dM). +Proposition A.3 (Continuity in pretrained model). Assume Assumption A.2, and assume that +there exists a constant L > 0 such that for any T Y +0 ∈ TY +0 , +T Y +0 (x1, y1) − T Y +0 (x2, y2) ≤ L (∥x1 − x2∥XT + ∥y1 − y2∥YS) , +for all (x1, y1), (x2, y2) ∈ XT × YS. Assume also that there exist some L′ > 0 and p ≥ 1 such that +the output transfer risk satisfies +��EO(h1) − EO(h2) +�� ≤ L′Wp(h1#Law(XT ), h2#Law(XT ))p +for all h1, h2 ∈ I. Then C(µ, ·) is continuous on (AS, dM) for any fixed µ ∈ P(XS). +Proof of Proposition A.3. Take any T X +0 +∈ TX +0 and T Y +0 +∈ TY +0 . For any f1, f2 ∈ (AS, dM), denote +their corresponding intermediate model as f1 +I and f2 +I , respectively. Then we have +|EO(f1 +I ) − EO(f2 +I )| ≤ L′Wp(f1 +I #Law(XT ), f2 +I #Law(XT ))p += L′ +inf +π∈Π(f1 +I #Law(XT ),f2 +I #Law(XT )) +� +YT ×YT +∥x − y∥p +YT π(dx, dy) +≤ L′ +inf +γ∈Π(Law(XT ),Law(XT )) +� +XT ×XT +∥T Y +0 (x, f1(T X +0 (x))) − T Y +0 (y, f2(T X +0 (y)))∥2 +YT π(dx, dy) +≤ 2p−1LpL′ +� +inf +γ∈Π(Law(XT ),Law(XT )) +� +XT ×XT +∥x − y∥p +XT dπ(dx, dy) + dM(f1, f2)p +� += 2p−1LpL′ [Wp(Law(XT ), Law(XT ))p + dM(f1, f2)p] = 2p−1LpdM(f1, f2)p. +The rest of the proof is similar to that of Proposition A.2. +This proposition shows that transfer risk will change continuously along with the modification +in the pretrained model. As seen from the proof, the sensitivity of transfer risk with respect to +the change in pretrained model is determined by three factors: (1) the Lipschitz constant inherited +from the C function in Definition 3.3, (2) the choice of output transport risk EO, and (3) the family +of output transport mappings TY +0 . In practice, one may control the sensitivity of the transfer risk +through careful choices of those quantities. +Propositions A.2 and A.3 lead to the following results. +Proposition A.4. Suppose the conditions in Proposition A.3 hold. Then the transfer risk C as in +Definition 3.3 is continuous on (S, dS). +Propositions A.2 – A.4 reveals that under a given target task, transfer risk is continuously +influenced by both the changes in the source input and the pretrained model. Therefore, transfer +risk is to evaluate the suitability of performing transfer learning and the appropriate choice of given +source tasks for a target task. +22 + +B +Tranfer Risk and Regret with Gaussian Data +In this section, we will revisit the example in Section 3.4. The proof of Proposition 3.4 will also be +presented in this section. In the following discussion, for any spaces X and Y, we use the notation +YX to denote the set of all the functions from X to Y. +More specifically, consider a transfer learning problem in linear regression where the source and +target data are sampled from two Gaussian distributions respectively. +B.1 +Basic case +Let us first focus on the case where both data sources are of the same dimension. For the source +task S, the input and the output spaces are XS = Rd and YS = R, respectively. The source data +(XS, YS) ∈ XS × YS is Gaussian distributed such that (XS, YS) ∼ N(µS, ΣS) with +µS = +�µS,X +µS,Y +� +, +ΣS = +� ΣS,X +ΣS,XY +ΣS,Y X +ΣS,Y +� +, +(26) +where µS,Y and ΣS,Y ∈ R, µS,X and ΣS,XY ∈ Rd, ΣS,Y X = Σ⊤ +S,XY , and ΣS,X ∈ Rd×d. Take the set +of admissible source models AS to be the set of functions f : Rd �→ R. For any f ∈ AS, let the +source loss function be +LS(f) = E∥YS − f(XS)∥2 +2. +(27) +Then the optimal source model +f∗ +S ∈ arg min +f∈AS +LS(f) +(28) +is given by +f∗ +S(x) = w⊤ +S x + bs, +(29) +where +wS = Σ−1 +S,XΣS,XY ∈ Rd, +bS = µS,Y − ΣS,Y XΣ−1 +S,XµS,X ∈ R. +(30) +Suchf∗ +S is then used as the pretrained model for the following target task T, where the target +input and output spaces are the same as in the source task, XT = XS and YT = YS. The target +data (XT , YT ) ∈ XT × YT follows a different Gaussian distribution from that in the source data such +that (XT , YT ) ∼ N(µT , ΣT ), with +µT = +�µT,X +µT,Y +� +, +ΣT = +� ΣT,X +ΣT,XY +ΣT,Y X +ΣT,Y +� +, +(31) +where µT,Y and ΣT,Y ∈ R, µT,X and ΣT,XY ∈ Rd, ΣT,Y X = Σ⊤ +T,XY , and ΣT,X ∈ Rd×d. +The set of admissible target models is the same as the in the source task such that AT = AS. +For any f ∈ AT , let the target loss function be LT (f) = E∥YT − f(XT )∥2 +2. Then similarly to the +source task, the optimal target model f∗ +T is given by +f∗ +T (x) = w⊤ +T x + bT , +∀x ∈ Rd, +(32) +where +wT = Σ−1 +T,XΣT,XY , +bT = µT,Y − ΣT,Y XΣ−1 +T,XµT,X. +(33) +The corresponding output distribution is then given by +PT = E[Y |X] = N(w⊤ +T µT,X + bT , w⊤ +T ΣT,XwT ) = N(µT , w⊤ +T ΣT,XwT ). +(34) +23 + +To initiate transfer learning from the source task S to the target task T, consider the sets of +input and output transport mappings TX = {f|f : XT → XS} and TY = {f|f : XT × YS → YT }, +with corresponding sets of initial transport mappings TX +0 = {idXT }, TY +0 = {idYS}. Then the set of +intermediate models I is a singleton with I = {fST : fST = f∗ +S}. +Given the optimal models in both the source task and the target task, specified by (29)-(30) and +(32)-(33), since the data distribution in the target task is given by (31), we have +PST = f∗ +S#N(µT,X, ΣT,X) = N(w⊤ +S µT,X + bS, w⊤ +S ΣT,XwS). +(35) +Notice that PT ≪ PST , therefore the Lebesgue decomposition leads to PT = ˜PT , such that +d˜PT (y) +dPST (y) = hST (y) = +� +w⊤ +S ΣT,XwS +w⊤ +T ΣT,XwT +exp +�[y − (w⊤ +S µT,X + bS)]2 +2w⊤ +S ΣT,XwS +− [y − (w⊤ +T µT,X + b)]2 +w⊤ +T ΣT,XwT +� +. +(36) +Direct computation leads to the following result. +• The KL-based output transfer risk is given by +EO +KL(fST ) =1 +2 +� +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +− log +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +− 1 ++ +� +µT,Y − µS,Y − ΣS,Y XΣ−1 +S,X (µT,X − µS,X) +�2 +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +� +� +� +� +� +. +• The Wasserstein-based output transfer risk is given by +EO +W (fST ) = +� +µT,Y − µS,Y − ΣS,Y XΣ−1 +S,X (µT,X − µS,X) +�2 ++ +�� +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY − +� +ΣT,Y XΣ−1 +T,XΣT,XY +�2 +. +The computation shows that +• The risk in transfer learning is due to the discrepancy in the data distributions between source +and target tasks, even when the source and target data are of matching dimensions and follow +the same family of distributions. +• In particular, in both the KL and the Wasserstein cases, the output transfer risk can be +decomposed into two parts, one being the variance terms errorv,· determined by the covariance +matrices of the source and target data, and the other being the bias terms errorb,· dependent +on the difference between the expectations of µT and µS. +To see this, write +EO +KL(fST ) = errorv,KL(S, T) + errorb,KL(S, T), +(37) +EO +W (fST ) = errorv,W (S, T) + errorb,W (S, T), +(38) +24 + +where +errorv,KL(S, T) = 1 +2 +� +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +− log +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +− 1 +� +, +errorb,KL(S, T) = +� +µT,Y − µS,Y − ΣS,Y XΣ−1 +S,X (µT,X − µS,X) +�2 +2ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +; +errorv,W (S, T) = +�� +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY − +� +ΣT,Y XΣ−1 +T,XΣT,XY +�2 +, +errorb,W (S, T) = +� +µT,Y − µS,Y − ΣS,Y XΣ−1 +S,X (µT,X − µS,X) +�2 +. +• The KL-based variance term +errorv,KL = h +� +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY +� +, +with the function h : (0, ∞) → R such that h(x) = 1 +2(x − log x − 1) for any x > 0, which is +strictly convex and reaches its minimum value 0 at x = 1. Thus, for both the KL- and the +Wasserstein-based output transfer risks, their variance risk components vanish if and only if +ΣT,Y XΣ−1 +T,XΣT,XY = ΣS,Y XΣ−1 +S,XΣT,XΣ−1 +S,XΣS,XY . +• The bias risk components errorb,KL(S, T) and errorb,W (S, T) remain strictly positive unless +the weighted difference between the expectations µT and µS is 0. +Regret analysis. +By direct computation, one can show that the regret (20) for this linear transfer +leaning problem is given by +R(S, T) = ∥Σ +1 +2 (wT − wS)∥2 +2 + +� +µT,Y − µS,Y − ΣS,Y XΣ−1 +S,X (µT,X − µS,X) +�2 +. +(39) +We denote the first term in (39) as +ˆ +errorv(S, T) := ∥Σ +1 +2 (wT − wS)∥2 +2, and denote the second term in +(39) as +ˆ +errorb(S, T) := +� +µT,Y − µS,Y − ΣS,Y XΣ−1 +S,X (µT,X − µS,X) +�2 +. +Recall from (19) that the Wasserstein-based transfer risk for this problem is defined as CW (S, T) = +EO +W (fST ) in (38). Meanwhile, it can be easily verified by comparing (38) and (39) that +R(S, T) = CW (S, T) + 2 +� +∥Σ1/2 +T,XwT ∥2∥Σ1/2 +T,XwS∥2 − ⟨Σ1/2 +T,XwT , Σ1/2 +T,XwS⟩ +� +. +(40) +Proposition 3.4 is an immediate consequence of (40) and the Cauchy–Schwarz inequality. +Remark B.1. Proposition 3.4 suggests that for evaluating a transfer learning scheme as in (7), +transfer risk provides a proper initial indication of its effectiveness, especially when eliminating +unsuitable candidate pretrained models or source tasks if the transfer risk is large. Further examining +the decomposition of the transfer CKL and CW as well as the regret R, we notice that +• A vanishing bias term in transfer risks is equivalent to a vanishing bias term in regret, i.e., +ˆ +errorb(S, T) = 0 ⇐⇒ errorb,KL(S, T) = errorb,W (S, T) = 0 +25 + +• A vanishing variance term in transfer risk is necessary for a vanishing variance term in regret, +i.e., +ˆ +errorv(S, T) = 0 =⇒ errorv,KL(S, T) = errorv,W (S, T) = 0. +• The residual term 2 +� +∥Σ +1 +2 +T,XwT ∥2∥Σ +1 +2 +T,XwS∥2 − ⟨Σ +1 +2 +T,XwT , Σ +1 +2 +T,XwS⟩ +� +in (40) depends entirely +on the source and target covariance matrices ΣS and ΣT is due to the variance term in the +learning objective difference. Therefore, when CW (S, T) = 0 (or CKL(S, T) = 0), the training +process is to reduce the angular distance between Σ1/2 +T,XwS and Σ1/2 +T,XwT caused by the discrepancy +in these two covariance matrices. +B.2 +Case with feature augmentation +Let us now consider the case with feature augmentation. That is, compared with the input data +in the source task, the target task includes more input information in the form of a higher input +dimension. We will see that potential extra transfer risk as a result of the extra augmented input +information as well as its benefit to eliminate the bias risk. +Take the same source task S as in the basic case; for the target task T, let the input space +XT = Rd+k with k ∈ N+, let the output space be the same as in the target task T such that +YT = YS = R. Since the transfer learning problem has a feature augmentation, let us first define a +projection XT from T X +0 +to XS such that +T X +0 (x) = +� +Id +... +Od×k +� +x, +∀x ∈ XT . +Then the target data (XT , YT ) ∈ XT × YT satisfies that T X +0 (XT ) = XS and YT = YS. That is, +(XT , YT ) is given by a Gaussian distribution N(µT , ΣT ) with µT and ΣT in the same form as in (31), +where +µT,X = +�µS,X +µA,X +� +, µT,Y = µS,Y ; +ΣT,X = +� ΣS,X +ΣAS,X +Σ⊤ +AS,X +ΣA,X +� +, ΣT,XY = +�ΣS,XY +ΣA,XY +� +, ΣT,Y = ΣS,Y . +Here µA,X ∈ Rk denotes the expectation of the augmented variable ˜XT such that X⊤ +T = +� +T X +0 (XT )⊤ +˜X⊤ +T +� +; +in the above covariance matrix ΣT , ΣAS,X = Cov(XS, ˜XT ) ∈ Rd×k, ΣA,X = V ar( ˜XT ) ∈ Rk×k, and +ΣA,XY = Cov( ˜XT , YT ) ∈ Rk. The optimal linear model f∗ +T is again given by (32)-(33) with the +optimal parameters wT and bT re-computed under the above modified target data distribution. The +corresponding output distribution PT is of the form (34) with updated parameters as in f∗ +T . +To initialize the transfer learning problem from S to T, consider TX +0 = {T X +0 }, TY +0 = {idYS}, +TX = {f|f : XT → XS}, and TY = {f|f : XT × YS → YT }. The set of intermediate models I is still +singleton, with I = {fST : fST = f∗ +S ◦ T X +0 }. Clearly, PST = Law(f∗ +S(XS)), with PT ≪ Law(f( +SXS)). +Now we have +• The KL-based output transfer risks are given by +EO +KL(fST ) = 1 +2 +� +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣS,XY +− log +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣS,XY +− 1 +� +; +(41) +• The Wasserstein-based output transfer risk is +EO +W (fST ) = +�� +ΣT,Y XΣ−1 +T,XΣT,XY − +� +ΣS,Y XΣ−1 +S,XΣS,XY +�2 +. +(42) +26 + +Comparing the basic case and this feature augmentation case, we see +• The extra input information enables the particular choice of the initial input and output +transport mappings, T X +0 +and idYS, which in turn eliminates the bias risk component in both +the KL- and Wasserstein-based output risk. +• Both output transfer risks come from their corresponding variance risk component. Take the +KL-based output transfer risk in (41) as an example. We see that +EO +KL(fST ) = errorv,KL(S, T) = h +� +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣS,XY +� +. +This suggests that the challenge of applying transfer learning with feature augmentation lies +mainly at the uncertainty from the augmented variable ˜XT . +• In particular, if one assumes that the added input information ˜XT is uncorrelated with the +existing input data XS, then +ΣT,Y XΣ−1 +T,XΣT,XY +ΣS,Y XΣ−1 +S,XΣS,XY += 1 + +ΣA,Y XΣ−1 +A,XΣA,XY +ΣS,Y XΣ−1 +S,XΣS,XY +. +That is, when one introduces new features that are uncorrelated with the existing ones, the +variance risk component is always positive unless these new features are also uncorrelated with +the output. +B.3 +Case with augmented output space +Let us now consider the case with an extra prediction task, i.e., a transfer learning problem with +augmented output space. In this case, we will see extra transfer risk with two major contributing +factors, one being the unforeseeable correlation between the input and the extra output information +and the other being the necessary initialization procedure due to the extra task in the target task. +To see this, let us consider a source task slightly modified from the basic case, where the output +space in S is allowed be of dimension bigger that 1, that is, YS = Rl with l ∈ N+. Then the source +data (XS, YS) is given by a Gaussian distribution N(µS, ΣS) with µS and ΣS as in (26) except that +µS,Y ∈ Rl, ΣS,XY ∈ Rd×l and ΣS,Y ∈ Rl×l. Again the optimal linear model f∗ +S is given by (29)-(30) +with optimal parameters wS and bS re-computed under the above modified source data distribution. +For the target task, let XT = XS and YT = Rl+k with k ∈ N+. Since the transfer learning +problem has an extra learning task with the same input data, the target data (XT , YT ) ∈ XT × YT +satisfies that XT = XS and Y ⊤ +T = +� +YS +YA +�⊤ from some random variable YA ∈ RK. Let us assume +that the joint distribution of the input and output variables (XT , YT ) follows a Gaussian distribution +N(µT , ΣT ) with µT and ΣT in the same form as in (31), where +µT,Y = +�µS,Y +µA,Y +� +, +µT,X = µS,X; +ΣT,Y = +� ΣS,Y +ΣAS,Y +Σ⊤ +AS,Y +ΣA,Y +� +, +ΣT,Y X = +�ΣS,Y X +ΣA,Y X +� +, +ΣT,XY = +� +ΣS,XY +ΣA,XY +� +, +ΣT,X = ΣS,X. +Here µA,Y = E[YA] ∈ Rk denotes the expectation of YA, ΣAS,Y = Cov(YS, YA) ∈ Rl×k, ΣA,Y = +V ar(YA) ∈ Rk×k and ΣA,XY = Σ⊤ +A,Y X = Cov(XS, YA) ∈ Rd×k. Then again the optimal linear model +27 + +f∗ +T is in the form of (32) with parameters +w⊤ +T = ΣT,Y XΣ−1 +T,X = +� +ΣS,Y XΣ−1 +S,X +ΣA,Y XΣ−1 +S,X +� += +� +w⊤ +S +ΣA,Y XΣ−1 +S,X +� +, +bT = µT,Y − ΣT,Y XΣ−1 +T,XµT,X = +�µS,Y +µA,Y +� +− +� +ΣS,Y XΣ−1 +S,XµS,X +ΣA,Y XΣ−1 +S,XµS,X +� += +� +bS +µA,Y − ΣA,Y XΣ−1 +S,XµS,X +� +. +Correspondingly, PT = E[YT |XT ] = N (µ1, Σ1), where +µ1 = w⊤ +T µT,X + bT = +�w⊤ +S µS,X + bS +µA,Y +� +, +Σ1 = w⊤ +T ΣT,XwT = +�w⊤ +S ΣS,XwS +w⊤ +S ΣA,XY +ΣA,Y XwS +ΣA,Y XΣ−1 +S,XΣA,XY +� +. +To initialize the transfer learning, consider the sets of input and output transport mappings +TX = {f|f : XT → XS} and TY = {f|f : XT ×YS → YT }, as well as the sets of initial input transport +mappings TX +0 = {idXT }. For the initial output mapping, in order to handle the newly added prediction +task from XT = XS to YA, let us define an initial function f0 : Rd → Rk as f0(x) = w⊤ +0 x + b0 +for any x ∈ Rd with fixed w0 ∈ Rk×d and b0 ∈ Rk. The set of initial output transport mappings +is given by TY +0 = {T Y +0 : XT × YS → YT |T Y +0 (x, y) = +� +y⊤ +... +f0(x)⊤ +�⊤ +}. Once again, the set of +intermediate models I is a singleton with I = {fST : fST (x) = T Y +0 (x, f∗ +S(x)), ∀x ∈ XT }. The +probability distribution PST of the intermediate model fST is given by PST = N(µ2, Σ2), where +µ2 = +�w⊤ +S +w⊤ +0 +� +µT,X + +�bS +b0 +� += +�w⊤ +S µS,X + bS +w⊤ +0 µS,X + b0 +� +, +Σ2 = +�w⊤ +S +w⊤ +0 +� +ΣT,X +� +wS +w0 +� += +�w⊤ +S ΣS,XwS +w⊤ +S ΣS,Xw0 +w⊤ +0 ΣS,XwS +w⊤ +0 ΣS,Xw0 +� +. +We have again PT ≪ PST , and +• The KL- and Wasserstein-based output transfer risks are given by +EO +KL(fST ) = 1 +2 +� +Tr(Σ−1 +2 Σ1) − log det(Σ1) +det(Σ2) − (k + l) + (µ1 − µ2)⊤Σ−1 +2 (µ1 − µ2) +� +; +(43) +• The Wasserstein-based output transfer risk is given by +EO +W (fST ) = ∥µ1 − µ2∥2 +2 + Tr +� +Σ1 + Σ2 − 2 +� +Σ +1 +2 +1 Σ2Σ +1 +2 +1 +� 1 +2 +� +. +(44) +The analysis shows that with the augmented output space, the output transfer risks vanish if the +initialization function f0 can neutralize the uncertainty brought by the correlation between the input +XS and the additional output information YA. +To see this, take the example of the KL-based output transfer risk in (43), and decompose +EO +KL(fST ) in (43) into its variance and bias components as in (37), with +errorv,KL(S, T) = 1 +2Tr(Σ−1 +2 Σ1) − log det(Σ1) +det(Σ2) − (k + l), +errorb,KL(S, T) = 1 +2(µ1 − µ2)⊤Σ−1 +2 (µ1 − µ2). +Now, we see that +28 + +• If 0 < λ1 ≤ · · · ≤ λk+l are the eigenvalues of Σ−1 +2 Σ1, and if Σ1 and Σ2 are invertible. Then +the variance term can be written as +errorv(S, T) = +k+l +� +i=1 +(λi − log λi − 1) ≥ 0, +which vanishes if and only if λi’s are all equal to 1 such that Σ1 = Σ2. +• The difference between the expectations of PT and PST is given by +µ1 − µ2 = +� +0 +µA,Y − w⊤ +0 µS,X − b0 +� +. +Therefore, the error between the expected augmented output µA,Y and w⊤ +0 µS,X + b0 derived +from the chosen initialization f0 is the main contributor to a strictly positive bias risk component +errorb,KL. +29 + diff --git a/sNFJT4oBgHgl3EQfcCyS/content/tmp_files/load_file.txt b/sNFJT4oBgHgl3EQfcCyS/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..741cfa0dd4ca796a67c9078e5db5fc55f62214e5 --- /dev/null +++ b/sNFJT4oBgHgl3EQfcCyS/content/tmp_files/load_file.txt @@ -0,0 +1,1027 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf,len=1026 +page_content='Feasibility and Transferability of Transfer Learning: A Mathematical Framework Haoyang Cao ∗ Haotian Gu † Xin Guo ‡ Mathieu Rosenbaum § January 26, 2023 Abstract Transfer learning is an emerging and popular paradigm for utilizing existing knowledge from previous learning tasks to improve the performance of new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Despite its numerous empirical successes, theoretical analysis for transfer learning is limited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this paper we build for the first time, to the best of our knowledge, a mathematical framework for the general procedure of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Our unique reformulation of transfer learning as an optimization problem allows for the first time, analysis of its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Additionally, we propose a novel concept of transfer risk to evaluate transferability of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Our numerical studies using the Office-31 dataset demonstrate the potential and benefits of incorporating transfer risk in the evaluation of transfer learning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 1 Introduction The basic idea of transfer learning is simple: it is to leverage knowledge from a well-studied learning problem, known as the source task, to improve the performance of a new learning problem with similar features, known as the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning has seen success in a variety of field, including natural language processing (Ruder et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Sung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2022), sentiment analysis (Jiang and Zhai, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019), computer vision (Deng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wang and Deng, 2018), activity recognition (Cook et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2018), medical data analysis (Zeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Kim et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2022), bio-informatics (Hwang and Kuang, 2010), finance (Leal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Rosenbaum and Zhang, 2021), recommendation system (Pan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019), and fraud detection (Lebichot et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' See also review papers (Pan and Yang, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning is a versatile and enduring paradigm in the rapidly changing AI landscape where new machine learning techniques and tools mushroom with a breakneck speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Despite its empirical successes, studies on transfer learning are primarily based on trial-and-error heuristics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Virtually there are neither basic theoretical frameworks for the general procedure of transfer learning, nor studies on the fundamental issue of it feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Existing theoretical works of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Earlier theoretical works for transfer learning tend to focus on specific learning problems, such as classification, and derive upper bounds of ∗Centre de Mathématiques Appliquées, Ecole Polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Email: haoyang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='cao@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='edu †Department of Mathematics, UC Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Email: haotian_gu@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='edu ‡Department of Industrial Engineering & Operations Research, UC Berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Email: xinguo@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='edu §Centre de Mathématiques Appliquées, Ecole Polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Email: mathieu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='rosenbaum@polytechnique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='edu 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='11542v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='LG] 27 Jan 2023 generalization error under different measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' There are the VC-dimension of the hypothesis space adopted in (Blitzer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2007), total variation distance in (Ben-David et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2010), f-divergence in (Harremoës and Vajda, 2011), Jensen-Shannon divergence in (Zhao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019), H-score in (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019), mutual information in (Bu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2020), and more recently X 2-divergence in (Tong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2021), and variations of optimal transport cost in (Tan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Another line of theoretical studies interprets transferability for transfer learning as a measurement of similarity between the source and the target data using various divergences, such as low-rank common information in (Saenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2010), KL-divergence in (Ganin and Lempitsky, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Tzeng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2017), l2-distance in (Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2014), and the optimal transport cost in (Courty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Our work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this paper, we address the issues of feasibility and transferability for transfer learning through rigorous and comprehensive mathematical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We build, for the first time to the best of our knowledge, a mathematical framework for the general procedure of transfer learning, identifying its three key steps and components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We reformulate this three-step transfer learning procedure as an optimization problem, enabling us to analyze, for the first time, its feasibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This is accomplished via analyzing the well- definedness of the corresponding optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Additionally, we propose a novel concept of transfer risk to evaluate the transferability of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Our form of transfer risk accounts for both the compatibility between the output and the input data and the compatibility between the models in the source and the target tasks, allowing for the study of the trade-off between the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This novel notion of transfer risk generalizes earlier works on transferability, including the H-score proposed in a particular classification setting in (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019) and (Saenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2014) on the relation between source and target inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the special case of linear regression with Gaussian data, we show that the regret in the learning problem can be lower bounded by Wasserstein-based transfer risk, which in turn is useful for prescreening unsuitable candidate pretrained models or source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Our numerical studies using the Office-31 dataset show the consistency of the transfer risk with existing statistical metrics in evaluating the performance of transfer learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and demonstrate the potential and benefit of adopting transfer risk to improve computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2 Mathematical Framework and Feasibility of Transfer Learning In this section, we will establish necessary concepts and a mathematical framework for the entire procedure of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We will then reformulate transfer learning as an optimization problem, the well-definedness of which yields the feasibility of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For ease of exposition and without loss of generality, we will focus on a supervised setting, with a source task S and a target task T on a probability space (Ω, F, P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 Mathematical Framework for Transfer Learning Target task T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the target task T, denote XT and YT as its input and output spaces, respectively, and (XT , YT ) as a pair of XT × YT -valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here, (XT , ∥ · ∥XT ) and (YT , ∥ · ∥YT ) are 2 Banach spaces with norms ∥ · ∥XT and ∥ · ∥YT , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let LT : YT × YT → R be a real-valued function, and assume that the learning objective for the target task is min f∈AT LT (fT ) = min fT ∈AT E[LT (YT , fT (XT ))], (1) where LT (fT ) is a loss function that measures a model fT : XT → YT for the target task T, and AT denotes the set of target models such that AT ⊂ {fT |fT : XT → YT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2) Take the image classification task as an example, XT is a space containing images as high dimensional vectors, YT is a space containing image labels, (XT , YT ) is a pair of random variables satisfying the empirical distribution of target images and their corresponding labels, and LT is the cross-entropy loss function between the actual label YT and the predicted label fT (XT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For the image classification task using neural networks, AT will depend on the neural network architecture as well as the constraints applied to the network parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let f∗ T denote the optimizer for the optimization problem (1), and PT = Law(f∗ T (XT )) for the probability distribution of its output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the model distribution PT depends on three factors: LT , the conditional distribution Law(YT |XT ), and the marginal distribution Law(XT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Note that in direct learning, this optimizer f∗ T ∈ AT is solved directly by analyzing the optimization problem (1), whereas in transfer learning, one leverages knowledge from the source task to facilitate the search of f∗ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Source task S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the source task S, denote XS and YS as the input and output spaces of the source task, respectively, and (XS, YS) as a pair of XS × YS-valued random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here, (XS, ∥ · ∥XS) and (YS, ∥ · ∥YS) are Banach spaces with norms ∥ · ∥XS and ∥ · ∥YS, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let LS : YS × YS → R be a real-valued function and let us assume that the learning objective for the source task is min fS∈AS LS(fS) = min f∈AS E[LS(YS, fS(XS))], (3) where LS(fS) is the loss function for a model fS : XS → YS for the source task S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here AS denotes the set of source task models such that AS ⊂ {fS|fS : XS → YS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (4) Moreover, denote the optimal solution for this optimization problem (3) as f∗ S, and the probability distribution of the output of f∗ S by PS = Law(f∗ S(XS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Meanwhile, similar as the target model, the model distribution PS will depend on the function LS, the conditional distribution Law(YS|XS), and the marginal distribution Law(XS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Back to the image classification example, the target task may only contain images of items in an office environment, the source task may have more image samples from a richer dataset, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', ImageNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Meanwhile, XS and YS may have different dimensions compared with XT and YT , since the image resolution and the class number vary from task to task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Similar to the admissible set AT in the target task, AS depends on the task description, and f∗ S is usually a deep neural network with parameters pretrained using the source data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In transfer learning, the optimal model f∗ S for the source task is also referred to as a pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The essence of transfer learning is to utilize this pretrained model f∗ S in the source task to accomplish the optimization objective (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We now define this procedure in three steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3 Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Input transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Since XT is not necessarily contained by the source input space XS, the first step is therefore to make an appropriate adaptation to the target input XT ∈ XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the example of image classification, popular choices for input transport may include resizing, cropping, rotation, and grayscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We define this adaptation as an input transport mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 (Input transport mapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A function T X ∈ {finput|finput : XT → XS} (5) is called an input transport mapping with respect to the source and target task pair (S, T) if it takes any data point in the target input space XT and maps it into the source input space XS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' With an input transport mapping T X, the first step of transfer learning can be represented as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' XT ∋ XT Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Input transport by T X �−−−−−−−−−−−−−−−−−−−→ T X(XT ) ∈ XS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In a class of transfer learning called domain adaption, it is assumed that the difference between the source input distribution Law(XS) and target input distribution Law(XT ) is the only factor to motivate the transfer, while the labeling function of the source and target tasks stays the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (See also Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 for more details on domain adaptation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, once a proper input transport mapping T X is found, transfer learning is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 is thus consistent with (Courty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2017), in which domain adaption is formulated as an optimal transport from the target input to the source input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For most transfer learning problems, however, one needs both a transport mapping for the input and a transport mapping for the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For instance, the labeling function for different classes of computer vision tasks, such as object detection, instance segmentation, and image classification, can vary greatly and depend on the specific task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Hence, the following two more steps are required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Applying pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' After applying an input transport mapping T X to the target input XT , the pretrained model f∗ S will take the transported data T X(XT ) ∈ XS as an input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' That is, XS ∋ T X(XT ) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Apply f∗ S �−−−−−−−−−−→ (f∗ S ◦ T X)(XT ) ∈ YS, where (f∗ S ◦ T X)(XT ) denotes the corresponding output of the pretrained model f∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Note here the composed function f∗ S ◦ T X ∈ {fint|fint : XT → YS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Output transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' After utilizing the pretrained model f∗ S, the resulting model f∗ S ◦ T X may, however, still be inadequate for the target model: one may need to map the YS-valued output into the target output space YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Hence, it is necessary to define an output transport mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 (Output transport mapping).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A function T Y ∈ {foutput|foutput : XT × YS → YT } (6) is called an output transport mapping with respect to the source and target task pair (S, T) if, for an optimal source model f∗ S : XS → YS, the composed function T Y (·, f∗ S(·)) ∈ AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now, this third and the final step in transfer learning can be expressed as XT × YS ∋ (XT , (f∗ S ◦ T X)(XT )) Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Output transport by T Y �−−−−−−−−−−−−−−−−−−−−→ T Y � XT , (f∗ S ◦ T X)(XT ) � ∈ YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 4 For the image classification task with transfer learning, the optimal source model usually consists of the first few layers of the neural network for feature extraction, and the output transport mapping is the subsequent prediction layers that map the features from the optimal source model to the target output labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' An output transport mapping can also be viewed as an operation to tailor the optimal source model into a suitable target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For instance, in (Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2022), a large language model is a collection of optimal pretrained transformer models and each model consists of a multi-head self-attention layer and feed-forward layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Thus, the output transport mapping is the structure pruning with distillation operation applied to each optimal transformer model, where pruning reduces the original transformer model to a simplified sub-model which is more suitable for the corresponding down-stream tasks, and where distillation ensures the proper knowledge is passed from the source model down to the target model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Combining these three steps, transfer learning can be presented by the following diagram, XS ∋ XS Pretrained model f∗ S from (3) ====================⇒ f∗ S(XS) ∈ YS T X��� ���T Y XT ∋ XT Direct learning (1) − − − − − − − → f∗ T ∈arg min f∈AT LT (fT ) f∗ T (XT ) ∈ YS (7) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 Optimization Formulation and Feasibility of Transfer Learning In summary, transfer learning aims to find an appropriate pair of input and output transport mappings T X and T Y , where the input transport mapping T X translates the target input XT back to the source input space XS in order to utilize the optimal source model f∗ S, and the output transport mapping T Y transforms a YS-valued model to a YT -valued model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This is in contrast to the direct learning, where the optimal model f∗ T is derived by solving the optimization problem in the target task (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In other words, transfer learning is the following optimization problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 (Transfer learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The three-step transfer learning procedure presented in (7) is to solve the optimization problem min T X∈TX,T Y ∈TY LT � T Y (·, (f∗ S ◦ T X)(·)) � = E � LT � YT , T Y (XT , (f∗ S ◦ T X)(XT )) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (8) Here, TX and TY are proper sets of transport mappings such that � T Y (·, (f∗ S ◦ T X)(·))|T X ∈ TX, T Y ∈ TY � ⊂ AT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, when XS = XT (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' YS = YT ), the identity mapping idX(x) = x (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' idY (x, y) = y) is included in TX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' TY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This optimization reformulation of the three-step transfer learning procedure provides potentially a unified framework to analyze the impact and implications of various transfer learning techniques, including resizing, cropping, pruning, and distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Moreover, it enables us to analyze the feasibility of transfer learning, which we establish in terms of the following well-definedness of the corresponding optimization problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Under suitable choices of loss functions for LT and appropriate compactness as- sumptions, there exists optimal solutions for optimization problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 5 Detailed assumptions and proof for Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 is deferred to Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The procedure of solving this optimization problem is often referred to as fine-tuning in the literature of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' It is to choose some initial transport mappings T X 0 ∈ TX 0 ⊂ TX and T Y 0 ∈ TY 0 ⊂ TY to derive an intermediate model fST ∈ AT with fST (x) = T Y 0 (x, (f∗ S ◦ T X 0 )(x)), ∀x ∈ XT , (9) with the set of possible intermediate models denoted as I = � T Y 0 (·, (f∗ S ◦ T X 0 )(·)) ��T X 0 ∈ TX 0 , T Y 0 ∈ TY 0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (10) This fine-tuning procedure allows for computationally efficient evaluation of transferability in terms of transfer risk, to be introduced in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 Examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Consider a transfer learning task in image classification using the Office-31 (Saenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2010) benchmark dataset, which consists of images from three domains: Amazon (A), Webcam (W) and DSLR (D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In total, the dataset contains 4110 images of 31 categories of objects typically found in an office environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Samples from the Office-31 dataset are shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Figure 1: Samples from Office-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The neural network architecture for the image classification task is shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' It sequentially consists of: 1) a data-preprocessing module which resizes a input image to 3 × 244 × 244 dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2) ResNet50 as a feature extractor whose output is a 2048-dimensional feature vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and 3) a two-layer neural network which maps a 2048-dimensional feature vector to a 31-dimensional probability vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 6 Amazon DSLR WebcamFigure 2: Neural network architecture for Office-31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this example, the source task can be chosen from any of three domains (A, D, or W), with XS = R3×244×244 being the space of resized image samples from the source domain, and YS = ∆31 := {p ∈ R31 : 31 � 1 pi = 1, pi ≥ 0, ∀1 ≤ i ≤ 31} being the space of image class labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Similarly, for any target task (A, D, or W), XT = XS = R3×244×244 is the space of resized image samples from the target domain, and YT = YS = ∆31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For both the source and the target tasks, the loss function LS = LT is chosen to be the cross entropy between the actual label and the predicted label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' As introduced in Figure 2, the set of source models are given by AS = {fNN ◦ fRes : XS → YS|fNN ∈ NN31 2048, fRes ∈ Res2048 3×244×244}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here Res2048 3×244×244 denotes all ResNet50 architectures with 3×244×244-dimensional input and 2048- dimensional output, and NN31 2048 denotes all two-layer neural networks which map a 2048-dimensional feature vector to a 31-dimensional probability vector in YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The source model f∗ Res,S and f∗ NN,S is obtained by solving the source task optimization (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' To transfer the source model to the target task, the pretrained ResNet50 model f∗ Res,S will be fixed, while the last two-layer classifier fNN ∈ NN31 2048 will be fine-tuned using part of the data from the target domain (XT , YT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The input transport set TX in this example is a singleton set whose element is the identity mapping on R3×244×244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Meanwhile, the set of output transport mappings is given by TY = {fNN ◦ f∗ Res,S : XT → YT |fNN ∈ NN31 2048}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (11) The transfer learning task is formulated as min T Y ∈TY E � LT � YT , T Y (XT ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 7 Resize: Feature Extractor: Source Fully Connected Source 3x244x244 ResNet50 Feature Layer Label Source Task Finetuning Target Task Target Feature Target LabelNote the formulation is slightly simpler than (8) because in this particular example, the output transport in TY takes inputs from XT instead of XT × YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Furthermore, in this example, there is no additional constraint on intermediate models defined in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, the set I defined in (10) is equivalent to TY in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Domain adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This class of transfer learning problem considers the case where the output variable for the source and target tasks coincides, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', YS = YT = Y ∈ Y, and there exists some one-to-one input transport T X such that T X(XT ) = XS almost surely (Courty et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here we define the family of admissible (initial) output transport mappings as TY 0 = TY = {idY}, where idY denotes the identity mapping on Y;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and define the family of admissible (initial) input transport mappings as TX 0 = TX = {T X : XT → XS | T is one-to-one}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then I = {f∗ S ◦ T|T ∈ TX 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' When the loss functions for the source and the target tasks are also in the same form such that LS = LT = L : Y × Y → R, it can be shown that the optimal source model and optimal target model satisfy the relation f∗ T = f∗ S ◦ T X, where f∗ := arg min f:X·→Y E[L(Y, f(X·))].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' From the transfer learning perspective, T X is also the optimal solution to the optimization problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, the transfer learning model f∗ T = f∗ S ◦ T X is equivalent to the optimal model from the direct learning, while solving the transfer learning problem (8) may require much less data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3 Transfer Risk and Transferability of Transfer Learning Given the mathematical framework and after the feasibility analysis of transfer learning, we will now propose a novel notion of transfer risk, to analyze the effectiveness and the appropriateness of transfer learning over the set of all intermediate models I given by (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 Transfer Risk The idea is to re-interpret the transfer learning framework (7) in a sequential manner: the mapping T X first transports Law(XT ) to some probability distribution Law(T X(XT ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' then, applying the pretrained model f∗ S for the optimization problem (3) yields the distribution ˜PS = Law(f∗ S(T X(XT ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Finally, an output transport mapping T Y , together with the target input XT , transports the distribution ˜PS to PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' That is, the transfer learning scheme can be viewed as the composition of the following two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (Psuedo) Domain adaption, which can also be seen as optimal transport from Law(XT ) to Law(XS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Optimal transport from ˜PS = Law(f∗ S(T X(XT ))) to PT over T Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In other words, in parallel to the three-step procedure in transfer learning, there are two major sources of transfer risk for a fixed intermediate model fST ∈ I: the risk that measures the mismatch between the output distributions of the intermediate model fST and the optimal target model f∗ T , and the risk reflecting the difference between the transported target input and the source input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let us first define the risk associated the output transport mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 (Output transport risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let EO : AT → R be a real-valued function on the set of target models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any fST ∈ I ⊂ AT , EO(fST ) is called an output transport risk of intermediate model fST if it satisfies 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' EO(fST ) ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', transfer learning always incurs a non-negative effort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' EO(fST ) = 0 if and only if PT = PST , where PT := Law(fT (XT )) and PST := Law(fST (XT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' That is, the output transport risk vanishes when the intermediate model fST completely recovers the distribution of the optimal target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Clearly, the smaller this output risk, the more effective the transfer scheme with the intermediate model fST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We next define the risk associated with the input transfer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 (Input transfer risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let EI : TX → R be a real-valued function on the set of input transport mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Given an import transport mapping T X 0 ∈ TX 0 ⊂ TX, EI(T X 0 ) is called an input transport risk if it satisfies 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' EI(T X 0 ) ≥ 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', transfer learning always incurs a non-negative effort;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' EI(T X 0 ) = 0 if and only if T X 0 #Law(XT ) = Law(XS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The smaller this input risk, the higher the similarity between the transported target input T X 0 (XT ) and the source input XS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Note that these definitions of risks involve the sets of initial transport mappings TX 0 and TY 0 , instead of the sets of all possible transport mappings TX and TY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' These reduced sets allow for efficient evaluation of transfer risk prior to starting the full-scale transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Both the input transfer risk and the output transfer risk are functions characterizing the divergence between probability distributions, and their exact forms can be task dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Nevertheless, there is a key difference between these two forms of risks: in the output transport risk, PT , the output distribution of the optimal target model, is unknown, and no prior knowledge about f∗ T is assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, analyzing the output transport risk is decisively more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' See more detailed discussions in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We are now ready to propose the notion of transfer risk by considering all intermediate models in I, in order to measure the effectiveness of a transfer learning framework (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 (Transfer risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For a transfer learning procedure characterized by the 6-tuple (S, T, TX, TX 0 , TY , TY 0 ) in (8), the transfer risk of the transfer learning framework (8) from source task S to target task T is defined as C(S, T) = inf fST ∈I C(S, T|fST ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (12) Here, for a given fST = T Y 0 (·, (f∗ S ◦ T X 0 )(·)) ∈ I, C(S, T|fST ) is called model-specific transfer risk such that C(S, T|fST ) ≥ 0 with the following properties: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let C : R × R → R with C(0, 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' C(S, T|fST ) = C(EO(fST ), EI(T X 0 )) is non-decreasing in EO(fST ) under any fixed EI(T X 0 ) and non-decreasing in EI(T X 0 ) under any fixed E(fST );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' C(S, T|fST ) is Lipschitz in the sense that for any other transfer problem characterized by ( ¯S, ¯T, ¯TX, ¯TX 0 , ¯TY , ¯TY 0 ) and one of its intermediate models ¯fST = ¯T Y 0 (·, ( ¯f∗ S ◦ ¯T X 0 )(·)) ∈ ¯I, there exists a constant L > 0 such that |C(S, T|fST ) − C( ¯S, ¯T| ¯fST )| ≤ L(|EO(fST ) − EO( ¯fST )| + |EI(T X 0 ) − EI( ¯T X 0 )|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 9 The expression of this Lipschitz property in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 is to emphasize the dependence of transfer risk on a given transfer learning problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This Lipschitz property is satisfied when the function C in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 is Lipschitz continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' One simple example of the model-specific transfer risk is Cλ(S, T|fST ) = EO(fST ) + λEI(T X 0 ), (13) where λ > 0 is a pre-specified parameter modulating the weight of the input transport in the transfer learning problem (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer risk in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 unifies the analysis of the risk from both the input and the output transport mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' It allows for studying the trade-off between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Moreover, two of its key components, the input and the output transfer risks in Definitions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 generalize earlier works on transferability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For instance, the H-score proposed in (Bao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2019) addresses transferability of a particular classification setting and can be incorporated into the output transfer risk in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Earlier works on the relation between source and target inputs such as (Saenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ganin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Long et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', 2014) correspond to the special case in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 with T X 0 being the identity mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Furthermore, one can establish the following properties of transfer risk: a) there is zero transfer risk if the source and the target tasks are identical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and b) transfer risk is continuous in the input distribution and robust with respect to the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (See the exact mathematical statement and analysis of these properties in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The continuity of the transfer risk in terms of the changes in the input and the pretrained model is useful to exclude a priori inappropriate source tasks when compared against existing viable source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 Examples We now revisit some examples in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 and their associated transfer risks based on Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, we will illustrate how the two key components of the transfer risk, namely, the input transport risk EI and the output transport risk EO, are embedded in transfer learning for a given intermediate model fST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer risk in domain adaption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Recall the domain adaptation problem in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3, and consider the case where the transfer risk is independent of the output transport risk, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', the input risk EI(T X 0 ) completely determine the transfer risk: C(S, T|fST ) = EI(T X 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this case, there exists a one-to-one input mapping T X ∈ TX 0 such that T X(XT ) = XS almost surely, implying T X#Law(XT ) = Law(XS) and consequently C(S, T) = C(S, T|f∗ S ◦ T X) = EI(T X) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, vanishing input transport risk is a necessary condition for the domain adaptation framework to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Thus, the input transport risk may be adopted to check the viability of domain adaptation on certain tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer risk in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Recall the image classification problem introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Fix a source task S and a target task T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Since the input transport set TX in this problem is a singleton set, the input transport risk EI is a constant depending on Law(XS) and Law(XT ), with the output transport risk denoted as EO(fST ) for any fST ∈ I in (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3, the model-specific transfer risk C(S, T|fST ) = C(EI, EO(fST )) for some appropriate function C : R2 → R satisfying conditions stated in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, since the function C is non-decreasing 10 with respect to EO(fST ), minimizing C(S, T|fST ) over fST ∈ I is equivalent to minimizing EO(fST ) over fST ∈ I: arg min fST ∈I C(S, T|fST ) = arg min fST ∈I EO(fST ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' And consequently, C(S, T) = min fST ∈I C(S, T|fST ) = C(EI, min fST ∈I EO(fST )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 Transfer Risk and Choices of Divergence Functions Clearly, different learning tasks may require different choices of divergence functions for assessment of transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this section, we present two forms of transfer risks based on two divergence functions, and analyze their properties and relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' KL-based output transport risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For learning tasks such as the classification problem, one may use cross-entropy as the loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Specifically, let PT = ˜PT + P0 be its unique Lebesgue decomposition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', for any measurable set B ⊂ YT , there exists some function hST : YT → R+ such that ˜PT (B) = � B hST dPST , with P0 singular with respect to PST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the KL-based output risk can be defined as EO KL(fST ) := DKL(˜PT ∥PST ) + H(P0), where H(P0) is the entropy function of P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For a classification problem over K ∈ N classes with cross entropy as the training loss, for any fST ∈ I, K � i=1 log pST (i) ≤ H(PT , PST ) − H(Law(YT ), PST ) ≤ − K � i=1 log pST (i), where pST denotes the probability mass function for PST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Note that H(Law(YT ), PST ) is indeed the cross-entropy loss for the classifier fST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, in actual training, one may use H(Law(YT ), PST ) ± �K i=1 log pST (i) to replace EO KL(fST ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wasserstein-based output transport risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For learning problems such as GANs or supervised learning with domain adaption, Wasserstein and related distances are popular choices to measure the distance between the generative distribution and the target distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, a Wassertein- based output risk is a natural choice related to such learning targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' More specifically, for p ≥ 1, let Pp(YT ) be the set of probability measures over YT such that � RdO,T ∥x∥p YT dµ(x) < ∞, ∀µ ∈ Pp(YT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The Wasserstein-based output risk is defined as EO W (fST ) := Wp(PST , PT )p := inf γ∈Π(PST ,PT ) � RdO,T ×RdO,T ∥x − y∥p YT dγ(dx, dy), (14) for some suitable choice of p ≥ 1, where Π(PST , PT ) denotes the set of couplings of probability measures PST and PT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Analogy to Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 is the following property for EO W (fST ), based on the triangle inequality of the Wasserstein distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 11 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The Wasserstein-based output risk EO W in (14) is upper bounded in the following sense: EO W (fST ) ≤ 2p−1[Wp(PST , Law(YT ))p + Wp(PT , Law(YT ))p].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now, consider any intermediate model fST , then Talagrand’s inequality (Talagrand, 1996) gives EI W (T X 0 ) ≤ 2EI KL(T Y 0 ), EO W (fST ) ≤ 2EO KL(fST ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, the linear transfer risk defined in (13) satisfies Cλ W (S, T|fST ) := EO W (fST ) + λ · EI W (T X 0 ) ≤ 2Cλ KL(S, T|fST ) := 2(EO KL(fST ) + λ · EI KL(T X 0 )) (15) Such a relation between KL- and Wasserstein-based linear transfer risks (15) gives the following proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Consider transfer risk in linear form as in (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Suppose YT is a finite-dimensional Euclidean space and PT ≪ PST .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then for a given transfer learning problem (S, T, TX, TY , T0 X, T0 Y ), CW (S, T) ≤ 2CKL(S, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 Transfer Risk and Regret We will establish the connection between the transfer risk (12) and the transfer learning performance through a linear regression example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Consider a source task S and a target task T with the same input space XS = XT = Rd and the same input space YS = YT = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Both source and target data satisfy two (d + 1)-dimensional Gaussian distributions: (X·, Y·) ∼ N(µ·, Σ·) with µ· = �µ·,X µ·,Y � , Σ· = � Σ·,X Σ·,XY Σ·,Y X Σ·,Y � , (16) where µ·,Y and Σ·,Y ∈ R, µ·,X and Σ·,XY ∈ Rd, Σ·,Y X = Σ⊤ ,XY , and Σ·,X ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Define the sets of admissible source and target models AS = AT = {f : Rd → R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any f ∈ AS = AT , define the loss function as LS(f) = E∥YS − f(XS)∥2 2, LT (f) = E∥YT − f(XT )∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (17) Under such a setting, the optimal source and target models are obtained by direct computations: f∗ (x) = w⊤ x + b· with w· = Σ−1 ,XΣ·,XY , b· = µ·,Y − Σ·,Y XΣ−1 ,Xµ·,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (18) Transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Take the above linear regression example, and consider a simple setting where the input (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' output) transport set TX (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' TY ) is a singleton set only containing the identical mapping on Rd (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then, the transfer learning scheme (8) is equivalent to directly applying the optimal source model f∗ S to the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Consequently, the intermediate model set I in (10) is also a singleton set with I = {f∗ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now, define the transfer risk in this linear regression problem as the Wassersteinn-based output transport risk as in (14): CW (S, T) = CW (S, T|f∗ S) = EO W (f∗ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (19) 12 Regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Next, define the notion of regret as the gap between the transfer learning and the direct learning: R(S, T) := LT (f∗ S) − LT (f∗ T ) (20) Then, the following proposition shows that the transfer risk serves as a lower bound of the regret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For transfer learning in linear regression with Gaussian data, the regret with respect to the chosen intermediate model R(S, T) in (20) is lower bounded by the Wasserstein-based transfer risk in (19), CW (S, T) ≤ R(S, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 suggests that in evaluating the transfer learning scheme (8), transfer risk provides a proper initial indication of its effectiveness, especially for eliminating unsuitable candidate pretrained models or source tasks if the transfer risk is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4, together with detailed analysis of transfer risk and regret with Gaussian data, is in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 4 Numerical Experiments with Office-31 In this section, we will demonstrate the correlation between the performance of the transfer learning scheme (8) and the transfer risk (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3), through numerical experimentation using the Office-31 dataset for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 Experiment Set-up Recall the neural network architecture for the experiment introduced in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For each pair of the source and the target tasks, the source model is first trained using the source data, and then the fully connected layer of the pretrained model is fine tuned using half of the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The performance of the model is measured by the classification accuracy using the remaining of the target data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now let us define the explicit form of transfer risk for this example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Fix a source- target pair (S, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Recall that the input transport risk EI is a constant since the input transport set TX is a singleton set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' More specifically, we define the input transport risk as EI := W1(Law(XS), Law(XT )), (21) which is the Wasserstein-1 distance between the empirical distribution of (resized) source images Law(XS) and the empirical distribution of (resized) target images Law(XT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Meanwhile, for any fST ∈ I (11), we define the output transport risk as EO(fST ) = W1(PST , PT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Furthermore, as discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2, the transfer risk is given by C(S, T) = C(EI, min fST ∈I EO(fST )), (22) for some function C : R2 → R satisfying the regularity conditions in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Note the the optimal target distribution PT in the definition of EO(fST ) is unknown a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Thus, as suggested by Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2, we approximate EO(fST ) by W1(PST , Law(YT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Denote the approximated output transfer risk as �EO = min fST ∈I W1(PST , Law(YT )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (23) Finding �EO (23) is an optimization problem over a neural network function class fST ∈ I (11), which is solved by gradient descent in the numerical experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Finally, the (approximated) transfer risk is obtained by plugging �EO into (22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 13 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 Numerical Result Three different domains in Office-31 (A, D, and W) lead to 3 × 2 = 6 source-target pairs in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The accuracy, the input transport risk EI (21), and the output transport risk �EO (23) for each pair of source and target tasks are reported in the first three rows of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here the input transport risk is rescaled by a constant factor to achieve the same scale as the other metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In order to compute the transfer risk C in (22) given EI in (21) and �EO in (23), an appropriate form of function C in (22) need to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this experiment, we search C from the class of second order polynomials, so as to maximize the (absolute value of) correlation between the transfer learning accuracy and the transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, we define the risk in the following form: C(S, T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='31 · EI + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='92 · � �EO�2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (24) Transfer risks for all source-target pair are reported in the last row of Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Metric\\Task A-W A-D W-A W-D D-A D-W Accuracy 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='9% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='9% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='5% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='6% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='8% Input Risk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='148 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='263 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='148 Output Risk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='428 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='380 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='545 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='543 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='412 Transfer Risk 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='214 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='330 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='052 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='353 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='201 Table 1: Accuracy and transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Accuracy v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Figure 3 demonstrates a significant negative correlation between the transfer learning accuracy and the transfer risk: the higher the risk, the lower the transfer learning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For example, it can be observed from Figure 3 that transfer learning between DSLR and Webcam (D-W or W-D) results in low risk and high accuracy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' while transfer learning from those domains to Amazon (D-A or W-A) is risky and suffers from low accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Those numerical findings demonstrate the potential of transfer risk as an informative metric for the effectiveness of transfer learning task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 14 Figure 3: Accuracy and transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Computational benefit of transfer risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this numerical experiment on the Office-31 dataset, assessing transfer risk is computationally efficient and guaranteed by the early-stopping trick in deep learning: for each source-target pair, the optimization problem (23) is solved by running the gradient descent for a small and fixed number (∼10) of epochs, while the transfer learning problem is solved until the accuracy converges, which may take up to 100 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This early stopping trick is essentially equivalent to shrinking the search space of the output mapping from TY in (11) to some smaller class of neural networks TY 0 ⊂ TY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Indeed, as emphasized in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1, computing transfer risk (12) is to solve an optimization problem over the sets TX 0 and TY 0 , which can be much smaller than the function classes TX and TY involved in the transfer learning problem (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This reduction of the function classes demonstrates the potential and benefit of adopting transfer risk for computational efficiency: one can first perform the much easier computing task of the transfer risk, and then assess whether or not to resort to the full-scale and more computationally intense form of transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 5 Conclusion This paper establishes a mathematical framework for transfer learning, and addresses issues of feasibility and transferability through rigorous and comprehensive mathematical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A novel concept of transfer risk is introduced, which not only generalizes existing notions for transferability but also provides a unified framework for future studies on the impact and implications of various transfer learning techniques, including resizing, cropping, pruning, and distillation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 15 Tranfer Accuracy V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer Risk A-W A-D W-A Accuracy of transfer learning W-D 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='9 D-A D-W 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='5 Transfer risk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='31 * input risk + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='92 * (output risk)2References Bao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zamir, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Guibas, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' An information- theoretic approach to transferability in task transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In 2019 IEEE International Conference on Image Processing, pages 2309–2313.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ben-David, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Blitzer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Crammer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Kulesza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Pereira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Vaughan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A theory of learning from different domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Machine learning, 79(1):151–175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Blitzer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Crammer, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Kulesza, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Pereira, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Wortman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Learning bounds for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 20th International Conference on Neural Information Processing Systems, volume 20, page 129–136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Curran Associates Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Bu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zou, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Veeravalli, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Tightening mutual information-based bounds on generalization error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE Journal on Selected Areas in Information Theory, 1(1):121–130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Cook, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Feuz, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Krishnan, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning for activity recognition: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Knowledge and Information Systems, 36:537–556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Courty, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Flamary, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Tuia, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Rakotomamonjy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Optimal transport for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE Transactions on Pattern Analysis and Machine Intelligence, 39(9):1853–1865.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Dong, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Socher, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Li, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Fei-Fei, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' ImageNet: A large-scale hierarchical image database.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 248–255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Deng, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhang, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Marchi, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Schuller, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Sparse autoencoder-based feature transfer learning for speech emotion recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 2013 Humaine Association Conference on Affective Computing and Intelligent Interaction, pages 511–516.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Devlin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Chang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Lee, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Toutanova, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' BERT: Pre-training of deep bidirectional transformers for language understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, volume 1, pages 4171–4186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ganin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Lempitsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Unsupervised domain adaptation by backpropagation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 32nd International Conference on Machine Learning, volume 37, pages 1180–1189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ganin, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Ustinova, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Ajakan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Germain, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Larochelle, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Laviolette, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Marchand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Lempitsky, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Domain-adversarial training of neural networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Journal of Machine Learning Research, 17(59):1–35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Harremoës, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Vajda, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' On pairs of f-divergences and their joint range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE Transactions on Information Theory, 57(6):3230–3235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Hwang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Kuang, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A heterogeneous label propagation algorithm for disease gene discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 2010 SIAM International Conference on Data Mining, pages 583–594.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' SIAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Jiang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Zhai, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Instance weighting for domain adaptation in nlp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 45th Annual Meeting of the Association of Computational Linguistics, pages 264–271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 16 Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Cosa-Linan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Santhanam, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Jannesari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Maros, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Ganslandt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning for medical image classification: A literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' BMC Medical Imaging, 22(1):69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Leal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Laurière, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Lehalle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Learning a functional control for high-frequency finance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' arXiv preprint arXiv:2006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='09611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Lebichot, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Le Borgne, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', He-Guelton, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Oblé, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Bontempi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Deep-learning domain adaptation techniques for credit cards fraud detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Recent Advances in Big Data and Deep Learning: Proceedings of the 2019 INNS Big Data and Deep Learning Conference, pages 78–88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Shi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Ji, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Jia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A survey of sentiment analysis based on transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE Access, 7:85401–85412.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Long, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Cao, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Jordan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Learning transferable features with deep adaptation networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 32nd International Conference on Machine Learning, volume 37, pages 97–105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Long, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Ding, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Sun, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Yu, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer joint matching for unsupervised domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 1410–1417.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Pan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A survey on transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE Transactions on Knowledge and Data Engineering, 22(10):1345–1359.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Pan, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Xiang, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Liu, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning in collaborative filtering for sparsity reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 24th AAAI Conference on Artificial Intelligence, volume 24, pages 230–235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Rosenbaum, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Deep calibration of the quadratic rough heston model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' arXiv preprint arXiv:2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='01611.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Ruder, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Peters, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Swayamdipta, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Wolf, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning in natural language processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 2019 Conference of the North American Chapter of the Association for Computational Linguistics: Tutorials, pages 15–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Association for Computational Linguistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Saenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Kulis, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Fritz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Adapting visual category models to new domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 11th European Conference on Computer Vision, pages 213–226.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Sung, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Cho, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Bansal, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Vl-adapter: Parameter-efficient transfer learning for vision-and-language tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 5227–5237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Talagrand, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transportation cost for gaussian and other product measures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Geometric & Functional Analysis GAFA, 6(3):587–600.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Tan, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Sun, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Kong, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhang, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A survey on deep transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In International Conference on Artificial Neural Networks, pages 270–279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Springer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 17 Tan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' OTCE: A transferability metric for cross-domain cross- task representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 15779–15788.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Tong, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Xu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Huang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Zheng, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A mathematical framework for quantifying transferability in multi-source transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 35th International Conference on Neural Information Processing Systems, volume 34, pages 26103–26116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Curran Associates, Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Tzeng, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Hoffman, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Saenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Darrell, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Adversarial discriminative domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pages 7167–7176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Kikuchi, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Yi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zou, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhou, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Guo, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Transfer learning for retinal vascular disease detection: A pilot study with diabetic retinopathy and retinopathy of prematurity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' arXiv preprint arXiv:2201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='01250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Chen, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Hu, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Peng, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Philip, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Stratified transfer learning for cross-domain activity recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 2013 IEEE International Conference on Pervasive Computing and Communications, pages 1–10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' IEEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Wang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' and Deng, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Deep visual domain adaptation: A survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Neurocomputing, 312:135–153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Xia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhong, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Chen, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Structured pruning learns compact and accurate models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics, volume 1, pages 1513–1528.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Yuan, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Yao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Benatallah, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Darec: deep domain adaptation for cross-domain recommendation via transferring rating patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 28th International Joint Conference on Artificial Intelligence, pages 4227–4233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' AAAI Press.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Zeng, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Li, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Fei, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Yu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Pan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Wang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Automatic icd-9 coding via deep transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Neurocomputing, 324:43–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Zhao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Des Combes, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and Gordon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' On learning invariant representations for domain adaptation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In Proceedings of the 36th International Conference on Machine Learning, volume 97, pages 7523–7532.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' PMLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Zhuang, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Qi, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Duan, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Xi, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Zhu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', Xiong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', and He, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A comprehensive survey on transfer learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proceedings of the IEEE, 109(1):43–76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 18 Appendix A Mathematical Proofs A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 We will show that the optimization problem (8) is well-defined in the sense that an optimal pair of transport mappings (T X,∗, T Y,∗) for (8) is obtainable, under certain regularity conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' More specifically, we will focus on the following type of loss function LT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 (Proper loss function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let (X, Y ) be a pair of XT × YT -valued random variables with Law(XT , YT ) ∈ P(XT × YT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A loss functional LT over AT is said to be proper with respect to (X, Y ) if there exist a corresponding function LT : YT × YT → R bounded from below such that for any f ∈ AT , LT (f) = E[LT (Y, f(X))] = E[E[LT (Y, f(X))|X]];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' moreover, the function ˜LT : YT → R given by ˜LT (y) = E[LT (Y, Y ′)|Y ′ = y], ∀y ∈ YT , is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Examples of proper loss functions include mean squared error and KL-divergence and more generally the Bregman divergence, assuming that the first and second moments of Y conditioned on Y ′ = y is continuous with respect to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Without loss of generality, we shall in this section assume the input transport set TX contains all functions from XT to XS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We then specify the following assumptions for the well-definedness of (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assume the following regularity conditions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' LT is a proper loss functional with respect to (XT , YT );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' the image f∗ S(XS) is compact in (YS, ∥ · ∥YS);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' the set TY is such that the following set of functions ˜TY = { ˜T Y : XT → YT | ∃T Y ∈ TY s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' ˜T Y (x) = inf y∈f∗ S(XS) ˜LT (T Y (x, y)), ∀x ∈ XT } is compact in ({f|f : XT → YT }, ∥·∥∞), where for any f : XT → YT , ∥f∥∞ := supx∈XT ∥f(x)∥YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The proper choice of loss functions for LT is fairly general and includes the mean squared error, the KL-divergence, and more generally the Bregman divergence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' the compactness assump- tions can be fairly flexible as long as the target optimal model f∗ T can be written as f∗ T (x) = T Y (x, f∗ S(T X(x))), ∀x ∈ XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This compactness condition can be implemented by choosing a particular family of activation functions or imposing boundaries restrictions to weights and biases when constructing machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now we are ready to prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 under Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Since LT is proper, there exists a function LT : YT × YT → R such that inf (y,y′)∈YT ×YT LT (y, y′) > −∞, 19 and LT (T Y (·, (f∗ S ◦ T X)(·))) = E[LT (YT , T Y (XT , (f∗ S ◦ T X)(XT )))], ∀T X ∈ TX, T X ∈ TX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, for the function ˜LT (·) = E[LT (Y, Y ′)|Y ′ = ·], there exists m ∈ R such that ˜LT (y) ≥ m for any y ∈ YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now fix any T Y ∈ TY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The continuity of ˜LT and the continuity of T Y (x, ·) for each x ∈ XT guarantee the continuity of ˜LT (T y(x, ·)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Together with the compactness of f∗ S(XS), we have that for any x ∈ XT , Mx = arg min y∈f∗ S(XS) ˜LT (T Y (x, y)) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, for any T Y , one can construct ˜T X ∈ TX such that ˜T X(x) ∈ Mx T Y for any x ∈ XT and min T X∈TX LT (T Y (·, (f∗ S ◦ T X)(·))) = E[˜LT ( ˜T Y (XT ))] =: ˜LT ( ˜T Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The continuity of the new loss functional ˜LT comes from the continuity of the function ˜L, and the particular choice of the function space ({f|f : XT → YT }, ∥ · ∥∞), where {f|f : XT → YT } contains all functions from XT to YT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Since ˜TY is compact in ({f|f : XT → YT }, ∥ · ∥∞), the minimum over ˜TY is attained at some ˜T Y,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' According to the definition of ˜TY , there exists T Y,∗ ∈ TY such that ˜T Y,∗(·) = infy∈f∗ S(XS T Y,∗(·, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let T X,∗ be the ˜T X ∈ TX corresponding to T Y,∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any T X ∈ TX and T Y ∈ TY , we have LT (T Y (·, (f∗ S ◦ T X)(·))) ≥ LT (T Y (·, (f∗ S ◦ ˜T X)(·))) = ˜LT ( ˜T Y (·)) ≥ ˜LT ( ˜T Y,∗(·)) = LT (T Y,∗(·, (f∗ S ◦ T X,∗))(·)) ≥ min T X∈TX,T Y ∈TY LT � T Y (·, (f∗ S ◦ T X)(·)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, the transfer learning problem (8) is well-defined and it attains its minimum at (T X,∗, T Y,∗) described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' If one removes the compactness assumptions in Assumption (A), then a sufficiently rich family of output transport mappings is needed, such that the target optimal model f∗ T can be written as f∗ T (x) = T Y (x, f∗ S(T X(x))), ∀x ∈ XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' However, it is often difficult to verify if the set TY is sufficiently rich, due to the construction of neural networks as well as the choices of optimization algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The compactness conditions, on the other hand, can be implemented through choosing a particular family of activation functions or imposing boundaries restrictions to weights and biases when constructing machine learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 Properties of Transfer risk In this section, the mathematical properties of transfer risk (12) will be studied under mild assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the following discussion, we will fix a target task T and explore how transfer risk is affected by the choice of source task S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' There are two vital pieces of information obtained from the source task S based on the optimization problem in (8), and transfer risks in (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' One is the probability distribution of source input Law(XS), and the other is the pretrained model f∗ S in (3) Therefore, we can characterize source task S by (Law(XS), f∗ S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' More specifically, given a target task T and the source input and output spaces XS and YS, we can define a corresponding set of pretrained source tasks S ⊂ P(XS) × AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Without ambiguity on the target task T, we denote C(S, T) = C(S) = C(µ, f) for any S = (µ, f) ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For the 20 set of probability measures over XS, P(XS), we can adopt a metric function D : P(XS)×P(XS) → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then for the set of functions AS, fix a sufficiently large constant M > 0 and define the following metric: ∀f1, f2 ∈ AS, dM(f1, f2) := min{M, sup x∈XS ∥f1(x) − f2(x)∥YS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then for any S1, S2 ∈ S such that S1 = (µ1, f1) and S2 = (µ2, f2), define dS(S1, S2) := D(µ1, µ2) + dM(f1, f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (25) It is easy to verify that dS is a metric over S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the following discussion on continuity, the next assumption is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 ensures that the choice of input transfer risk is consistent with the metric dS in (25) defined between source tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any input transport mapping T X 0 ∈ TX 0 , assume the input transfer risk EI(T X 0 ) take the form EI(T X 0 ) := D(T X 0 #Law(XT ), Law(XS)), where D : P(XS) × P(XS) → R is the distance function appearing in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' By definition, the following degenerate case holds immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 (Zero transfer risk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Suppose XT = XS, YT = YS and the target task T ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then C(T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now, we consider source tasks S1, S2 ∈ S that differ only in the input distribution, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', S1 f = (µ1, f) and S2 f = (µ2, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then we have the following continuity property for C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 (Continuity in input distribution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assume Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Fix f ∈ AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' C(·, f) is continuous on (P(XS), D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Fix an arbitrary ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Take any µ ∈ P(XS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then we first establish the lower semi-continuity: For any T X 0 ∈ TX 0 and T Y 0 ∈ TY 0 , let fI denote the corresponding intermediate model from source model f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' By Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3, we have C(µ, f) − 1 2ϵ < C(µ, f|fI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' By triangle inequality of D and the Lipschitz property of C(µ, f|fI), take δ = ϵ 2L for any µ′ ∈ Bδ(µ) ⊂ P(XS), C(µ, f|fI) ≤ C(µ′, f|fI) + Lδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Notice that the choice of δ is independent of T X 0 and T Y 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, C(µ, f) − ϵ < C(µ′, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' fI) ⇒ C(µ, f) − ϵ < C(µ′, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now we show the upper semi-continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' From the infimum nature of C, there exists ¯T X 0 ∈ TX 0 and ¯T Y 0 ∈ TY 0 , with corresponding intermediate model ¯fI, such that C(µ, f| ¯fI) < C(µ, f) + 1 2ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Again, by triangle inequality of D and the Lipschitz property of C(µ, f|fI), take δ = ϵ 2L for any µ′ ∈ Bδ(µ) ⊂ P(XS), C(µ, f| ¯fI) ≥ C(µ′, f| ¯fI) − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then we have C(µ′, f) ≤ C(µ′, f| ¯fI) < C(µ, f) + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Hence, we conclude that C(·, f) is continuous on (P(XS), D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 21 This proposition shows that transfer risk will change continuously along with any modification in source input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Its proof indicates that the sensitivity of transfer risk with respect to the change in source input distribution depends on the Lipschitz constant L of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, one can modulate this sensitivity by carefully designing the C function in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For instance, for linear transfer risk Cλ in (13), the sensitivity can be controlled by varying the value of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Next, consider source tasks S1, S2 ∈ S that differ only in the pretrained model, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', S1 µ = (µ, f1) and S2 µ = (µ, f2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then we have the robustness of the transferability in terms of the continuity of C(µ, ·) in pretrained model f ∈ (AS, dM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 (Continuity in pretrained model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assume Assumption A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2, and assume that there exists a constant L > 0 such that for any T Y 0 ∈ TY 0 , T Y 0 (x1, y1) − T Y 0 (x2, y2) ≤ L (∥x1 − x2∥XT + ∥y1 − y2∥YS) , for all (x1, y1), (x2, y2) ∈ XT × YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Assume also that there exist some L′ > 0 and p ≥ 1 such that the output transfer risk satisfies ��EO(h1) − EO(h2) �� ≤ L′Wp(h1#Law(XT ), h2#Law(XT ))p for all h1, h2 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then C(µ, ·) is continuous on (AS, dM) for any fixed µ ∈ P(XS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proof of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Take any T X 0 ∈ TX 0 and T Y 0 ∈ TY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any f1, f2 ∈ (AS, dM), denote their corresponding intermediate model as f1 I and f2 I , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then we have |EO(f1 I ) − EO(f2 I )| ≤ L′Wp(f1 I #Law(XT ), f2 I #Law(XT ))p = L′ inf π∈Π(f1 I #Law(XT ),f2 I #Law(XT )) � YT ×YT ∥x − y∥p YT π(dx, dy) ≤ L′ inf γ∈Π(Law(XT ),Law(XT )) � XT ×XT ∥T Y 0 (x, f1(T X 0 (x))) − T Y 0 (y, f2(T X 0 (y)))∥2 YT π(dx, dy) ≤ 2p−1LpL′ � inf γ∈Π(Law(XT ),Law(XT )) � XT ×XT ∥x − y∥p XT dπ(dx, dy) + dM(f1, f2)p � = 2p−1LpL′ [Wp(Law(XT ), Law(XT ))p + dM(f1, f2)p] = 2p−1LpdM(f1, f2)p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The rest of the proof is similar to that of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This proposition shows that transfer risk will change continuously along with the modification in the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' As seen from the proof, the sensitivity of transfer risk with respect to the change in pretrained model is determined by three factors: (1) the Lipschitz constant inherited from the C function in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3, (2) the choice of output transport risk EO, and (3) the family of output transport mappings TY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In practice, one may control the sensitivity of the transfer risk through careful choices of those quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 lead to the following results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Suppose the conditions in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the transfer risk C as in Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 is continuous on (S, dS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 – A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 reveals that under a given target task, transfer risk is continuously influenced by both the changes in the source input and the pretrained model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, transfer risk is to evaluate the suitability of performing transfer learning and the appropriate choice of given source tasks for a target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 22 B Tranfer Risk and Regret with Gaussian Data In this section, we will revisit the example in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 will also be presented in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In the following discussion, for any spaces X and Y, we use the notation YX to denote the set of all the functions from X to Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' More specifically, consider a transfer learning problem in linear regression where the source and target data are sampled from two Gaussian distributions respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1 Basic case Let us first focus on the case where both data sources are of the same dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For the source task S, the input and the output spaces are XS = Rd and YS = R, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The source data (XS, YS) ∈ XS × YS is Gaussian distributed such that (XS, YS) ∼ N(µS, ΣS) with µS = �µS,X µS,Y � , ΣS = � ΣS,X ΣS,XY ΣS,Y X ΣS,Y � , (26) where µS,Y and ΣS,Y ∈ R, µS,X and ΣS,XY ∈ Rd, ΣS,Y X = Σ⊤ S,XY , and ΣS,X ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Take the set of admissible source models AS to be the set of functions f : Rd �→ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any f ∈ AS, let the source loss function be LS(f) = E∥YS − f(XS)∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (27) Then the optimal source model f∗ S ∈ arg min f∈AS LS(f) (28) is given by f∗ S(x) = w⊤ S x + bs, (29) where wS = Σ−1 S,XΣS,XY ∈ Rd, bS = µS,Y − ΣS,Y XΣ−1 S,XµS,X ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (30) Suchf∗ S is then used as the pretrained model for the following target task T, where the target input and output spaces are the same as in the source task, XT = XS and YT = YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The target data (XT , YT ) ∈ XT × YT follows a different Gaussian distribution from that in the source data such that (XT , YT ) ∼ N(µT , ΣT ), with µT = �µT,X µT,Y � , ΣT = � ΣT,X ΣT,XY ΣT,Y X ΣT,Y � , (31) where µT,Y and ΣT,Y ∈ R, µT,X and ΣT,XY ∈ Rd, ΣT,Y X = Σ⊤ T,XY , and ΣT,X ∈ Rd×d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The set of admissible target models is the same as the in the source task such that AT = AS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For any f ∈ AT , let the target loss function be LT (f) = E∥YT − f(XT )∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then similarly to the source task, the optimal target model f∗ T is given by f∗ T (x) = w⊤ T x + bT , ∀x ∈ Rd, (32) where wT = Σ−1 T,XΣT,XY , bT = µT,Y − ΣT,Y XΣ−1 T,XµT,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (33) The corresponding output distribution is then given by PT = E[Y |X] = N(w⊤ T µT,X + bT , w⊤ T ΣT,XwT ) = N(µT , w⊤ T ΣT,XwT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (34) 23 To initiate transfer learning from the source task S to the target task T, consider the sets of input and output transport mappings TX = {f|f : XT → XS} and TY = {f|f : XT × YS → YT }, with corresponding sets of initial transport mappings TX 0 = {idXT }, TY 0 = {idYS}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the set of intermediate models I is a singleton with I = {fST : fST = f∗ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Given the optimal models in both the source task and the target task, specified by (29)-(30) and (32)-(33), since the data distribution in the target task is given by (31), we have PST = f∗ S#N(µT,X, ΣT,X) = N(w⊤ S µT,X + bS, w⊤ S ΣT,XwS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (35) Notice that PT ≪ PST , therefore the Lebesgue decomposition leads to PT = ˜PT , such that d˜PT (y) dPST (y) = hST (y) = � w⊤ S ΣT,XwS w⊤ T ΣT,XwT exp �[y − (w⊤ S µT,X + bS)]2 2w⊤ S ΣT,XwS − [y − (w⊤ T µT,X + b)]2 w⊤ T ΣT,XwT � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (36) Direct computation leads to the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The KL-based output transfer risk is given by EO KL(fST ) =1 2 � ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY − log ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY − 1 + � µT,Y − µS,Y − ΣS,Y XΣ−1 S,X (µT,X − µS,X) �2 ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY � � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The Wasserstein-based output transfer risk is given by EO W (fST ) = � µT,Y − µS,Y − ΣS,Y XΣ−1 S,X (µT,X − µS,X) �2 + �� ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY − � ΣT,Y XΣ−1 T,XΣT,XY �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The computation shows that The risk in transfer learning is due to the discrepancy in the data distributions between source and target tasks, even when the source and target data are of matching dimensions and follow the same family of distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, in both the KL and the Wasserstein cases, the output transfer risk can be decomposed into two parts, one being the variance terms errorv,· determined by the covariance matrices of the source and target data, and the other being the bias terms errorb,· dependent on the difference between the expectations of µT and µS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' To see this, write EO KL(fST ) = errorv,KL(S, T) + errorb,KL(S, T), (37) EO W (fST ) = errorv,W (S, T) + errorb,W (S, T), (38) 24 where errorv,KL(S, T) = 1 2 � ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY − log ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY − 1 � , errorb,KL(S, T) = � µT,Y − µS,Y − ΣS,Y XΣ−1 S,X (µT,X − µS,X) �2 2ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' errorv,W (S, T) = �� ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY − � ΣT,Y XΣ−1 T,XΣT,XY �2 , errorb,W (S, T) = � µT,Y − µS,Y − ΣS,Y XΣ−1 S,X (µT,X − µS,X) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The KL-based variance term errorv,KL = h � ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY � , with the function h : (0, ∞) → R such that h(x) = 1 2(x − log x − 1) for any x > 0, which is strictly convex and reaches its minimum value 0 at x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Thus, for both the KL- and the Wasserstein-based output transfer risks, their variance risk components vanish if and only if ΣT,Y XΣ−1 T,XΣT,XY = ΣS,Y XΣ−1 S,XΣT,XΣ−1 S,XΣS,XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The bias risk components errorb,KL(S, T) and errorb,W (S, T) remain strictly positive unless the weighted difference between the expectations µT and µS is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Regret analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' By direct computation, one can show that the regret (20) for this linear transfer leaning problem is given by R(S, T) = ∥Σ 1 2 (wT − wS)∥2 2 + � µT,Y − µS,Y − ΣS,Y XΣ−1 S,X (µT,X − µS,X) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (39) We denote the first term in (39) as ˆ errorv(S, T) := ∥Σ 1 2 (wT − wS)∥2 2, and denote the second term in (39) as ˆ errorb(S, T) := � µT,Y − µS,Y − ΣS,Y XΣ−1 S,X (µT,X − µS,X) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Recall from (19) that the Wasserstein-based transfer risk for this problem is defined as CW (S, T) = EO W (fST ) in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Meanwhile, it can be easily verified by comparing (38) and (39) that R(S, T) = CW (S, T) + 2 � ∥Σ1/2 T,XwT ∥2∥Σ1/2 T,XwS∥2 − ⟨Σ1/2 T,XwT , Σ1/2 T,XwS⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (40) Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 is an immediate consequence of (40) and the Cauchy–Schwarz inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='4 suggests that for evaluating a transfer learning scheme as in (7), transfer risk provides a proper initial indication of its effectiveness, especially when eliminating unsuitable candidate pretrained models or source tasks if the transfer risk is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Further examining the decomposition of the transfer CKL and CW as well as the regret R, we notice that A vanishing bias term in transfer risks is equivalent to a vanishing bias term in regret, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', ˆ errorb(S, T) = 0 ⇐⇒ errorb,KL(S, T) = errorb,W (S, T) = 0 25 A vanishing variance term in transfer risk is necessary for a vanishing variance term in regret, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', ˆ errorv(S, T) = 0 =⇒ errorv,KL(S, T) = errorv,W (S, T) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The residual term 2 � ∥Σ 1 2 T,XwT ∥2∥Σ 1 2 T,XwS∥2 − ⟨Σ 1 2 T,XwT , Σ 1 2 T,XwS⟩ � in (40) depends entirely on the source and target covariance matrices ΣS and ΣT is due to the variance term in the learning objective difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, when CW (S, T) = 0 (or CKL(S, T) = 0), the training process is to reduce the angular distance between Σ1/2 T,XwS and Σ1/2 T,XwT caused by the discrepancy in these two covariance matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='2 Case with feature augmentation Let us now consider the case with feature augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' That is, compared with the input data in the source task, the target task includes more input information in the form of a higher input dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We will see that potential extra transfer risk as a result of the extra augmented input information as well as its benefit to eliminate the bias risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Take the same source task S as in the basic case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' for the target task T, let the input space XT = Rd+k with k ∈ N+, let the output space be the same as in the target task T such that YT = YS = R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Since the transfer learning problem has a feature augmentation, let us first define a projection XT from T X 0 to XS such that T X 0 (x) = � Id .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Od×k � x, ∀x ∈ XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the target data (XT , YT ) ∈ XT × YT satisfies that T X 0 (XT ) = XS and YT = YS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' That is, (XT , YT ) is given by a Gaussian distribution N(µT , ΣT ) with µT and ΣT in the same form as in (31), where µT,X = �µS,X µA,X � , µT,Y = µS,Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' ΣT,X = � ΣS,X ΣAS,X Σ⊤ AS,X ΣA,X � , ΣT,XY = �ΣS,XY ΣA,XY � , ΣT,Y = ΣS,Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here µA,X ∈ Rk denotes the expectation of the augmented variable ˜XT such that X⊤ T = � T X 0 (XT )⊤ ˜X⊤ T � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' in the above covariance matrix ΣT , ΣAS,X = Cov(XS, ˜XT ) ∈ Rd×k, ΣA,X = V ar( ˜XT ) ∈ Rk×k, and ΣA,XY = Cov( ˜XT , YT ) ∈ Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The optimal linear model f∗ T is again given by (32)-(33) with the optimal parameters wT and bT re-computed under the above modified target data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The corresponding output distribution PT is of the form (34) with updated parameters as in f∗ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' To initialize the transfer learning problem from S to T, consider TX 0 = {T X 0 }, TY 0 = {idYS}, TX = {f|f : XT → XS}, and TY = {f|f : XT × YS → YT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The set of intermediate models I is still singleton, with I = {fST : fST = f∗ S ◦ T X 0 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Clearly, PST = Law(f∗ S(XS)), with PT ≪ Law(f( SXS)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now we have The KL-based output transfer risks are given by EO KL(fST ) = 1 2 � ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣS,XY − log ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣS,XY − 1 � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (41) The Wasserstein-based output transfer risk is EO W (fST ) = �� ΣT,Y XΣ−1 T,XΣT,XY − � ΣS,Y XΣ−1 S,XΣS,XY �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (42) 26 Comparing the basic case and this feature augmentation case, we see The extra input information enables the particular choice of the initial input and output transport mappings, T X 0 and idYS, which in turn eliminates the bias risk component in both the KL- and Wasserstein-based output risk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Both output transfer risks come from their corresponding variance risk component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Take the KL-based output transfer risk in (41) as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We see that EO KL(fST ) = errorv,KL(S, T) = h � ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣS,XY � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' This suggests that the challenge of applying transfer learning with feature augmentation lies mainly at the uncertainty from the augmented variable ˜XT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In particular, if one assumes that the added input information ˜XT is uncorrelated with the existing input data XS, then ΣT,Y XΣ−1 T,XΣT,XY ΣS,Y XΣ−1 S,XΣS,XY = 1 + ΣA,Y XΣ−1 A,XΣA,XY ΣS,Y XΣ−1 S,XΣS,XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' That is, when one introduces new features that are uncorrelated with the existing ones, the variance risk component is always positive unless these new features are also uncorrelated with the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='3 Case with augmented output space Let us now consider the case with an extra prediction task, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=', a transfer learning problem with augmented output space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' In this case, we will see extra transfer risk with two major contributing factors, one being the unforeseeable correlation between the input and the extra output information and the other being the necessary initialization procedure due to the extra task in the target task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' To see this, let us consider a source task slightly modified from the basic case, where the output space in S is allowed be of dimension bigger that 1, that is, YS = Rl with l ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the source data (XS, YS) is given by a Gaussian distribution N(µS, ΣS) with µS and ΣS as in (26) except that µS,Y ∈ Rl, ΣS,XY ∈ Rd×l and ΣS,Y ∈ Rl×l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Again the optimal linear model f∗ S is given by (29)-(30) with optimal parameters wS and bS re-computed under the above modified source data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For the target task, let XT = XS and YT = Rl+k with k ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Since the transfer learning problem has an extra learning task with the same input data, the target data (XT , YT ) ∈ XT × YT satisfies that XT = XS and Y ⊤ T = � YS YA �⊤ from some random variable YA ∈ RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Let us assume that the joint distribution of the input and output variables (XT , YT ) follows a Gaussian distribution N(µT , ΣT ) with µT and ΣT in the same form as in (31), where µT,Y = �µS,Y µA,Y � , µT,X = µS,X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' ΣT,Y = � ΣS,Y ΣAS,Y Σ⊤ AS,Y ΣA,Y � , ΣT,Y X = �ΣS,Y X ΣA,Y X � , ΣT,XY = � ΣS,XY ΣA,XY � , ΣT,X = ΣS,X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Here µA,Y = E[YA] ∈ Rk denotes the expectation of YA, ΣAS,Y = Cov(YS, YA) ∈ Rl×k, ΣA,Y = V ar(YA) ∈ Rk×k and ΣA,XY = Σ⊤ A,Y X = Cov(XS, YA) ∈ Rd×k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then again the optimal linear model 27 f∗ T is in the form of (32) with parameters w⊤ T = ΣT,Y XΣ−1 T,X = � ΣS,Y XΣ−1 S,X ΣA,Y XΣ−1 S,X � = � w⊤ S ΣA,Y XΣ−1 S,X � , bT = µT,Y − ΣT,Y XΣ−1 T,XµT,X = �µS,Y µA,Y � − � ΣS,Y XΣ−1 S,XµS,X ΣA,Y XΣ−1 S,XµS,X � = � bS µA,Y − ΣA,Y XΣ−1 S,XµS,X � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Correspondingly, PT = E[YT |XT ] = N (µ1, Σ1), where µ1 = w⊤ T µT,X + bT = �w⊤ S µS,X + bS µA,Y � , Σ1 = w⊤ T ΣT,XwT = �w⊤ S ΣS,XwS w⊤ S ΣA,XY ΣA,Y XwS ΣA,Y XΣ−1 S,XΣA,XY � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' To initialize the transfer learning, consider the sets of input and output transport mappings TX = {f|f : XT → XS} and TY = {f|f : XT ×YS → YT }, as well as the sets of initial input transport mappings TX 0 = {idXT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' For the initial output mapping, in order to handle the newly added prediction task from XT = XS to YA, let us define an initial function f0 : Rd → Rk as f0(x) = w⊤ 0 x + b0 for any x ∈ Rd with fixed w0 ∈ Rk×d and b0 ∈ Rk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The set of initial output transport mappings is given by TY 0 = {T Y 0 : XT × YS → YT |T Y 0 (x, y) = � y⊤ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' f0(x)⊤ �⊤ }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Once again, the set of intermediate models I is a singleton with I = {fST : fST (x) = T Y 0 (x, f∗ S(x)), ∀x ∈ XT }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The probability distribution PST of the intermediate model fST is given by PST = N(µ2, Σ2), where µ2 = �w⊤ S w⊤ 0 � µT,X + �bS b0 � = �w⊤ S µS,X + bS w⊤ 0 µS,X + b0 � , Σ2 = �w⊤ S w⊤ 0 � ΣT,X � wS w0 � = �w⊤ S ΣS,XwS w⊤ S ΣS,Xw0 w⊤ 0 ΣS,XwS w⊤ 0 ΣS,Xw0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' We have again PT ≪ PST , and The KL- and Wasserstein-based output transfer risks are given by EO KL(fST ) = 1 2 � Tr(Σ−1 2 Σ1) − log det(Σ1) det(Σ2) − (k + l) + (µ1 − µ2)⊤Σ−1 2 (µ1 − µ2) � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (43) The Wasserstein-based output transfer risk is given by EO W (fST ) = ∥µ1 − µ2∥2 2 + Tr � Σ1 + Σ2 − 2 � Σ 1 2 1 Σ2Σ 1 2 1 � 1 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' (44) The analysis shows that with the augmented output space, the output transfer risks vanish if the initialization function f0 can neutralize the uncertainty brought by the correlation between the input XS and the additional output information YA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' To see this, take the example of the KL-based output transfer risk in (43), and decompose EO KL(fST ) in (43) into its variance and bias components as in (37), with errorv,KL(S, T) = 1 2Tr(Σ−1 2 Σ1) − log det(Σ1) det(Σ2) − (k + l), errorb,KL(S, T) = 1 2(µ1 − µ2)⊤Σ−1 2 (µ1 − µ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Now, we see that 28 If 0 < λ1 ≤ · · · ≤ λk+l are the eigenvalues of Σ−1 2 Σ1, and if Σ1 and Σ2 are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Then the variance term can be written as errorv(S, T) = k+l � i=1 (λi − log λi − 1) ≥ 0, which vanishes if and only if λi’s are all equal to 1 such that Σ1 = Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' The difference between the expectations of PT and PST is given by µ1 − µ2 = � 0 µA,Y − w⊤ 0 µS,X − b0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' Therefore, the error between the expected augmented output µA,Y and w⊤ 0 µS,X + b0 derived from the chosen initialization f0 is the main contributor to a strictly positive bias risk component errorb,KL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/sNFJT4oBgHgl3EQfcCyS/content/2301.11542v1.pdf'} diff --git a/t9E1T4oBgHgl3EQfjwTU/content/tmp_files/2301.03267v1.pdf.txt b/t9E1T4oBgHgl3EQfjwTU/content/tmp_files/2301.03267v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3f57bbfb2184a500bdaf63c5cff32632f9bc341f --- /dev/null +++ b/t9E1T4oBgHgl3EQfjwTU/content/tmp_files/2301.03267v1.pdf.txt @@ -0,0 +1,2846 @@ +arXiv:2301.03267v1 [physics.plasm-ph] 9 Jan 2023 +Non-modal kinetic theory of the stability of the compressed-sheared plasma +flows generated by the inhomogeneous microscale turbulence in the tokamak +edge plasma +V. S. Mikhailenko,1, a) V. V. Mikhailenko,2, b) and Hae June Lee3, c) +1)Plasma Research Center, Pusan National University, Busan 46241, South Korea +2)BK21 FOUR Information Technology, Pusan National University, Busan 46241, +South Korea +3)Department of Electrical Engineering, Pusan National University, Busan 46241, +South Korea +(Dated: 10 January 2023) +A nonmodal kinetic theory of the stability of the two-dimensional compressed-sheared mesoscale plasma +flows, generated by the radially inhomogeneous electrostatic ion cyclotron parametric microturbulence in the +pedestal plasma with a sheared poloidal flow, is developed. This theory reveals that the separate spatially +uniform Fourier modes of the electrostatic responses of the ions and of the electrons on the mesoscale con- +vective flows are determined only in the frames of references moved with velocities of the ion and electron +convective flows. In the laboratory frame, these modes are observed as the compressed-sheared modes with +time dependent wave numbers. The integral equation, which governs the separate Fourier mode of the electro- +static potential of the plasma species responses on the mesoscale convective flows, is derived. In this equation, +the effects of the compressing and shearing of the convective flows are revealed as the time dependence of +the finite ion Larmor radius effect. The solution of this equation for the kinetic drift instability displays the +nonmodal transformation of the potential to the zero frequency cell-like perturbation when time elapsed. +PACS numbers: 52.35.Ra, 52.35.Kt +I. +INTRODUCTION +The linear theory of the interaction of the fast waves +(FW) with tokamak plasma predicts1,2 that the injection +of FW may be the efficient method for the electron heat- +ing and current drive to aid in steady-state non-inductive +tokamak operation. +These predictions has been con- +firmed for the propagation and absorption of FWs in the +hot core tokamak plasma bounded by the last closed flux +surface (LCFS) on numerous tokamak devices over the +past half century. These experiments, however, demon- +strated that the efficiency of the FW heating and cur- +rent drive reduces by the FW power lost, which occurs +in the near-antenna layer of the cold low density scrape- +off layer (SOL) tokamak plasma. The FW heating ex- +periments on the National Spherical Torus eXperiment +(NSTX) showed3,4 that around 30% to more than 60% +of the FW energy lost directly in the SOL. The bursts +of the ions with energy above 20 keV, experimentally +observed5 in SOL following FW injection, and the devel- +opment of the parametric instabilities in SOL6, predicted +earlier theoretically7,8, were considered as the main chan- +nels of the FW absorption in SOL. The development of +the ion cyclotron (IC) quasimode decay instability was +considered5,6 as the main nonlinear process responsible +for the absorption of the FW power in SOL plasma and +a)E-mail:vsmikhailenko@pusan.ac.kr +b)E-mail: vladimir@pusan.ac.kr +c)E-mail: haejune@pusan.ac.kr +of the anomalous heating of ions in SOL. The analysis of +the turbulent heating of ions by the IC parametric tur- +bulence, powered by the IC quasimode decay instability, +was given in Ref.9 on the base of the numerical solution +of the dispersion equation for the IC parametric instabil- +ities driven by FW, and on the base of quasilinear theory +for ion distribution function, which accounted for the in- +teraction of ions with IC parametric turbulence powered +by the IC quasimode decay instability. The derived es- +timates for the turbulent ion heating rates revealed that +the absorption of the FW energy by ions in SOL is a +weak effect, which provides only negligibly small heating +of cold ions in SOL and can not be responsible for the +observed generation of the high energy ions. +The FW power loss in SOL was investigated in +Refs.10,11 by using the numerical full wave simula- +tion code AORSA (all-orders spectral algorithm)12,13, +in which the edge plasma beyond LCFS is included in +the solution domain. This simulation displays that the +dominant loss of the FW power occurs in the edge of +the tokamak plasma, where the plasma density and the +amplitude of the FW field strongly change on the ra- +dial intermediate spatial scale (mesoscale) between the +macroscale of the spatial inhomogeneity length of the +FW field in the bulk of the tokamak plasma, and the mi- +croscale commensurable with the wavelength of the para- +metric instabilities of the IC perturbations. The theory +of the microscale parametric turbulence of the inhomo- +geneous plasma driven by the strong inhomogeneous on +the mesoscale FW, was developed in Ref.14. This the- +ory reveals the effect of the formation of the radial and +poloidal convective flows of such a plasma caused by the + +2 +mesoscale spatial inhomogeneity of the microscale IC or +drift turbulence. The radial and poloidal convective flow +velocity components are proportional to the gradient of +the spectral intensity of the electric field of the micro- +turbulence. This result gives the possible explanation of +the FW heating experiment on the National Spherical +Torus eXperiment (NSTX)3,4,15, where it was found that +a significant part of the FW power loss occurs due to the +anomalous convective flow of the collisionless dense hot +plasma from the tokamak edge to the cold low density +SOL plasma. +The pedestal region, where plasma density profile has +largest radial gradient, is the most preferable region in +tokamaks for the development of the convective flows, +driven by the spatially inhomogeneous microturbulence. +The inherent component of the pedestal plasma is the +sheared poloidal flow, in which the drift type instabili- +ties, responsible for the anomalous transport of plasma, +are suppressed when their growth rates are less than the +flow velocity shearing rate16. It was proved in Refs.17–19 +that the basic point in understanding the processes of the +instabilities and turbulence evolution in plasma sheared +flows is the proper treatment of the persistent deforma- +tion of the perturbations by the sheared flows. This ef- +fect, which is completely ignored in the canonical stabil- +ity theory, where the perturbations are considered having +a static structure of a plane wave ∼ exp (ik · r − iωt), is +involved in the nonmodal kinetic theory, developed in +Refs.17–19, grounded on the methodology of the sheared +modes. It was found17,18 that in the sheared flow, the +separate spatial Fourier mode with a static spatial struc- +ture ∼ exp (ikxx + ikyy + ikzz) can be determined only +in the frame convected with a sheared flow. In the labo- +ratory frame, this mode is observed as the sheared mode +with time dependent structure resulted from the continu- +ous distortion with time the perturbation by the sheared +flow. This distortion grows with time and forms a time- +dependent nonmodal process which is investigated as the +initial value problem. +In Ref.20, the Vlasov equations, which govern the ion +and electron mesoscale convective flows with radially in- +homogeneous flow velocities in the poloidal sheared flow +were derived. These equations predict the generation of +the sheared poloidal convective flow and of the radial +compressed flow with radial flow velocity gradient. The +hydrodynamic theory of the mesoscale convective flows, +derived as the moments of the obtained Vlasov equation, +reveals the radial compressed convective flow as the dom- +inant factor in the formation of the steep pedestal den- +sity profile with density gradient exponentially growing +with time. The focus of this paper is the development +of the nonmodal kinetic theory of the stability of the +two-dimensional compressed-sheared convective flow. In +Sec. II, we present basic equations and their transforma- +tions. In Sec. III, we develop the nonmodal approach +to the kinetic theory of the compressed-sheared convec- +tive flows. In this theory we derived new spatial refer- +ence coordinates in which the distribution functions of +the unperturbed convective sheared-compressed flows is +stationary. The stability of such a distribution functions +of the convected plasma species against the development +of the short scale instabilities is given in Sec. IV employ- +ing the developed compressed-sheared modes approach. +The Conclusions are given in Sec. V. +II. +BASIC EQUATIONS AND TRANSFORMATIONS +Our theory is based on the Vlasov-Poisson system in +a slab geometry approximation where x, y, z directions +are viewed as corresponding to the radial, poloidal and +toroidal directions, respectively, of the toroidal coordi- +nate system. +Within this approximation, the Vlasov +equation for the velocity distribution function Fα of the +poloidal sheared flow of α plasma species (α = i for ions +and α = e for electrons) in the FW field with coordinates +r = (x, y, z) has a form +∂Fα (v, r, t) +∂t ++ v∂Fα (v, r, t) +∂r ++ eα +mα +� +E0x (x) + E1 (x, t) + ˜E (r, t) ++1 +c [v × (B0 + B1 (r, t))] +� ∂Fα (v, r, t) +∂v += 0. +(1) +This equation contains the inhomogeneous radial electric +field E0x (x), which governs the basic poloidal sheared +flow, the FW electric field E1 (x, t), the electric field +˜E (r, t) of the self-consistent plasma response on FW, +the uniform plasma-confining magnetic field B0 directed +along coordinate z, and FW magnetic field B1 (r, t). For +the edge layer of the tokamak plasma, this equation con- +tains two disparate spatial inhomogeneity lengths, which +are introduced by the FW field and by plasma parame- +ters. In the edge plasma, the spatial inhomogeneity of +E0x (x) and of FW fields are commensurable with a spa- +tial inhomogeneity length of the pedestal plasma density. +These mesoscale spatial inhomogeneity lengths on order +of the pedestal width are much less than the the inho- +mogeneity scale lengths of FW and of the plasma param- +eters in the plasma core, but are much larger than the +radial wavelengths of the IC parametric and drift micro- +turbulence. Electric field ˜E (r, t), being the microscale +responc of the inhomogeneous plasma on the inhomoge- +neous FW and E0x (x) fields, contains micro- and meso- +spatial scales. Our theory of the mesoscale plasma evo- +lution caused by the mesoscale inhomogeneities of the +microturbulence, involves the treatments on the micro- +and mesoscales. In our theory20, we introduced jointly +with variables r = (x, y, z) and time t for the microscale +fast variations on time of the order of the FW period or +of the period of the IC microturbulence, the slow time +T = εt, and the slow spatial variables X = εx, Y = εy, +where the dimensionless parameter ε ≪ 1, for the de- +scription of the slow evolutionary mesoscale processes in +the pedestal region. With these microscale and mesoscale + +3 +variables, electric field E0x depends only on X. The FW +fields E1, B1, determined as +E1 (X, t) = E1x (X) cos ω0t + E1y (X) sin ω0t, +(2) +B1 (X, t) = c +ω0 +dE1y (X) +dX +cos ω0t ez. +(3) +depend on slow mesoscale X and fast time t. The elec- +tric field ˜E (r, X, t) depends on the spatial micro- and +mesoscale variables and on the fast time. This field is +determined by the Poisson equation +∇ · ˜E (r, X, t) = 4π +� +α=i,e +eα +� +fα (v, r, X, t) dv, +(4) +in which fα is the fluctuating part of the distribution +function Fα, fα = Fα−F0α, where F0α is the equilibrium +distribution function. +It is obvious that it is not possible to apply directly +to the Vlasov-Poisson system (1), (4) with spatially in- +homogeneous oscillating FW fields the methods of the +solutions known for the investigations of the stability of +a plasma in static equilibrium. Any microscale perturba- +tions of the ion and electron densities are convected by +FW field with inhomogeneous on the mesoscale veloci- +ties, different for ions and electrons, and oscillating with +frequency of the FW field. It was found in Ref.14 that +the spatially inhomogeneous FW field may be excluded +from the Vlasov equation (1) by the transformation of +the velocity v and the position r = (x, y, z) variables of +the Vlasov equation (1) to new velocity vi and position +ri = (xi, yi, z) variables determined in the convected ref- +erence flow, which moves relative to the laboratory frame +with the velocity Vi (Xi, t) of ion in the FW field, given +by the equation +dVi (Xi, t) +dt += ei +mα +(E1 (Xi + εRix (Xi, t) , t) ++1 +c [Vi × B0] + 1 +c [Vi × B1 (Xi + εRix (Xi, t) , t)] +� +,(5) +with initial value Vi (Xi, t = t0 = 0) = 0. The Vlasov +equation (1) with new variables vi, Xi, contains the elec- +tric FW field only in terms on the order of |Ri/LE| ≪ 1, +where |Ri| is the amplitude of the ion displacement in +the spatially inhomogeneous FW field with spatial inho- +mogeneity scale length LE. These terms are negligibly +small14 for the conditions of the FW tokamak plasma +heating and may be neglected. +Without these terms, +the Vlasov equation in the frame convected with velocity +Vi (Xi, t) has a form as for a static equilibria without the +external FW field. That equation for ions, +∂Fi (vi, ri, Xi, t) +∂t ++ vi +∂Fi +∂ri ++ ei +mic [vi × B0] ∂Fi +∂vi ++ ei +mi +˜Ei (ri, Xi, t) ∂Fi (vi, ri, Xi, t) +∂vi += 0, +(6) +and similar equation for electrons, determined in the elec- +tron reference flow, and the Poisson equation for the elec- +tric field +∇ · ˜Ei (ri, Xi, t) = 4π +� +α=i,e +eα +� +fα (vα, rα, Xα, t) dvα,(7) +determined in the ion reference flow, compose the system +of equations for the investigation of the microscale tur- +bulence in FW field. The mesoscale variables Xi and Xe +are presented in this system as parameters. +At the time above the inverse growth rate of the IC +instabilities, t ≫ γ−1 (k) > |ω−1 (k) | the microscale IC +turbulence attains the steady state. +At this state the +electric field ˜Ei of the electrostatic two dimensional IC +parametric microturbulence, directed almost across the +magnetic field B0, may be presented in the ion reference +flow in the form +˜Ei (ri, Xi, t) = +� +n +˜Ei (ri, Xi, n, t) += +1 +(2π)2 +� +n +1 +2 +� +dk +� +˜Ei (k, Xi, n) eiψn(ri,Xi,t) ++˜E∗ +i (k, Xi, n) e−iψn(ri,Xi,t)� +, +(8) +where +ψn (ri, Xi, t) = −iωn (k) t + ikri + iθ (k) , +(9) +i. +e. +as a linear superposition of the electric fields +of IC perturbations with frequencies ωn (k) = nωci + +δωn (k) with wave vectors k directed across the mag- +netic field and with mesoscale position dependent am- +plitudes ˜Ei (k, Xi, n) and with phases fast changed with +time on the microscales. Reality of ˜Ei (ri, Xi, t) is in- +sured without introducing negative frequencies by the +addition of the complex conjugate terms with amplitudes +˜E∗ +i (k, Xi, n). The integration over k is performed over +wave vectors of the linearly unstable IC perturbations, +and θ (k) is their initial phase. In the electron frame, +oscillating relative to the ion frame, this electric field is +determined by Eq. (8) with species subscript i changed +on e. The electric field ˜Ee (re, Xe, t) in variables ri, Xi +has a form +˜Ee (re, Xe, t) = +� +n +˜Ee (re, Xe, n, t) += +1 +(2π)2 +� +n +1 +2 +� +dk +� +˜Ei (k, Xi, n) +∞ +� +p=−∞ +Jp (aie) +×eiψn(ri,Xi,t)−ip(ω0t+δie(k,Xi)) ++˜E∗ +i (k, Xi, n) +∞ +� +p=−∞ +Jp (aie) +×e−iψn(ri,Xi,t)+ip(ω0t+δie(k,Xi))� +, +(10) +where Jp (aie) is the first kind Bessel function of order +p with argument aie. Parameters aie ∼ kξie, where ξie +is the amplitude of the relative displacement of electrons +relative to ions in FW field, and δie were determined in +Ref.9. + +4 +III. +THE KINETIC THEORY OF THE MESOSCALE +COMPRESSED-SHEARED CONVECTIVE FLOWS +For the investigation on the slow time scale T the +mesoscale evolution of the poloidal plasma sheared flow +with microscale turbulence not suppressed by the sheared +flow, we transform the Vlasov-Poisson system from the +microscale to the mesoscale variables using the relations +xi = +1 +εXi, yi = +1 +εYi and t = +1 +εT variables. +With +mesoscale variables Eq. (6) becomes +ε∂Fi (vi, Xi, Yi, T, ε) +∂T ++ εvix +∂Fi +∂Xi ++ εviy +∂Fi +∂Yi ++ ei +mic [vi × B0] ∂Fi +∂vi ++ ei +mi +˜Ei (Xi, Yi, T, ε) ∂Fi +∂vi += 0, +(11) +The electric field ˜Ei (Xi, Yi, T, ε) is determined by Eq. +(8), in which phase ψn is determined in the form +ψn (Xi, Yi, T, ε) += −i1 +ε (ωn (k) T − ikxXi − ikyYi) + iθ (k) . +(12) +The similar transformations should be performed for the +electron Vlasov equation. +On the nonlinear stage of +the IC parametric microturbulence evolution at the time +above the inverse growth rate of the IC instabilities, +t ≫ γ−1 (k) > |ω−1 (k) |, electric field (8) becomes the +random function of the initial phase θ (k). The motion of +ions and electrons in this field has a form of the random +scattering of particles by the turbulent electric field and +mimics to the thermal motion of particles. +The average effect of the mesoscale inhomogeneity of +the microscale IC turbulence on the mesoscale evolu- +tion of the ion and electron distribution functions of +the poloidal sheared flow was considered in Ref.20. The +central point in this theory is the transformation of +the velocity vi and coordinates Xi and Yi to the new +microturbulence-associated velocity field ˜vi and coordi- +nates ˜Xi, ˜Yi determined by the relations20 +˜vi = vi − ˜Vi (Xi, Yi, T, ε) , +(13) +˜Xi = Xi − +t +� +t0 +˜Vix (Xi, Yi, T1, ε) dT1, +(14) +˜Yi = Yi − +T +� +T0 +˜Viy (Xi, Yi, T1, ε) dT1, +(15) +or by their inverse, +vi = ˜vi + ˜Ui +� +˜Xi, ˜Yi, T, ε +� +, +(16) +Xi = ˜Xi + ˜Rix +� +˜Xi, ˜Yi, T, ε +� += += ˜Xi + +T +� +T0 +˜Uix +� +˜Xi, ˜Yi, T1, ε +� +dT1, +(17) +Yi = Y − V ′ +0XT += ˜Yi + +T +� +T0 +˜Uiy +� +˜Xi, ˜Yi, T1, ε +� +dT1, +(18) +where V ′ +0 is the velocity shear of the poloidal flow velocity +V0 (X) = V ′ +0X directed along coordinate Y . The velocity +˜Vi (Xi, Yi, T, ε) is determined by the Euler equation +ε∂ ˜Vi +∂T + ε ˜Vix +∂ ˜Vi (Xi, Yi, T, ε) +∂Xi += ei +mi +� +˜Ei (Xi, Yi, T, ε) + 1 +c +� +˜Vi × B0 +�� +(19) +as +the +velocity +of +an +ion +in +the +electric +field +˜Ei (Xi, Yi, T, ε) of the IC parametric turbulence, where +variables Xi and Yi are determined in the frame of ref- +erences which moves with the velocity of an ion in the +FW field in the poloidal sheared flow20. +In variables +� +˜Xi, ˜Yi, T +� +, the convective nonlinear part ˜Vix +∂ +∂Xi of the +operator +∂ +∂T + ˜Vix +∂ +∂Xi of Eq. (19) vanishes and this op- +erator is transformed to the linear one, +∂ +∂T . Then, Eq. +(19) becomes the ordinary differential equation +ε d +dT +˜Ui +� +˜Xi, ˜Yi, T, ε +� += ei +mi +� +˜Ei +� +˜Xi + ˜Rix +� +˜Xi, ˜Yi, T, ε +� +, ˜Yi, T, ε +� ++1 +c +� +˜Ui +� +˜Xi, ˜Yi, T, ε +� +× B0 +�� +. +(20) +for ˜Ui +� +˜Xi, ˜Yi, T, ε +� += ˜Vi (Xi, Yi, T, ε). The solution to +Eq. (20) may be easily derived20 for the case of the small +displacement, +��� ˜Rix +��� ≪ L ˜ +E, of an ion in the inhomoge- +neous electric field ˜Ei. With the approximation for the +amplitude ˜Ei (k, Xi, n) of the n-th harmonic of the IC +electric field in Eq. (20) +˜Ei +� +k, ˜Xi + ˜Rix +� +˜Xi, ˜Yi, T, ε +� +, n +� +≈ ˜Ei +� +k, ˜Xi, n +� +,(21) +the +solution to +Eq. +(20) +with +the +initial value +˜Ui +� +˜Xi, ˜Yi, T0 = 0 +� += 0 is +˜Uix +� +˜Xi, ˜Yi, T, ε +� += +ei +εmi +� +n +T +� +0 +dT1 +× +� +˜Eix +� +˜Xi, ˜Yi, n, T, ε +� +cos 1 +εωci (T − T1) ++ ˜Eiy +� +˜Xi, ˜Yi, n, T, ε +� +sin 1 +εωci (T − T1) +� +, +(22) + +5 +˜Uiy +� +˜Xi, ˜Yi, T, ε +� += +ei +εmi +� +n +T +� +0 +dT1 +× +� +− ˜Eix +� +˜Xi, ˜Yi, n, T, ε +� +sin 1 +εωci (T − T1) ++ ˜Eiy +� +˜Xi, ˜Yi, n, T, ε +� +cos 1 +εωci (T − T1) +� +, +(23) +With variables ˜vi, +˜Xi, ˜Yi, Zi, T, ε, where Zi += +εz, +the +Vlasov +equation +for +the +distribution +function +Fi +� +˜vi, ˜Xi, ˜Yi, Zi, T, ε +� +of ions in the sheared poloidal flow +for time T ≫ τcorr ∼ γ−1 becomes +ε∂Fi +∂T + ε˜vix +∂Fi +∂ ˜Xi ++ ε (˜viy − V ′ +0T ˜vix) ∂Fi +∂ ˜Yi ++ εviz +∂Fi +∂Zi +− ε˜vix +T +� +0 +∂ +∂Xi +˜Vix (Xi, Yi, T1, ε) dT1 +∂Fi +∂ ˜Xi +− ε˜vix +T +� +0 +∂ +∂Xi +˜Viy (Xi, Yi, T1, ε) dT1 +∂Fi +∂ ˜Yi +− ε ˜Uix +� +˜Xi, ˜Yi, T, ε +� +T +� +0 +∂ +∂Xi +˜Vix (Xi, Yi, T1, ε) dT1 +∂Fi +∂ ˜Xi +− ε ˜Uix +� +˜Xi, ˜Yi, T, ε +� + +V ′ +0T + +T +� +0 +∂ +∂Xi +˜Viy (Xi, Yi, T1, ε) dT1 + + ∂Fi +∂ ˜Yi ++ ωci˜viy +∂Fi +∂˜vix +− ωci˜vix +∂Fi +∂˜viy +− ε ei +mi +� ∂ +∂ ˜Xi +ϕi +� +˜ri + ˜Ri, ˜Xi, ˜Yi, t +� +− V ′ +0T ∂ +∂ ˜Yi +ϕi +� +˜ri + ˜Ri, ˜Xi, ˜Yi, t +�� ∂Fi +∂˜vix +− ε ei +mi +∂ +∂ ˜Yi +ϕi +� +˜ri + ˜Ri, ˜Xi, ˜Yi, T +� ∂Fi +∂˜viy +− ε ei +mi +∂ +∂Zi +ϕi +� +˜ri + ˜Ri, ˜Xi, ˜Yi, t +� ∂Fi +∂viz += 0. +(24) +The electrostatic potential ϕi in Eq. (24) depends on the +micro- and mesoscales and can be expressed in the form +ϕi +� +˜ri + ˜Ri, ˜Xi, ˜Yi, t +� += ˜ϕi +� +˜ri + ˜Ri, ˜Xi, t +� ++ Φi +� +˜Xi, ˜Yi, Zi, T +� +, +(25) +where ˜ϕi is the electrostatic potential of the microscale +turbulence, +˜Ei +� +˜Xi, ˜Yi, T, ε +� += −∇riϕi +� +˜ri + ˜Ri, ˜Xi, t +� +, +(26) +and Φi +� +˜Xi, ˜Yi, Zi, T +� +is the potential which determines +the electric field of the plasma response on the mesoscale +convective flows, +¯Ei +� +˜Xi, ˜Yi, Zi, T +� += −∇Φi +� +˜Xi, ˜Yi, Zi, T +� +. +(27) +The Vlasov equation for the ion distribution function +¯Fi +� +˜vi, ˜Xi, ˜Yi, Zi, T, ε +� +, averaged over the microscale ini- +tial phases for a time t ≫ τcorr ∼ γ−1, is +∂ ¯Fi +∂T + ˜vix +∂ ¯Fi +∂ ˜Xi ++ (˜viy − V ′ +0T ˜vix) ∂ ¯Fi +∂ ˜Yi +− ¯Uix +� +˜Xi +� ∂ ¯Fi +∂ ˜Xi +− ¯Uiy +� +˜Xi +� ∂ ¯Fi +∂ ˜Yi ++ viz +∂ ¯Fi +∂Zi ++ 1 +εωci˜viy +∂ ¯Fi +∂˜vix +− 1 +εωci +∂ ¯Fi +∂˜viy +− ei +mi + + +∂Φi +� +˜Xi, ˜Yi, Zi, T +� +∂ ˜Xi +− V ′ +0T +∂Φi +� +˜Xi, ˜Yi, Zi, T +� +∂ ˜Yi + + ∂Fi +∂˜vix +− ei +mi +∂Φi +� +˜Xi, ˜Yi, Zi, T +� +∂ ˜Yi +∂Fi +∂˜viy +− ei +mi +∂Φi +� +˜Xi, ˜Yi, Zi, T +� +∂Zi +∂Fi +∂viz += 0, +(28) + +6 +where the velocities ¯Uix +� +˜Xi +� +and ¯Uiy +� +˜Xi +� +are +¯Uix +� +˜Xi +� += +� +˜Uix +� +˜Xi, ˜Yi, T, ε +� +T +� +0 +∂ +∂Xi +˜Vix (Xi, Yi, T1, ε) dT1 +� +, +(29) +¯Uiy +� +˜Xi +� += +� +˜Uix +� +˜Xi, ˜Yi, T, ε +� +T +� +0 +∂ +∂Xi +˜Viy (Xi, Yi, T1, ε) dT1 +� +. +(30) +The details of the calculation of ¯Uix +� +˜Xi +� +and ¯Uiy +� +˜Xi +� +for the arbitrary electric field ˜Ei are given in Ref.14. +For the electric field (8), the velocities ¯Uix +� +˜Xi, T +� +and +¯Uiy +� +˜Xi, T +� +are presented in the Appendix. +The electrostatic potential Φi +� +˜Xi, ˜Yi, Zi, T +� +of the +plasma response on the mesoscale convective flows is gov- +erned by the Poisson equation +∂2Φi +� +˜Xi, ˜Yi, Zi, T, ε +� +∂ ˜ +X2 +i ++ +∂2Φi +� +˜Xi, ˜Yi, Zi, T, ε +� +∂ ˜ +Y 2 +i ++ +∂2Φi +� +˜Xi, ˜Yi, Zi, T, ε +� +∂Z2 +i += −4π +� +α=i,e +eαnα +� +˜Xα, ˜Yα, Zα, T, ε +� +. +(31) +in +which +nα +� +˜Xα, ˜Yα, Zα, T, ε +� += +� +d˜vα ¯fα +� +˜vα, ˜Xα, ˜Yα, Zα, T, ε +� +is +the +density +perturbation, +and +¯fα +� +˜vα, ˜Xα, ˜Yα, Zα, T, ε +� += +¯Fα (˜vα⊥, φ, vz, ξα, ηα, Zα, T, ε) − +¯Fα0 +is +the +pertur- +bation of the equilibrium distribution function ¯Fα0 of +the convected plasma species α. +The Vlasov equation for the average electron distribu- +tion ¯Fe +� +˜ve, ˜Xe, ˜Ye, Ze, T +� +, where ˜Xe, ˜Ye are determined +by Eqs. +(14), (15) with ion species subscript changed +on the electron species subscript, has a form similar to +Eq. (28). The velocities ¯Uex +� +˜Xe +� +and ¯Uey +� +˜Xe +� +are de- +termined in this equation by Eqs. (29), (30) in which +velocities ˜Uex +� +˜re, ˜Xe, T +� +and ˜Uey +� +˜re, ˜Xe, T +� +are deter- +mined by Eqs. (22), (23) with the turbulent electric field +˜Ee +� +˜re, ˜Xe, T +� +, given by Eq. (10). +The Vlasov equations (28) for ¯Fi +� +˜vi, ˜Xi, ˜Yi, Zi, T +� +, +and the similar equation for +¯Fe +� +˜ve, ˜Xe, ˜Ye, Ze, T +� +, and the Poisson equation (31) +compose the Vlasov-Poisson system, which governs the +kinetic mesoscale evolution of a plasma under the average +action of the spatially inhomogeneous microturbulence. +As a first step to solution of the system of Eqs. (28), +(31), we find the characteristics of the operator +D +DT = ∂ +∂T − ¯Uix +� +˜Xi +� +∂ +∂ ˜Xi +− ¯Uiy +� +˜Xi +� ∂ +∂ ˜Yi +, +(32) +which are determined by the system +dT = − +d ˜Xi +¯Uix +� +˜Xi +� = − +d ˜Yi +¯Uiy +� +˜Xi +�. +(33) +For deriving the simplest solution to system (33), which +reveals the effects of the spatial inhomogeneity of the con- +vective flow velocities, we use in Eq. (33) the expansions +¯Uix +� +˜Xi +� += ¯U (0) +ix + ¯U ′ +ix +� +˜X(0) +i +� � +˜Xi − ˜X(0) +i +� +, +(34) +and +¯Uiy +� +˜Xi +� += ¯U (0) +iy + ¯U ′ +iy +� +˜X(0) +i +� � +˜Xi − ˜X(0) +i +� +(35) +at the vicinity of an arbitrary coordinate ˜X(0) +i +, and con- +sider the case of the uniform velocity compressing rate, +¯U ′ +ix = const, and of the uniform velocity shearing rate, +¯U ′ +iy = const. The solution to system (33) for this case +has a form20 +ˇXi = +1 +¯U ′ +ix +�� +¯U (0) +ix + ¯U ′ +ix +� +˜Xi − ˜X(0) +i +�� +e +¯U′ +ixT +− ¯U (0) +ix +� +, +(36) +and +ˇYi = ˜Yi + +� +¯U (0) +iy − ¯U (0) +ix +¯U ′ +iy +¯U ′ +ix +� +T +− +¯U ′ +iy +� ¯U ′ +ix +�2 +� +¯U (0) +ix + ¯U ′ +ix +� +˜Xi − ˜X(0) +i +�� +, +(37) +where ˇXi and ˇYi are the integrals of system (33) with +expansions (34), (35). Note, that at T = 0, ˇXi = ˜Xi − +˜X(0) +i +. In what follows, we put X0 = 0 for simplicity. +With variables ˇXi, ˇYi, Zi, T, vz, ε and with ˜vi⊥, φ, de- +termined by relations ˜vix = ˜vi⊥ cos φ and ˜viy = ˜vi⊥ sin φ, +the Vlasov equation (28) obtains the form, which does +not contain the explicit dependence on ˜Xi, + +7 +∂ +∂T +¯Fi +� +˜vi⊥, φ, vz, ˇXi, ˇYi, Zi, T, ε +� ++ ˜vi⊥ cos φ e +¯U′ +ixT ∂ ¯Fi +∂ ˇXi ++ +� +˜vi⊥ sin φ − ˜vi⊥ cos φ +� +V ′ +0T + +¯U ′ +iy +¯U ′ +ix +�� +∂ ¯Fi +∂ ˇYi ++ viz +∂ ¯Fi +∂Zi +− ωci +∂ ¯Fi +ε∂φ +− ei +mi +� +e +¯U′ +ixT ∂Φi +� ˇXi, ˇYi, Zi, T +� +∂ ˇXi +− +� +V ′ +0T + +¯U ′ +iy +¯U ′ +ix +� +∂Φi +� ˇXi, ˇYi, Zi, T +� +∂ ˇYi +� +∂ ¯Fi +∂ˇvix +− ei +mi +∂Φi +� ˇXi, ˇYi, Zi, T +� +∂ ˇYi +∂ ¯Fi +∂˜viy +− ei +mi +∂Φi +� ˇXi, ˇYi, Zi, T +� +∂Zi +∂ ¯Fi +∂viz += 0. +(38) +In Eq. +(38), the effects the spatial inhomogeneity of +the sheared and convected flows is transformed to the +time domain. +These effects are presented by the lin- +early growing with time V ′ +0T coefficient originated from +the basic poloidal sheared flow, and of the exponentially +growing with time coefficient e ¯U′ +ixT originated from the +compressed convective flow. These coefficients reveal the +effects of the continuous distortion of the perturbations +in the sheared and compressed flows17–19. It was found +in Ref.20 that the compressing rate ¯U ′ +ix of the radial ve- +locity of the compressed flow and the shearing rate V ′ +0 of +the poloidal sheared flow velocity are commensurable for +the tokamak edge condition, and both are much less than +the ion cyclotron frequency ωci. Equation (38) displays +that the exponentially growing with time effect of the +compressed flow is the dominant factor in the mesoscale +temporal evolution at time T > +� ¯U ′ +ix +�−1 of the plasma +with a radially inhomogeneous turbulence. Therefore the +small parameter ε in Eq. (38), which determines a ratio +of the micro- to meso- scales, is naturally to define as +equal to ε = +¯U′ +ix +ωci . +The next substantial simplification of Eq. (38) arises +with transformation of the part of Eq. (38), which does +not contain the electrostatic potential Φi +� ˇXi, ˇYi, T +� +, by +employing the characteristic equations +dT = −εdφ +ωci += dZi +vz += +d ˇXi +˜vi⊥ cos φ e ¯U′ +ixT += +d ˇYi +˜vi⊥ sin φ − ˜vi⊥ cos φ +� +V ′ +0T + +¯U′ +iy +¯U′ +ix +�. +(39) +The solutions to Eqs. (39) are given by the relations +ˇXi = ξi − ˜vi⊥ε +ωci +e +¯U′ +ixT sin +� +φ1 − 1 +εωciT +� ++ O +�ε ¯U ′ +ix +ωci +≪ 1 +� +, +(40) +ˇYi = ηi + ˜vi⊥ε +ωci +cos +� +φ1 − 1 +εωciT +� ++ ˜vi⊥ε +ωci +� +V ′ +0T + +¯U ′ +iy +¯U ′ +ix +� +sin +� +φ1 − 1 +εωciT +� ++ O +� +ε V ′ +0 +ωci +≪ 1 +� +, +(41) +φ = φ1 − 1 +εωciT, +(42) +Zi = Zi1 + vzT. +(43) +The integrals ξi and ηi in Eqs. (40) and (41) are the guid- +ing center coordinates in the compressed-sheared convec- +tive flow. In coordinates ξi, ηi, φ1, Zi1, Eq. (38) has a +simple form +∂ +∂T +¯Fi (˜vi⊥, φ1, vz, ξi, ηi, Zi, T, ε) + ei +mi +ωci +ε˜vi⊥ +�∂Φi +∂φ1 +∂ ¯Fi +∂˜vi⊥ +− ∂Φi +∂˜vi⊥ +∂ ¯Fi +∂φ1 +� ++ +eiε +miωci +e +¯U′ +ixT +�∂Φi +∂ξi +∂ ¯Fi +∂ηi +− ∂Φi +∂ηi +∂ ¯Fi +∂ξi +� +− ei +mi +∂Φi +∂Zi1 +∂ ¯Fi +∂vz += 0. +(44) +The solution to Eq. (44) for the ion distribution function +¯Fi we derive in the form ¯Fi (˜vi⊥, φ, vz, ξi, ηi, Zi1, T, ε) = +¯Fi0 + ¯fi (˜vi⊥, φ, vz, ξi, ηi, Zi1, T, ε), where ¯Fi0 is the ion +distribution function ¯Fi0 of the unperturbed ion convec- +tive flow, and ¯fi is the perturbation of ¯Fi0 caused by the +ions respond on the plasma convective flows. The equa- +tion for ¯Fi0 follows from Eq. (44), in which potential Φi +of the electrostatic response of plasma on the mesoscale +convective flows is excluded. This equation, +∂ ¯Fi0/∂T = 0, +(45) +reveals that with the guiding center coordinates ξi, ηi, + +8 +the unperturbed ion distribution function +¯Fi0 of the +compressed-sheared ion flow is stationary. The solution +to Eq. (45) for the inhomogeneous ion component along +coordinate ˜Xi is an arbitrary function ¯Fi0 = ¯Fi0 (˜vi, ξi). +With the initial Maxwellian distribution +¯Fi0 +� +˜vi, ˜Xi +� += +ni0 +� +˜Xi +� +� +2πv2 +T i +� +˜Xi +��3/2 e +− +˜v2 +i +2v2 +T i( ˜ +Xi) , +(46) +given for time T = 0, at which ξi ≈ ˇXi = ˜Xi (it follows +from Eq. (36) and from the estimate ˜vi/ωci ∼ vT i/ωci ≪ +(LE, Lni)) the solution for ¯Fi0 will have a form (46), in +which the ion equilibrium density and the ion thermal ve- +locity are equal to ni0 (ξi) and vT i (ξi) respectively. Note, +that with variable ˜Xi the ion density ni0 is the time de- +pendent, +ni0 (ξi) = ni0 +� 1 +¯U ′ +ix +�� +¯U (0) +ix + ¯U ′ +ix ˜Xi +� +e +¯U′ +ixT +− ¯U (0) +ix +�� +. +(47) +The same dependences on ˜Xi and T has the ion ther- +mal velocity vT i (ξi) in Eq. (46). Equation (46) reveals, +that the time dependent inhomogeneous ion density of +the convective flow, as it is with coordinate ˜Xi, becomes +the steady spatially inhomogeneous ion density distribu- +tion along characteristic ξi at any time T . +IV. +THE COMPRESSED-SHEARED MODES +APPROACH TO THE STABILITY THEORY OF THE +MESOSCALE CONVECTIVE FLOWS +In this section, we consider the microscale respond of +the ions and electrons, determined by the functions ¯fi +and ¯fe on the generation of the mesoscale convective +flows. +It follows from Eq. (44), that the Vlasov equation for +the perturbation +¯fi (˜vi⊥, φ, viz, ξi, ηi, Zi1, T, ε) of ¯Fi0 (˜vi, ξi) has a simple +form in guiding center coordinates ξi and ηi, +∂ +∂T +¯fi (˜vi⊥, φ1, viz, ξi, ηi, Zi1, T, ε) = − ei +mi +ωci +ε˜vi⊥ +∂Φi +∂φ1 +∂ ¯Fi0 +∂˜vi⊥ ++ +eiε +miωci +e +¯U′ +ixT ∂Φi +∂ηi +∂ ¯Fi0 +∂ξi ++ ei +mi +∂Φi +∂Zi1 +∂ ¯Fi0 +∂viz +. +(48) +The Vlasov equation (48) for ¯fi with given unperturbed +ion distribution function ¯Fi0, the equation for the pertur- +bation ¯fe (˜ve⊥, φ1, vz, ξe, ηe, Ze1, T, ε) of the electron dis- +tribution similar to Eq. (48), and the Poisson equation +(35) for the potential Φi +� ˇXi, ˇYi, Zi1, T, ε +� +in coordinates +ˇXi, ˇYi +e2 ¯U′ +ixT ∂2Φi +∂ ˇX2 +i +− 2e +¯U′ +ixT +� +V ′ +0T + +¯U ′ +iy +¯U ′ +ix +� +∂2Φi +∂ ˇXi∂ ˇYi ++ + +1 + +� +V ′ +0T + +¯U ′ +iy +¯U ′ +ix +�2 + ∂2Φi +∂ ˇY 2 +i ++ ∂2Φi +∂Z2 +i1 += −4π +� +eini +� ˇXi, ˇYi, Zi1, T ; ε +� +−|e|ne +� ˇXe, ˇYe, Ze1, T ; ε +�� +. +(49) +compose the system of equations for the investigations +of the stability of the mesoscale convective flows. +In +this section, we consider the stability of the compressed- +sheared flows against the development of the low fre- +quency microscale instabilities. For that goal, we trans- +form Eqs. +(48), (49) to the microscale time t = +T +ε , +the microscale spatial coordinates xi = 1 +εXi, yi = 1 +εYi +and to microscale coordinates guiding center coordinates +ˇξi = ξi +ε , ˇηi = ηi +ε . With time t and microscale coordinates +ˇξi, ˇηi, the solution to Eq. (48) with known distribution +¯Fi0 is +¯fi = ei +mi +t +� +0 +dt1 +� +e ¯U′ +ixt +ωci +∂Φi +∂ˇηi +∂ ¯Fi0 +∂ ˇξi +− ωci +˜vi⊥ +∂Φi +∂φ1 +∂ ¯Fi0 +∂˜vi⊥ ++ ∂Φi +∂Zi1 +∂ ¯Fi0 +∂vz +� +, +(50) +where the prime in ¯U ′ +ix, ¯U ′ +iy and V ′ +0 denotes in this sec- +tion the derivatives of ¯Uix, ¯Uiy and V0 with respect to +the microscale coordinate ˜xi = +˜ +Xi +ε . Equation (50), as +well as Eq. (44), do not contain the spatial inhomogene- +ity originated from the inhomogeneity of the convective +flows velocities. Therefore, by the Fourier transforming +the potential Φi (ˇxi, ˇyi, zi1, t) over the microscale spatial +coordinates ˇxi, ˇyi, +Φi (ˇxi, ˇyi, zi1, t, ) = +1 +(2π)3 +� +dkˇxidkˇyidkz +× Φi (kˇxi, kˇyi, kz, t) ei(kˇxi ˇxi+kˇ +yi ˇyi+kzzi) +(51) +we will derive from Eqs. (48) and (49) the equation for +the separate spatial Fourier mode Φi (kˇxi, kˇyi, kz, t) of the +microscale plasma response on the mesoscale compressed- +sheared convective flows. With coordinates ˇξi, ˇηi, used +in Eq. (50), potential Φi +�ˇξi, ˇηi, zi1, t +� +is determined by +the relation +Φi (ˇxi, ˇηi, zi1, t) = +1 +(2π)3 +� +dkˇxidkˇyidkZ +× Φi (kˇxi, kˇyi, kz, t) ei(kˇxi ˇξi+kˇ +yi ˇηi+kzzi1) +× exp +� +−iki⊥ (t) ˜vi⊥ +ωci +sin (φ − ωcit − δ (t)) +� += +� +dkˇxidkˇyidkzΦi (kˇxi, kˇyi, kz, t) +× ei(kˇxi ˇξi+kˇ +yi ˇηi+kzzi1) +× +∞ +� +n=−∞ +Jn +�ki⊥ (t) ˜vi⊥ +ωci +� +× e−in(φ1−ωcit−δ(t)), +(52) + +9 +in which Jn is the Bessel function of the first kind of the +order n and the wave number component ki⊥ (t) across +the magnetic field grows with time due to the distort- +ing of the wave structure by the compressed and sheared +flows, +k2 +i⊥ (t) = +� +kˇxie +¯U′ +ixt − kˇyi +� +V ′ +0t + +¯U ′ +iy +¯U ′ +ix +��2 ++ k2 +ˇyi,(53) +and +tan δi (t) = kˇyi +� +kˇxie +¯U′ +ixt − kˇyi +� +V ′ +0t + +¯U ′ +iy +¯U ′ +ix +��−1 +.(54) +The solution (50) with potential (52) is +¯fi +� +˜vi⊥, φ1, viz, ˇξi, ˇηi, zi1, t +� += i ei +mi +t +� +0 +dt1 +× +� +dkˇxidkˇyidkzΦi (kˇxi, kˇyi, kz, t1) +× ei(kˇxi ˇξi+kˇ +yi ˇηi+kzzi) +× +∞ +� +n=−∞ +Jn +�ki⊥ (t1) ˜vi⊥ +ωci +� +e−in(φ1−ωcit1−δ(t1)) +× +�kˇyi +ωci +e +¯U′ +ixt1 ∂ ¯Fi0 +∂ ˇξi ++ nωci +˜vi⊥ +∂ ¯Fi0 +∂˜vi⊥ ++ kz +∂ ¯Fi0 +∂viz +� +. +(55) +In what follows, we consider the stability of the mi- +croscale perturbations of the convective flows with wave- +length much less than the plasma inhomogeneity scale +length L ˇ +Xi, for which |ki⊥Lˇxi| ≫ 1 and the Fourier +transform of ¯fi over ˇxi can be performed in the local +approximation. The Fourier transformed microscale per- +turbation of the ion density, determined within this ap- +proximation, is +ni (kˇxi, kˇyi, kz, t) = i2πei +mi +∞ +� +n=−∞ +t +� +0 +dt1Φi (kˇxi, kˇyi, kz, t1) +∞ +� +−∞ +dviz +∞ +� +0 +d˜vi⊥˜vi⊥ +× +∞ +� +n=−∞ +Jn +�ki⊥ (t) ˜vi⊥ +ωci +� +Jn +�ki⊥ (t1) ˜vi⊥ +ωci +� +e−ikzviz(t−t1)−in(ωci(t−t1)−δ(t)+δ(t1)) +× +�kˇyi +ωci +e +¯U′ +ixt ∂ ¯Fi0 +∂ ˇξi ++ nωci +˜vi⊥ +∂ ¯Fi0 +∂˜vi⊥ ++ kz +∂ ¯Fi0 +∂viz +� +. +(56) +For the Maxwellian distribution ¯Fi0 +�˜vi, ˇξi +� +of ions with +initial value (46) in the case of the uniform ion tempera- +ture Eq. (56) gives +ni (kˇxi, kˇyi, kz, t) = in0i +�ˇξ +� +ei +Ti +× +∞ +� +n=−∞ +t +� +0 +dt1Φi (kˇxi, kˇyi, kz, t1) +× In +� +ki⊥ (t) ki⊥ (t1) ρ2 +i +� +e− 1 +2 ρ2 +i(k2 +i⊥(t)+k2 +i⊥(t1)) +× e− 1 +2 k2 +zvz(t−t1)2−in(ωci(t−t1)−δ(t)+δ(t1)) +× +� +kˇyivdie +¯U′ +ixt − nωci + ik2 +zv2 +T i (t − t1) +� +. +(57) +where vdα = (cTα/eαB0)d ln n0(ˇxi)/dˇxi is the ion (α = i) +and electron (α = e) diamagnetic velocity, ρi is the ther- +mal ion Larmor radius, and In is the modified Bessel +function of the first kind and order n. The Fourier trans- +form ne (kˇxe, kˇye, kz, t) of the microscale perturbation of +the electron density, performed in the electron frame with +coordinates ˇxe, ˇye, ze1, t, is determined in the same way +as it is given by Eqs. (34)-(58) for ni (kˇxi, kˇyi, kz, t) with +changed ion on electron species subscripts. The derived +ion and electron density perturbations are employed in +the Poisson equation, Fourier transformed over the vari- +ables ˇxi, ˇyi. Therefore ne (kˇxe, kˇye, kz, t) for the Poisson +equation should be recalculated in the variables ˇxi, ˇyi of +the ion frame. For this goal, we derive the relations be- +tween variables ˇxi, ˇyi and ˇxe, ˇye. Because the difference +between ˜xi and ˜xe, as well as between ˜yi and ˜ye, are on +the order of the microscale displacements of the ions rela- +tive to electrons in FW field, which are on the order of or +less than the wavelength of the microscale perturbations, +we can use relations ˜xi = ˜xe, and ˜yi = ˜ye with Eqs. (40), +(41) and obtain on this way the relations +ˇxe (ˇxi, t) = +1 +¯U ′ex +� +e +¯U′ +ext +� +¯U (0) +ex + +¯U ′ +ex +¯U ′ +ix +�� +¯U (0) +ix + ¯U ′ +ixˇxi +� +e− ¯U′ +ixt − ¯U (0) +ix +�� +− ¯U (0) +ex +� +, +(58) + +10 +ˇye (ˇyi, ˇxi, t) = ˇyi − +�� +¯U (0) +iy − ¯U (0) +ix +¯U ′ +iy +¯U ′ +ix +� +− +� +¯U (0) +ey − ¯U (0) +ex +¯U ′ +ey +¯U ′ex +�� +t ++ +¯U (0) +ix +¯U ′ +ix +� ¯U ′ +iy +¯U ′ +ix +− +¯U ′ +ey +¯U ′ex +� +e− ¯U′ +ixt + +¯U ′ +ey +¯U ′ex +� ¯U (0) +ix +¯U ′ +ix +− +¯U (0) +ex +¯U ′ex +� ++ +� ¯U ′ +iy +¯U ′ +ix +− +¯U ′ +ey +¯U ′ex +� +ˇxie− ¯U′ +ixt. +(59) +Note, the relations for ˇxi (ˇxe, t) and for ˇyi (ˇye, ˇxe, t) are +derived by changing species subscripts i ⇆ e in Eqs. (58), +(59). +The equation for ne (kˇxe, kˇye, kz, t), similar to (56) for +ni, contains the Fourier transform Φe (kˇxe, kˇye, kz, t1) of +the potential Φe (ˇxe, ˇye, z, t1). The connection relation of +Φe (kˇxe, kˇye, kz, t1) with Φi (kˇxi, kˇyi, kz, t1) follows from +the relation +Φe (kˇxe, kˇye, kz, t1) = +� +dˇxe +� +dˇyeΦ (ˇxe, ˇye, kz, t1) e−i(kˇxe ˇxe+kˇ +ye ˇye) += +1 +(2π)2 +� +dkˇxi +� +dkˇyiΦi +� +kˇxi, k ˇYi, kz, t1 +� � +dˇxi +� +dˇyi +∂ (ˇxe, ˇye) +∂ (ˇxi, ˇyi) +× ei(kˇxi −kˇxe)ˇxi+i(kˇ +yi −kˇ +ye)ˇyi−ikˇxe (ˇxe−ˇxi)−ikˇ +ye (ˇye−ˇyi) += Φi (kˇxe + kˇxeb1x (t1) + kˇyeb1y (t1) , kˇye, kz, t1) e( ¯U′ +ex− ¯U′ +ix)t1e−ikˇxe b0x(t1)−ikˇ +ye b0y(t1). +(60) +In Eq. (60), +∂ (ˇxe, ˇye) +∂ (ˇxi, ˇyi) = e( ¯U′ +ex− ¯U′ +ix)t1 +(61) +is the Jacobian of the transformation ˇxe, ˇye to ˇxi, ˇyi, and +the relations +ˇxe (ˇxi, t1) − ˇxi = b0x (t1) + b1x (t1) ˇxi +(62) +and +ˇye (ˇyi, ˇxi, t1) − ˇyi = b0y (t1) + b1y (t1) ˇxi, +(63) +where +b0x (t1) = +1 +¯U ′ex +e +¯U′ +ext1 +� +¯U (0) +ex + ¯U (0) +ix +¯U ′ +ex +¯U ′ +ix +� +e− ¯U′ +ixt1 − 1 +�� +− +¯U (0) +ix +¯U ′ +ix +, +(64) +b1x (t1) = +� +e( ¯U′ +ex− ¯U′ +ix)t1 − 1 +� +, +(65) +b0y (t1) = +�� +¯U (0) +ey − ¯U (0) +ex +¯U ′ +ey +¯U ′ex +� +− +� +¯U (0) +iy − ¯U (0) +ix +¯U ′ +iy +¯U ′ +ix +�� +t1 ++ +¯U (0) +ix +¯U ′ +ix +� ¯U ′ +iy +¯U ′ +ix +− +¯U ′ +ey +¯U ′ex +� +e− ¯U′ +ixt1 ++ +¯U ′ +ey +¯U ′ex +� ¯U (0) +ix +¯U ′ +ix +− +¯U (0) +ex +¯U ′ex +� +, +(66) +b1y (t1) = +� ¯U ′ +iy +¯U ′ +ix +− +¯U ′ +ey +¯U ′ex +� +e− ¯U′ +ixt1, +(67) +were used. +Now we determine the relation between the Fourier +transform ne (kˇxe, kˇye, kz, t) of the electron density per- +turbation ne (ˇxe, ˇye, ze, t), performed in the electron +frame with variables ˇxe, ˇye, with the Fourier transform +n(i) +e (kˇxi, kˇyi, kz, t) of ne (ˇxe, ˇye, Ze, t), performed in the +ion frame with variables ˇxi, ˇyi. +n(i) +e (kˇxi, kˇyi, kz, t) = +� +dˇxi +� +dˇyie−i(kˇxi ˇxi+kˇ +yi ˇyi)ne (ˇxe, ˇye, kz, t) += +� +dˇxe +� +dˇyene (ˇxe, ˇye, kz, t) ∂ (ˇxi, ˇyi) +∂ (ˇxe, ˇye)e−ikˇxi ˇxe−ikˇ +yi ˇye−ikˇxi (ˇxi−ˇxe)−ikˇ +yi (ˇyi−ˇye) += +� +dˇxe +� +dˇyene (ˇxe, ˇye, kz, t) e( ¯U′ +ix− ¯U′ +ex)t +× e−ikˇxi ˇxe−ikˇ +yi ˇye−ikˇxi (a0x(t)+a1x(t)ˇxe)−ikˇ +yi (a0y(t)+a1y(t)ˇxe) += e( ¯U′ +ix− ¯U′ +ex)te−ikˇxi a0x(t)−ikˇ +yi a0y(t)ne (kˇxi (1 + a1x (t)) + kˇyia1y (t) , kˇyi, kz, t) , +(68) + +11 +where the relations +ˇxi (ˇxe, t) − ˇxe = a0x (t) + a1x (t) ˇxe +(69) +and +ˇyi (ˇye, ˇxe, t) − ˇye = a0y (t) + a1y (t) ˇxe, +(70) +where used. The functions a0x (t), a1x (t), a0y (t), and +a1y (t) are determined by the functions b0x (t), b1x (t), +b0y (t), and b1y (t), respectively, by changing species sub- +scripts i ⇆ e in Eqs. (59) -(64). +By +replacing +kˇxe +and +kˇye +in +Eq. +(64) +on +kˇxi (1 + a1x (t)) + kˇyia1y (t) and kˇyi, which, as it follows +from Eq. (64), are the new wave numbers conjugate with +coordinates ˇxe and ˇye in n(i) +e (kˇxi, kˇyi, kz, t), we derive the +following relation for n(i) +e : +n(i) +e (kˇxi, kˇyi, kz, t) = 2iπe +me +e−ikˇxia0x(t)−ikˇ +yi a0y(t) +t +� +0 +dt1e( ¯U′ +ix− ¯U′ +ex)(t−t1) +× Φi ((kˇxi (1 + a1x (t)) + kˇyia1y (t1)) (1 + b1x (t1)) + kˇyib1y (t1) , kˇyi, kz, t1) +× e−i(kˇxi (1+a1x(t))+kˇ +yi a1y(t1))b0x(t1)−ikˇ +yi b0y(t1) +× +∞ +� +0 +dve⊥ve⊥ +∞ +� +−∞ +dveze−ikzvez(t−t1) +� kˇyi +ωce +e +¯U′ +ext1 ∂ ¯Fe +∂ ˇξe ++ kz +∂ ¯Fe +∂vez +� +. +(71) +Equation (71) is valid for the perturbations with fre- +quency much less than the electron cyclotron frequency +and with wavelength across the magnetic field much +larger than the thermal electron Larmor radius. +The Poisson equation (49) for Φi, Fourier transformed +over ˇxi and ˇyi, + +k2 +ˇxie2 ¯U′ +ixt − 2e +¯U′ +ixt +� +V ′ +0t + +¯U ′ +iy +¯U ′ +ix +� +kˇxikˇyi + + +1 + +� +V ′ +0t + +¯U ′ +iy +¯U ′ +ix +�2 + k2 +ˇyi + k2 +z + + Φi (kˇxi, kˇyi, kz, t) += 4π +� +eini (kˇxi, kˇyi, kz, t) − |e|n(i) +e (kˇxe (kˇxi, kˇyi, t) , kˇyi, kz, t) +� +, +(72) +where ni and n(i) +e +are determined by Eqs. +(54) and +(71) respectively, is the equation which determines the +temporal evolution of the single spatial Fourier mode +Φi (kˇxi, kˇyi, kz, t) in the compressed-sheared flow. +Now we consider the particular cases for Eq.(72), in +which ni and n(i) +e +are determined by Eqs. (57) and (71). +1. In the case of the currentless compressed-sheared +flow +¯U (0) +ix += +¯U (0) +ex , +¯U ′ +ix += +¯U ′ +ex, and +¯U (0) +iy += +¯U (0) +ey , +¯U ′ +iy += +¯U ′ +ey. +It follows from Eqs. +(62)-(67) that +in this case b0x (t1) += +a0x (t) += +0, +b1x (t1) += +a1x (t) = 0, and b0y (t1) = a0y (t) = 0, b1y (t1) = +a1y (t) += +0. +Therefore in this case +ˇxi += +ˇxe, +ˇyi = ˇye, and Φe (kˇxe, kˇye, kz, t) = Φi (kˇxi, kˇyi, kz, t) and +n(i) +e (kˇxi, kˇyi, kz, t) = ne (kˇxe, kˇye, kz, t). Equation (72) in +this case has a form +λ2 +Di +� +k2 +i⊥ (t) + k2 +z +� +Φi (kˇxi, kˇyi, kz, t) = +∞ +� +n=−∞ +t +� +t0 +dt1Φi (kˇxi, kˇyi, kz, t1) In +� +ki⊥ (t) ki⊥ (t1) ρ2 +i +� +× e− 1 +2 ρ2 +i(k2 +i⊥(t)+k2 +i⊥(t1)) � +ikˇyivdie +¯U′ +ixt1 − inωci − k2 +zv2 +T i (t − t1) +� +e− 1 +2 k2 +zv2 +T i(t−t1)2−in(ωci(t−t1)−δ(t)+δ(t1)) += Ti +Te +t +� +t0 +dt1Φi (kˇxi, kˇyi, kz, t1) e− 1 +2 k2 +zv2 +T e(t−t1)2 � +ikˇyivdee +¯U′ +ext1 − k2 +zv2 +T e (t − t1) +� +. +(73) +where t0 ≥ 0, λDi(e) is the ion (electron) Debye length, +and +Ain (t, t1) = In +� +ki⊥ (t) ki⊥ (t1) ρ2 +i +� + +12 +× e− 1 +2 ρ2 +i(k2 +i⊥(t)+k2 +i⊥(t1)). +(74) +Equation (73) was derived for the first time in Ref.18 +for the poloidal sheared plasma flow without the con- +vective flows (i. +e. +for the case ¯U ′ +ix = ¯U ′ +ex = 0 and +¯U ′ +iy = ¯U ′ +ey = 0). In that case, the poloidal velocity shear +manifests as a time- dependence of Ain (t, t1) function +which determines effect of the finite ion Larmor radius. +The solution of Eq. (73), derived for the kinetic drift +instability in the poloidal sheared flow, displays18 the +nonmodal effect of the reduction with time of the fre- +quency and of the growth rate of this instability caused +by the flow velocity shear. By the integration by parts +of the first term on the right part of Eq. +(73), this +equation for the low frequency perturbations, for which +dΦ/dt ≪ ωciΦ, may be presented in the form similar to +Eq. (25) of Ref.18, +t +� +t0 +dt1 +d +dt1 +� +Φi (kˇxi, kˇyi, kz, t1) +� +1 + Ti +Te +− Ain (t, t1) +�� +− i +t +� +t0 +dt1Φikˇyivdie +¯U′ +ext1Ain (t, t1) += Ti +Te +t +� +t0 +�dΦi +dt1 ++ ikˇyivdee +¯U′ +ext1Φi +� +e− 1 +2 k2 +zv2 +T e(t−t1)2 +(75) +We found that in the time domain (t, t0), in which +ki⊥ (t1) ρi ≫ 1, the temporal evolution of the potential +Φi in the compressed flow, predicted by the solution to +Eq. (75), resembles the temporal evolution of the po- +tential Φi in the poloidal sheared flow: the potential Φi +gradually becomes a zero- frequency cell-like perturba- +tion when time elapsed. +V. +CONCLUSIONS +A nonmodal kinetic theory of the stability of the two- +dimensional compressed-sheared mesoscale plasma flows, +generated by the radially inhomogeneous electrostatic +ion cyclotron parametric microturbulence in the pedestal +plasma with a sheared poloidal flow, is developed. This +theory reveals that the separate spatially uniform Fourier +modes of the electrostatic responses of the ions and of +the electrons on the mesoscale convective flows are de- +termined only in the frames of references moved with +velocities of the ion and electron convective flows. +In +the laboratory frame, these modes are observed as the +compressed-sheared modes with time dependent wave +numbers. The integral equation, which governs the sepa- +rate Fourier mode of the electrostatic potential of the +plasma species responses on the mesoscale convective +flows, is derived. +In this equation, the effects of the +compressing and shearing of the convective flows are re- +vealed as the time dependence of the finite ion Larmor +radius effect. The solution of this equation for the kinetic +drift instability displays the nonmodal transformation of +the potential to the zero frequency cell-like perturbation +when time elapsed. +ACKNOWLEDGMENTS +This work was supported by National R&D Program +through the National Research Foundation of Korea +(NRF) funded by the Ministry of Education, Science and +Technology (Grant No. +NRF-2018R1D1A3B07051247) +and BK21 FOUR, the Creative Human Resource Educa- +tion and Research Programs for ICT Convergence in the +4th Industrial Revolution. +DATA AVAILABILITY +The data that support the findings of this study are +available from the corresponding author upon reasonable +request. +Appendix A: Velocities ˜Uix +� +˜ +Xi +� +and ˜Uiy +� +˜ +Xi +� +of the +convective flows +For the electric field ˜Ei, given by Eq. (8), velocities +¯Uix +� +˜Xi +� +and ¯Uiy +� +˜Xi +� +, after long calculation similar to +performed in Ref.14, are determined by the following re- +lations: +¯Uix +� +˜Xi +� += +1 +2ωci +e2 +i +m2 +i +1 +4π2 +� +n +� +dk +� +ai1 (k, n) ˜Eix +� +k, ˜Xi, n +� +∂ +∂ ˜Xi +� +˜E∗ +iy +� +k, ˜Xi, n +�� + +13 ++ai2 (k, n) ˜Eiy +� +k, ˜Xi, n +� +∂ +∂ ˜Xi +� +˜E∗ +ix +� +k, ˜Xi, n +��� +(A1) +and +¯U (0) +iy +� +˜Xi +� += − 1 +4ωci +e2 +i +m2 +i +1 +4π2 +� +n +� +dk +� +ai1 (k, n) +∂ +∂ ˜Xi +��� ˜Eix +� +k, ˜Xi, n +���� +2 +− ai2 (k, n) +∂ +∂Xi +��� ˜Eiy +� +k, ˜Xi, n +���� +2� +(A2) +where the asterisk in Eq. (A1) implies the operation of complex conjugate. The coefficients ai1 (k, n) and ai2 (k, n) +are determined as +ai1 (k, n) = +� +ωci +ωn (k) (ωci + ωn (k))2 + +ωci +ωn (k) (ωci − ωn (k))2 + +1 +(ω2 +ci − ω2n (k)) +� +, +(A3) +and +ai2 (k) = +� +1 +(ωci + ωn (k))2 + +1 +(ωci − ωn (k))2 + +1 +(ω2 +ci − ω2n (k)) +� +. +(A4) +The velocities of electrons ¯Uex +� +˜Xi +� +and ¯Uey +� +˜Xi +� +in the +ion frame, with electric field Ee +� +ˆri, ˜Xi, t +� +determined by +Eq. (10), are +¯Uex +� +˜Xi +� +≈ c2 +B2 +0 +1 +4π2 +� +n +� +dk ˜Eix +� +k, ˜Xi, n +� +× +∂ +∂ ˜Xi +� +˜E∗ +iy +� +k, ˜Xi, n +�� +∞ +� +p=−∞ +J2 +p +� +aie +� +k, ˜Xi +�� +× +1 +Ωp +� +k, ˜Xi +�, +(A5) +and +¯Uey +� +˜Xi +� +≈ −1 +2 +c2 +B2 +0 +1 +4π2 +� +n +� +dk ∂ +∂ ˜Xi +��� ˜Eix +� +k, ˜Xi, n +���� +2 +× +∞ +� +p=−∞ +J2 +p +� +aie +� +k, ˜Xi +�� +1 +Ωp +� +k, ˜Xi +�, +(A6) +where limit |ωce| ≫ Ωp ∼ ωci was used. +1M. Ono, ”High harmonic fast waves in high beta plasmas”, Phys. +Plasmas 2,4075 (1995). +2S. C. Chiu, V. S. Chan, R. W. Harvey, M. Porkolab, ”Theory of +fast wave current drive for tokamak plasmas”, Nucl. Fusion 29, +2175 (1989). +3R. J. Perkins, +J. C. Hosea, +G. J. Kramer, +J. -W. Ahn, +R. E. Bell, A. Diallo, S. Gerhardt, T. K. Gray, D. L. Green, +E. F. Jaeger, M. A. Jaworski, B. P. LeBlanc, A. McLean, +R. Maingi, C. K. Phillips, L. Roquemore, P. M. Ryan, S. Sab- +bagh, G. Taylor, J. R. Wilson, ”High-Harmonic Fast-Wave Power +Flow along Magnetic Field Lines in the Scrape-Off Layer of +NSTX,” Phys. Rev. Lett. 109, 045001 (2012). +4R. J. Perkins, J. -W. Ahn, R. E. Bell, A. Diallo, S. Gerhardt, +T. K. Gray, D. L. Green, E. F. Jaeger, J. C. Hosea, M. A. Ja- +worski, B. P. LeBlanc, G. J. Kramer, A. McLean, R. Maingi, +C. K. Phillips, M. Podest‘a, L. Roquemore, P. M. Ryan, S. Sab- +bagh, F. Scotti, G. Taylor, J. R. Wilson, ”Fast-wave power flow +along SOL field lines in NSTX and the associated power deposi- +tion profile across the SOL in front of the antenna,” Nucl. Fusion +53, 083025(2013). +5D. C. Pace, R. I. Pinsker, W. W. Heidbrink, R. K. Fisher, +M. A. Van Zeeland, M. E. Austin, G. R. McKee, and M. Garc´ia- +Mu˜noz, ”Scrape-off layer ion acceleration during fast wave injec- +tion in the DIII-D tokamak”, Nucl. Fusion 52, 063019 (2012). +6J. R. Wilson, S. Bernabei, T. Biewer, S. Diem, J. Hosea, +B. LeBlanc, C. K. Phillips, P. Ryan, and D. W. Swain. ”Para- +metric Decay During HHFW on NSTX”. AIP Conference Pro- +ceedings 787, 66 (2005). +7M. Porkolab, ”Parametric processes in magnetically confined +CTR plasmas,” Nucl. Fusion 18, 367 (1978). +8M. Porkolab, ”Parametric instabilities in the tokamak edge +plasma in the ion cyclotron heating regimes,” Fusion Engineering +and Design 12, 93 (1990). +9V. S. Mikhailenko, V. V. Mikhailenko, Hae June Lee, ”The ion +cyclotron parametric instabilities and the anomalous heating of +ions in the tokamak edge plasma in the fast wave heating regime”, +Phys. Plasmas 27, 052508 (2020). +10N. Bertelli, E. F. Jaeger, J. C. Hosea, C. K. Phillips, L. Berry, +S. P. Gerhardt, D. Green, B. LeBlanc, R. J. Perkins, P. M. Ryan, +G. Taylor, E. J. Valeo, J. R. Wilson, ”Full wave simulations +of fast wave heating losses in the scrape-off layer of NSTX and +NSTX-U,” Nucl. Fusion 54, 083004 (2014). +11N. Bertelli, E. F. Jaeger, J. C. Hosea, C. K. Phillips, L. Berry, +P. T. Bonoli, S. P. Gerhardt, D. Green, B. LeBlanc, R. J. Perkins, +C. M. Qin, R. I. Pinsker, R. Prater, P. M. Ryan, G. Taylor, +E. J. Valeo, J. R. Wilson, J .C. Wright, X. J. Zhang, ”Full wave +simulations of fast wave efficiency and power losses in the scrape- +off layer of tokamak plasmas in mid/high harmonic and minority +heating regimes,” Nucl. Fusion 56, 016019 (2016). +12E. F. Jaeger, L. A. Berry, E. D’Azevedo, D. B. Batchelor, and +M. D. Carter, ”All-orders spectral calculation of radio-frequency +heating in two-dimensional toroidal plasmas”, Phys. Plasmas 8, +1573 (2001). +13D. L. Green, L. A. Berry, G. Chen, P. M. Ryan, J. M. Canik, +E. F. Jaeger, ”Predicting High Harmonic Ion Cyclotron Heating +Efficiency in Tokamak Plasmas”, Phys. Rev. Lett. 107, 145001 +(2011). +14V. S. Mikhailenko, V. V. Mikhailenko, Hae June Lee, ”Ion cy- +clotron parametric turbulence and anomalous convective trans- +port of the inhomogeneous plasma in front of the fast wave an- +tenna,” Phys. Plasmas 28, 042304 (2021). +15J. C. Hosea, R. E. Bell, E. Feibush, R. W. Harvey, E. F. Jaeger, +B. +P.LeBlanc, +R. +Maingi, +C. +K.Phillips, +L. +Roquemore, +P. M. Ryan, G. Taylor, K. Tritz, E. J. Valeo, J. Wilgen, J. R, Wil- +son, and the NSTX Team,”Recent Fast Wave Coupling and Heat- +ing Studies on NSTX, with Possible Implications for ITER”, AIP + +14 +Conf. Proc. 1187, 105 (2009). +16K. H. Burrell, ”Effects of ExB velocity shear and magnetic shear +on turbulence and transport in magnetic confinement”,Phys. +Plasmas 4, 1499 (1997) +17V. S. Mikhailenko, V. V. Mikhailenko, K. N. Stepanov, ”Tur- +bulence evolution in plasma shear flows”, Plasma Fusion Res.5, +S2015 (2010). +18V. S. Mikhailenko, V. V. Mikhailenko, K. N. Stepanov, ”Renor- +malized non-modal theory of the kinetic drift instability of +plasma shear flows”, Phys. Plasmas 18, 062103 (2011). +19V. V. Mikhailenko, V. S. Mikhailenko, Hae June Lee, ”Non- +modal theory of the kinetic ion temperature gradient driven in- +stability of a plasma shear flows across the magnetic field”, Phys. +Plasmas 23, 062115 (2016). +20V. S. Mikhailenko, V. V. Mikhailenko, Hae June Lee, ”Anoma- +lous convective transport of the tokamak edge plasma, caused by +the inhomogeneous ion cyclotron parametric turbulence”, Phys. +Plasmas 29, 072301 (2022). + diff --git a/t9E1T4oBgHgl3EQfjwTU/content/tmp_files/load_file.txt b/t9E1T4oBgHgl3EQfjwTU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..39885951e2d2d6265f239598be5f98f66c313fbe --- /dev/null +++ b/t9E1T4oBgHgl3EQfjwTU/content/tmp_files/load_file.txt @@ -0,0 +1,820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf,len=819 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='03267v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='plasm-ph] 9 Jan 2023 Non-modal kinetic theory of the stability of the compressed-sheared plasma flows generated by the inhomogeneous microscale turbulence in the tokamak edge plasma V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko,1, a) V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' b) and Hae June Lee3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' c) 1)Plasma Research Center,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Pusan National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Busan 46241,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' South Korea 2)BK21 FOUR Information Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Pusan National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Busan 46241,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' South Korea 3)Department of Electrical Engineering,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Pusan National University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Busan 46241,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' South Korea (Dated: 10 January 2023) A nonmodal kinetic theory of the stability of the two-dimensional compressed-sheared mesoscale plasma flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' generated by the radially inhomogeneous electrostatic ion cyclotron parametric microturbulence in the pedestal plasma with a sheared poloidal flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This theory reveals that the separate spatially uniform Fourier modes of the electrostatic responses of the ions and of the electrons on the mesoscale con- vective flows are determined only in the frames of references moved with velocities of the ion and electron convective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In the laboratory frame, these modes are observed as the compressed-sheared modes with time dependent wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The integral equation, which governs the separate Fourier mode of the electro- static potential of the plasma species responses on the mesoscale convective flows, is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In this equation, the effects of the compressing and shearing of the convective flows are revealed as the time dependence of the finite ion Larmor radius effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The solution of this equation for the kinetic drift instability displays the nonmodal transformation of the potential to the zero frequency cell-like perturbation when time elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' PACS numbers: 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='Ra, 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='Kt I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' INTRODUCTION The linear theory of the interaction of the fast waves (FW) with tokamak plasma predicts1,2 that the injection of FW may be the efficient method for the electron heat- ing and current drive to aid in steady-state non-inductive tokamak operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These predictions has been con- firmed for the propagation and absorption of FWs in the hot core tokamak plasma bounded by the last closed flux surface (LCFS) on numerous tokamak devices over the past half century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These experiments, however, demon- strated that the efficiency of the FW heating and cur- rent drive reduces by the FW power lost, which occurs in the near-antenna layer of the cold low density scrape- off layer (SOL) tokamak plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The FW heating ex- periments on the National Spherical Torus eXperiment (NSTX) showed3,4 that around 30% to more than 60% of the FW energy lost directly in the SOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The bursts of the ions with energy above 20 keV, experimentally observed5 in SOL following FW injection, and the devel- opment of the parametric instabilities in SOL6, predicted earlier theoretically7,8, were considered as the main chan- nels of the FW absorption in SOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The development of the ion cyclotron (IC) quasimode decay instability was considered5,6 as the main nonlinear process responsible for the absorption of the FW power in SOL plasma and a)E-mail:vsmikhailenko@pusan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='kr b)E-mail: vladimir@pusan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='kr c)E-mail: haejune@pusan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='kr of the anomalous heating of ions in SOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The analysis of the turbulent heating of ions by the IC parametric tur- bulence, powered by the IC quasimode decay instability, was given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='9 on the base of the numerical solution of the dispersion equation for the IC parametric instabil- ities driven by FW, and on the base of quasilinear theory for ion distribution function, which accounted for the in- teraction of ions with IC parametric turbulence powered by the IC quasimode decay instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The derived es- timates for the turbulent ion heating rates revealed that the absorption of the FW energy by ions in SOL is a weak effect, which provides only negligibly small heating of cold ions in SOL and can not be responsible for the observed generation of the high energy ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The FW power loss in SOL was investigated in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='10,11 by using the numerical full wave simula- tion code AORSA (all-orders spectral algorithm)12,13, in which the edge plasma beyond LCFS is included in the solution domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This simulation displays that the dominant loss of the FW power occurs in the edge of the tokamak plasma, where the plasma density and the amplitude of the FW field strongly change on the ra- dial intermediate spatial scale (mesoscale) between the macroscale of the spatial inhomogeneity length of the FW field in the bulk of the tokamak plasma, and the mi- croscale commensurable with the wavelength of the para- metric instabilities of the IC perturbations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The theory of the microscale parametric turbulence of the inhomo- geneous plasma driven by the strong inhomogeneous on the mesoscale FW, was developed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This the- ory reveals the effect of the formation of the radial and poloidal convective flows of such a plasma caused by the 2 mesoscale spatial inhomogeneity of the microscale IC or drift turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The radial and poloidal convective flow velocity components are proportional to the gradient of the spectral intensity of the electric field of the micro- turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This result gives the possible explanation of the FW heating experiment on the National Spherical Torus eXperiment (NSTX)3,4,15, where it was found that a significant part of the FW power loss occurs due to the anomalous convective flow of the collisionless dense hot plasma from the tokamak edge to the cold low density SOL plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The pedestal region, where plasma density profile has largest radial gradient, is the most preferable region in tokamaks for the development of the convective flows, driven by the spatially inhomogeneous microturbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The inherent component of the pedestal plasma is the sheared poloidal flow, in which the drift type instabili- ties, responsible for the anomalous transport of plasma, are suppressed when their growth rates are less than the flow velocity shearing rate16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It was proved in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='17–19 that the basic point in understanding the processes of the instabilities and turbulence evolution in plasma sheared flows is the proper treatment of the persistent deforma- tion of the perturbations by the sheared flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This ef- fect, which is completely ignored in the canonical stabil- ity theory, where the perturbations are considered having a static structure of a plane wave ∼ exp (ik · r − iωt), is involved in the nonmodal kinetic theory, developed in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='17–19, grounded on the methodology of the sheared modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It was found17,18 that in the sheared flow, the separate spatial Fourier mode with a static spatial struc- ture ∼ exp (ikxx + ikyy + ikzz) can be determined only in the frame convected with a sheared flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In the labo- ratory frame, this mode is observed as the sheared mode with time dependent structure resulted from the continu- ous distortion with time the perturbation by the sheared flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This distortion grows with time and forms a time- dependent nonmodal process which is investigated as the initial value problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='20, the Vlasov equations, which govern the ion and electron mesoscale convective flows with radially in- homogeneous flow velocities in the poloidal sheared flow were derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These equations predict the generation of the sheared poloidal convective flow and of the radial compressed flow with radial flow velocity gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The hydrodynamic theory of the mesoscale convective flows, derived as the moments of the obtained Vlasov equation, reveals the radial compressed convective flow as the dom- inant factor in the formation of the steep pedestal den- sity profile with density gradient exponentially growing with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The focus of this paper is the development of the nonmodal kinetic theory of the stability of the two-dimensional compressed-sheared convective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' II, we present basic equations and their transforma- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' III, we develop the nonmodal approach to the kinetic theory of the compressed-sheared convec- tive flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In this theory we derived new spatial refer- ence coordinates in which the distribution functions of the unperturbed convective sheared-compressed flows is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The stability of such a distribution functions of the convected plasma species against the development of the short scale instabilities is given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' IV employ- ing the developed compressed-sheared modes approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Conclusions are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' BASIC EQUATIONS AND TRANSFORMATIONS Our theory is based on the Vlasov-Poisson system in a slab geometry approximation where x, y, z directions are viewed as corresponding to the radial, poloidal and toroidal directions, respectively, of the toroidal coordi- nate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Within this approximation, the Vlasov equation for the velocity distribution function Fα of the poloidal sheared flow of α plasma species (α = i for ions and α = e for electrons) in the FW field with coordinates r = (x, y, z) has a form ∂Fα (v, r, t) ∂t + v∂Fα (v, r, t) ∂r + eα mα � E0x (x) + E1 (x, t) + ˜E (r, t) +1 c [v × (B0 + B1 (r, t))] � ∂Fα (v, r, t) ∂v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (1) This equation contains the inhomogeneous radial electric field E0x (x), which governs the basic poloidal sheared flow, the FW electric field E1 (x, t), the electric field ˜E (r, t) of the self-consistent plasma response on FW, the uniform plasma-confining magnetic field B0 directed along coordinate z, and FW magnetic field B1 (r, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' For the edge layer of the tokamak plasma, this equation con- tains two disparate spatial inhomogeneity lengths, which are introduced by the FW field and by plasma parame- ters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In the edge plasma, the spatial inhomogeneity of E0x (x) and of FW fields are commensurable with a spa- tial inhomogeneity length of the pedestal plasma density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These mesoscale spatial inhomogeneity lengths on order of the pedestal width are much less than the the inho- mogeneity scale lengths of FW and of the plasma param- eters in the plasma core, but are much larger than the radial wavelengths of the IC parametric and drift micro- turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Electric field ˜E (r, t), being the microscale responc of the inhomogeneous plasma on the inhomoge- neous FW and E0x (x) fields, contains micro- and meso- spatial scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Our theory of the mesoscale plasma evo- lution caused by the mesoscale inhomogeneities of the microturbulence, involves the treatments on the micro- and mesoscales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In our theory20, we introduced jointly with variables r = (x, y, z) and time t for the microscale fast variations on time of the order of the FW period or of the period of the IC microturbulence, the slow time T = εt, and the slow spatial variables X = εx, Y = εy, where the dimensionless parameter ε ≪ 1, for the de- scription of the slow evolutionary mesoscale processes in the pedestal region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With these microscale and mesoscale 3 variables, electric field E0x depends only on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The FW fields E1, B1, determined as E1 (X, t) = E1x (X) cos ω0t + E1y (X) sin ω0t, (2) B1 (X, t) = c ω0 dE1y (X) dX cos ω0t ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (3) depend on slow mesoscale X and fast time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The elec- tric field ˜E (r, X, t) depends on the spatial micro- and mesoscale variables and on the fast time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This field is determined by the Poisson equation ∇ · ˜E (r, X, t) = 4π � α=i,e eα � fα (v, r, X, t) dv, (4) in which fα is the fluctuating part of the distribution function Fα, fα = Fα−F0α, where F0α is the equilibrium distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It is obvious that it is not possible to apply directly to the Vlasov-Poisson system (1), (4) with spatially in- homogeneous oscillating FW fields the methods of the solutions known for the investigations of the stability of a plasma in static equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Any microscale perturba- tions of the ion and electron densities are convected by FW field with inhomogeneous on the mesoscale veloci- ties, different for ions and electrons, and oscillating with frequency of the FW field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It was found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='14 that the spatially inhomogeneous FW field may be excluded from the Vlasov equation (1) by the transformation of the velocity v and the position r = (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' z) variables of the Vlasov equation (1) to new velocity vi and position ri = (xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' z) variables determined in the convected ref- erence flow,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' which moves relative to the laboratory frame with the velocity Vi (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) of ion in the FW field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' given by the equation dVi (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) dt = ei mα (E1 (Xi + εRix (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) +1 c [Vi × B0] + 1 c [Vi × B1 (Xi + εRix (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t)] � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='(5) with initial value Vi (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t = t0 = 0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Vlasov equation (1) with new variables vi, Xi, contains the elec- tric FW field only in terms on the order of |Ri/LE| ≪ 1, where |Ri| is the amplitude of the ion displacement in the spatially inhomogeneous FW field with spatial inho- mogeneity scale length LE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These terms are negligibly small14 for the conditions of the FW tokamak plasma heating and may be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Without these terms, the Vlasov equation in the frame convected with velocity Vi (Xi, t) has a form as for a static equilibria without the external FW field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' That equation for ions, ∂Fi (vi, ri, Xi, t) ∂t + vi ∂Fi ∂ri + ei mic [vi × B0] ∂Fi ∂vi + ei mi ˜Ei (ri, Xi, t) ∂Fi (vi, ri, Xi, t) ∂vi = 0, (6) and similar equation for electrons, determined in the elec- tron reference flow, and the Poisson equation for the elec- tric field ∇ · ˜Ei (ri, Xi, t) = 4π � α=i,e eα � fα (vα, rα, Xα, t) dvα,(7) determined in the ion reference flow, compose the system of equations for the investigation of the microscale tur- bulence in FW field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The mesoscale variables Xi and Xe are presented in this system as parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' At the time above the inverse growth rate of the IC instabilities, t ≫ γ−1 (k) > |ω−1 (k) | the microscale IC turbulence attains the steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' At this state the electric field ˜Ei of the electrostatic two dimensional IC parametric microturbulence, directed almost across the magnetic field B0, may be presented in the ion reference flow in the form ˜Ei (ri, Xi, t) = � n ˜Ei (ri, Xi, n, t) = 1 (2π)2 � n 1 2 � dk � ˜Ei (k, Xi, n) eiψn(ri,Xi,t) +˜E∗ i (k, Xi, n) e−iψn(ri,Xi,t)� , (8) where ψn (ri, Xi, t) = −iωn (k) t + ikri + iθ (k) , (9) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' as a linear superposition of the electric fields of IC perturbations with frequencies ωn (k) = nωci + δωn (k) with wave vectors k directed across the mag- netic field and with mesoscale position dependent am- plitudes ˜Ei (k, Xi, n) and with phases fast changed with time on the microscales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Reality of ˜Ei (ri, Xi, t) is in- sured without introducing negative frequencies by the addition of the complex conjugate terms with amplitudes ˜E∗ i (k, Xi, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The integration over k is performed over wave vectors of the linearly unstable IC perturbations, and θ (k) is their initial phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In the electron frame, oscillating relative to the ion frame, this electric field is determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (8) with species subscript i changed on e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The electric field ˜Ee (re, Xe, t) in variables ri, Xi has a form ˜Ee (re, Xe, t) = � n ˜Ee (re, Xe, n, t) = 1 (2π)2 � n 1 2 � dk � ˜Ei (k, Xi, n) ∞ � p=−∞ Jp (aie) ×eiψn(ri,Xi,t)−ip(ω0t+δie(k,Xi)) +˜E∗ i (k, Xi, n) ∞ � p=−∞ Jp (aie) ×e−iψn(ri,Xi,t)+ip(ω0t+δie(k,Xi))� , (10) where Jp (aie) is the first kind Bessel function of order p with argument aie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Parameters aie ∼ kξie, where ξie is the amplitude of the relative displacement of electrons relative to ions in FW field, and δie were determined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 4 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' THE KINETIC THEORY OF THE MESOSCALE COMPRESSED-SHEARED CONVECTIVE FLOWS For the investigation on the slow time scale T the mesoscale evolution of the poloidal plasma sheared flow with microscale turbulence not suppressed by the sheared flow, we transform the Vlasov-Poisson system from the microscale to the mesoscale variables using the relations xi = 1 εXi, yi = 1 εYi and t = 1 εT variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With mesoscale variables Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (6) becomes ε∂Fi (vi, Xi, Yi, T, ε) ∂T + εvix ∂Fi ∂Xi + εviy ∂Fi ∂Yi + ei mic [vi × B0] ∂Fi ∂vi + ei mi ˜Ei (Xi, Yi, T, ε) ∂Fi ∂vi = 0, (11) The electric field ˜Ei (Xi, Yi, T, ε) is determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (8), in which phase ψn is determined in the form ψn (Xi, Yi, T, ε) = −i1 ε (ωn (k) T − ikxXi − ikyYi) + iθ (k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (12) The similar transformations should be performed for the electron Vlasov equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' On the nonlinear stage of the IC parametric microturbulence evolution at the time above the inverse growth rate of the IC instabilities, t ≫ γ−1 (k) > |ω−1 (k) |, electric field (8) becomes the random function of the initial phase θ (k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The motion of ions and electrons in this field has a form of the random scattering of particles by the turbulent electric field and mimics to the thermal motion of particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The average effect of the mesoscale inhomogeneity of the microscale IC turbulence on the mesoscale evolu- tion of the ion and electron distribution functions of the poloidal sheared flow was considered in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The central point in this theory is the transformation of the velocity vi and coordinates Xi and Yi to the new microturbulence-associated velocity field ˜vi and coordi- nates ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi determined by the relations20 ˜vi = vi − ˜Vi (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (13) ˜Xi = Xi − t � t0 ˜Vix (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (14) ˜Yi = Yi − T � T0 ˜Viy (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (15) or by their inverse,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' vi = ˜vi + ˜Ui � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (16) Xi = ˜Xi + ˜Rix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � = = ˜Xi + T � T0 ˜Uix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � dT1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (17) Yi = Y − V ′ 0XT = ˜Yi + T � T0 ˜Uiy � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � dT1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (18) where V ′ 0 is the velocity shear of the poloidal flow velocity V0 (X) = V ′ 0X directed along coordinate Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The velocity ˜Vi (Xi, Yi, T, ε) is determined by the Euler equation ε∂ ˜Vi ∂T + ε ˜Vix ∂ ˜Vi (Xi, Yi, T, ε) ∂Xi = ei mi � ˜Ei (Xi, Yi, T, ε) + 1 c � ˜Vi × B0 �� (19) as the velocity of an ion in the electric field ˜Ei (Xi, Yi, T, ε) of the IC parametric turbulence, where variables Xi and Yi are determined in the frame of ref- erences which moves with the velocity of an ion in the FW field in the poloidal sheared flow20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In variables � ˜Xi, ˜Yi, T � , the convective nonlinear part ˜Vix ∂ ∂Xi of the operator ∂ ∂T + ˜Vix ∂ ∂Xi of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (19) vanishes and this op- erator is transformed to the linear one, ∂ ∂T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (19) becomes the ordinary differential equation ε d dT ˜Ui � ˜Xi, ˜Yi, T, ε � = ei mi � ˜Ei � ˜Xi + ˜Rix � ˜Xi, ˜Yi, T, ε � , ˜Yi, T, ε � +1 c � ˜Ui � ˜Xi, ˜Yi, T, ε � × B0 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (20) for ˜Ui � ˜Xi, ˜Yi, T, ε � = ˜Vi (Xi, Yi, T, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (20) may be easily derived20 for the case of the small displacement, ��� ˜Rix ��� ≪ L ˜ E, of an ion in the inhomoge- neous electric field ˜Ei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With the approximation for the amplitude ˜Ei (k, Xi, n) of the n-th harmonic of the IC electric field in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (20) ˜Ei � k, ˜Xi + ˜Rix � ˜Xi, ˜Yi, T, ε � , n � ≈ ˜Ei � k, ˜Xi, n � ,(21) the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (20) with the initial value ˜Ui � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T0 = 0 � = 0 is ˜Uix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � = ei εmi � n T � 0 dT1 × � ˜Eix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � cos 1 εωci (T − T1) + ˜Eiy � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � sin 1 εωci (T − T1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (22) 5 ˜Uiy � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � = ei εmi � n T � 0 dT1 × � − ˜Eix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � sin 1 εωci (T − T1) + ˜Eiy � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � cos 1 εωci (T − T1) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (23) With variables ˜vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' where Zi = εz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' the Vlasov equation for the distribution function Fi � ˜vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � of ions in the sheared poloidal flow for time T ≫ τcorr ∼ γ−1 becomes ε∂Fi ∂T + ε˜vix ∂Fi ∂ ˜Xi + ε (˜viy − V ′ 0T ˜vix) ∂Fi ∂ ˜Yi + εviz ∂Fi ∂Zi − ε˜vix T � 0 ∂ ∂Xi ˜Vix (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1 ∂Fi ∂ ˜Xi − ε˜vix T � 0 ∂ ∂Xi ˜Viy (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1 ∂Fi ∂ ˜Yi − ε ˜Uix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � T � 0 ∂ ∂Xi ˜Vix (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1 ∂Fi ∂ ˜Xi − ε ˜Uix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � \uf8eb \uf8edV ′ 0T + T � 0 ∂ ∂Xi ˜Viy (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1 \uf8f6 \uf8f8 ∂Fi ∂ ˜Yi + ωci˜viy ∂Fi ∂˜vix − ωci˜vix ∂Fi ∂˜viy − ε ei mi � ∂ ∂ ˜Xi ϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t � − V ′ 0T ∂ ∂ ˜Yi ϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t �� ∂Fi ∂˜vix − ε ei mi ∂ ∂ ˜Yi ϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂Fi ∂˜viy − ε ei mi ∂ ∂Zi ϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t � ∂Fi ∂viz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (24) The electrostatic potential ϕi in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (24) depends on the micro- and mesoscales and can be expressed in the form ϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t � = ˜ϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t � + Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (25) where ˜ϕi is the electrostatic potential of the microscale turbulence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Ei � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � = −∇riϕi � ˜ri + ˜Ri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (26) and Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � is the potential which determines the electric field of the plasma response on the mesoscale convective flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ¯Ei � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � = −∇Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (27) The Vlasov equation for the ion distribution function ¯Fi � ˜vi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' averaged over the microscale ini- tial phases for a time t ≫ τcorr ∼ γ−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' is ∂ ¯Fi ∂T + ˜vix ∂ ¯Fi ∂ ˜Xi + (˜viy − V ′ 0T ˜vix) ∂ ¯Fi ∂ ˜Yi − ¯Uix � ˜Xi � ∂ ¯Fi ∂ ˜Xi − ¯Uiy � ˜Xi � ∂ ¯Fi ∂ ˜Yi + viz ∂ ¯Fi ∂Zi + 1 εωci˜viy ∂ ¯Fi ∂˜vix − 1 εωci ∂ ¯Fi ∂˜viy − ei mi \uf8eb \uf8ed ∂Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂ ˜Xi − V ′ 0T ∂Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂ ˜Yi \uf8f6 \uf8f8 ∂Fi ∂˜vix − ei mi ∂Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂ ˜Yi ∂Fi ∂˜viy − ei mi ∂Φi � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂Zi ∂Fi ∂viz = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (28) 6 where the velocities ¯Uix � ˜Xi � and ¯Uiy � ˜Xi � are ¯Uix � ˜Xi � = � ˜Uix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � T � 0 ∂ ∂Xi ˜Vix (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (29) ¯Uiy � ˜Xi � = � ˜Uix � ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˜Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � T � 0 ∂ ∂Xi ˜Viy (Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Yi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε) dT1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (30) The details of the calculation of ¯Uix � ˜Xi � and ¯Uiy � ˜Xi � for the arbitrary electric field ˜Ei are given in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' For the electric field (8), the velocities ¯Uix � ˜Xi, T � and ¯Uiy � ˜Xi, T � are presented in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The electrostatic potential Φi � ˜Xi, ˜Yi, Zi, T � of the plasma response on the mesoscale convective flows is gov- erned by the Poisson equation ∂2Φi � ˜Xi, ˜Yi, Zi, T, ε � ∂ ˜ X2 i + ∂2Φi � ˜Xi, ˜Yi, Zi, T, ε � ∂ ˜ Y 2 i + ∂2Φi � ˜Xi, ˜Yi, Zi, T, ε � ∂Z2 i = −4π � α=i,e eαnα � ˜Xα, ˜Yα, Zα, T, ε � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (31) in which nα � ˜Xα, ˜Yα, Zα, T, ε � = � d˜vα ¯fα � ˜vα, ˜Xα, ˜Yα, Zα, T, ε � is the density perturbation, and ¯fα � ˜vα, ˜Xα, ˜Yα, Zα, T, ε � = ¯Fα (˜vα⊥, φ, vz, ξα, ηα, Zα, T, ε) − ¯Fα0 is the pertur- bation of the equilibrium distribution function ¯Fα0 of the convected plasma species α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Vlasov equation for the average electron distribu- tion ¯Fe � ˜ve, ˜Xe, ˜Ye, Ze, T � , where ˜Xe, ˜Ye are determined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (14), (15) with ion species subscript changed on the electron species subscript, has a form similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The velocities ¯Uex � ˜Xe � and ¯Uey � ˜Xe � are de- termined in this equation by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (29), (30) in which velocities ˜Uex � ˜re, ˜Xe, T � and ˜Uey � ˜re, ˜Xe, T � are deter- mined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (22), (23) with the turbulent electric field ˜Ee � ˜re, ˜Xe, T � , given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Vlasov equations (28) for ¯Fi � ˜vi, ˜Xi, ˜Yi, Zi, T � , and the similar equation for ¯Fe � ˜ve, ˜Xe, ˜Ye, Ze, T � , and the Poisson equation (31) compose the Vlasov-Poisson system, which governs the kinetic mesoscale evolution of a plasma under the average action of the spatially inhomogeneous microturbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' As a first step to solution of the system of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (28), (31), we find the characteristics of the operator D DT = ∂ ∂T − ¯Uix � ˜Xi � ∂ ∂ ˜Xi − ¯Uiy � ˜Xi � ∂ ∂ ˜Yi , (32) which are determined by the system dT = − d ˜Xi ¯Uix � ˜Xi � = − d ˜Yi ¯Uiy � ˜Xi �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (33) For deriving the simplest solution to system (33), which reveals the effects of the spatial inhomogeneity of the con- vective flow velocities, we use in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (33) the expansions ¯Uix � ˜Xi � = ¯U (0) ix + ¯U ′ ix � ˜X(0) i � � ˜Xi − ˜X(0) i � , (34) and ¯Uiy � ˜Xi � = ¯U (0) iy + ¯U ′ iy � ˜X(0) i � � ˜Xi − ˜X(0) i � (35) at the vicinity of an arbitrary coordinate ˜X(0) i , and con- sider the case of the uniform velocity compressing rate, ¯U ′ ix = const, and of the uniform velocity shearing rate, ¯U ′ iy = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The solution to system (33) for this case has a form20 ˇXi = 1 ¯U ′ ix �� ¯U (0) ix + ¯U ′ ix � ˜Xi − ˜X(0) i �� e ¯U′ ixT − ¯U (0) ix � , (36) and ˇYi = ˜Yi + � ¯U (0) iy − ¯U (0) ix ¯U ′ iy ¯U ′ ix � T − ¯U ′ iy � ¯U ′ ix �2 � ¯U (0) ix + ¯U ′ ix � ˜Xi − ˜X(0) i �� , (37) where ˇXi and ˇYi are the integrals of system (33) with expansions (34), (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Note, that at T = 0, ˇXi = ˜Xi − ˜X(0) i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In what follows, we put X0 = 0 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With variables ˇXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇYi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' vz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε and with ˜vi⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' de- termined by relations ˜vix = ˜vi⊥ cos φ and ˜viy = ˜vi⊥ sin φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' the Vlasov equation (28) obtains the form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' which does not contain the explicit dependence on ˜Xi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 7 ∂ ∂T ¯Fi � ˜vi⊥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' vz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇYi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � + ˜vi⊥ cos φ e ¯U′ ixT ∂ ¯Fi ∂ ˇXi + � ˜vi⊥ sin φ − ˜vi⊥ cos φ � V ′ 0T + ¯U ′ iy ¯U ′ ix �� ∂ ¯Fi ∂ ˇYi + viz ∂ ¯Fi ∂Zi − ωci ∂ ¯Fi ε∂φ − ei mi � e ¯U′ ixT ∂Φi � ˇXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇYi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂ ˇXi − � V ′ 0T + ¯U ′ iy ¯U ′ ix � ∂Φi � ˇXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇYi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂ ˇYi � ∂ ¯Fi ∂ˇvix − ei mi ∂Φi � ˇXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇYi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂ ˇYi ∂ ¯Fi ∂˜viy − ei mi ∂Φi � ˇXi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇYi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T � ∂Zi ∂ ¯Fi ∂viz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (38) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (38), the effects the spatial inhomogeneity of the sheared and convected flows is transformed to the time domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These effects are presented by the lin- early growing with time V ′ 0T coefficient originated from the basic poloidal sheared flow, and of the exponentially growing with time coefficient e ¯U′ ixT originated from the compressed convective flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' These coefficients reveal the effects of the continuous distortion of the perturbations in the sheared and compressed flows17–19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It was found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='20 that the compressing rate ¯U ′ ix of the radial ve- locity of the compressed flow and the shearing rate V ′ 0 of the poloidal sheared flow velocity are commensurable for the tokamak edge condition, and both are much less than the ion cyclotron frequency ωci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Equation (38) displays that the exponentially growing with time effect of the compressed flow is the dominant factor in the mesoscale temporal evolution at time T > � ¯U ′ ix �−1 of the plasma with a radially inhomogeneous turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Therefore the small parameter ε in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (38), which determines a ratio of the micro- to meso- scales, is naturally to define as equal to ε = ¯U′ ix ωci .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The next substantial simplification of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (38) arises with transformation of the part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (38), which does not contain the electrostatic potential Φi � ˇXi, ˇYi, T � , by employing the characteristic equations dT = −εdφ ωci = dZi vz = d ˇXi ˜vi⊥ cos φ e ¯U′ ixT = d ˇYi ˜vi⊥ sin φ − ˜vi⊥ cos φ � V ′ 0T + ¯U′ iy ¯U′ ix �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (39) The solutions to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (39) are given by the relations ˇXi = ξi − ˜vi⊥ε ωci e ¯U′ ixT sin � φ1 − 1 εωciT � + O �ε ¯U ′ ix ωci ≪ 1 � , (40) ˇYi = ηi + ˜vi⊥ε ωci cos � φ1 − 1 εωciT � + ˜vi⊥ε ωci � V ′ 0T + ¯U ′ iy ¯U ′ ix � sin � φ1 − 1 εωciT � + O � ε V ′ 0 ωci ≪ 1 � , (41) φ = φ1 − 1 εωciT, (42) Zi = Zi1 + vzT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (43) The integrals ξi and ηi in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (40) and (41) are the guid- ing center coordinates in the compressed-sheared convec- tive flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In coordinates ξi, ηi, φ1, Zi1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (38) has a simple form ∂ ∂T ¯Fi (˜vi⊥, φ1, vz, ξi, ηi, Zi, T, ε) + ei mi ωci ε˜vi⊥ �∂Φi ∂φ1 ∂ ¯Fi ∂˜vi⊥ − ∂Φi ∂˜vi⊥ ∂ ¯Fi ∂φ1 � + eiε miωci e ¯U′ ixT �∂Φi ∂ξi ∂ ¯Fi ∂ηi − ∂Φi ∂ηi ∂ ¯Fi ∂ξi � − ei mi ∂Φi ∂Zi1 ∂ ¯Fi ∂vz = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (44) The solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (44) for the ion distribution function ¯Fi we derive in the form ¯Fi (˜vi⊥, φ, vz, ξi, ηi, Zi1, T, ε) = ¯Fi0 + ¯fi (˜vi⊥, φ, vz, ξi, ηi, Zi1, T, ε), where ¯Fi0 is the ion distribution function ¯Fi0 of the unperturbed ion convec- tive flow, and ¯fi is the perturbation of ¯Fi0 caused by the ions respond on the plasma convective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The equa- tion for ¯Fi0 follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (44), in which potential Φi of the electrostatic response of plasma on the mesoscale convective flows is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This equation, ∂ ¯Fi0/∂T = 0, (45) reveals that with the guiding center coordinates ξi, ηi, 8 the unperturbed ion distribution function ¯Fi0 of the compressed-sheared ion flow is stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (45) for the inhomogeneous ion component along coordinate ˜Xi is an arbitrary function ¯Fi0 = ¯Fi0 (˜vi, ξi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With the initial Maxwellian distribution ¯Fi0 � ˜vi, ˜Xi � = ni0 � ˜Xi � � 2πv2 T i � ˜Xi ��3/2 e − ˜v2 i 2v2 T i( ˜ Xi) , (46) given for time T = 0, at which ξi ≈ ˇXi = ˜Xi (it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (36) and from the estimate ˜vi/ωci ∼ vT i/ωci ≪ (LE, Lni)) the solution for ¯Fi0 will have a form (46), in which the ion equilibrium density and the ion thermal ve- locity are equal to ni0 (ξi) and vT i (ξi) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Note, that with variable ˜Xi the ion density ni0 is the time de- pendent, ni0 (ξi) = ni0 � 1 ¯U ′ ix �� ¯U (0) ix + ¯U ′ ix ˜Xi � e ¯U′ ixT − ¯U (0) ix �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (47) The same dependences on ˜Xi and T has the ion ther- mal velocity vT i (ξi) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Equation (46) reveals, that the time dependent inhomogeneous ion density of the convective flow, as it is with coordinate ˜Xi, becomes the steady spatially inhomogeneous ion density distribu- tion along characteristic ξi at any time T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' THE COMPRESSED-SHEARED MODES APPROACH TO THE STABILITY THEORY OF THE MESOSCALE CONVECTIVE FLOWS In this section, we consider the microscale respond of the ions and electrons, determined by the functions ¯fi and ¯fe on the generation of the mesoscale convective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (44), that the Vlasov equation for the perturbation ¯fi (˜vi⊥, φ, viz, ξi, ηi, Zi1, T, ε) of ¯Fi0 (˜vi, ξi) has a simple form in guiding center coordinates ξi and ηi, ∂ ∂T ¯fi (˜vi⊥, φ1, viz, ξi, ηi, Zi1, T, ε) = − ei mi ωci ε˜vi⊥ ∂Φi ∂φ1 ∂ ¯Fi0 ∂˜vi⊥ + eiε miωci e ¯U′ ixT ∂Φi ∂ηi ∂ ¯Fi0 ∂ξi + ei mi ∂Φi ∂Zi1 ∂ ¯Fi0 ∂viz .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (48) The Vlasov equation (48) for ¯fi with given unperturbed ion distribution function ¯Fi0, the equation for the pertur- bation ¯fe (˜ve⊥, φ1, vz, ξe, ηe, Ze1, T, ε) of the electron dis- tribution similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (48), and the Poisson equation (35) for the potential Φi � ˇXi, ˇYi, Zi1, T, ε � in coordinates ˇXi, ˇYi e2 ¯U′ ixT ∂2Φi ∂ ˇX2 i − 2e ¯U′ ixT � V ′ 0T + ¯U ′ iy ¯U ′ ix � ∂2Φi ∂ ˇXi∂ ˇYi + \uf8eb \uf8ed1 + � V ′ 0T + ¯U ′ iy ¯U ′ ix �2\uf8f6 \uf8f8 ∂2Φi ∂ ˇY 2 i + ∂2Φi ∂Z2 i1 = −4π � eini � ˇXi, ˇYi, Zi1, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε � −|e|ne � ˇXe, ˇYe, Ze1, T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ε �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (49) compose the system of equations for the investigations of the stability of the mesoscale convective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In this section, we consider the stability of the compressed- sheared flows against the development of the low fre- quency microscale instabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' For that goal, we trans- form Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (48), (49) to the microscale time t = T ε , the microscale spatial coordinates xi = 1 εXi, yi = 1 εYi and to microscale coordinates guiding center coordinates ˇξi = ξi ε , ˇηi = ηi ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With time t and microscale coordinates ˇξi, ˇηi, the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (48) with known distribution ¯Fi0 is ¯fi = ei mi t � 0 dt1 � e ¯U′ ixt ωci ∂Φi ∂ˇηi ∂ ¯Fi0 ∂ ˇξi − ωci ˜vi⊥ ∂Φi ∂φ1 ∂ ¯Fi0 ∂˜vi⊥ + ∂Φi ∂Zi1 ∂ ¯Fi0 ∂vz � , (50) where the prime in ¯U ′ ix, ¯U ′ iy and V ′ 0 denotes in this sec- tion the derivatives of ¯Uix, ¯Uiy and V0 with respect to the microscale coordinate ˜xi = ˜ Xi ε .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Equation (50), as well as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (44), do not contain the spatial inhomogene- ity originated from the inhomogeneity of the convective flows velocities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Therefore, by the Fourier transforming the potential Φi (ˇxi, ˇyi, zi1, t) over the microscale spatial coordinates ˇxi, ˇyi, Φi (ˇxi, ˇyi, zi1, t, ) = 1 (2π)3 � dkˇxidkˇyidkz × Φi (kˇxi, kˇyi, kz, t) ei(kˇxi ˇxi+kˇ yi ˇyi+kzzi) (51) we will derive from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (48) and (49) the equation for the separate spatial Fourier mode Φi (kˇxi, kˇyi, kz, t) of the microscale plasma response on the mesoscale compressed- sheared convective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' With coordinates ˇξi, ˇηi, used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (50),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' potential Φi �ˇξi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' zi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t � is determined by the relation Φi (ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇηi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' zi1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) = 1 (2π)3 � dkˇxidkˇyidkZ × Φi (kˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) ei(kˇxi ˇξi+kˇ yi ˇηi+kzzi1) × exp � −iki⊥ (t) ˜vi⊥ ωci sin (φ − ωcit − δ (t)) � = � dkˇxidkˇyidkzΦi (kˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) × ei(kˇxi ˇξi+kˇ yi ˇηi+kzzi1) × ∞ � n=−∞ Jn �ki⊥ (t) ˜vi⊥ ωci � × e−in(φ1−ωcit−δ(t)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (52) 9 in which Jn is the Bessel function of the first kind of the order n and the wave number component ki⊥ (t) across the magnetic field grows with time due to the distort- ing of the wave structure by the compressed and sheared flows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' k2 i⊥ (t) = � kˇxie ¯U′ ixt − kˇyi � V ′ 0t + ¯U ′ iy ¯U ′ ix ��2 + k2 ˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='(53) and tan δi (t) = kˇyi � kˇxie ¯U′ ixt − kˇyi � V ′ 0t + ¯U ′ iy ¯U ′ ix ��−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (54) The solution (50) with potential (52) is ¯fi � ˜vi⊥, φ1, viz, ˇξi, ˇηi, zi1, t � = i ei mi t � 0 dt1 × � dkˇxidkˇyidkzΦi (kˇxi, kˇyi, kz, t1) × ei(kˇxi ˇξi+kˇ yi ˇηi+kzzi) × ∞ � n=−∞ Jn �ki⊥ (t1) ˜vi⊥ ωci � e−in(φ1−ωcit1−δ(t1)) × �kˇyi ωci e ¯U′ ixt1 ∂ ¯Fi0 ∂ ˇξi + nωci ˜vi⊥ ∂ ¯Fi0 ∂˜vi⊥ + kz ∂ ¯Fi0 ∂viz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (55) In what follows, we consider the stability of the mi- croscale perturbations of the convective flows with wave- length much less than the plasma inhomogeneity scale length L ˇ Xi, for which |ki⊥Lˇxi| ≫ 1 and the Fourier transform of ¯fi over ˇxi can be performed in the local approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Fourier transformed microscale per- turbation of the ion density, determined within this ap- proximation, is ni (kˇxi, kˇyi, kz, t) = i2πei mi ∞ � n=−∞ t � 0 dt1Φi (kˇxi, kˇyi, kz, t1) ∞ � −∞ dviz ∞ � 0 d˜vi⊥˜vi⊥ × ∞ � n=−∞ Jn �ki⊥ (t) ˜vi⊥ ωci � Jn �ki⊥ (t1) ˜vi⊥ ωci � e−ikzviz(t−t1)−in(ωci(t−t1)−δ(t)+δ(t1)) × �kˇyi ωci e ¯U′ ixt ∂ ¯Fi0 ∂ ˇξi + nωci ˜vi⊥ ∂ ¯Fi0 ∂˜vi⊥ + kz ∂ ¯Fi0 ∂viz � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (56) For the Maxwellian distribution ¯Fi0 �˜vi, ˇξi � of ions with initial value (46) in the case of the uniform ion tempera- ture Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (56) gives ni (kˇxi, kˇyi, kz, t) = in0i �ˇξ � ei Ti × ∞ � n=−∞ t � 0 dt1Φi (kˇxi, kˇyi, kz, t1) × In � ki⊥ (t) ki⊥ (t1) ρ2 i � e− 1 2 ρ2 i(k2 i⊥(t)+k2 i⊥(t1)) × e− 1 2 k2 zvz(t−t1)2−in(ωci(t−t1)−δ(t)+δ(t1)) × � kˇyivdie ¯U′ ixt − nωci + ik2 zv2 T i (t − t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (57) where vdα = (cTα/eαB0)d ln n0(ˇxi)/dˇxi is the ion (α = i) and electron (α = e) diamagnetic velocity, ρi is the ther- mal ion Larmor radius, and In is the modified Bessel function of the first kind and order n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Fourier trans- form ne (kˇxe, kˇye, kz, t) of the microscale perturbation of the electron density, performed in the electron frame with coordinates ˇxe, ˇye, ze1, t, is determined in the same way as it is given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (34)-(58) for ni (kˇxi, kˇyi, kz, t) with changed ion on electron species subscripts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The derived ion and electron density perturbations are employed in the Poisson equation, Fourier transformed over the vari- ables ˇxi, ˇyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Therefore ne (kˇxe, kˇye, kz, t) for the Poisson equation should be recalculated in the variables ˇxi, ˇyi of the ion frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' For this goal, we derive the relations be- tween variables ˇxi, ˇyi and ˇxe, ˇye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Because the difference between ˜xi and ˜xe, as well as between ˜yi and ˜ye, are on the order of the microscale displacements of the ions rela- tive to electrons in FW field, which are on the order of or less than the wavelength of the microscale perturbations, we can use relations ˜xi = ˜xe, and ˜yi = ˜ye with Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (40),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (41) and obtain on this way the relations ˇxe (ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) = 1 ¯U ′ex � e ¯U′ ext � ¯U (0) ex + ¯U ′ ex ¯U ′ ix �� ¯U (0) ix + ¯U ′ ixˇxi � e− ¯U′ ixt − ¯U (0) ix �� − ¯U (0) ex � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (58) 10 ˇye (ˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) = ˇyi − �� ¯U (0) iy − ¯U (0) ix ¯U ′ iy ¯U ′ ix � − � ¯U (0) ey − ¯U (0) ex ¯U ′ ey ¯U ′ex �� t + ¯U (0) ix ¯U ′ ix � ¯U ′ iy ¯U ′ ix − ¯U ′ ey ¯U ′ex � e− ¯U′ ixt + ¯U ′ ey ¯U ′ex � ¯U (0) ix ¯U ′ ix − ¯U (0) ex ¯U ′ex � + � ¯U ′ iy ¯U ′ ix − ¯U ′ ey ¯U ′ex � ˇxie− ¯U′ ixt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (59) Note, the relations for ˇxi (ˇxe, t) and for ˇyi (ˇye, ˇxe, t) are derived by changing species subscripts i ⇆ e in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (58), (59).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The equation for ne (kˇxe, kˇye, kz, t), similar to (56) for ni, contains the Fourier transform Φe (kˇxe, kˇye, kz, t1) of the potential Φe (ˇxe, ˇye, z, t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The connection relation of Φe (kˇxe, kˇye, kz, t1) with Φi (kˇxi, kˇyi, kz, t1) follows from the relation Φe (kˇxe, kˇye, kz, t1) = � dˇxe � dˇyeΦ (ˇxe, ˇye, kz, t1) e−i(kˇxe ˇxe+kˇ ye ˇye) = 1 (2π)2 � dkˇxi � dkˇyiΦi � kˇxi, k ˇYi, kz, t1 � � dˇxi � dˇyi ∂ (ˇxe, ˇye) ∂ (ˇxi, ˇyi) × ei(kˇxi −kˇxe)ˇxi+i(kˇ yi −kˇ ye)ˇyi−ikˇxe (ˇxe−ˇxi)−ikˇ ye (ˇye−ˇyi) = Φi (kˇxe + kˇxeb1x (t1) + kˇyeb1y (t1) , kˇye, kz, t1) e( ¯U′ ex− ¯U′ ix)t1e−ikˇxe b0x(t1)−ikˇ ye b0y(t1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (60) In Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (60),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ∂ (ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇye) ∂ (ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇyi) = e( ¯U′ ex− ¯U′ ix)t1 (61) is the Jacobian of the transformation ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇye to ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' and the relations ˇxe (ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t1) − ˇxi = b0x (t1) + b1x (t1) ˇxi (62) and ˇye (ˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t1) − ˇyi = b0y (t1) + b1y (t1) ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (63) where b0x (t1) = 1 ¯U ′ex e ¯U′ ext1 � ¯U (0) ex + ¯U (0) ix ¯U ′ ex ¯U ′ ix � e− ¯U′ ixt1 − 1 �� − ¯U (0) ix ¯U ′ ix ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (64) b1x (t1) = � e( ¯U′ ex− ¯U′ ix)t1 − 1 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (65) b0y (t1) = �� ¯U (0) ey − ¯U (0) ex ¯U ′ ey ¯U ′ex � − � ¯U (0) iy − ¯U (0) ix ¯U ′ iy ¯U ′ ix �� t1 + ¯U (0) ix ¯U ′ ix � ¯U ′ iy ¯U ′ ix − ¯U ′ ey ¯U ′ex � e− ¯U′ ixt1 + ¯U ′ ey ¯U ′ex � ¯U (0) ix ¯U ′ ix − ¯U (0) ex ¯U ′ex � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (66) b1y (t1) = � ¯U ′ iy ¯U ′ ix − ¯U ′ ey ¯U ′ex � e− ¯U′ ixt1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (67) were used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Now we determine the relation between the Fourier transform ne (kˇxe, kˇye, kz, t) of the electron density per- turbation ne (ˇxe, ˇye, ze, t), performed in the electron frame with variables ˇxe, ˇye, with the Fourier transform n(i) e (kˇxi, kˇyi, kz, t) of ne (ˇxe, ˇye, Ze, t), performed in the ion frame with variables ˇxi, ˇyi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' n(i) e (kˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) = � dˇxi � dˇyie−i(kˇxi ˇxi+kˇ yi ˇyi)ne (ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) = � dˇxe � dˇyene (ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) ∂ (ˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇyi) ∂ (ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇye)e−ikˇxi ˇxe−ikˇ yi ˇye−ikˇxi (ˇxi−ˇxe)−ikˇ yi (ˇyi−ˇye) = � dˇxe � dˇyene (ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) e( ¯U′ ix− ¯U′ ex)t × e−ikˇxi ˇxe−ikˇ yi ˇye−ikˇxi (a0x(t)+a1x(t)ˇxe)−ikˇ yi (a0y(t)+a1y(t)ˇxe) = e( ¯U′ ix− ¯U′ ex)te−ikˇxi a0x(t)−ikˇ yi a0y(t)ne (kˇxi (1 + a1x (t)) + kˇyia1y (t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (68) 11 where the relations ˇxi (ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) − ˇxe = a0x (t) + a1x (t) ˇxe (69) and ˇyi (ˇye,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) − ˇye = a0y (t) + a1y (t) ˇxe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (70) where used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The functions a0x (t), a1x (t), a0y (t), and a1y (t) are determined by the functions b0x (t), b1x (t), b0y (t), and b1y (t), respectively, by changing species sub- scripts i ⇆ e in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (59) -(64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' By replacing kˇxe and kˇye in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (64) on kˇxi (1 + a1x (t)) + kˇyia1y (t) and kˇyi, which, as it follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' are the new wave numbers conjugate with coordinates ˇxe and ˇye in n(i) e (kˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' we derive the following relation for n(i) e : n(i) e (kˇxi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t) = 2iπe me e−ikˇxia0x(t)−ikˇ yi a0y(t) t � 0 dt1e( ¯U′ ix− ¯U′ ex)(t−t1) × Φi ((kˇxi (1 + a1x (t)) + kˇyia1y (t1)) (1 + b1x (t1)) + kˇyib1y (t1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kˇyi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' kz,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' t1) × e−i(kˇxi (1+a1x(t))+kˇ yi a1y(t1))b0x(t1)−ikˇ yi b0y(t1) × ∞ � 0 dve⊥ve⊥ ∞ � −∞ dveze−ikzvez(t−t1) � kˇyi ωce e ¯U′ ext1 ∂ ¯Fe ∂ ˇξe + kz ∂ ¯Fe ∂vez � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (71) Equation (71) is valid for the perturbations with fre- quency much less than the electron cyclotron frequency and with wavelength across the magnetic field much larger than the thermal electron Larmor radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The Poisson equation (49) for Φi, Fourier transformed over ˇxi and ˇyi, \uf8eb \uf8edk2 ˇxie2 ¯U′ ixt − 2e ¯U′ ixt � V ′ 0t + ¯U ′ iy ¯U ′ ix � kˇxikˇyi + \uf8eb \uf8ed1 + � V ′ 0t + ¯U ′ iy ¯U ′ ix �2\uf8f6 \uf8f8 k2 ˇyi + k2 z \uf8f6 \uf8f8 Φi (kˇxi, kˇyi, kz, t) = 4π � eini (kˇxi, kˇyi, kz, t) − |e|n(i) e (kˇxe (kˇxi, kˇyi, t) , kˇyi, kz, t) � , (72) where ni and n(i) e are determined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (54) and (71) respectively, is the equation which determines the temporal evolution of the single spatial Fourier mode Φi (kˇxi, kˇyi, kz, t) in the compressed-sheared flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Now we consider the particular cases for Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (72), in which ni and n(i) e are determined by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (57) and (71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In the case of the currentless compressed-sheared flow ¯U (0) ix = ¯U (0) ex , ¯U ′ ix = ¯U ′ ex, and ¯U (0) iy = ¯U (0) ey , ¯U ′ iy = ¯U ′ ey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' It follows from Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (62)-(67) that in this case b0x (t1) = a0x (t) = 0, b1x (t1) = a1x (t) = 0, and b0y (t1) = a0y (t) = 0, b1y (t1) = a1y (t) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Therefore in this case ˇxi = ˇxe, ˇyi = ˇye, and Φe (kˇxe, kˇye, kz, t) = Φi (kˇxi, kˇyi, kz, t) and n(i) e (kˇxi, kˇyi, kz, t) = ne (kˇxe, kˇye, kz, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Equation (72) in this case has a form λ2 Di � k2 i⊥ (t) + k2 z � Φi (kˇxi, kˇyi, kz, t) = ∞ � n=−∞ t � t0 dt1Φi (kˇxi, kˇyi, kz, t1) In � ki⊥ (t) ki⊥ (t1) ρ2 i � × e− 1 2 ρ2 i(k2 i⊥(t)+k2 i⊥(t1)) � ikˇyivdie ¯U′ ixt1 − inωci − k2 zv2 T i (t − t1) � e− 1 2 k2 zv2 T i(t−t1)2−in(ωci(t−t1)−δ(t)+δ(t1)) = Ti Te t � t0 dt1Φi (kˇxi, kˇyi, kz, t1) e− 1 2 k2 zv2 T e(t−t1)2 � ikˇyivdee ¯U′ ext1 − k2 zv2 T e (t − t1) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (73) where t0 ≥ 0, λDi(e) is the ion (electron) Debye length, and Ain (t, t1) = In � ki⊥ (t) ki⊥ (t1) ρ2 i � 12 × e− 1 2 ρ2 i(k2 i⊥(t)+k2 i⊥(t1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (74) Equation (73) was derived for the first time in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='18 for the poloidal sheared plasma flow without the con- vective flows (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' for the case ¯U ′ ix = ¯U ′ ex = 0 and ¯U ′ iy = ¯U ′ ey = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In that case, the poloidal velocity shear manifests as a time- dependence of Ain (t, t1) function which determines effect of the finite ion Larmor radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (73), derived for the kinetic drift instability in the poloidal sheared flow, displays18 the nonmodal effect of the reduction with time of the fre- quency and of the growth rate of this instability caused by the flow velocity shear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' By the integration by parts of the first term on the right part of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (73), this equation for the low frequency perturbations, for which dΦ/dt ≪ ωciΦ, may be presented in the form similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (25) of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='18, t � t0 dt1 d dt1 � Φi (kˇxi, kˇyi, kz, t1) � 1 + Ti Te − Ain (t, t1) �� − i t � t0 dt1Φikˇyivdie ¯U′ ext1Ain (t, t1) = Ti Te t � t0 �dΦi dt1 + ikˇyivdee ¯U′ ext1Φi � e− 1 2 k2 zv2 T e(t−t1)2 (75) We found that in the time domain (t, t0), in which ki⊥ (t1) ρi ≫ 1, the temporal evolution of the potential Φi in the compressed flow, predicted by the solution to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (75), resembles the temporal evolution of the po- tential Φi in the poloidal sheared flow: the potential Φi gradually becomes a zero- frequency cell-like perturba- tion when time elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' CONCLUSIONS A nonmodal kinetic theory of the stability of the two- dimensional compressed-sheared mesoscale plasma flows, generated by the radially inhomogeneous electrostatic ion cyclotron parametric microturbulence in the pedestal plasma with a sheared poloidal flow, is developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' This theory reveals that the separate spatially uniform Fourier modes of the electrostatic responses of the ions and of the electrons on the mesoscale convective flows are de- termined only in the frames of references moved with velocities of the ion and electron convective flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In the laboratory frame, these modes are observed as the compressed-sheared modes with time dependent wave numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The integral equation, which governs the sepa- rate Fourier mode of the electrostatic potential of the plasma species responses on the mesoscale convective flows, is derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' In this equation, the effects of the compressing and shearing of the convective flows are re- vealed as the time dependence of the finite ion Larmor radius effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The solution of this equation for the kinetic drift instability displays the nonmodal transformation of the potential to the zero frequency cell-like perturbation when time elapsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ACKNOWLEDGMENTS This work was supported by National R&D Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' NRF-2018R1D1A3B07051247) and BK21 FOUR, the Creative Human Resource Educa- tion and Research Programs for ICT Convergence in the 4th Industrial Revolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Appendix A: Velocities ˜Uix � ˜ Xi � and ˜Uiy � ˜ Xi � of the convective flows For the electric field ˜Ei, given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (8), velocities ¯Uix � ˜Xi � and ¯Uiy � ˜Xi � , after long calculation similar to performed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='14, are determined by the following re- lations: ¯Uix � ˜Xi � = 1 2ωci e2 i m2 i 1 4π2 � n � dk � ai1 (k, n) ˜Eix � k, ˜Xi, n � ∂ ∂ ˜Xi � ˜E∗ iy � k, ˜Xi, n �� 13 +ai2 (k, n) ˜Eiy � k, ˜Xi, n � ∂ ∂ ˜Xi � ˜E∗ ix � k, ˜Xi, n ��� (A1) and ¯U (0) iy � ˜Xi � = − 1 4ωci e2 i m2 i 1 4π2 � n � dk � ai1 (k, n) ∂ ∂ ˜Xi ��� ˜Eix � k, ˜Xi, n ���� 2 − ai2 (k, n) ∂ ∂Xi ��� ˜Eiy � k, ˜Xi, n ���� 2� (A2) where the asterisk in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (A1) implies the operation of complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' The coefficients ai1 (k, n) and ai2 (k, n) are determined as ai1 (k, n) = � ωci ωn (k) (ωci + ωn (k))2 + ωci ωn (k) (ωci − ωn (k))2 + 1 (ω2 ci − ω2n (k)) � , (A3) and ai2 (k) = � 1 (ωci + ωn (k))2 + 1 (ωci − ωn (k))2 + 1 (ω2 ci − ω2n (k)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (A4) The velocities of electrons ¯Uex � ˜Xi � and ¯Uey � ˜Xi � in the ion frame, with electric field Ee � ˆri, ˜Xi, t � determined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' (10), are ¯Uex � ˜Xi � ≈ c2 B2 0 1 4π2 � n � dk ˜Eix � k, ˜Xi, n � × ∂ ∂ ˜Xi � ˜E∗ iy � k, ˜Xi, n �� ∞ � p=−∞ J2 p � aie � k, ˜Xi �� × 1 Ωp � k, ˜Xi �, (A5) and ¯Uey � ˜Xi � ≈ −1 2 c2 B2 0 1 4π2 � n � dk ∂ ∂ ˜Xi ��� ˜Eix � k, ˜Xi, n ���� 2 × ∞ � p=−∞ J2 p � aie � k, ˜Xi �� 1 Ωp � k, ˜Xi �, (A6) where limit |ωce| ≫ Ωp ∼ ωci was used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 1M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ono, ”High harmonic fast waves in high beta plasmas”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 2,4075 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 2S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Chiu, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Chan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Harvey, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Porkolab, ”Theory of fast wave current drive for tokamak plasmas”, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fusion 29, 2175 (1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 3R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Perkins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Hosea, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Kramer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' -W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ahn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Diallo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Gerhardt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Gray, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Green, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaworski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' LeBlanc, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' McLean, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Maingi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Phillips, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Roquemore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Sab- bagh, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wilson, ”High-Harmonic Fast-Wave Power Flow along Magnetic Field Lines in the Scrape-Off Layer of NSTX,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 109, 045001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 4R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Perkins, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' -W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ahn, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Diallo, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Gerhardt, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Gray, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Green, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Hosea, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ja- worski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' LeBlanc, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Kramer, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' McLean, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Maingi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Phillips, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Podest‘a, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Roquemore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Sab- bagh, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Scotti, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Taylor, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wilson, ”Fast-wave power flow along SOL field lines in NSTX and the associated power deposi- tion profile across the SOL in front of the antenna,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fusion 53, 083025(2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 5D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Pace, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Pinsker, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Heidbrink, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fisher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Van Zeeland, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Austin, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' McKee, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Garc´ia- Mu˜noz, ”Scrape-off layer ion acceleration during fast wave injec- tion in the DIII-D tokamak”, Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fusion 52, 063019 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 6J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wilson, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bernabei, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Biewer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Diem, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Hosea, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' LeBlanc, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Phillips, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Swain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' ”Para- metric Decay During HHFW on NSTX”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' AIP Conference Pro- ceedings 787, 66 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 7M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Porkolab, ”Parametric processes in magnetically confined CTR plasmas,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fusion 18, 367 (1978).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 8M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Porkolab, ”Parametric instabilities in the tokamak edge plasma in the ion cyclotron heating regimes,” Fusion Engineering and Design 12, 93 (1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 9V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, Hae June Lee, ”The ion cyclotron parametric instabilities and the anomalous heating of ions in the tokamak edge plasma in the fast wave heating regime”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 27, 052508 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 10N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bertelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Hosea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Phillips, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Berry, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Gerhardt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Green, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' LeBlanc, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Perkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Taylor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Valeo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wilson, ”Full wave simulations of fast wave heating losses in the scrape-off layer of NSTX and NSTX-U,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fusion 54, 083004 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 11N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bertelli, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Hosea, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Phillips, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Berry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bonoli, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Gerhardt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Green, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' LeBlanc, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Perkins, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Qin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Pinsker, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Prater, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Taylor, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Valeo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wilson, J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wright, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Zhang, ”Full wave simulations of fast wave efficiency and power losses in the scrape- off layer of tokamak plasmas in mid/high harmonic and minority heating regimes,” Nucl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Fusion 56, 016019 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 12E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Berry, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' D’Azevedo, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Batchelor, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Carter, ”All-orders spectral calculation of radio-frequency heating in two-dimensional toroidal plasmas”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 8, 1573 (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 13D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Green, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Berry, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Chen, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Canik, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, ”Predicting High Harmonic Ion Cyclotron Heating Efficiency in Tokamak Plasmas”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 107, 145001 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 14V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, Hae June Lee, ”Ion cy- clotron parametric turbulence and anomalous convective trans- port of the inhomogeneous plasma in front of the fast wave an- tenna,” Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 28, 042304 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 15J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Hosea, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Bell, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Feibush, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Harvey, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Jaeger, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='LeBlanc, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Maingi, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='Phillips, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Roquemore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Ryan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Taylor, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Tritz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Valeo, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Wilgen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' R, Wil- son, and the NSTX Team,”Recent Fast Wave Coupling and Heat- ing Studies on NSTX, with Possible Implications for ITER”, AIP 14 Conf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 1187, 105 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 16K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Burrell, ”Effects of ExB velocity shear and magnetic shear on turbulence and transport in magnetic confinement”,Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 4, 1499 (1997) 17V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Stepanov, ”Tur- bulence evolution in plasma shear flows”, Plasma Fusion Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content='5, S2015 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 18V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Stepanov, ”Renor- malized non-modal theory of the kinetic drift instability of plasma shear flows”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 18, 062103 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 19V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, Hae June Lee, ”Non- modal theory of the kinetic ion temperature gradient driven in- stability of a plasma shear flows across the magnetic field”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 23, 062115 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' 20V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Mikhailenko, Hae June Lee, ”Anoma- lous convective transport of the tokamak edge plasma, caused by the inhomogeneous ion cyclotron parametric turbulence”, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} +page_content=' Plasmas 29, 072301 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/t9E1T4oBgHgl3EQfjwTU/content/2301.03267v1.pdf'} diff --git a/tNAyT4oBgHgl3EQf0PnV/vector_store/index.pkl b/tNAyT4oBgHgl3EQf0PnV/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..7adadc5309ae68b6ac586e6e4f0f750142185f53 --- /dev/null +++ b/tNAyT4oBgHgl3EQf0PnV/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:4ce1c5f3caa8fd6255a3ceaf1c798b0b9d44fab7f4603772f7188a03e314479b +size 139454 diff --git a/u9E3T4oBgHgl3EQf-QtO/vector_store/index.faiss b/u9E3T4oBgHgl3EQf-QtO/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..e52d892cc26ac3444a1636a5720f08dc71567df8 --- /dev/null +++ b/u9E3T4oBgHgl3EQf-QtO/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:7a566d3664ac5f51d1ed4d61333a1ab216fc60bcc96e13bb4118c78e40544510 +size 10223661 diff --git a/v9AyT4oBgHgl3EQfaffQ/content/tmp_files/2301.00245v1.pdf.txt b/v9AyT4oBgHgl3EQfaffQ/content/tmp_files/2301.00245v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1e6d5812c2611501f7662fb0cd64a5414b17741e --- /dev/null +++ b/v9AyT4oBgHgl3EQfaffQ/content/tmp_files/2301.00245v1.pdf.txt @@ -0,0 +1,1077 @@ + + + +Reprogrammable holograms from maskless surface photo- +morphing + + +Francesco Reda,1 Marcella Salvatore,1,2 Marco Astarita,3 Fabio Borbone4 and +Stefano L. Oscurato1,2,* + +1 Physics Department “E. Pancini”, University of Naples “Federico II”, Complesso Universitario +di Monte Sant’Angelo, via Cinthia 21, 80126, Naples, Italy +2 Centro Servizi Metrologici e tecnologici Avanzati (CeSMA), University of Naples “Federico II”, +Complesso Universitario di Monte Sant’Angelo, Via Cintia 21, 80126, Naples, Italy +3 Physics Department, Politecnico di Milano, 20133, Milan, Italy +4 Department of Chemical Sciences, University of Naples “Federico II”, Complesso Universitario +di Monte Sant’Angelo, Via Cintia, 80126 Naples, Italy +*Corresponding author: stefanoluigi.oscurato@unina.it + +Abstract + +Holographic technologies have the potentiality to impact our everyday life in many sectors +including science, education, entertainment, art, and healthcare. Although holographic screens and +projectors are part of common imagination since long time, they are still at initial stages of +development and integration. Recent achievements of metasurface and flat optics research gave an +unprecedented strength to this field, overcoming critical aspects as efficiency, size and flexibility +of conventional optics and liquid crystal technologies. However, although diffractive and +metasurface holographic projectors with advanced functionalities and improved efficiencies are +continuously reported, they are static devices, requiring demanding, burdensome, and irreversible +manufacturing processes. Here we report an all-optical and single-step lithographic framework for +the fabrication of diffractive holographic projectors directly on the surface of a photo-morphable +polymer film. Real-time optimization during the accurate surface patterning and fully structural +reconfigurability allowed for the first prototype of a fully reprogrammable pixel-less +morphological projector, opening new routes for holographic image displaying and optical data +sharing. + + + +1. Introduction + +Light-modulating planar devices can empower many emerging technologies as virtual and +augmented reality (1–3), optical wireless communication (4, 5), green energy harvesting (6, 7), +opening also to the next‐generation of displays and holographic projectors (8). Despite holograms +can be implemented through addressable liquid crystals on silicon (LCOS) devices (9–12), +diffractive optical elements (13–15) and metasurfaces (16–18) are increasingly gaining interest for +holographic applications, due to their ability to generate arbitrary optical fields from the +modulation of an incident light beam through a ultra-compact and planar device. Untied from +electronics, efficiency, and size limitations of LCOS displays, planar holographic devices promise +a greater possibility of miniaturization while maintaining higher efficiencies and light modulation +capabilities. In addition to images projection, planar holographic devices can also represent a valid +platform for optical information storing, encryption and sharing (19–21). Nevertheless, as also +valid for holographic displays, those technologies intrinsically require optical supports able to be +fully erased and rewritten, a milestone only partially achieved with several limitations by tunable +metasurfaces (22–24). However, these features come at the expense of realization of complex +surface geometries at light (sub-)wavelength scale, where the manufacturing process, typically +leading to static devices, (25, 26) can pose severe performance and/or economic limitations. +Optical lithography is among the most used surface patterning techniques (27) for the +fabrication of planar optical devices. Starting with the irradiation of a photoresist by means of a +structured illumination pattern produced by a mask, the typical workflow requires additional post- +exposure chemical, physical and mechanical processes, through which the desired surface pattern +is finally transferred to the operating device (27, 28). The multileveled surface patterns needed for +optimal functionality of a holographic device can even require several iterations of this scheme +(13). Maskless methods, where the multistep mask exposure is replaced digital-based projection +of spatially structured intensity patterns over the photoresist surface, can offer however greater +control and flexibility for the realization of the complex lateral geometry and the grayscale +modulation (29). Both deformable micromirror devices (DMDs) (30, 31) and LCOS (32–34) were +explored as programmable spatial light modulators to achieve digital maskless surface patterning +for optical devices manufacturing. In addition, even non-optical maskless approaches as particle +beam fabrication methods (35, 36) and scanning probe lithography (37, 38) have been reported for +the accurate optical device manufacturing, but these methods suffers of reduced throughput, +increased costs and energetic impact with respect to optical techniques (26, 27). +Here, we demonstrate the direct all-optical maskless fabrication of fully reconfigurable +diffractive holographic devices, implemented as thin structured transmissive phase retarders +realized on the surface of a reprogrammable dielectric material. To this aim, a digital holographic +optical scheme is used to generate and project a grayscale spatially structured intensity distribution +of light on an azobenzene-containing polymer film, whose surface locally deforms according to + +the irradiated spatial light distribution. In this way, the structured surface of the operating optical +device is directly produced without any additional lithographic step. The photomechanical process +responsible for the direct surface morphing of the polymer is intrinsically reversible, allowing the +update of the fabricated surface geometry at will. Compared to other optical maskless techniques, +our approach allows to fully exploit the possibility of arbitrary spatiotemporal modulation of the +holographic writing beam and the integration of the lithographic system with a real-time optical +characterization setup allowing to evaluate the device performance already during the fabrication. +The all-optical system is used here to realize operating optical configurations and devices with +advanced and optimized functionalities, including reprogrammable grayscale holograms with +improved visibility and tunable axial position, and a high-density optical encryption scheme able +to temporally split secreted holographic information. Representing a new state-of-the-art as a +reprogrammable all-optical fabrication framework for custom multileveled flat optical devices, +our approach can assist the development of next generation photonics, starting from devices +prototyping, testing and assembly until to their large-scale distribution. +2. Results +2.1. Direct holographic surface structuration +To elucidate the main features of our direct holographic maskless surface patterning scheme, +schematically represented in Fig. 1A, we first demonstrate the realization of simple arbitrary +binary pattern on the surface of an azobenzene-containing polymer thin film (herein referred to +as azo-resist to highlight its functionality as lithographic material). This class of amorphous +materials exhibit the unique property of stable surface reliefs formation under low-intensity +structured UV-visible light irradiation (39, 40) as a consequence of a directional material transport +initiated by the azobenzene chromophores hosted in the polymeric matrix (41–43), with a +mechanism still to be fully unveiled (44) . Due to the sensitivity to both the intensity and the +polarization of the irradiated light, the surface reliefs on azopolymer films enable a direct vectorial +lithography, exploited in many configurations, including interference, high-fusing, near-field, and +pure structured polarization illumination (45–54) . +In the irradiation of a circularly polarized light patten 𝐼𝑊(𝑥, 𝑦) in a low-focusing regime, the +spatiotemporal evolution of the surface morphology ℎ(𝑥, 𝑦, 𝑡) can be phenomenologically +described as (54, 55): + +ℎ(𝑥, 𝑦, 𝑡) = ∇2[𝐼𝑊(𝑥, 𝑦)] ∙ ℎ0(𝑡) +(1) + + + +In a low intensity regime, the relief depth ℎ0 increases approximately linearly with the +exposure time (ℎ0(𝑡) = 𝑐 ∙ 𝑡 , where 𝑐 is a phenomenological inscription efficiency constant), +while the material flows from the high intensity region toward dark areas (as schematized in Fig. + +1B), forming then a surface relief pattern with the same geometry as the illuminating intensity +𝐼𝑊(𝑥, 𝑦). +In our maskless optical lithographic scheme, we used a phase-only Computer-Generated +Hologram (CGH) system to fully exploit the direct surface structuration process described by +eq.(1). In this configuration (see also Materials and Methods), arbitrary grayscale illumination +patterns 𝐼𝑊(𝑥, 𝑦), originated by a computer-controlled phase-only Spatial Light Modulator (SLM), +can be directly transferred to the entire illuminated area of the polymer surface in a single exposure +step. + + +Fig. 1. Holographic structuration of azo-resist surface. A Graphical representation of the holographic inscription +scheme. Writing beam, with arbitrary shaped intensity profile is directly projected over the azo-resist surface by an +objective. B Light triggered mass migration occurring at surface of amorphous azopolymer films under structured +illumination absorption, leading to stable surface geometries ℎ(𝑥, 𝑦, 𝑡). C Design and reconstruction of a QR code +shaped holographic pattern. The experimental intensity pattern is the result of time averaging of the holographic +sequence, allowing to reduce speckle noise effects. D Atomic force microscope micrograph of the structured surface +collected right after the exposure step. Red scale bar, both in panels C and D corresponds to a physical size of 20 𝜇𝑚 +on the sample. E Height distribution probability (orange plot) compared with the intensity probability distribution +(sky blue plot) of the holographic beam. Each point of the line plot represents the probability 𝑃𝑖 of having a fixed +height value ℎ𝑖 in the AFM image corresponding to the implemented intensity level 𝐼(𝑤)𝑖. +To demonstrate our ability in arbitrary direct surface patterning, we designed a two-levels QR +code as an 8-bit two-dimensional image, from which the illuminating light pattern 𝐼𝑊(𝑥, 𝑦) is +calculated (56) (Fig. 1C top). The generated holographic writing pattern is projected on the surface + +4 +B +h(x,y,t) +Azo-resist +Holographicpattern +n +0.1 +0.2 +h (um)of the azo-resist by means of a long-working distance microscope objective, where a relief pattern +ℎ(𝑥, 𝑦) directly appears. Additional details about the writing holographic design, the illumination +homogeneity improvement, and the resolution of our configuration can be found in Methods +section and in Fig. S4. + Fig. 1D shows the Atomic Force Microscope (AFM) micrograph of the polymer film surface +after being exposed to the holographic pattern for 𝑡 = 20 𝑠. AFM image is collected right after the +exposure step without any additional post-exposure process. The surface relief pattern faithfully +reproduces the target image, and, as expected from eq. (1), is the complementary of the +illuminating hologram. +To extend the visual comparison to a quantitative analysis needed in the fabrication of +complex relief pattern from the design of a diffractive phase-modulating mask acting as +holographic projector, we characterize eventual mismatch errors between the target and the +experimental surface morphology described in Fig. 1. To this aim, we retrieved the height +distribution of the surface form the topographic image, with a sampling interval of 0.387 𝜇𝑚 +determined by the pixel size of the AFM scan (see also Methods). The distribution, shown in Fig. +1E, must be compared with the target one, in which there are only two equally weighted levels +corresponding to the black (𝐼𝐵) and white (𝐼𝑊) pixels of the image. Despite the presence of two +narrow bands in the distribution extracted from the optical image of the hologram (blue curve in +Fig. 1E), confirming the high contrast in the writing binary pattern, the topographic distribution +(orange curve) of the two heigh levels appears broadened. The origin of such structural mismatch +resides in the relief smoothing at illumination edges with sharp contrast jumps (see also Fig. S5), +as predicted for the light-induced material transport phenomenology described by eq. (1). In our +previous implementation of this lithographic method, we circumvented this issue by limiting +quantitative design to smooth sinusoidal surfaces (46, 57). However, sharp features could +potentially be encoded in the design of a suitable optimized holographic pattern associated to the +target image, providing eventually a narrower topographical distribution when transferred on the +azo-resist film. As further detailed below, the all-optical scheme used here to fabricate and +simultaneously characterize the diffractive optical components allows the minimization of the +effects on optical performances originated by similar fabrication-design mismatches inherent to +the simplified description of material transport in hologram design. +Even in the simplistic linear response relief design used here, the results in Fig. 1 fully +demonstrate the potentialities of our scheme as a direct maskless holographic technique for the +arbitrary structuration of the surfaces at the microscale. The fidelity of the surface pattern can be +further demonstrated by to the possibility of effectively read the binary QR code (by any camera +QR code reading software) from the topographic data, rendered as two-dimensional image with a +linear colormap (Fig. 1D). + +2.2. Holographic morphological projectors: design, optimization, and fabrication +For the design of the azopolymer-based morphological holographic projectors we leverage +the results of the scalar diffraction theory (10). While conventional projection displays exploit +amplitude-modulating pixels to locally and selectively block part of the incident light to form +images, a diffractive holographic projector can be implemented as a phase-only planar device for +a coherent monochromatic light modulation (10, 12), able to reconstruct a desired light pattern +without making use of absorption phenomena. Phase-only holographic plates, named kinoforms +(58), can implement the proper modulating complex transmission function 𝑡(𝑥, 𝑦) = +exp (𝑖𝜑(𝑥, 𝑦)) as local thickness variations ℎ(𝑥, 𝑦) of a dielectric material (Fig. 2), which +influence the optical path traveled by an input monochromatic field 𝑈𝑖𝑛(𝑥, 𝑦, 𝑧𝑖𝑛) (see Materials +and Methods). The phase mask 𝜑(𝑥, 𝑦) is typically referred to as kinoform (58). +According to the diffraction theory, in the case of far-field propagation 𝑧 ≫ 𝑧𝑖𝑛 (Fraunhofer +approximation, where), the emerging modulated field 𝑈𝑜𝑢𝑡(𝑥, 𝑦, 𝑧) is two-dimensional spatial +Fourier transform of the beam modulated at the kinoform plane, resulting in a reconstructed image +𝐼𝑜𝑢𝑡(𝑥, 𝑦) determined by the relation (10): + + +𝐼𝑜𝑢𝑡(𝑥, 𝑦, 𝑧) = |𝐹𝑇[𝑈𝑖𝑛(𝑥, 𝑦, 0) ∙ 𝑒𝑖𝜑(𝑥,𝑦)]| +2 + +(2) +An analogous result can be also found between the two focal planes of a thin lens, reducing +the image reconstruction to finite distances (10). By inversion of eq. (2), the kinoform 𝜑(𝑥, 𝑦), +and the relative mask surface relief pattern ℎ(𝑥, 𝑦) for any given target holographic image +𝐼𝑜𝑢𝑡(𝑥, 𝑦) could be potentially calculated. However, for a phase-only modulator, the kinoform can +be retrieved only through iterative algorithms (8). Fig. 2 schematically shows this process for the +case for the desired output image 𝐼𝑜𝑢𝑡 representing the Greek letter “π”, where the conventional +Gerchberg–Saxton (GS) algorithm (59) is used as iterative Fourier transform algorithm (IFTA) to +retrieve the kinoform 𝜑(𝑥, 𝑦). +Once the kinoform is calculated, all the challenges involved in the fabrication of the +holographic projector are shifted to manufacturing level. Optimal image reconstruction requires +an accurate transfer of the designed phase mask, including the position of the phase discontinuities +(lateral pattern) and the value of local and maximum phase delays, in the proper surface relief +pattern. Any defect arising in this process deteriorates the hologram quality, causing the reduction +in the diffraction efficiency and the appearance of spurious contributions in the target holographic +image, consisting of an unmodulated optical component (DC term) and several shifted and scaled +replicas of the desired intensity pattern (ghost or false images) (60). These contributions can +overlap in the reconstruction plane, requiring eventually an off-axis design for the hologram (Fig. +2), which reduces the available target image domain by half of the field of view (61). However, +even in the case of a defect-free lateral pattern transfer, a deviation from a full 2𝜋 modulation + +depth, associated with eventual total relief heigh errors induced in the dielectric structured surface, +still cause the emergence of the spurious holographic terms. To reduce this effect, an ideal optimal +modulation depth of ℎ0 = 𝜆/(𝑛 − 1) should be realized. This condition simultaneously grants the +maximization of the diffraction efficiency in the target holographic image and ghost hologram +suppression (see also Supplementary Information). +In our direct lithographic scheme, the surface relief pattern ℎ(𝑥, 𝑦) and the modulation depth +ℎ0 can be independently controlled by the digital holographic design and by the exposure time, +respectively. Then, the generalization of the inscription scheme of Fig. 1 to the projection of a +grayscale structured light pattern with the geometry of a calculated kinoform 𝜑(𝑥, 𝑦) can lead to +the fabrication of optimized morphological holographic projectors directly as a surface relief +pattern on the dielectric azo-resist film. + + +Fig. 2. Design of holographic morphological projectors. Target intensity 𝐼𝑜𝑢𝑡 is used to retrieve, by GS iterative +algorithm, the proper phase map 𝜑(𝑥, 𝑦) to be implemented as dielectric height modulated phase retarder. The material +with refractive index 𝑛 is assumed to be immersed in a surrounding medium with refractive index 𝑛𝑠. When +illuminated with monochromatic light, with wavevector 𝑘 = 2𝜋/𝜆, the phase retarder (kinoform) produces a diffracted +beam depending on the optical delay accumulated by the light passing through the structured surface. The kinoform +allows the reconstruction of the target holographic image defined during the design and additional spurious diffraction +orders to be suppressed by tuning the total modulation depth ℎ0. + +To this aim, we first characterized the ability of our lithographic scheme in encoding multiple +discrete intensity levels of light in a single holographic pattern, useful to calibrate the response our +system for the generation of the complex grayscale pattern required by a kinoform fabrication (see +Fig. S6). +Then, we directly inscribe on the azopolymer film the grayscale surface profile ℎ(𝑥, 𝑦) of the +kinoform calculated for the reconstruction of far-field holographic image of the Greek letter “π”. + +元 +元 +Image +DC +Ghost +t(x,y)In this process, the 8-bits (256 levels) digitally calculated kinoform 𝜑(𝑥, 𝑦) is converted into a +gray-scale holographic pattern 𝐼𝑊(𝑥, 𝑦), which induces the correspondent relief pattern ℎ(𝑥, 𝑦) on +the azo-resist surface (Fig. 3). For the analysis of the lateral pattern and the determination of the +total height excursion ℎ0 of the produced surface relief, we performed SEM and AFM analysis +after the exposure process. The SEM analysis (Fig. 3) confirms a correct position-matching of the +phase discontinuity in the kinoform, granting a global correct relief lateral geometry. Fig. 3A +shows, instead, the three-dimensional topographic micrograph of a portion of a typical azo-resist +kinoform surface, evidencing the continuous heigh variation in the pattern, encoded in the +grayscale writing holographic pattern (Fig. 3). + +Fig. 3. Fabrication and optimization of azopolymer holographic projectors implemented as kinoforms. The middle +panel shows the grayscale holographic pattern reproducing the kinoform design and the resulting SEM image of the +structured surface after the exposure. A Atomic force microscope (AFM) scan of a quarter portion of the structured +surface (100 𝑋 100 𝜇𝑚) collected right after the exposure process. B Full modulation depth ℎ0 as function of the total +exposure time. Experimental data are fitted with the model trend ℎ0 = 𝑐 ∙ 𝑡, allowing for the experimental +determination of the surface inscription efficiency 𝑐 = 10.5 ± 0.5 𝑛𝑚/𝑠. Blue axis shows the implemented phase +depth for a probe wavelength 𝜆𝑃. C Diffraction pattern acquired at the optimal exposure time, maximizing the +diffracted light power effectively shaped in the target holographic image. D Experimental trend of the diffraction +efficiency reconstructed during the inscription process. Trends are the results of five independent exposures: the +average value for the experimental diffraction efficiency at each exposure time is represented by a solid line. The +shadow represents the punctual standard deviation. + +To quantitatively evaluate the quality of the fabricated surface relief pattern with respect to the +design, we retrieved the surface height distribution from the AFM analysis. The topographic +distribution is then transformed in a phase delay distribution (by eq. S1) and compared with the +phase distribution extracted from the designed phase map, (additional details are presented in Fig. +S7-9). We used the Root Mean Square Error (RMSE) to quantitatively definethe average mismatch + +Holographic image +Hologram +0.2mm +DC +0.904 +Surface +50μm +0errors occurred during the fabrication step. The analysis, repeated for different exposure times, +provided a constant RMSE, ensuring that any topographical mismatch, related to the hologram +design and to the material response, is not worsened by increasing the surface modulation depth +ℎ0(𝑡) to reach the target ℎ0. From the height distributions obtained with fixed illumination +parameters at different exposure times, we also determined an experimental estimation of the +writing efficiency parameter 𝑐 entering in eq. (1). We extracted the full relief modulation range +from the retrieved distributions to estimate the total modulation depth ℎ0(𝑡), whose experimental +results are provided in Fig. 3B. Those results allowed the empirical definition of the exposure time +that provides the optimal 2𝜋 modulation depth in the kinoform for the probe light wavelength of +𝜆𝑝 = 632.8 𝑛𝑚. A total exposure time of 𝑡 = 86 𝑠 is sufficient for optimal inscription of the +considered kinoform fabrication to in our experimental conditions. +Nevertheless, this off-line structural characterization roadmap does not guarantee a standardization +of the manufacturing process. A new calibration step would be necessary for each different relief +geometry and illumination parameters, leading to a time consuming and multi-step workflow. +However, the surface relief pattern developing on the azo-resist can be characterized directly +during the surface structuration, providing a real-time feedback on the writing process. Despite +different techniques based on mechanical (62) and optical (45) real-time topographic investigation +have been successfully proposed, they do not directly characterize the optical performances of the +diffractive surface. On the contrary, the all-optical lithographic scheme proposed here easily +allows the direct evaluation of the optimized writing parameters from the analysis of the +developing holographic diffraction pattern (46, 63), to act also on specific aspects relevant for +applications, as the suppression of the ghost holograms. +To this aim, we illuminated the developing morphological holographic plate on the azo-resist film +with an additional laser beam at the probe wavelength 𝜆𝑝 during the surface writing step. The +developing diffraction pattern is continuously recorded with a CCD, at a repetition rate of 5Hz, +during the exposure (Fig. 3C). For each of the acquired frames, we evaluated in real-time the +relative diffraction efficiencies 𝜂𝑖 in the target holographic image, and in the spurious terms (DC +order and the ghost image) (Fig. S10). +Fig. 3D summarizes the experimental results for five independent kinoform fabrications. The +optimal exposure time (𝑡𝑜𝑝𝑡 = 103 ± 1 s) was chosen such that the light power diffracted in the +holographic target image is maximized. In this condition, in experimental efficiency 𝜂+1 = +0.60 ± 0.02 was obtained. We also observed a relative transmissivity (|𝑡(𝑥, 𝑦|2) equal to 0.96 for +the final developed surface (Fig. S11), demonstrating also minimal influence of possible +unfavorable light scattering sources produced by the lithographic process. Our approach +demonstrates the big advantages offered by a single-step and all-optical structuration technique, +allowing the tuning of the optimal exposure parameters in real time, which leads to a fully working + +device right after its inscription without the need of further time-consuming surface analysis or +preliminary calibration procedures. +The off-axis hologram design, analyzed here mainly to highlight the characteristics of our holo- +lithographic scheme has a fundamental limitation in practice due to the presence of ghost +holograms simultaneous to the target holographic image. In every physical device with +unavoidable structural mismatches in the kinoform fabrication, this imposes a having for the +exploitable holographic plane and a physical filtering process for the spurious terms. +However, in many applications such as augmented reality and wearable holographic projectors, +the holographic image could be formed in a very specific plane of the optical axis, which typically +coincides with the observer's eye or with a detector sensor (1). When appropriately designed and +fabricated, a holographic plate operating in this configuration allows to overlook the presence of +any other spurious diffraction order, relaxing also eventual design constrains. +An additional advantage of kinoform-based holographic projectors is the possibility to encode +multiple optical functionalities in the same substrate, multiplexing, during the design, the optical +properties that two or more phase masks would have exhibited individually. Multiplexing has no +impact in terms of calculation resources during the design step, and it can easily explored by the +unique combination of our material and the holographic setup (64). +Starting from the target phase mask, e.g. resulting from kinoform calculation, an additional proper +phase mask can be superimposed to produce an axial shift of the target holographic image with +respect to the ghost and DC orders (Fig. 4A). This task can be achieved in an equivalent way by +making the light passing in an additional lens of focal length f, so the kinoform 𝜑(𝑥, 𝑦) must be +multiplexed with the phase shift produced by a thin lens (10), equal to 𝜑𝐿(𝑥, 𝑦) = 𝜋/𝜆𝑓(𝑥2 + 𝑦2). +As the phase of the beam after passing through the phase mask is required to be modulo 2𝜋, the +resulting multiplexed phase map (65) 𝜑𝑀, to be converted in the holographic writing pattern, is +𝜑𝑀 = (𝜑 + 𝜑𝐿)𝑚𝑜𝑑(2𝜋). Form the Fourier transform relation (eq. (2)) it can be easily +demonstrated, using the generalized Fourier analysis (61), that each diffraction order 𝑖 is axially +splitted along the optical axis and it is reconstructed in a different plane located at 𝑧 = 𝑖 ∙ Δ𝑧, where +𝑧 = 0 denotes the reconstruction plane of the kinoform without the additional lens phase map. The +distance Δ𝑧 is function of the focal length 𝑓, which determines the axial separation between the +holographic image and the other (spurious) orders (Fig. 4B). +Fig. 4C shows a SEM image of the surface relief pattern inscribed on the azo-resist surface using +such multiplexed kinoform design. The corresponding diffraction pattern in the target +reconstruction plane is presented in Fig. 4D. In this plane of the optical axis, only the target +holographic image was clearly visible, while the out of focus DC and ghost terms contributed only +with negligible background in the image. + + + +Fig. 4. Design, fabrication, and optimization of multiplexed kinoforms. A The resulting kinoform, from a GS algorithm +performed on the on-axis image of the letter pi, is multiplexed with a spherical phase profile. The new phase profile +is used to encode the different intensity levels of the writing beam. B Representation of the diffractive behavior of a +multiplexed kinoform. When illuminated with monochromatic coherent light, different diffractive orders are axially +reconstructed on shifted planes. Assuming that 𝑧 = 0 is the plane where the holographic pattern is reconstructed +without the multiplexing process, each diffraction order 𝑖 is reconstructed at 𝑧 = 𝑖 ∙ Δ𝑧. C SEM image of the +azopolymer surface after the exposure to the holographic beam for 𝑡𝑜𝑝𝑡 = 120 𝑠. D Resulting diffraction pattern +acquired at 𝑧 = Δ𝑧 E Experimental trend of the pattern visibility reconstructed during the inscription process as result +of five independent exposures: the average value for the pattern visibility at each exposure time is represented by a +solid line. The shadow represents the punctual standard deviation. +As we could not simultaneously access to all the diffracted orders during the surface developing +to define the relative diffraction efficiency in the holograms, we used the image visibility 𝒱 as +quality estimator for the light pattern in the target reconstruction plane (additional details are +presented in Fig. S12). Similarly to the previous case, the real-time control of this parameter +allowed us to directly optimize the exposure time 𝑡𝑜𝑝𝑡 = 120 ± 1 𝑠 for maximum visibility of +𝒱𝑚𝑎𝑥 = 0.83 ± 0.03 (Fig. 4E). This high contrast image was also the result of an independent +tuning of the multiplexed focal length 𝑓, chosen, according to our setup resolution limit, to +maximize orders separation and subsequently the holographic image contrast (Fig. S13). + + +元 +C +Surface +0.5 mm +50 μm +t=120s +Fig. 5. Fully reprogrammable kinoform for time average image quality improvement and data storing and sharing. A +Reprogrammable holographic projector: after surface pattering and holographic image acquisition, morphology can +be completely restored to pristine flat state, allowing for a new patterning step. Quality enhanced experimental images +are the result of the time averaging of multiple holographic patterns. Full resolution images are provided in Fig. S14. +B Experimental results of holograms time averaging for speckle noise effects reduction. On the left is showed the +grayscale pattern acquired after a single exposure step while on the right the same pattern is reconstructed as time +average of ten independent exposures over the same azopolymer area. C Experimental results of the holographic data +storing and sharing. Holographic patterns are plotted with a rainbow colormap. Blue-indigo, green-yellow and orange- +red colors are respectively related to three possible intensity levels encoding three digital logic states. Experimental +images are converted from an analog to digital map for information readout. Word “HELLO” is reconstructed after a +first step of surface writing loop followed by a second multiple exposure step allowing for the reconstruction of the +second part of the message, “WORLD”. +As additional requisite for the use of morphological holographic projectors in real photonics +applications, ranging from optical cryptography to holographic refreshable displays, the surface +morphology should be completely reversible and reprogrammable on demand. One of the +interesting features of azopolymers is that when illuminated with unstructured light in the +chromophore absorption band (see also Fig. S3), the pristine flat surface can be optically restored, +allowing multiple and reversible patterning cycles (46, 66). Fig. 5A schematically shows this all- +optical reprogrammable surface structing process. +One of the features of dynamic holographic platforms (e.g. LCOS SLMs or DMDs) is that the +temporal coordinate can be exploited to produce effective holographic patterns with either +enhanced lateral complexity (64) or higher image quality (67). In these processes, the final +holographic image is the result of the temporal average of the individual patterns that are +instantaneously produced by the dynamically changing diffractive device. The unique reversible +photo-mechanical properties of the azopolymer used here as can be exploited to achieve similar + +Single frame +10frames average +0.5 mm +0.5 mmeffects. To demonstrate an example practical relevance for our dynamically-evolvign +morphological holographic projectors, we repeatedly reprogram the kinoform written on the +surface of the azo-resist to produce a time-averaged holographic diffracted image with a reduced +speckle noise, intrinsically associated to the kinoform design with a IFT algorithm (68, 69). +Fig S.14 in the Supplementary Information shows the details of the characterization of the +holograms recorded in a typical dynamical kinoform reconfiguration experiment. The procedure +for the improved average holographic image started by irradiating the pristine azopolymer surface +with a holographic writing kinoform (in the multiplexed design). After an inscription process +providing optimized visibility in the diffracted holographic pattern, an image 𝐼(𝑥, 𝑦, 1) of the +hologram was collected by the CCD and stored as single frame of a holographic projection movie. +At this stage the surface was completely (optically) erased, and the same area of the azo-resist was +exposed with a new independently calculated holographic writing pattern, characterized by an +independent random distribution of speckle grains. This loop was iterated, acquiring the relative +holographic image 𝐼(𝑥, 𝑦, 𝑖) each time. After 𝑁 = 10 writing/erasing steps, the time averaged +holographic image was calculated as 〈𝐼(𝑥, 𝑦)〉 = (𝑁)−1 ∑ +𝐼(𝑥, 𝑦, 𝑖) +𝑁 +𝑖=1 +. As expected, the averaged +image is characterized by a speckle severity reduced by a factor 1 √𝑁 +⁄ +, as demonstrated in Fig. +S14 for three different target holographic images. This artificial image improvement trough a time +averaging process is the same as that performed by an ideal “slow eye or detector”, which has a +time response much higher than the typical surface reconfiguration time (~ 120 𝑠 in our +experimental condition). Despite still far from the refresh rates achievable with other dynamical +systems, these results allow us to include for the first versatile dynamical modulation capabilities +for applications with a planar optical diffractive component. +As additional proof, we show the speckle noise time filtering for a three-level target holographic +image. Fig. 5B shows the diffraction pattern 𝐼(𝑥, 𝑦) and the corresponding time averaged +holographic pattern 〈𝐼(𝑥, 𝑦)〉 representing the image of a cube, where each of the three displayed +faces encode a different diffracted intensity level. The grayscale nature of the hologram became +visually clear only once the that the speckle noise contrast reduction is performed (see also Fig. +S15), with a significant improvement with respect to a single holographic image. +Morphological reprogrammable devices able to also encode grayscale optical information can +represent a valid platform to store encrypted optical information. The use of non-binary bits of +light can increase the storage capacity, while simultaneously reducing the required space on the +physical support (70). We used our azopolymer as a morphological holographic memory support +where the visual information was encrypted in the surface topography. The secret message, +displaying the word “HELLO”, was converted into a ternary base where each letter is codified +into three different trit (ternary digit), each assuming three separate logic states. The trits that +defines each letter of the word have been arranged in rows to form a three-level grayscale image, +where each level corresponds to one of the three possible logic states. The details of the designed + +ternary alphabet are discussed in Fig. S16. When this image was used to define surface morphology +and transferred to the azo-resist surface, all the original information was encrypted by the Fourier +transform algorithm, therefore, information readout is possible only optically by means of a proper +optical setup (Fig. 5C). Fourier-transform coding also offers the advantage that if part of the +surface would be damaged or destroyed, reading the secreted information would still be +theoretically possible. We finally completely erased and reshaped the surface geometry to share +the second part of the secreted message composed by the word “WORLD”. This temporal +holographic splitting of the message enhances encryption capabilities and information sharing +security. Additionally, it prospects azopolymer structured films as promising reversible high- +density memory substrates. We further estimated that by a single surface illumination process, +with the defined architecture, we are capable of simultaneously encoding 3,125 bytes of +information in a secreted hologram. +3. Discussion and Conclusions + +Our direct all-optical maskless lithography, using azopolymers as photoresist, represents the +state of the art as fabrication technique of fully reversible diffractive flat optical elements with +arbitrary holographic pattern reconstruction capabilities. In the simple case of binary modulation +for the writing beam demonstrated in this work, we proved our ability to faithfully transfer, in a +pure optical process, complex bidimensional geometries as a two-level surface modulation of an +azobenzene-containing polymer film. This process, in another perspective, can also be interpreted +as a form of information storing if the target image (e.g. the QR code image) is seen as the +information to be encrypted as surface morphology on the azopolymeric film. An additional +morphological analysis of the surface, right after the exposure process, demonstrated no significant +information losses in the morphological information transfer, even considering the differential +light response of our material to the writing illumination. +As additional milestone of such method, we extended and scaled our approach for the +realization of diffractive kinoforms, where complex lateral geometries with grayscale modulation +depth are simultaneously required. The additional possibility to test the devices functionality +during the fabrication process provides a cost-effective design and prototyping of operating +diffractive optical devices, implemented as azopolymer phase retarders. We characterized both the +surface morphology and its diffractive behavior right during the exposure, investigating the +quantization and pixelation effects and non-linear responses of the material to the structuring +technique, enhancing their relevant impact during the device optimization and fabrication. Our +approach led to the realization of pixel-free morphological holographic projectors, ensuring high +efficiency and ultra-compact devices, whose depth results comparable with the operating light +wavelength. The opposite happens in conventional digital devices, where the discrete nature of the +pixels limits the spatial resolution and the addressable phase sampling, while simultaneously + +generating spurious periodic replication of the reconstructed image, with a consequent overall +efficiency loss. Despite simple, morphological encoding design of dielectric diffractive surfaces +totally changes the perspective when holographic projectors are also compared to traditional wide +displays. First, the complex modulation provided by the realized kinoform has an almost unitary +transmittance, resulting in a lossless structuring of light. Furthermore, the Fourier relationship +linking the modulation and the image reconstruction plane is non-local, meaning that each of the +point of the kinoform will contribute to form the entire holographic image. In other words, a +kinoform preserves the information content in all its parts, consequently breaking or damaging the +device will not compromise at all the holographic image reconstruction. +Additionally, as the azopolymer surface can be optically restored to the flat pristine state in +place, multiple writing/erasing cycles can be performed on time scales of few minutes. As, up to +now, no material and structuration method combination for such dynamically changing surfaces +exist, our approach represent the state of the art for reversible, all-optical custom flat optical +devices fabrication. This possibility allowed time-averaged enhanced quality holographic images +and paved the way for the fabrication of morphological reshapable devices able to encode optical +information with both morphological and temporal encryption. As valid every time that +information needs to be stored on a physical support, the main requests for the substrate are time +stability and reversibility. On the other side also the encoding process is required to be highly +controlled, as any critical issue may result in information degradation or even in its loss. We +demonstrated that azopolymers, when illuminated with digital reconstructed intensity patterns of +light, can meet those requirements. For the first time we showed that azopolymer unique optical +properties can also be exploited to implement a new class of photonics devices with several +applications ranging from wearable holographic projectors and displays to high quality supports +for data storing, encryption and sharing. Even if still at a primitive level, this approach already +makes evident the benefits that can completely change our prospective for holographic displays, +optical data storage, and encryption, opening also to practical applications in emerging +technologies as VR\AR displays and wearable devices. + + + + + +Materials and Methods + +Experimental setup +The experimental configuration for the azopolymer surface relief inscription is based on a +phase-only Computer-Generated Holograms (CGHs) scheme. Its schematic representation is +shown in Fig. S2. A laser diode source (Cobolt Calypso) produces a TEM00 beam at wavelength +λ=491 nm which, after a beam expander (lenses L1 and L2), is phase-modulated by a computer- +controlled reflective phase-only Spatial Light Modulator (SLM, Holoeye Pluto). The modulated +beam is propagated through a 4f lenses system with the input plane located in the SLM plane. The +output plane coincides with the back focal plane of an infinity-corrected long-working distance +50X objective (Mitutoyo), with numerical aperture NA=0.55. The focal lengths of the lenses L3 +(300 mm) and L4 (175 mm) are chosen to maximize the spatial resolution in the hologram +reconstruction plane. This choice also defines the diameter (~200 μm) of the accessible circular +area in the objective front focal plane, which can be used to structure the azopolymer surface in a +single illumination step. The position of the sample near the objective focal region is accurately +controlled by means of a x-y-z translation stage. Average intensity in the range 12.7-14.0 W⁄cm2 +and circular polarization are used for the structuration of the azopolymer surface. To reduce the +speckle noise contrast effects (67), the holographic illumination over the azopolymer surface is the +result of the time average of several holographic patterns generated from different kinoforms. Each +pattern is reconstructed after an independent design from the same target image, initializing the +algorithm with random phase. The SLM refresh time (30 Hz for this work) is faster than the +azopolymer response so that the effective illumination profile is the temporal average of the +illumination profile associated with each of the many independent kinoforms sent in sequence to +the modulator. For visual inspection, and proper focusing of the holographic pattern on the +photoresponsive surface, a 70/30 beam splitter, placed in the light-path, redirects the light +retroreflected by the surface and re-collimated through the objective toward a tube lens (with focal +length equal to 200 mm). This lens forms an image of the holographic pattern in its second focal +plane, where a “DCC3240M Thorlabs” CCD camera is positioned. During the exposure, an +additional diode laser beam at 405 nm illuminates the photoresist film from the substrate side. The +beam has circular polarization and different intensity levels depending on its intended function. +When the intensity is 0.6 W⁄cm2, the beam favors the surface structuring process, acting as a +writing assisting beam. At intensity higher than 0.9 W⁄cm2, its absorption causes the erasure of +previously inscribed surface structures, acting as an erasing beam. Further characterizations about +assisting/erasing beam are described in a previous work (46). An additional He-Ne laser beam, at +632.8 nm, is used as sample back-illumination source to test diffraction behavior of the modulated +surface during the structuration process. The beam splitter also allows the collection of part of this +light without interfering with the writing process, Fig. S3. The image of the surface is projected + +on the back focal plane of the tube lens and coupled by means of a mirror (mounted on a flip +mount) to an additional 2f system composed by the lens L5 (300 mm). Fourier transform image is +captured with an additional CCD camera. + +Azo-resist synthesis +The photoresponsive material used in this work is an azobenzene-containing polymer +(azopolymer) in amorphous state, Fig. S3. All reagents were purchased from Merck and used +without further purification. The azopolymer was synthesized, purified and characterized as +previously reported (Mw = 27000; phase sequence: Glass 67 °C Nematic 113 °C Isotropic; max += 350 nm) (46, 54, 71). The solution for film deposition was prepared by dissolving 70 mg of the +polymer in 0.50 ml of 1,1,2,2-tetrachloroethane and filtered on 0.2 µm PTFE membrane filters. +The desired film thickness (typically 1.5 ± 0.1 𝜇𝑚) was obtained by spin coating the solution on +24x60 mm cover slides at 300 rpm for 4 minutes. In the final stage, the samples were kept under +vacuum at room temperature for 24 h to remove solvent traces. Molecular structural formula and +the absorbance in the UV-visible are provided in Fig. S3. + +Morphological characterization of structured surfaces +Topographic characterization of inscribed azopolymer surface reliefs is performed using +AFM and SEM. For AFM measurements, a WITec Alpha RS300 microscope is used. The AFM +is operated in tapping mode using a cantilever with 75 kHz resonance frequency and nominal force +constant of 2.8 N/m. AFM tips (Arrow FM type from Nano World), with nominal radius of +curvature of ≈10 nm, are used. The maximum scanned area has a size of 100 × 100 μm2, acquired +with resolution of 500 points per lines and 500 lines per scan. For each AFM the minimum of the +topography is set to zero to extract the height distribution 𝑃𝑗, representing the probability to find a +pixel in the image with a height value between ℎ𝑗 and ℎ𝑗+1 where ℎ𝑗 = 𝑗∆ℎ. Here 𝑗 ranges from +zero to 𝑁 − 1 where 𝑁 is the number of occupied bins in each image, while ∆ℎ = 10𝑛𝑚 represents +a reasonably choice for the fixed bin width. Each height distribution is normalized to match the +condition ∑ 𝑃𝑖 +𝑁 += 1. The expected value ℎ̅ = ∑ ℎ𝑖𝑃𝑖 +𝑁 + and variance 𝜎2 = ∑ (ℎ𝑖 − ℎ̅) +2 𝑃𝑖 +𝑁 + are +extracted for each distribution. To retrieve an estimation of the modulation depth ℎ0 we consider +the discrete integral function 𝐼(𝑛) = ∑ +𝑃𝑖 +𝑛 +𝑖=0 +. Firstly, we define ℎ0 = ℎ𝑘 where 𝑘 satisfies the +relation 𝐼(𝑘) = 1. In that case we represent the total modulation depth as the full dispersion range +of the distribution 𝑃𝑖. Since, due to our material behavior, the height distribution is not uniform, +we also estimate the modulation depth as ℎ0 = ℎ𝑎 − ℎ𝑏 where 𝑎 and 𝑏 satisfy respectively 𝐼(𝑎) = +0.95 and 𝐼(𝑏) = 0.05; with this assumption ℎ0 represents the range, uniformly distributed around +the median of the distribution, where there is the 0.90 of probability to find a fixed value of ℎ𝑖, see +Fig. S7. + +Scanning electron microscopy (SEM) images are acquired with a field emission gun (FEG–SEM) +FEI/ThermoFisher Nova NanoSEM 450 microscope. Samples are sputtered with a layer of Au/Pd +using a Denton Vacuum Desk V TSC coating system prior to observation. + +Iterative Fourier transform algorithms +Despite the simple Fourier relation, optical modulation cannot be retrieved by simply +inverting the equation (2). To design the proper phase mask able to lossless transform a given input +light distribution into a desired light pattern, an iterative Fourier transform algorithm (IFTA) has +been used. In this class of algorithms, the optical field is bounced back and forth between two +planes related by a Fourier transform, applying specific constraints to the retrieved fields at each +iteration. The used algorithm for diffractive kinoforms design is the Gerchberg-Saxton algorithm +(72). This algorithm can be easily implemented with modern computing capabilities, and once a +digital representation of 𝐼𝑜𝑢𝑡 is provided by a grayscale 8-bit digital image it returns a digital +representation of the phase map 𝜑(𝑥, 𝑦). When the desired light distribution is only constrained in +a limited region of space, as for the holographic writing beam, the complex amplitude outside this +area can be arbitrary chosen or left free to vary, allowing to increase the light hologram quality. +This possibility is typically referred to as amplitude freedom. We used this approach to generate +the writing holograms for the azopolymer structuration by a mixed region amplitude freedom +(MRAF) algorithm (56). We implemented both GS and MRAF algorithms in MATLAB, using the +Fast Fourier Transform (FFT) algorithm. + + + + + + + +References + +1. C. Chang, K. Bang, G. Wetzstein, B. Lee, L. Gao, Toward the next-generation VR/AR optics: a review of +holographic near-eye displays from a human-centric perspective. Optica. 7, 1563 (2020). +2. B. C. Kress, I. Chatterjee, Waveguide combiners for mixed reality headsets: a nanophotonics design +perspective. Nanophotonics. 10, 41–74 (2020). +3. Z. He, X. Sui, G. Jin, L. Cao, Progress in virtual reality and augmented reality based on holographic display. +Appl. Opt. 58, A74 (2019). +4. M. Uysal, H. Nouri, "Optical wireless communications — An emerging technology" in 2014 16th +International Conference on Transparent Optical Networks (ICTON) (IEEE, Graz, Austria, 2014; +http://ieeexplore.ieee.org/document/6876267/), pp. 1–7. +5. L. Xu, M. Rahmani, Programmable structured surfaces can change the future of wireless communications. +Light Sci Appl. 11, 196 (2022). +6. A. Peter Amalathas, M. Alkaisi, Nanostructures for Light Trapping in Thin Film Solar Cells. Micromachines. +10, 619 (2019). +7. I. E. Khodasevych, L. Wang, A. Mitchell, G. Rosengarten, Micro- and Nanostructured Surfaces for Selective +Solar Absorption. Advanced Optical Materials. 3, 852–881 (2015). +8. K. Matsushima, Introduction to Computer Holography: Creating Computer-Generated Holograms as the +Ultimate 3D Image (Springer, Cham, Switzerland, 1st ed. 2020 edition., 2020). +9. C. Slinger, C. Cameron, M. Stanley, Computer-generated holography as a generic display technology. +Computer. 38, 46–53 (2005). +10. J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005). +11. D. Pi, J. Liu, Y. Wang, Review of computer-generated hologram algorithms for color dynamic holographic +three-dimensional display. Light Sci Appl. 11, 231 (2022). +12. E. Buckley, "Computer-Generated Phase-Only Holograms for Real-Time Image Display" in Advanced +Holography - Metrology and Imaging, I. Naydenova, Ed. (InTech, 2011; +http://www.intechopen.com/books/advanced-holography-metrology-and-imaging/computer-generated-phase- +only-holograms-for-real-time-image-display). +13. D. C. O’Shea, T. J. Suleski, A. D. Kathman, D. W. Prather, Diffractive Optics: Design, Fabrication, and Test +(SPIE Press, 2004). +14. B. C. Kress, P. Meyrueis, Applied digital optics: from micro-optics to nanophotonics (Wiley, Chichester, U.K., +2009; https://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=302622). +15. H. Wang, Y. Liu, Q. Ruan, H. Liu, R. J. H. Ng, Y. S. Tan, H. Wang, Y. Li, C. Qiu, J. K. W. Yang, Off‐Axis +Holography with Uniform Illumination via 3D Printed Diffractive Optical Elements. Advanced Optical +Materials. 7, 1900068 (2019). +16. Q. Jiang, G. Jin, L. Cao, When metasurface meets hologram: principle and advances. Adv. Opt. Photon. 11, 518 +(2019). + +17. G. Lee, J. Sung, B. Lee, Recent advances in metasurface hologram technologies (Invited paper). ETRI Journal. +41, 10–22 (2019). +18. Y. Hu, X. Luo, Y. Chen, Q. Liu, X. Li, Y. Wang, N. Liu, H. Duan, 3D-Integrated metasurfaces for full-colour +holography. Light Sci Appl. 8, 86 (2019). +19. P. Georgi, Q. Wei, B. Sain, C. Schlickriede, Y. Wang, L. Huang, T. Zentgraf, Optical secret sharing with +cascaded metasurface holography. Sci. Adv. 7, eabf9718 (2021). +20. K. T. P. Lim, H. Liu, Y. Liu, J. K. W. Yang, Holographic colour prints for enhanced optical security by +combined phase and amplitude control. Nat Commun. 10, 25 (2019). +21. G. Qu, W. Yang, Q. Song, Y. Liu, C.-W. Qiu, J. Han, D.-P. Tsai, S. Xiao, Reprogrammable meta-hologram for +optical encryption. Nat Commun. 11, 5484 (2020). +22. L. Yan, J. Xiao, T. Plaskocinski, M. Biabanifard, S. Persheyev, M. Askari, A. D. Falco, Two-tier manipulation +of holographic information. Opt. Express. 30, 19145 (2022). +23. K. Du, H. Barkaoui, X. Zhang, L. Jin, Q. Song, S. Xiao, Optical metasurfaces towards multifunctionality and +tunability. Nanophotonics. 11, 1761–1781 (2022). +24. S. C. Malek, H.-S. Ee, R. Agarwal, Strain Multiplexed Metasurface Holograms on a Stretchable Substrate. +Nano Lett. 17, 3641–3645 (2017). +25. P. van Assenbergh, E. Meinders, J. Geraedts, D. Dodou, Nanostructure and Microstructure Fabrication: From +Desired Properties to Suitable Processes. Small. 14, 1703401 (2018). +26. J. A. Liddle, G. M. Gallatin, Lithography, metrology and nanomanufacturing. Nanoscale. 3, 2679–2688 (2011). +27. E. Sharma, R. Rathi, J. Misharwal, B. Sinhmar, S. Kumari, J. Dalal, A. Kumar, Evolution in Lithography +Techniques: Microlithography to Nanolithography. Nanomaterials. 12, 2754 (2022). +28. S. Okazaki, High resolution optical lithography or high throughput electron beam lithography: The technical +struggle from the micro to the nano-fabrication evolution. Microelectronic Engineering. 133, 23–35 (2015). +29. K. Jain, T. J. Dunn, J. M. Hoffman, Seamless, maskless lithography system using spatial light modulator +(2001). +30. S. Huang, M. Li, L. Wang, Y. Su, Y. Liang, Precise fabrication of large-area microstructures by digital oblique +scanning lithography strategy and stage self-calibration technique. Appl. Phys. Express. 12, 096501 (2019). +31. H.-L. Chien, Y.-H. Chiu, Y.-C. Lee, Maskless lithography based on oblique scanning of point array with digital +distortion correction. Optics and Lasers in Engineering. 136, 106313 (2021). +32. L. Huang, K. Xu, D. Yuan, J. Hu, X. Wang, S. Xu, Sub-wavelength patterned pulse laser lithography for +efficient fabrication of large-area metasurfaces. Nat Commun. 13, 5823 (2022). +33. M. S. Khan, R. lachmayer, B. Roth, Maskless lithography for versatile and low cost fabrication of polymer +based micro optical structures. OSA Continuum. 3, 2808 (2020). +34. C. Zhang, Y. Hu, W. Du, P. Wu, S. Rao, Z. Cai, Z. Lao, B. Xu, J. Ni, J. Li, G. Zhao, D. Wu, J. Chu, K. +Sugioka, Optimized holographic femtosecond laser patterning method towards rapid integration of high-quality +functional devices in microchannels. Sci Rep. 6, 33281 (2016). + +35. V.-C. Su, C. H. Chu, G. Sun, D. P. Tsai, Advances in optical metasurfaces: fabrication and applications +[Invited]. Opt. Express. 26, 13148 (2018). +36. M. Altissimo, E-beam lithography for micro-/nanofabrication. Biomicrofluidics. 4, 026503 (2010). +37. N. Lassaline, R. Brechbühler, S. J. W. Vonk, K. Ridderbeek, M. Spieser, S. Bisig, B. le Feber, F. T. Rabouw, +D. J. Norris, Optical Fourier surfaces. Nature. 582, 506–510 (2020). +38. S. T. Howell, A. Grushina, F. Holzner, J. Brugger, Thermal scanning probe lithography—a review. Microsyst +Nanoeng. 6, 21 (2020). +39. S. L. Oscurato, M. Salvatore, P. Maddalena, A. Ambrosio, From nanoscopic to macroscopic photo-driven +motion in azobenzene-containing materials. Nanophotonics. 7, 1387–1422 (2018). +40. K. Kim, H. Park, K. J. Park, S. H. Park, H. H. Kim, S. Lee, Light-Directed Soft Mass Migration for +Micro/Nanophotonics. Advanced Optical Materials. 7, 1900074 (2019). +41. F. Borbone, S. Luigi Oscurato, S. D. Sorbo, F. Pota, M. Salvatore, F. Reda, P. Maddalena, R. Centore, A. +Ambrosio, Enhanced photoinduced mass migration in supramolecular azopolymers by H-bond driven positional +constraint. Journal of Materials Chemistry C. 9, 11368–11375 (2021). +42. J. M. Ilnytskyi, V. Toshchevikov, M. Saphiannikova, Modeling of the photo-induced stress in azobenzene +polymers by combining theory and computer simulations. Soft Matter. 15, 9894–9908 (2019). +43. B. Yadav, J. Domurath, K. Kim, S. Lee, M. Saphiannikova, Orientation Approach to Directional +Photodeformations in Glassy Side-Chain Azopolymers. J. Phys. Chem. B. 123, 3337–3347 (2019). +44. S. Loebner, B. Yadav, N. Lomadze, N. Tverdokhleb, H. Donner, M. Saphiannikova, S. Santer, Local Direction +of Optomechanical Stress in Azobenzene Containing Polymers During Surface Relief Grating Formation. +Macro Materials & Eng. 307, 2100990 (2022). +45. H. Rekola, A. Berdin, C. Fedele, M. Virkki, A. Priimagi, Digital holographic microscopy for real-time +observation of surface-relief grating formation on azobenzene-containing films. Sci Rep. 10, 19642 (2020). +46. S. L. Oscurato, F. Reda, M. Salvatore, F. Borbone, P. Maddalena, A. Ambrosio, Shapeshifting Diffractive +Optical Devices. Laser & Photonics Reviews, 2100514 (2022). +47. S. L. Oscurato, F. Reda, M. Salvatore, F. Borbone, P. Maddalena, A. Ambrosio, Large-Scale Multiplexed +Azopolymer Gratings with Engineered Diffraction Behavior. Advanced Materials Interfaces. 2101375, 1–9 +(2021). +48. Y. Lim, B. Kang, S. J. Hong, H. Son, E. Im, J. Bang, S. Lee, A Field Guide to Azopolymeric Optical Fourier +Surfaces and Augmented Reality. Advanced Functional Materials. 31, 2104105 (2021). +49. J. Strobelt, D. Stolz, M. Leven, M. V. Soelen, L. Kurlandski, H. Abourahma, D. J. McGee, Optical +microstructure fabrication using structured polarized illumination. Opt. Express. 30, 7308 (2022). +50. M. Salvatore, F. Borbone, S. L. Oscurato, Deterministic Realization of Quasicrystal Surface Relief Gratings on +Thin Azopolymer Films. Advanced Materials Interfaces. 7, 1902118 (2020). +51. R. Bachelot, F. H’Dhili, D. Barchiesi, G. Lerondel, R. Fikri, P. Royer, N. Landraud, J. Peretti, F. Chaput, G. +Lampel, J.-P. Boilot, K. Lahlil, Apertureless near-field optical microscopy: A study of the local tip field +enhancement using photosensitive azobenzene-containing films. Journal of Applied Physics. 94, 2060–2072 +(2003). + +52. C. Hubert, A. Rumyantseva, G. Lerondel, J. Grand, S. Kostcheev, L. Billot, A. Vial, R. Bachelot, P. Royer, S. +Chang, S. K. Gray, G. P. Wiederrecht, G. C. Schatz, Near-Field Photochemical Imaging of Noble Metal +Nanostructures. Nano Lett. 5, 615–619 (2005). +53. A. Porfirev, S. Khonina, N. Ivliev, A. Meshalkin, E. Achimova, A. Forbes, Writing and reading with the +longitudinal component of light using carbazole-containing azopolymer thin films. Sci Rep. 12, 3477 (2022). +54. A. Ambrosio, L. Marrucci, F. Borbone, A. Roviello, P. Maddalena, Light-induced spiral mass transport in azo- +polymer films under vortex-beam illumination. Nat Commun. 3, 1–9 (2012). +55. A. Ambrosio, P. Maddalena, L. Marrucci, Molecular Model for Light-Driven Spiral Mass Transport in +Azopolymer Films. Phys. Rev. Lett. 110, 146102 (2013). +56. M. Pasienski, B. DeMarco, A high-accuracy algorithm for designing arbitrary holographic atom traps. Opt. +Express, OE. 16, 2176–2190 (2008). +57. F. Reda, M. Salvatore, F. Borbone, P. Maddalena, A. Ambrosio, S. L. Oscurato, Varifocal diffractive lenses for +multi-depth microscope imaging. Opt. Express, OE. 30, 12695–12711 (2022). +58. L. B. Lesem, P. M. Hirsch, J. A. Jordan, The Kinoform: A New Wavefront Reconstruction Device. IBM J. Res. +& Dev. 13, 150–155 (1969). +59. R. W. Gerchberg, A practical algorithm for the determination of the phase from image and diffraction plane +pictures. Optik. 35, 237–246 (1972). +60. I. Moreno, J. Campos, C. Gorecki, M. J. Y. M. J. Yzuel, Effects of Amplitude and Phase Mismatching Errors in +the Generation of a Kinoform for Pattern Recognition. Jpn. J. Appl. Phys. 34, 6423 (1995). +61. T.-C. Poon, Ed., Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, +New York, 2006th edition., 2006). +62. J. Jelken, C. Henkel, S. Santer, Solving an old puzzle: fine structure of diffraction spots from an azo-polymer +surface relief grating. Appl. Phys. B. 125, 218 (2019). +63. F. Reda, M. Salvatore, F. Borbone, P. Maddalena, S. L. Oscurato, Accurate Morphology-Related Diffraction +Behavior of Light-Induced Surface Relief Gratings on Azopolymers. ACS Materials Lett. 4, 953–959 (2022). +64. M. Salvatore, F. Reda, F. Borbone, P. Maddalena, S. L. Oscurato, "Multiplexed surface reliefs on azopolymer +thin films" in Photosensitive Materials and their Applications II (SPIE, 2022; +https://www.spiedigitallibrary.org/conference-proceedings-of-spie/12151/121510A/Multiplexed-surface-reliefs- +on-azopolymer-thin-films/10.1117/12.2621640.full), vol. 12151, pp. 77–83. +65. J. Leach, K. Wulff, G. Sinclair, P. Jordan, J. Courtial, L. Thomson, G. Gibson, K. Karunwi, J. Cooper, Z. J. +Laczik, M. Padgett, Interactive approach to optical tweezers control. Appl. Opt. 45, 897 (2006). +66. J. Vapaavuori, R. H. A. Ras, M. Kaivola, C. G. Bazuin, A. Priimagi, From partial to complete optical erasure of +azobenzene–polymer gratings: effect of molecular weight. J. Mater. Chem. C. 3, 11011–11016 (2015). +67. S. L. Oscurato, M. Salvatore, F. Borbone, P. Maddalena, A. Ambrosio, Computer-generated holograms for +complex surface reliefs on azopolymer films. Scientific Reports. 9, 6775 (2019). +68. L. Chen, S. Tian, H. Zhang, L. Cao, G. Jin, Phase hologram optimization with bandwidth constraint strategy for +speckle-free optical reconstruction. Opt. Express. 29, 11645 (2021). + +69. A. E. G. Madsen, R. L. Eriksen, J. Glückstad, Comparison of state-of-the-art Computer Generated Holography +algorithms and a machine learning approach. Optics Communications. 505, 127590 (2022). +70. J. Zhang, A. Čerkauskaitė, R. Drevinskas, A. Patel, M. Beresna, P. G. Kazansky, "Eternal 5D data storage by +ultrafast laser writing in glass" in, U. Klotzbach, K. Washio, C. B. Arnold, Eds. (San Francisco, California, +United States, 2016; http://proceedings.spiedigitallibrary.org/proceeding.aspx?doi=10.1117/12.2220600), p. +97360U. +71. S. L. Oscurato, F. Borbone, P. Maddalena, A. Ambrosio, Light-Driven Wettability Tailoring of Azopolymer +Surfaces with Reconfigured Three-Dimensional Posts. ACS Appl. Mater. Interfaces. 9, 30133–30142 (2017). +72. R. W. Gerchberg, A practical algorithm for the determination of the phase from image and diffraction plane +pictures. Optik. 35, 237–246 (1972). + + + + + + + +Supplementary Material for + +Reprogrammable holograms from maskless surface photo- +morphing + + +Francesco Reda,1 Marcella Salvatore,1,2 Marco Astarita,3 Fabio Borbone4 and +Stefano L. Oscurato1,2,* + +1 Physics Department “E. Pancini”, University of Naples “Federico II”, Complesso Universitario +di Monte Sant’Angelo, via Cinthia 21, 80126, Naples, Italy +2 Centro Servizi Metrologici e tecnologici Avanzati (CeSMA), University of Naples “Federico II”, +Complesso Universitario di Monte Sant’Angelo, Via Cintia 21, 80126, Naples, Italy +3 Physics Department, Politecnico di Milano, 20133, Milan, Italy +4 Department of Chemical Sciences, University of Naples “Federico II”, Complesso Universitario +di Monte Sant’Angelo, Via Cintia, 80126 Naples, Italy +*Corresponding author: stefanoluigi.oscurato@unina.it + + + +Diffraction properties of holographic morphological projectors. + +Phase-only holographic plates are typically represented by a complex transmission function +𝑡(𝑥, 𝑦) describing in the scalar approximation of wave optics, the wavefront phase modulation of +an incident monochromatic optical field, at wavelength 𝜆 passing through the device. Isotropic +dielectric phase retarders implement phase modulation as result of local variations in thickness +ℎ(𝑥, 𝑦) and refractive index 𝑛(𝑥, 𝑦, 𝜆) of the device, whereby the planar modulation function can +be written as: + +𝑡(𝑥, 𝑦) = exp [𝑖𝜑(𝑥, 𝑦)] = exp [𝑖 2𝜋 +𝜆 (𝑛(𝑥, 𝑦, 𝜆) − 𝑛𝑠)ℎ(𝑥, 𝑦)] + +(S1) +representing the local phase delay 𝜑(𝑥, 𝑦) accumulated by the light due to optical path variation +imposed by the plate immersed in a surrounding material whose refractive index is 𝑛𝑠. When a +kinoform, supposed to be at 𝑧 = 0, is illuminated by the light field 𝑈𝑖𝑛(𝑥, 𝑦, 0), the resulting + +complex field 𝑈𝑜𝑢𝑡(𝑥, 𝑦, 𝑧) formed due to Fraunhofer (𝑧 ≫ 0) diffraction is the two-dimensional +spatial Fourier transform of the modulated beam at the kinoform plane, and the reconstructed +image 𝐼𝑜𝑢𝑡 is determined by the relation: + + +𝐼𝑜𝑢𝑡(𝑥, 𝑦, 𝑧) = |𝐹𝑇[𝑈𝑖𝑛(𝑥, 𝑦, 0) ∙ 𝑡(𝑥, 𝑦, 0)]|2 +(S2) + +The phase encoding process, from the phase design to its implementation by lithography, leads to +a device whose real transmittance 𝑡𝑟𝑒𝑎𝑙, is a function g of the designed phase 𝑡𝑟𝑒𝑎𝑙(𝑥, 𝑦) = +𝑒𝑖𝑔[𝜑(𝑥,𝑦)]. To consider any possible deviation from this ideal case, the complex transmittance of +the real kinoform can be decomposed into a linear superposition of functions, clearly describing +the effects of phase mismatches. Assuming that the deformation is space invariant, the spatial +coordinates can be omitted and the function 𝑡𝑟𝑒𝑎𝑙 can be expanded in terms of its argument, +according to the generalized harmonic analysis (61): + + +𝑡𝑟𝑒𝑎𝑙 = ∑ 𝐺𝛼𝑒𝑖𝛼𝜑 ++∞ +𝛼=−∞ + +(S3) +where 𝐺𝛼 = ∫ +𝑡𝑟𝑒𝑎𝑙𝑒𝑖𝛼𝜑 +2𝜋 +0 +𝑑𝜑 and 𝛼 an integer. The 𝛼 = 1 term is the only one whose Fourier +transform results in an optical field with intensity 𝐼𝑜𝑢𝑡. The amount of optical power shaped in the +reconstructed intensity profile with respect to the total transmitted power, is equal to 𝜂1 = |𝐺1|2 +and it is equal to one only in the ideal case 𝑔(𝜑) = 𝜑. The other terms of the series, apart from the +term 𝛼 = 0 which determines an unmodulated optical component named DC term, contribute with +shifted and scaled replicas of the desired intensity pattern, known as ghosts or false images (8, 60). +The total reconstructed pattern is a weighted sum of the desired image, the DC term, and false +images, typically also spatially overlapped in the reconstruction plane and with a relative +efficiency 𝜂𝛼 = |𝐺𝛼|2. At best, once the geometry ℎ(𝑥, 𝑦) is fixed, 𝑔 is linear with the total surface +reliefs amplitude ℎ0, which has to be tuned in order to reach a fully 2𝜋 modulation depth. +According to equation (S1) this condition is achieved for ℎ0 = 𝜆/(𝑛(𝜆) − 𝑛𝑆); for our material at +the operating wavelength 𝜆 = 0.6328 𝜇𝑚, 𝑛(𝜆) = 1.696 and 𝑛𝑆 = 1 (for air immersed +kinoforms), the condition is satisfied for ℎ0 = 0.9092 𝜇𝑚. In this simple case if ℎ0 +∗ is the +implemented modulation depth, the ratio 𝑚 = ℎ0 +∗/ℎ0 denotes a mismatch parameter. Under those +conditions diffraction efficiency 𝜂𝛼 can be written as (60): + + +𝜂𝛼 = 𝑠𝑖𝑛𝑐2(𝑚 − 𝛼) +(S4) +The condition 𝑚 = 1 guarantees the maximum diffraction efficiency 𝜂1 = 1, ensuring that all the +incident optical power is effectively shaped in the reconstructed holographic pattern [Fig. S1]. +During the encoding process, more complex distortion effects can determine a non-linear form for +g. These certainly include quantization and pixelation effects and non-linear responses of the + +material to the structuring process that led to a kinoform in which phase mismatches are included +providing 𝜂1 ≠ 1 even if target modulation depth is reached. + +Supplementary figures + + + +Fig. S1: Theoretical diffraction efficiency from an ideal kinoform as function of the height mismatch error m. + + +Fig. S2: Schematic of the experimental setup. Beam expander - lenses 𝐿1 (f1=-50 mm) and 𝐿2 (f2=250 mm). SLM - +Holoeye Pluto, LCOS spatial light modulator, phase only (reflective). 4f configuration - lenses 𝐿3, (f3=300 mm) and +𝐿4 (f2=175 mm). QWP - quarter wave plate. BS - 70/30 beam splitter. Objective - 50X Mitutoyo Plan Apo Infinity +Corrected Long WD Objective. TL - tube lens (fTL=200 mm). Fourier transforming lens - lens 𝐿5, (f5=300 mm). CCD1/2 +- “DCC3240M Thorlabs” camera. + + +Fig. S3: Azopolymer optical characterization. A Molecular structural formula. B Absorbance in the UV-visible. Probe +wavelength 𝜆𝑝 = 633 𝑛𝑚 is chosen different from the writing beam and out of the absorption band of the material. +Refractive index at 𝜆𝑝 was measured via ellipsometry. + + +2Ne=1.696 +Assisting/erasing +(405 nm) +Wriling +Probe +(491 nm) +(633 nm) + + +Fig. S4: Holographic setup spatial resolution. A Holographic reconstruction of square shaped light pattern with lateral +size Δ. B Experimental squares size Δ as function of designed size Δ’. The slope 𝑏 = 0.376 ± 0.002 𝜇𝑚 of the fitted +line trend Δ = 𝑏Δ′ defines the calibration of physical dimensions of patterns in the polymer plane with respect to the +analytically designed target images. C Contrast of the holographic reconstructed square as function of lateral size Δ. +Contrast is defined as 𝐶 = (𝐼𝑊 + 𝐼𝐵)/(𝐼𝑊 − 𝐼𝐵) with 𝐼𝑊 and 𝐼𝐵 representing the average experimental intensity levels +corresponding to white and black areas of the target image, respectively. Resolution limit is reasonably set to Δ0 = +5𝑏 = 1.88 𝜇𝑚. Kinoforms resulting from design step are scaled by a factor 5 before being encoded into the +illumination pattern, ensuring the highest contrast for the holographic writing pattern maintaining a reasonably +reduced pixel size. + + + + +Fig. S5: Azopolymer response to an intensity structured field with designed lateral size Δ0 = 1.88 𝜇𝑚. + + + + + +Intensity +Surface +Fig. S6: Holographic setup intensity level modulation. A Holographic reconstruction of square shaped light pattern +with lateral size Δ0 = 1.88 𝜇𝑚 and linearly spaced gray levels. B Implemented intensity levels as function of the +addressed gray value in the target image. The line trend has a slope equal to 0.077 (𝑎. 𝑢. ). + + + + +Fig. S7: Height distribution and height modulation depth estimation. A Height distribution 𝑃𝑖 related to the AFM +micrograph presented in Fig. 3A. Full range dispersion, allowing for the experimental estimation of the total +modulation depth ℎ0 = 0.910 𝜇𝑚 is defined as ℎ0 = 𝑁 ∙ Δℎ, where 𝑁 = 91 is the number of occupied bins while +∆ℎ = 0.01 𝜇𝑚 is the fixed bin width. Implemented phase depth is considered for a probe wavelength 𝜆𝑃 = 0.6328 𝜇𝑚 +assuming a refractive index equal to 𝑛 = 1.696. B Integral function 𝐼(𝑛) = ∑ +𝑃𝑖 +𝑛 +𝑖=0 +. Experimental estimation of +ℎ0(90%) represents the height range, uniformly distributed around the median of the distribution, where there is the +0.90 of probability to find a corresponding height value in the AFM micrograph. + + +0.95 +0.5 +0.05 + +Fig. S8: Comparison between: A phase distribution probability in the target phase map resulting from GS algorithm, +B intensity distribution probability in the holographic pattern and C implemented phase distribution retrieved from +the AFM image. Implemented phase depth is considered for a probe wavelength 𝜆𝑃 = 0.6328 𝜇𝑚 assuming a +refractive index equal to 𝑛 = 1.696. For visual clarity, data are binned considering 𝑁 = 20. + + + + +Fig. S9: Temporal characterization of structured surface. A Height distribution for six different exposure times. Each +dot represents the expected value ℎ̅ and relative variance 𝜎2 for the corresponding distribution. B Root mean square +error defined as RMSE = √∑ (𝑃𝑖 − 𝑃̅)2 +𝑁 + as function of the total exposure time. 𝑃̅ represents the target uniform +distribution expected at different exposure times. C Full range modulation depth ℎ0 and 90% dispersion range as +function of the total exposure time. Implemented phase depth is considered for a probe wavelength 𝜆𝑃 = 0.6328 𝜇𝑚 +assuming a refractive index equal to 𝑛 = 1.696. + +t 20 s +t=40 s +t 60 s +t=80 s +t=100 s +t=120 s +-0.904 +2元 +Fig. S10: Experimental determination of diffraction efficiency 𝜂𝑖 determined by integrating the CCD signal over the +regions of interest delimited by the colored trace in the image. Green area corresponds to the holographic image +efficiency while light blue and orange area correspond to the DC order and ghost image, respectively. + + +Fig. S11: Kinoform transmittance over exposure time determined by integrating the CCD signal over the full sensor +size. + + +Fig. S12: Experimental determination of pattern visibility. Visibility is defined as 𝑉 = (𝐼𝑆𝑅 + 𝐼𝑁𝑅)/(𝐼𝑆𝑅 − 𝐼𝑁𝑅) where +𝐼𝑆𝑅 is the average intensity inside the signal region (green area) and 𝐼𝑁𝑅 is the average noise level outside the +holographic image (orange area). + +Fologra +t=40s +0.5mm +GhostOff axis +OnaxisNR +SR +0.5 mm + + +Fig. S13: Optimization of multiplexed spherical profile. A Axial shifting Δ𝑧 of the holographic image as function of +the spherical phase profile parameter 𝑓. B Maximum visibility achieved with different spherical phase profile +parameter 𝑓. Best value for multiplexed focal length is 𝑓 = 0.450 𝑚𝑚, allowing for high visibility and reasonable +orders separation. + + + +Fig. S14: Speckle contrast reduction process by holograms time averaging. A Average holographic pattern acquired +after 10 writing/erasing cycles representing the on-axis image of the Greek letter pi. B Speckle noise severity as +function of the number of averaged frames. Severity is defined as 𝜎/〈𝐼〉 where 〈𝐼〉 is the mean intensity and 𝜎 is its +standard deviation measured in the image, see also (67). C-D Average holographic pattern acquired after 10 +writing/erasing cycles representing the on-axis image of a music note and a smile, respectively. + +DCorder noise +.owresolutionA +0.6 +0.4 +0.2 +0.5mm +C +0.5mm +0.5mm +Fig. S15 Speckle analysis of a time averaged grayscale pattern. A Target image. B Resulting average holographic +pattern acquired after 10 writing/erasing cycles. C Comparison between the mean intensity level of three cube faces +for the single frame and the time average. D Comparison between the speckle severity of three cube faces for the +single frame and the time average. + +B +Top +Top +Side +Side +Front +Front +0.5mm + +Fig. S16: Look up table for optical encryption and decryption of text messages. + +Trit +Decimal +Char +- ++ +- \ No newline at end of file diff --git a/v9AyT4oBgHgl3EQfaffQ/content/tmp_files/load_file.txt b/v9AyT4oBgHgl3EQfaffQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..182b2e382824d2abf643839e6d5acb9836abad32 --- /dev/null +++ b/v9AyT4oBgHgl3EQfaffQ/content/tmp_files/load_file.txt @@ -0,0 +1,1354 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf,len=1353 +page_content='Reprogrammable holograms from maskless surface photo- morphing Francesco Reda,1 Marcella Salvatore,1,2 Marco Astarita,3 Fabio Borbone4 and Stefano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato1,2,* 1 Physics Department “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Pancini”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' University of Naples “Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Complesso Universitario di Monte Sant’Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' via Cinthia 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 80126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy 2 Centro Servizi Metrologici e tecnologici Avanzati (CeSMA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' University of Naples “Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Complesso Universitario di Monte Sant’Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Via Cintia 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 80126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy 3 Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Politecnico di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 20133,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Milan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy 4 Department of Chemical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' University of Naples “Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Complesso Universitario di Monte Sant’Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Via Cintia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 80126 Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy *Corresponding author: stefanoluigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='oscurato@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='it Abstract Holographic technologies have the potentiality to impact our everyday life in many sectors including science, education, entertainment, art, and healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Although holographic screens and projectors are part of common imagination since long time, they are still at initial stages of development and integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Recent achievements of metasurface and flat optics research gave an unprecedented strength to this field, overcoming critical aspects as efficiency, size and flexibility of conventional optics and liquid crystal technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' However, although diffractive and metasurface holographic projectors with advanced functionalities and improved efficiencies are continuously reported, they are static devices, requiring demanding, burdensome, and irreversible manufacturing processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Here we report an all-optical and single-step lithographic framework for the fabrication of diffractive holographic projectors directly on the surface of a photo-morphable polymer film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Real-time optimization during the accurate surface patterning and fully structural reconfigurability allowed for the first prototype of a fully reprogrammable pixel-less morphological projector, opening new routes for holographic image displaying and optical data sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Introduction Light-modulating planar devices can empower many emerging technologies as virtual and augmented reality (1–3), optical wireless communication (4, 5), green energy harvesting (6, 7), opening also to the next‐generation of displays and holographic projectors (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Despite holograms can be implemented through addressable liquid crystals on silicon (LCOS) devices (9–12), diffractive optical elements (13–15) and metasurfaces (16–18) are increasingly gaining interest for holographic applications, due to their ability to generate arbitrary optical fields from the modulation of an incident light beam through a ultra-compact and planar device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Untied from electronics, efficiency, and size limitations of LCOS displays, planar holographic devices promise a greater possibility of miniaturization while maintaining higher efficiencies and light modulation capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In addition to images projection, planar holographic devices can also represent a valid platform for optical information storing, encryption and sharing (19–21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nevertheless, as also valid for holographic displays, those technologies intrinsically require optical supports able to be fully erased and rewritten, a milestone only partially achieved with several limitations by tunable metasurfaces (22–24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' However, these features come at the expense of realization of complex surface geometries at light (sub-)wavelength scale, where the manufacturing process, typically leading to static devices, (25, 26) can pose severe performance and/or economic limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optical lithography is among the most used surface patterning techniques (27) for the fabrication of planar optical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Starting with the irradiation of a photoresist by means of a structured illumination pattern produced by a mask, the typical workflow requires additional post- exposure chemical, physical and mechanical processes, through which the desired surface pattern is finally transferred to the operating device (27, 28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The multileveled surface patterns needed for optimal functionality of a holographic device can even require several iterations of this scheme (13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maskless methods, where the multistep mask exposure is replaced digital-based projection of spatially structured intensity patterns over the photoresist surface, can offer however greater control and flexibility for the realization of the complex lateral geometry and the grayscale modulation (29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Both deformable micromirror devices (DMDs) (30, 31) and LCOS (32–34) were explored as programmable spatial light modulators to achieve digital maskless surface patterning for optical devices manufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In addition, even non-optical maskless approaches as particle beam fabrication methods (35, 36) and scanning probe lithography (37, 38) have been reported for the accurate optical device manufacturing, but these methods suffers of reduced throughput, increased costs and energetic impact with respect to optical techniques (26, 27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Here, we demonstrate the direct all-optical maskless fabrication of fully reconfigurable diffractive holographic devices, implemented as thin structured transmissive phase retarders realized on the surface of a reprogrammable dielectric material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To this aim, a digital holographic optical scheme is used to generate and project a grayscale spatially structured intensity distribution of light on an azobenzene-containing polymer film, whose surface locally deforms according to the irradiated spatial light distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In this way, the structured surface of the operating optical device is directly produced without any additional lithographic step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The photomechanical process responsible for the direct surface morphing of the polymer is intrinsically reversible, allowing the update of the fabricated surface geometry at will.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Compared to other optical maskless techniques, our approach allows to fully exploit the possibility of arbitrary spatiotemporal modulation of the holographic writing beam and the integration of the lithographic system with a real-time optical characterization setup allowing to evaluate the device performance already during the fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The all-optical system is used here to realize operating optical configurations and devices with advanced and optimized functionalities, including reprogrammable grayscale holograms with improved visibility and tunable axial position, and a high-density optical encryption scheme able to temporally split secreted holographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Representing a new state-of-the-art as a reprogrammable all-optical fabrication framework for custom multileveled flat optical devices, our approach can assist the development of next generation photonics, starting from devices prototyping, testing and assembly until to their large-scale distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Results 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Direct holographic surface structuration To elucidate the main features of our direct holographic maskless surface patterning scheme, schematically represented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1A, we first demonstrate the realization of simple arbitrary binary pattern on the surface of an azobenzene-containing polymer thin film (herein referred to as azo-resist to highlight its functionality as lithographic material).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This class of amorphous materials exhibit the unique property of stable surface reliefs formation under low-intensity structured UV-visible light irradiation (39, 40) as a consequence of a directional material transport initiated by the azobenzene chromophores hosted in the polymeric matrix (41–43), with a mechanism still to be fully unveiled (44) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Due to the sensitivity to both the intensity and the polarization of the irradiated light, the surface reliefs on azopolymer films enable a direct vectorial lithography, exploited in many configurations, including interference, high-fusing, near-field, and pure structured polarization illumination (45–54) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In the irradiation of a circularly polarized light patten 𝐼𝑊(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦) in a low-focusing regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' the spatiotemporal evolution of the surface morphology ℎ(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑡) can be phenomenologically described as (54,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 55): ℎ(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑡) = ∇2[𝐼𝑊(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦)] ∙ ℎ0(𝑡) (1) In a low intensity regime,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' the relief depth ℎ0 increases approximately linearly with the exposure time (ℎ0(𝑡) = 𝑐 ∙ 𝑡 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' where 𝑐 is a phenomenological inscription efficiency constant),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' while the material flows from the high intensity region toward dark areas (as schematized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1B), forming then a surface relief pattern with the same geometry as the illuminating intensity 𝐼𝑊(𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In our maskless optical lithographic scheme, we used a phase-only Computer-Generated Hologram (CGH) system to fully exploit the direct surface structuration process described by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In this configuration (see also Materials and Methods), arbitrary grayscale illumination patterns 𝐼𝑊(𝑥, 𝑦), originated by a computer-controlled phase-only Spatial Light Modulator (SLM), can be directly transferred to the entire illuminated area of the polymer surface in a single exposure step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Holographic structuration of azo-resist surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Graphical representation of the holographic inscription scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Writing beam, with arbitrary shaped intensity profile is directly projected over the azo-resist surface by an objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Light triggered mass migration occurring at surface of amorphous azopolymer films under structured illumination absorption, leading to stable surface geometries ℎ(𝑥, 𝑦, 𝑡).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C Design and reconstruction of a QR code shaped holographic pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The experimental intensity pattern is the result of time averaging of the holographic sequence, allowing to reduce speckle noise effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D Atomic force microscope micrograph of the structured surface collected right after the exposure step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Red scale bar, both in panels C and D corresponds to a physical size of 20 𝜇𝑚 on the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' E Height distribution probability (orange plot) compared with the intensity probability distribution (sky blue plot) of the holographic beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Each point of the line plot represents the probability 𝑃𝑖 of having a fixed height value ℎ𝑖 in the AFM image corresponding to the implemented intensity level 𝐼(𝑤)𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To demonstrate our ability in arbitrary direct surface patterning, we designed a two-levels QR code as an 8-bit two-dimensional image, from which the illuminating light pattern 𝐼𝑊(𝑥, 𝑦) is calculated (56) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1C top).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The generated holographic writing pattern is projected on the surface 4 B h(x,y,t) Azo-resist Holographicpattern n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2 h (um)of the azo-resist by means of a long-working distance microscope objective, where a relief pattern ℎ(𝑥, 𝑦) directly appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Additional details about the writing holographic design, the illumination homogeneity improvement, and the resolution of our configuration can be found in Methods section and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1D shows the Atomic Force Microscope (AFM) micrograph of the polymer film surface after being exposed to the holographic pattern for 𝑡 = 20 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' AFM image is collected right after the exposure step without any additional post-exposure process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The surface relief pattern faithfully reproduces the target image, and, as expected from eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (1), is the complementary of the illuminating hologram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To extend the visual comparison to a quantitative analysis needed in the fabrication of complex relief pattern from the design of a diffractive phase-modulating mask acting as holographic projector, we characterize eventual mismatch errors between the target and the experimental surface morphology described in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To this aim, we retrieved the height distribution of the surface form the topographic image, with a sampling interval of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='387 𝜇𝑚 determined by the pixel size of the AFM scan (see also Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The distribution, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1E, must be compared with the target one, in which there are only two equally weighted levels corresponding to the black (𝐼𝐵) and white (𝐼𝑊) pixels of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Despite the presence of two narrow bands in the distribution extracted from the optical image of the hologram (blue curve in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1E), confirming the high contrast in the writing binary pattern, the topographic distribution (orange curve) of the two heigh levels appears broadened.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The origin of such structural mismatch resides in the relief smoothing at illumination edges with sharp contrast jumps (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S5), as predicted for the light-induced material transport phenomenology described by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In our previous implementation of this lithographic method, we circumvented this issue by limiting quantitative design to smooth sinusoidal surfaces (46, 57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' However, sharp features could potentially be encoded in the design of a suitable optimized holographic pattern associated to the target image, providing eventually a narrower topographical distribution when transferred on the azo-resist film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As further detailed below, the all-optical scheme used here to fabricate and simultaneously characterize the diffractive optical components allows the minimization of the effects on optical performances originated by similar fabrication-design mismatches inherent to the simplified description of material transport in hologram design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Even in the simplistic linear response relief design used here, the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1 fully demonstrate the potentialities of our scheme as a direct maskless holographic technique for the arbitrary structuration of the surfaces at the microscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The fidelity of the surface pattern can be further demonstrated by to the possibility of effectively read the binary QR code (by any camera QR code reading software) from the topographic data, rendered as two-dimensional image with a linear colormap (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Holographic morphological projectors: design, optimization, and fabrication For the design of the azopolymer-based morphological holographic projectors we leverage the results of the scalar diffraction theory (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' While conventional projection displays exploit amplitude-modulating pixels to locally and selectively block part of the incident light to form images, a diffractive holographic projector can be implemented as a phase-only planar device for a coherent monochromatic light modulation (10, 12), able to reconstruct a desired light pattern without making use of absorption phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phase-only holographic plates, named kinoforms (58), can implement the proper modulating complex transmission function 𝑡(𝑥, 𝑦) = exp (𝑖𝜑(𝑥, 𝑦)) as local thickness variations ℎ(𝑥, 𝑦) of a dielectric material (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2), which influence the optical path traveled by an input monochromatic field 𝑈𝑖𝑛(𝑥, 𝑦, 𝑧𝑖𝑛) (see Materials and Methods).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The phase mask 𝜑(𝑥, 𝑦) is typically referred to as kinoform (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' According to the diffraction theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' in the case of far-field propagation 𝑧 ≫ 𝑧𝑖𝑛 (Fraunhofer approximation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' where),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' the emerging modulated field 𝑈𝑜𝑢𝑡(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑧) is two-dimensional spatial Fourier transform of the beam modulated at the kinoform plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' resulting in a reconstructed image 𝐼𝑜𝑢𝑡(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦) determined by the relation (10): 𝐼𝑜𝑢𝑡(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑧) = |𝐹𝑇[𝑈𝑖𝑛(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 0) ∙ 𝑒𝑖𝜑(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='𝑦)]| 2 (2) An analogous result can be also found between the two focal planes of a thin lens,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' reducing the image reconstruction to finite distances (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' By inversion of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (2), the kinoform 𝜑(𝑥, 𝑦), and the relative mask surface relief pattern ℎ(𝑥, 𝑦) for any given target holographic image 𝐼𝑜𝑢𝑡(𝑥, 𝑦) could be potentially calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' However, for a phase-only modulator, the kinoform can be retrieved only through iterative algorithms (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2 schematically shows this process for the case for the desired output image 𝐼𝑜𝑢𝑡 representing the Greek letter “π”, where the conventional Gerchberg–Saxton (GS) algorithm (59) is used as iterative Fourier transform algorithm (IFTA) to retrieve the kinoform 𝜑(𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Once the kinoform is calculated, all the challenges involved in the fabrication of the holographic projector are shifted to manufacturing level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optimal image reconstruction requires an accurate transfer of the designed phase mask, including the position of the phase discontinuities (lateral pattern) and the value of local and maximum phase delays, in the proper surface relief pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Any defect arising in this process deteriorates the hologram quality, causing the reduction in the diffraction efficiency and the appearance of spurious contributions in the target holographic image, consisting of an unmodulated optical component (DC term) and several shifted and scaled replicas of the desired intensity pattern (ghost or false images) (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' These contributions can overlap in the reconstruction plane, requiring eventually an off-axis design for the hologram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2), which reduces the available target image domain by half of the field of view (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' However, even in the case of a defect-free lateral pattern transfer, a deviation from a full 2𝜋 modulation depth, associated with eventual total relief heigh errors induced in the dielectric structured surface, still cause the emergence of the spurious holographic terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To reduce this effect, an ideal optimal modulation depth of ℎ0 = 𝜆/(𝑛 − 1) should be realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This condition simultaneously grants the maximization of the diffraction efficiency in the target holographic image and ghost hologram suppression (see also Supplementary Information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In our direct lithographic scheme, the surface relief pattern ℎ(𝑥, 𝑦) and the modulation depth ℎ0 can be independently controlled by the digital holographic design and by the exposure time, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Then, the generalization of the inscription scheme of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1 to the projection of a grayscale structured light pattern with the geometry of a calculated kinoform 𝜑(𝑥, 𝑦) can lead to the fabrication of optimized morphological holographic projectors directly as a surface relief pattern on the dielectric azo-resist film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Design of holographic morphological projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Target intensity 𝐼𝑜𝑢𝑡 is used to retrieve, by GS iterative algorithm, the proper phase map 𝜑(𝑥, 𝑦) to be implemented as dielectric height modulated phase retarder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The material with refractive index 𝑛 is assumed to be immersed in a surrounding medium with refractive index 𝑛𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When illuminated with monochromatic light, with wavevector 𝑘 = 2𝜋/𝜆, the phase retarder (kinoform) produces a diffracted beam depending on the optical delay accumulated by the light passing through the structured surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The kinoform allows the reconstruction of the target holographic image defined during the design and additional spurious diffraction orders to be suppressed by tuning the total modulation depth ℎ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To this aim, we first characterized the ability of our lithographic scheme in encoding multiple discrete intensity levels of light in a single holographic pattern, useful to calibrate the response our system for the generation of the complex grayscale pattern required by a kinoform fabrication (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Then, we directly inscribe on the azopolymer film the grayscale surface profile ℎ(𝑥, 𝑦) of the kinoform calculated for the reconstruction of far-field holographic image of the Greek letter “π”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 元 元 Image DC Ghost t(x,y)In this process, the 8-bits (256 levels) digitally calculated kinoform 𝜑(𝑥, 𝑦) is converted into a gray-scale holographic pattern 𝐼𝑊(𝑥, 𝑦), which induces the correspondent relief pattern ℎ(𝑥, 𝑦) on the azo-resist surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For the analysis of the lateral pattern and the determination of the total height excursion ℎ0 of the produced surface relief, we performed SEM and AFM analysis after the exposure process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The SEM analysis (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3) confirms a correct position-matching of the phase discontinuity in the kinoform, granting a global correct relief lateral geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3A shows, instead, the three-dimensional topographic micrograph of a portion of a typical azo-resist kinoform surface, evidencing the continuous heigh variation in the pattern, encoded in the grayscale writing holographic pattern (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fabrication and optimization of azopolymer holographic projectors implemented as kinoforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The middle panel shows the grayscale holographic pattern reproducing the kinoform design and the resulting SEM image of the structured surface after the exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Atomic force microscope (AFM) scan of a quarter portion of the structured surface (100 𝑋 100 𝜇𝑚) collected right after the exposure process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Full modulation depth ℎ0 as function of the total exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Experimental data are fitted with the model trend ℎ0 = 𝑐 ∙ 𝑡, allowing for the experimental determination of the surface inscription efficiency 𝑐 = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 𝑛𝑚/𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Blue axis shows the implemented phase depth for a probe wavelength 𝜆𝑃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C Diffraction pattern acquired at the optimal exposure time, maximizing the diffracted light power effectively shaped in the target holographic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D Experimental trend of the diffraction efficiency reconstructed during the inscription process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Trends are the results of five independent exposures: the average value for the experimental diffraction efficiency at each exposure time is represented by a solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The shadow represents the punctual standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To quantitatively evaluate the quality of the fabricated surface relief pattern with respect to the design, we retrieved the surface height distribution from the AFM analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The topographic distribution is then transformed in a phase delay distribution (by eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S1) and compared with the phase distribution extracted from the designed phase map, (additional details are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S7-9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We used the Root Mean Square Error (RMSE) to quantitatively definethe average mismatch Holographic image Hologram 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2mm DC 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='904 Surface 50μm 0errors occurred during the fabrication step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The analysis, repeated for different exposure times, provided a constant RMSE, ensuring that any topographical mismatch, related to the hologram design and to the material response, is not worsened by increasing the surface modulation depth ℎ0(𝑡) to reach the target ℎ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' From the height distributions obtained with fixed illumination parameters at different exposure times, we also determined an experimental estimation of the writing efficiency parameter 𝑐 entering in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We extracted the full relief modulation range from the retrieved distributions to estimate the total modulation depth ℎ0(𝑡), whose experimental results are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Those results allowed the empirical definition of the exposure time that provides the optimal 2𝜋 modulation depth in the kinoform for the probe light wavelength of 𝜆𝑝 = 632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='8 𝑛𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A total exposure time of 𝑡 = 86 𝑠 is sufficient for optimal inscription of the considered kinoform fabrication to in our experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nevertheless, this off-line structural characterization roadmap does not guarantee a standardization of the manufacturing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A new calibration step would be necessary for each different relief geometry and illumination parameters, leading to a time consuming and multi-step workflow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' However, the surface relief pattern developing on the azo-resist can be characterized directly during the surface structuration, providing a real-time feedback on the writing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Despite different techniques based on mechanical (62) and optical (45) real-time topographic investigation have been successfully proposed, they do not directly characterize the optical performances of the diffractive surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' On the contrary, the all-optical lithographic scheme proposed here easily allows the direct evaluation of the optimized writing parameters from the analysis of the developing holographic diffraction pattern (46, 63), to act also on specific aspects relevant for applications, as the suppression of the ghost holograms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To this aim, we illuminated the developing morphological holographic plate on the azo-resist film with an additional laser beam at the probe wavelength 𝜆𝑝 during the surface writing step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The developing diffraction pattern is continuously recorded with a CCD, at a repetition rate of 5Hz, during the exposure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For each of the acquired frames, we evaluated in real-time the relative diffraction efficiencies 𝜂𝑖 in the target holographic image, and in the spurious terms (DC order and the ghost image) (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3D summarizes the experimental results for five independent kinoform fabrications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The optimal exposure time (𝑡𝑜𝑝𝑡 = 103 ± 1 s) was chosen such that the light power diffracted in the holographic target image is maximized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In this condition, in experimental efficiency 𝜂+1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='60 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='02 was obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We also observed a relative transmissivity (|𝑡(𝑥, 𝑦|2) equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='96 for the final developed surface (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S11), demonstrating also minimal influence of possible unfavorable light scattering sources produced by the lithographic process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Our approach demonstrates the big advantages offered by a single-step and all-optical structuration technique, allowing the tuning of the optimal exposure parameters in real time, which leads to a fully working device right after its inscription without the need of further time-consuming surface analysis or preliminary calibration procedures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The off-axis hologram design, analyzed here mainly to highlight the characteristics of our holo- lithographic scheme has a fundamental limitation in practice due to the presence of ghost holograms simultaneous to the target holographic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In every physical device with unavoidable structural mismatches in the kinoform fabrication, this imposes a having for the exploitable holographic plane and a physical filtering process for the spurious terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=" However, in many applications such as augmented reality and wearable holographic projectors, the holographic image could be formed in a very specific plane of the optical axis, which typically coincides with the observer's eye or with a detector sensor (1)." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When appropriately designed and fabricated, a holographic plate operating in this configuration allows to overlook the presence of any other spurious diffraction order, relaxing also eventual design constrains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' An additional advantage of kinoform-based holographic projectors is the possibility to encode multiple optical functionalities in the same substrate, multiplexing, during the design, the optical properties that two or more phase masks would have exhibited individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Multiplexing has no impact in terms of calculation resources during the design step, and it can easily explored by the unique combination of our material and the holographic setup (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Starting from the target phase mask, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' resulting from kinoform calculation, an additional proper phase mask can be superimposed to produce an axial shift of the target holographic image with respect to the ghost and DC orders (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This task can be achieved in an equivalent way by making the light passing in an additional lens of focal length f, so the kinoform 𝜑(𝑥, 𝑦) must be multiplexed with the phase shift produced by a thin lens (10), equal to 𝜑𝐿(𝑥, 𝑦) = 𝜋/𝜆𝑓(𝑥2 + 𝑦2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As the phase of the beam after passing through the phase mask is required to be modulo 2𝜋, the resulting multiplexed phase map (65) 𝜑𝑀, to be converted in the holographic writing pattern, is 𝜑𝑀 = (𝜑 + 𝜑𝐿)𝑚𝑜𝑑(2𝜋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Form the Fourier transform relation (eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (2)) it can be easily demonstrated, using the generalized Fourier analysis (61), that each diffraction order 𝑖 is axially splitted along the optical axis and it is reconstructed in a different plane located at 𝑧 = 𝑖 ∙ Δ𝑧, where 𝑧 = 0 denotes the reconstruction plane of the kinoform without the additional lens phase map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The distance Δ𝑧 is function of the focal length 𝑓, which determines the axial separation between the holographic image and the other (spurious) orders (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4C shows a SEM image of the surface relief pattern inscribed on the azo-resist surface using such multiplexed kinoform design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The corresponding diffraction pattern in the target reconstruction plane is presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In this plane of the optical axis, only the target holographic image was clearly visible, while the out of focus DC and ghost terms contributed only with negligible background in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Design, fabrication, and optimization of multiplexed kinoforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A The resulting kinoform, from a GS algorithm performed on the on-axis image of the letter pi, is multiplexed with a spherical phase profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The new phase profile is used to encode the different intensity levels of the writing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Representation of the diffractive behavior of a multiplexed kinoform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When illuminated with monochromatic coherent light, different diffractive orders are axially reconstructed on shifted planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Assuming that 𝑧 = 0 is the plane where the holographic pattern is reconstructed without the multiplexing process, each diffraction order 𝑖 is reconstructed at 𝑧 = 𝑖 ∙ Δ𝑧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C SEM image of the azopolymer surface after the exposure to the holographic beam for 𝑡𝑜𝑝𝑡 = 120 𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D Resulting diffraction pattern acquired at 𝑧 = Δ𝑧 E Experimental trend of the pattern visibility reconstructed during the inscription process as result of five independent exposures: the average value for the pattern visibility at each exposure time is represented by a solid line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The shadow represents the punctual standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As we could not simultaneously access to all the diffracted orders during the surface developing to define the relative diffraction efficiency in the holograms, we used the image visibility 𝒱 as quality estimator for the light pattern in the target reconstruction plane (additional details are presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Similarly to the previous case, the real-time control of this parameter allowed us to directly optimize the exposure time 𝑡𝑜𝑝𝑡 = 120 ± 1 𝑠 for maximum visibility of 𝒱𝑚𝑎𝑥 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='83 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='03 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This high contrast image was also the result of an independent tuning of the multiplexed focal length 𝑓, chosen, according to our setup resolution limit, to maximize orders separation and subsequently the holographic image contrast (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 元 C Surface 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 mm 50 μm t=120s Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fully reprogrammable kinoform for time average image quality improvement and data storing and sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Reprogrammable holographic projector: after surface pattering and holographic image acquisition, morphology can be completely restored to pristine flat state, allowing for a new patterning step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Quality enhanced experimental images are the result of the time averaging of multiple holographic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Full resolution images are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Experimental results of holograms time averaging for speckle noise effects reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' On the left is showed the grayscale pattern acquired after a single exposure step while on the right the same pattern is reconstructed as time average of ten independent exposures over the same azopolymer area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C Experimental results of the holographic data storing and sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Holographic patterns are plotted with a rainbow colormap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Blue-indigo, green-yellow and orange- red colors are respectively related to three possible intensity levels encoding three digital logic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Experimental images are converted from an analog to digital map for information readout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Word “HELLO” is reconstructed after a first step of surface writing loop followed by a second multiple exposure step allowing for the reconstruction of the second part of the message, “WORLD”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As additional requisite for the use of morphological holographic projectors in real photonics applications, ranging from optical cryptography to holographic refreshable displays, the surface morphology should be completely reversible and reprogrammable on demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' One of the interesting features of azopolymers is that when illuminated with unstructured light in the chromophore absorption band (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S3), the pristine flat surface can be optically restored, allowing multiple and reversible patterning cycles (46, 66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 5A schematically shows this all- optical reprogrammable surface structing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' One of the features of dynamic holographic platforms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' LCOS SLMs or DMDs) is that the temporal coordinate can be exploited to produce effective holographic patterns with either enhanced lateral complexity (64) or higher image quality (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In these processes, the final holographic image is the result of the temporal average of the individual patterns that are instantaneously produced by the dynamically changing diffractive device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The unique reversible photo-mechanical properties of the azopolymer used here as can be exploited to achieve similar Single frame 10frames average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 mmeffects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To demonstrate an example practical relevance for our dynamically-evolvign morphological holographic projectors, we repeatedly reprogram the kinoform written on the surface of the azo-resist to produce a time-averaged holographic diffracted image with a reduced speckle noise, intrinsically associated to the kinoform design with a IFT algorithm (68, 69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='14 in the Supplementary Information shows the details of the characterization of the holograms recorded in a typical dynamical kinoform reconfiguration experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The procedure for the improved average holographic image started by irradiating the pristine azopolymer surface with a holographic writing kinoform (in the multiplexed design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' After an inscription process providing optimized visibility in the diffracted holographic pattern, an image 𝐼(𝑥, 𝑦, 1) of the hologram was collected by the CCD and stored as single frame of a holographic projection movie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' At this stage the surface was completely (optically) erased, and the same area of the azo-resist was exposed with a new independently calculated holographic writing pattern, characterized by an independent random distribution of speckle grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This loop was iterated, acquiring the relative holographic image 𝐼(𝑥, 𝑦, 𝑖) each time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' After 𝑁 = 10 writing/erasing steps, the time averaged holographic image was calculated as 〈𝐼(𝑥, 𝑦)〉 = (𝑁)−1 ∑ 𝐼(𝑥, 𝑦, 𝑖) 𝑁 𝑖=1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As expected, the averaged image is characterized by a speckle severity reduced by a factor 1 √𝑁 ⁄ , as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S14 for three different target holographic images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This artificial image improvement trough a time averaging process is the same as that performed by an ideal “slow eye or detector”, which has a time response much higher than the typical surface reconfiguration time (~ 120 𝑠 in our experimental condition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Despite still far from the refresh rates achievable with other dynamical systems, these results allow us to include for the first versatile dynamical modulation capabilities for applications with a planar optical diffractive component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As additional proof, we show the speckle noise time filtering for a three-level target holographic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 5B shows the diffraction pattern 𝐼(𝑥, 𝑦) and the corresponding time averaged holographic pattern 〈𝐼(𝑥, 𝑦)〉 representing the image of a cube, where each of the three displayed faces encode a different diffracted intensity level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The grayscale nature of the hologram became visually clear only once the that the speckle noise contrast reduction is performed (see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S15), with a significant improvement with respect to a single holographic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Morphological reprogrammable devices able to also encode grayscale optical information can represent a valid platform to store encrypted optical information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The use of non-binary bits of light can increase the storage capacity, while simultaneously reducing the required space on the physical support (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We used our azopolymer as a morphological holographic memory support where the visual information was encrypted in the surface topography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The secret message, displaying the word “HELLO”, was converted into a ternary base where each letter is codified into three different trit (ternary digit), each assuming three separate logic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The trits that defines each letter of the word have been arranged in rows to form a three-level grayscale image, where each level corresponds to one of the three possible logic states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The details of the designed ternary alphabet are discussed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When this image was used to define surface morphology and transferred to the azo-resist surface, all the original information was encrypted by the Fourier transform algorithm, therefore, information readout is possible only optically by means of a proper optical setup (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 5C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fourier-transform coding also offers the advantage that if part of the surface would be damaged or destroyed, reading the secreted information would still be theoretically possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We finally completely erased and reshaped the surface geometry to share the second part of the secreted message composed by the word “WORLD”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This temporal holographic splitting of the message enhances encryption capabilities and information sharing security.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Additionally, it prospects azopolymer structured films as promising reversible high- density memory substrates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We further estimated that by a single surface illumination process, with the defined architecture, we are capable of simultaneously encoding 3,125 bytes of information in a secreted hologram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Discussion and Conclusions Our direct all-optical maskless lithography, using azopolymers as photoresist, represents the state of the art as fabrication technique of fully reversible diffractive flat optical elements with arbitrary holographic pattern reconstruction capabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In the simple case of binary modulation for the writing beam demonstrated in this work, we proved our ability to faithfully transfer, in a pure optical process, complex bidimensional geometries as a two-level surface modulation of an azobenzene-containing polymer film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This process, in another perspective, can also be interpreted as a form of information storing if the target image (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' the QR code image) is seen as the information to be encrypted as surface morphology on the azopolymeric film.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' An additional morphological analysis of the surface, right after the exposure process, demonstrated no significant information losses in the morphological information transfer, even considering the differential light response of our material to the writing illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As additional milestone of such method, we extended and scaled our approach for the realization of diffractive kinoforms, where complex lateral geometries with grayscale modulation depth are simultaneously required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The additional possibility to test the devices functionality during the fabrication process provides a cost-effective design and prototyping of operating diffractive optical devices, implemented as azopolymer phase retarders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We characterized both the surface morphology and its diffractive behavior right during the exposure, investigating the quantization and pixelation effects and non-linear responses of the material to the structuring technique, enhancing their relevant impact during the device optimization and fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Our approach led to the realization of pixel-free morphological holographic projectors, ensuring high efficiency and ultra-compact devices, whose depth results comparable with the operating light wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The opposite happens in conventional digital devices, where the discrete nature of the pixels limits the spatial resolution and the addressable phase sampling, while simultaneously generating spurious periodic replication of the reconstructed image, with a consequent overall efficiency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Despite simple, morphological encoding design of dielectric diffractive surfaces totally changes the perspective when holographic projectors are also compared to traditional wide displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' First, the complex modulation provided by the realized kinoform has an almost unitary transmittance, resulting in a lossless structuring of light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Furthermore, the Fourier relationship linking the modulation and the image reconstruction plane is non-local, meaning that each of the point of the kinoform will contribute to form the entire holographic image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In other words, a kinoform preserves the information content in all its parts, consequently breaking or damaging the device will not compromise at all the holographic image reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Additionally, as the azopolymer surface can be optically restored to the flat pristine state in place, multiple writing/erasing cycles can be performed on time scales of few minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As, up to now, no material and structuration method combination for such dynamically changing surfaces exist, our approach represent the state of the art for reversible, all-optical custom flat optical devices fabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This possibility allowed time-averaged enhanced quality holographic images and paved the way for the fabrication of morphological reshapable devices able to encode optical information with both morphological and temporal encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' As valid every time that information needs to be stored on a physical support, the main requests for the substrate are time stability and reversibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' On the other side also the encoding process is required to be highly controlled, as any critical issue may result in information degradation or even in its loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We demonstrated that azopolymers, when illuminated with digital reconstructed intensity patterns of light, can meet those requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For the first time we showed that azopolymer unique optical properties can also be exploited to implement a new class of photonics devices with several applications ranging from wearable holographic projectors and displays to high quality supports for data storing, encryption and sharing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Even if still at a primitive level, this approach already makes evident the benefits that can completely change our prospective for holographic displays, optical data storage, and encryption, opening also to practical applications in emerging technologies as VR\\AR displays and wearable devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Materials and Methods Experimental setup The experimental configuration for the azopolymer surface relief inscription is based on a phase-only Computer-Generated Holograms (CGHs) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Its schematic representation is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A laser diode source (Cobolt Calypso) produces a TEM00 beam at wavelength λ=491 nm which, after a beam expander (lenses L1 and L2), is phase-modulated by a computer- controlled reflective phase-only Spatial Light Modulator (SLM, Holoeye Pluto).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The modulated beam is propagated through a 4f lenses system with the input plane located in the SLM plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The output plane coincides with the back focal plane of an infinity-corrected long-working distance 50X objective (Mitutoyo), with numerical aperture NA=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The focal lengths of the lenses L3 (300 mm) and L4 (175 mm) are chosen to maximize the spatial resolution in the hologram reconstruction plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This choice also defines the diameter (~200 μm) of the accessible circular area in the objective front focal plane, which can be used to structure the azopolymer surface in a single illumination step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The position of the sample near the objective focal region is accurately controlled by means of a x-y-z translation stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Average intensity in the range 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='7-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='0 W⁄cm2 and circular polarization are used for the structuration of the azopolymer surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To reduce the speckle noise contrast effects (67), the holographic illumination over the azopolymer surface is the result of the time average of several holographic patterns generated from different kinoforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Each pattern is reconstructed after an independent design from the same target image, initializing the algorithm with random phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The SLM refresh time (30 Hz for this work) is faster than the azopolymer response so that the effective illumination profile is the temporal average of the illumination profile associated with each of the many independent kinoforms sent in sequence to the modulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For visual inspection, and proper focusing of the holographic pattern on the photoresponsive surface, a 70/30 beam splitter, placed in the light-path, redirects the light retroreflected by the surface and re-collimated through the objective toward a tube lens (with focal length equal to 200 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This lens forms an image of the holographic pattern in its second focal plane, where a “DCC3240M Thorlabs” CCD camera is positioned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' During the exposure, an additional diode laser beam at 405 nm illuminates the photoresist film from the substrate side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The beam has circular polarization and different intensity levels depending on its intended function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When the intensity is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='6 W⁄cm2, the beam favors the surface structuring process, acting as a writing assisting beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' At intensity higher than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='9 W⁄cm2, its absorption causes the erasure of previously inscribed surface structures, acting as an erasing beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Further characterizations about assisting/erasing beam are described in a previous work (46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' An additional He-Ne laser beam, at 632.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='8 nm, is used as sample back-illumination source to test diffraction behavior of the modulated surface during the structuration process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The beam splitter also allows the collection of part of this light without interfering with the writing process, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The image of the surface is projected on the back focal plane of the tube lens and coupled by means of a mirror (mounted on a flip mount) to an additional 2f system composed by the lens L5 (300 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fourier transform image is captured with an additional CCD camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Azo-resist synthesis The photoresponsive material used in this work is an azobenzene-containing polymer (azopolymer) in amorphous state, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' All reagents were purchased from Merck and used without further purification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The azopolymer was synthesized, purified and characterized as previously reported (Mw = 27000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' phase sequence: Glass 67 °C Nematic 113 °C Isotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' \uf06cmax = 350 nm) (46, 54, 71).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The solution for film deposition was prepared by dissolving 70 mg of the polymer in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='50 ml of 1,1,2,2-tetrachloroethane and filtered on 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2 µm PTFE membrane filters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The desired film thickness (typically 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='1 𝜇𝑚) was obtained by spin coating the solution on 24x60 mm cover slides at 300 rpm for 4 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In the final stage, the samples were kept under vacuum at room temperature for 24 h to remove solvent traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Molecular structural formula and the absorbance in the UV-visible are provided in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Morphological characterization of structured surfaces Topographic characterization of inscribed azopolymer surface reliefs is performed using AFM and SEM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For AFM measurements, a WITec Alpha RS300 microscope is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The AFM is operated in tapping mode using a cantilever with 75 kHz resonance frequency and nominal force constant of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='8 N/m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' AFM tips (Arrow FM type from Nano World), with nominal radius of curvature of ≈10 nm, are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The maximum scanned area has a size of 100 × 100 μm2, acquired with resolution of 500 points per lines and 500 lines per scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For each AFM the minimum of the topography is set to zero to extract the height distribution 𝑃𝑗, representing the probability to find a pixel in the image with a height value between ℎ𝑗 and ℎ𝑗+1 where ℎ𝑗 = 𝑗∆ℎ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Here 𝑗 ranges from zero to 𝑁 − 1 where 𝑁 is the number of occupied bins in each image, while ∆ℎ = 10𝑛𝑚 represents a reasonably choice for the fixed bin width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Each height distribution is normalized to match the condition ∑ 𝑃𝑖 𝑁 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The expected value ℎ̅ = ∑ ℎ𝑖𝑃𝑖 𝑁 and variance 𝜎2 = ∑ (ℎ𝑖 − ℎ̅) 2 𝑃𝑖 𝑁 are extracted for each distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To retrieve an estimation of the modulation depth ℎ0 we consider the discrete integral function 𝐼(𝑛) = ∑ 𝑃𝑖 𝑛 𝑖=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Firstly, we define ℎ0 = ℎ𝑘 where 𝑘 satisfies the relation 𝐼(𝑘) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In that case we represent the total modulation depth as the full dispersion range of the distribution 𝑃𝑖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Since, due to our material behavior, the height distribution is not uniform, we also estimate the modulation depth as ℎ0 = ℎ𝑎 − ℎ𝑏 where 𝑎 and 𝑏 satisfy respectively 𝐼(𝑎) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='95 and 𝐼(𝑏) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='05;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' with this assumption ℎ0 represents the range, uniformly distributed around the median of the distribution, where there is the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='90 of probability to find a fixed value of ℎ𝑖, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Scanning electron microscopy (SEM) images are acquired with a field emission gun (FEG–SEM) FEI/ThermoFisher Nova NanoSEM 450 microscope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Samples are sputtered with a layer of Au/Pd using a Denton Vacuum Desk V TSC coating system prior to observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Iterative Fourier transform algorithms Despite the simple Fourier relation, optical modulation cannot be retrieved by simply inverting the equation (2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To design the proper phase mask able to lossless transform a given input light distribution into a desired light pattern, an iterative Fourier transform algorithm (IFTA) has been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In this class of algorithms, the optical field is bounced back and forth between two planes related by a Fourier transform, applying specific constraints to the retrieved fields at each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The used algorithm for diffractive kinoforms design is the Gerchberg-Saxton algorithm (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This algorithm can be easily implemented with modern computing capabilities, and once a digital representation of 𝐼𝑜𝑢𝑡 is provided by a grayscale 8-bit digital image it returns a digital representation of the phase map 𝜑(𝑥, 𝑦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When the desired light distribution is only constrained in a limited region of space, as for the holographic writing beam, the complex amplitude outside this area can be arbitrary chosen or left free to vary, allowing to increase the light hologram quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' This possibility is typically referred to as amplitude freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We used this approach to generate the writing holograms for the azopolymer structuration by a mixed region amplitude freedom (MRAF) algorithm (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' We implemented both GS and MRAF algorithms in MATLAB, using the Fast Fourier Transform (FFT) algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Bang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wetzstein, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gao, Toward the next-generation VR/AR optics: a review of holographic near-eye displays from a human-centric perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7, 1563 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kress, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chatterjee, Waveguide combiners for mixed reality headsets: a nanophotonics design perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nanophotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 10, 41–74 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' He, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sui, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Cao, Progress in virtual reality and augmented reality based on holographic display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 58, A74 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Uysal, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nouri, "Optical wireless communications —' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' An emerging technology" in 2014 16th International Conference on Transparent Optical Networks (ICTON) (IEEE, Graz, Austria, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='org/document/6876267/), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 1–7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xu, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rahmani, Programmable structured surfaces can change the future of wireless communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Light Sci Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 11, 196 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Peter Amalathas, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Alkaisi, Nanostructures for Light Trapping in Thin Film Solar Cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Micromachines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 10, 619 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Khodasevych, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Mitchell, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rosengarten, Micro- and Nanostructured Surfaces for Selective Solar Absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Advanced Optical Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3, 852–881 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Matsushima, Introduction to Computer Holography: Creating Computer-Generated Holograms as the Ultimate 3D Image (Springer, Cham, Switzerland, 1st ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2020 edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Slinger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Cameron, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Stanley, Computer-generated holography as a generic display technology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 38, 46–53 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Pi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, Review of computer-generated hologram algorithms for color dynamic holographic three-dimensional display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Light Sci Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 11, 231 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Buckley, "Computer-Generated Phase-Only Holograms for Real-Time Image Display" in Advanced Holography - Metrology and Imaging, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Naydenova, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (InTech, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='intechopen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='com/books/advanced-holography-metrology-and-imaging/computer-generated-phase- only-holograms-for-real-time-image-display).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' O’Shea, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Suleski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kathman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Prather, Diffractive Optics: Design, Fabrication, and Test (SPIE Press, 2004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kress, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Meyrueis, Applied digital optics: from micro-optics to nanophotonics (Wiley, Chichester, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' https://search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='ebscohost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='com/login.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='aspx?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='direct=true&scope=site&db=nlebk&db=nlabk&AN=302622).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ruan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ng, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Tan, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Li, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Qiu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yang, Off‐Axis Holography with Uniform Illumination via 3D Printed Diffractive Optical Elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Advanced Optical Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7, 1900068 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jiang, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jin, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Cao, When metasurface meets hologram: principle and advances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 11, 518 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sung, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, Recent advances in metasurface hologram technologies (Invited paper).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' ETRI Journal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 41, 10–22 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Luo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chen, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Li, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Duan, 3D-Integrated metasurfaces for full-colour holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Light Sci Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 8, 86 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Georgi, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wei, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sain, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Schlickriede, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Zentgraf, Optical secret sharing with cascaded metasurface holography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7, eabf9718 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yang, Holographic colour prints for enhanced optical security by combined phase and amplitude control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nat Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 10, 25 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Qu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yang, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Song, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Qiu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Han, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Tsai, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xiao, Reprogrammable meta-hologram for optical encryption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nat Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 11, 5484 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xiao, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Plaskocinski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Biabanifard, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Persheyev, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Askari, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Falco, Two-tier manipulation of holographic information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 30, 19145 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Du, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Barkaoui, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jin, Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Song, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xiao, Optical metasurfaces towards multifunctionality and tunability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nanophotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 11, 1761–1781 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Malek, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ee, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Agarwal, Strain Multiplexed Metasurface Holograms on a Stretchable Substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 17, 3641–3645 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' van Assenbergh, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Meinders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Geraedts, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Dodou, Nanostructure and Microstructure Fabrication: From Desired Properties to Suitable Processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 14, 1703401 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liddle, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gallatin, Lithography, metrology and nanomanufacturing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nanoscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3, 2679–2688 (2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sharma, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rathi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Misharwal, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sinhmar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kumari, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Dalal, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kumar, Evolution in Lithography Techniques: Microlithography to Nanolithography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nanomaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 12, 2754 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Okazaki, High resolution optical lithography or high throughput electron beam lithography: The technical struggle from the micro to the nano-fabrication evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Microelectronic Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 133, 23–35 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jain, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Dunn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hoffman, Seamless, maskless lithography system using spatial light modulator (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Huang, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Li, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Su, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Liang, Precise fabrication of large-area microstructures by digital oblique scanning lithography strategy and stage self-calibration technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 12, 096501 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chien, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chiu, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, Maskless lithography based on oblique scanning of point array with digital distortion correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optics and Lasers in Engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 136, 106313 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Huang, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yuan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hu, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xu, Sub-wavelength patterned pulse laser lithography for efficient fabrication of large-area metasurfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nat Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 13, 5823 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Khan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' lachmayer, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Roth, Maskless lithography for versatile and low cost fabrication of polymer based micro optical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' OSA Continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3, 2808 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hu, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Du, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wu, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rao, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Cai, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Xu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ni, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Li, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Zhao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wu, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chu, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sugioka, Optimized holographic femtosecond laser patterning method towards rapid integration of high-quality functional devices in microchannels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sci Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 6, 33281 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Su, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chu, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sun, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Tsai, Advances in optical metasurfaces: fabrication and applications [Invited].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 26, 13148 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Altissimo, E-beam lithography for micro-/nanofabrication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Biomicrofluidics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4, 026503 (2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lassaline, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Brechbühler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Vonk, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ridderbeek, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Spieser, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Bisig, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' le Feber, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rabouw, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Norris, Optical Fourier surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 582, 506–510 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Howell, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Grushina, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Holzner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Brugger, Thermal scanning probe lithography—a review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Microsyst Nanoeng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 6, 21 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, From nanoscopic to macroscopic photo-driven motion in azobenzene-containing materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nanophotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7, 1387–1422 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kim, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Park, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Park, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Park, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, Light-Directed Soft Mass Migration for Micro/Nanophotonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Advanced Optical Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7, 1900074 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Luigi Oscurato, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sorbo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Pota, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Reda, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Centore, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, Enhanced photoinduced mass migration in supramolecular azopolymers by H-bond driven positional constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Journal of Materials Chemistry C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 9, 11368–11375 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ilnytskyi, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Toshchevikov, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Saphiannikova, Modeling of the photo-induced stress in azobenzene polymers by combining theory and computer simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Soft Matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 15, 9894–9908 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yadav, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Domurath, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kim, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Saphiannikova, Orientation Approach to Directional Photodeformations in Glassy Side-Chain Azopolymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 123, 3337–3347 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Loebner, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yadav, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lomadze, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Tverdokhleb, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Donner, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Saphiannikova, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Santer, Local Direction of Optomechanical Stress in Azobenzene Containing Polymers During Surface Relief Grating Formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Macro Materials & Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 307, 2100990 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rekola, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Berdin, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fedele, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Virkki, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Priimagi, Digital holographic microscopy for real-time observation of surface-relief grating formation on azobenzene-containing films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sci Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 10, 19642 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Reda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, Shapeshifting Diffractive Optical Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Laser & Photonics Reviews, 2100514 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Reda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, Large-Scale Multiplexed Azopolymer Gratings with Engineered Diffraction Behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Advanced Materials Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2101375, 1–9 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lim, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hong, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Son, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Im, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Bang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lee, A Field Guide to Azopolymeric Optical Fourier Surfaces and Augmented Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Advanced Functional Materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 31, 2104105 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Strobelt, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Stolz, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Leven, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Soelen, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kurlandski, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Abourahma, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' McGee, Optical microstructure fabrication using structured polarized illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 30, 7308 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, Deterministic Realization of Quasicrystal Surface Relief Gratings on Thin Azopolymer Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Advanced Materials Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 7, 1902118 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Bachelot, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H’Dhili, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Barchiesi, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lerondel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fikri, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Royer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Landraud, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Peretti, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chaput, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lampel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Boilot, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lahlil, Apertureless near-field optical microscopy: A study of the local tip field enhancement using photosensitive azobenzene-containing films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Journal of Applied Physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 94, 2060–2072 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hubert, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rumyantseva, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lerondel, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Grand, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kostcheev, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Billot, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Vial, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Bachelot, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Royer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chang, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gray, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wiederrecht, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Schatz, Near-Field Photochemical Imaging of Noble Metal Nanostructures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nano Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 5, 615–619 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Porfirev, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Khonina, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ivliev, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Meshalkin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Achimova, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Forbes, Writing and reading with the longitudinal component of light using carbazole-containing azopolymer thin films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sci Rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 12, 3477 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Marrucci, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Roviello, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, Light-induced spiral mass transport in azo- polymer films under vortex-beam illumination.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Nat Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3, 1–9 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Marrucci, Molecular Model for Light-Driven Spiral Mass Transport in Azopolymer Films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 110, 146102 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Pasienski, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' DeMarco, A high-accuracy algorithm for designing arbitrary holographic atom traps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express, OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 16, 2176–2190 (2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Reda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, Varifocal diffractive lenses for multi-depth microscope imaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express, OE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 30, 12695–12711 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Lesem, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Hirsch, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jordan, The Kinoform: A New Wavefront Reconstruction Device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' IBM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' & Dev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 13, 150–155 (1969).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gerchberg, A practical algorithm for the determination of the phase from image and diffraction plane pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 35, 237–246 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Moreno, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Campos, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gorecki, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Yzuel, Effects of Amplitude and Phase Mismatching Errors in the Generation of a Kinoform for Pattern Recognition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jpn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 34, 6423 (1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Poon, Ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=', Digital Holography and Three-Dimensional Display: Principles and Applications (Springer, New York, 2006th edition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=', 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jelken, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Henkel, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Santer, Solving an old puzzle: fine structure of diffraction spots from an azo-polymer surface relief grating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 125, 218 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Reda, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, Accurate Morphology-Related Diffraction Behavior of Light-Induced Surface Relief Gratings on Azopolymers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' ACS Materials Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4, 953–959 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Reda, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, "Multiplexed surface reliefs on azopolymer thin films" in Photosensitive Materials and their Applications II (SPIE, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='spiedigitallibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='org/conference-proceedings-of-spie/12151/121510A/Multiplexed-surface-reliefs- on-azopolymer-thin-films/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2621640.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='full), vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 12151, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 77–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Leach, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Wulff, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Sinclair, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jordan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Courtial, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Thomson, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gibson, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Karunwi, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Cooper, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Laczik, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Padgett, Interactive approach to optical tweezers control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 45, 897 (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Vapaavuori, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ras, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kaivola, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Bazuin, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Priimagi, From partial to complete optical erasure of azobenzene–polymer gratings: effect of molecular weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3, 11011–11016 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Salvatore, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, Computer-generated holograms for complex surface reliefs on azopolymer films.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Scientific Reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 9, 6775 (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Chen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Tian, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Zhang, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Cao, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Jin, Phase hologram optimization with bandwidth constraint strategy for speckle-free optical reconstruction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Express.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 29, 11645 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Madsen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Eriksen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Glückstad, Comparison of state-of-the-art Computer Generated Holography algorithms and a machine learning approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optics Communications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 505, 127590 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Zhang, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Čerkauskaitė, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Drevinskas, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Patel, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Beresna, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kazansky, "Eternal 5D data storage by ultrafast laser writing in glass" in, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Klotzbach, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Washio, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Arnold, Eds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' (San Francisco, California, United States, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' http://proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='spiedigitallibrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='org/proceeding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='aspx?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='doi=10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='1117/12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2220600), p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 97360U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Borbone, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Maddalena, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Ambrosio, Light-Driven Wettability Tailoring of Azopolymer Surfaces with Reconfigured Three-Dimensional Posts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' ACS Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Interfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 9, 30133–30142 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Gerchberg, A practical algorithm for the determination of the phase from image and diffraction plane pictures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Optik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 35, 237–246 (1972).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Supplementary Material for Reprogrammable holograms from maskless surface photo- morphing Francesco Reda,1 Marcella Salvatore,1,2 Marco Astarita,3 Fabio Borbone4 and Stefano L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Oscurato1,2,* 1 Physics Department “E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Pancini”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' University of Naples “Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Complesso Universitario di Monte Sant’Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' via Cinthia 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 80126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy 2 Centro Servizi Metrologici e tecnologici Avanzati (CeSMA),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' University of Naples “Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Complesso Universitario di Monte Sant’Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Via Cintia 21,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 80126,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy 3 Physics Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Politecnico di Milano,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 20133,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Milan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy 4 Department of Chemical Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' University of Naples “Federico II”,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Complesso Universitario di Monte Sant’Angelo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Via Cintia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 80126 Naples,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Italy *Corresponding author: stefanoluigi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='oscurato@unina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='it Diffraction properties of holographic morphological projectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Phase-only holographic plates are typically represented by a complex transmission function 𝑡(𝑥, 𝑦) describing in the scalar approximation of wave optics, the wavefront phase modulation of an incident monochromatic optical field, at wavelength 𝜆 passing through the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Isotropic dielectric phase retarders implement phase modulation as result of local variations in thickness ℎ(𝑥, 𝑦) and refractive index 𝑛(𝑥, 𝑦, 𝜆) of the device, whereby the planar modulation function can be written as: 𝑡(𝑥, 𝑦) = exp [𝑖𝜑(𝑥, 𝑦)] = exp [𝑖 2𝜋 𝜆 (𝑛(𝑥, 𝑦, 𝜆) − 𝑛𝑠)ℎ(𝑥, 𝑦)] (S1) representing the local phase delay 𝜑(𝑥, 𝑦) accumulated by the light due to optical path variation imposed by the plate immersed in a surrounding material whose refractive index is 𝑛𝑠.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' When a kinoform,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' supposed to be at 𝑧 = 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' is illuminated by the light field 𝑈𝑖𝑛(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' the resulting complex field 𝑈𝑜𝑢𝑡(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑧) formed due to Fraunhofer (𝑧 ≫ 0) diffraction is the two-dimensional spatial Fourier transform of the modulated beam at the kinoform plane,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' and the reconstructed image 𝐼𝑜𝑢𝑡 is determined by the relation: 𝐼𝑜𝑢𝑡(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑧) = |𝐹𝑇[𝑈𝑖𝑛(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 0) ∙ 𝑡(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 0)]|2 (S2) The phase encoding process,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' from the phase design to its implementation by lithography,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' leads to a device whose real transmittance 𝑡𝑟𝑒𝑎𝑙,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' is a function g of the designed phase 𝑡𝑟𝑒𝑎𝑙(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑦) = 𝑒𝑖𝑔[𝜑(𝑥,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='𝑦)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' To consider any possible deviation from this ideal case, the complex transmittance of the real kinoform can be decomposed into a linear superposition of functions, clearly describing the effects of phase mismatches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Assuming that the deformation is space invariant, the spatial coordinates can be omitted and the function 𝑡𝑟𝑒𝑎𝑙 can be expanded in terms of its argument, according to the generalized harmonic analysis (61): 𝑡𝑟𝑒𝑎𝑙 = ∑ 𝐺𝛼𝑒𝑖𝛼𝜑 +∞ 𝛼=−∞ (S3) where 𝐺𝛼 = ∫ 𝑡𝑟𝑒𝑎𝑙𝑒𝑖𝛼𝜑 2𝜋 0 𝑑𝜑 and 𝛼 an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The 𝛼 = 1 term is the only one whose Fourier transform results in an optical field with intensity 𝐼𝑜𝑢𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The amount of optical power shaped in the reconstructed intensity profile with respect to the total transmitted power, is equal to 𝜂1 = |𝐺1|2 and it is equal to one only in the ideal case 𝑔(𝜑) = 𝜑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The other terms of the series, apart from the term 𝛼 = 0 which determines an unmodulated optical component named DC term, contribute with shifted and scaled replicas of the desired intensity pattern, known as ghosts or false images (8, 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The total reconstructed pattern is a weighted sum of the desired image, the DC term, and false images, typically also spatially overlapped in the reconstruction plane and with a relative efficiency 𝜂𝛼 = |𝐺𝛼|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' At best, once the geometry ℎ(𝑥, 𝑦) is fixed, 𝑔 is linear with the total surface reliefs amplitude ℎ0, which has to be tuned in order to reach a fully 2𝜋 modulation depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' According to equation (S1) this condition is achieved for ℎ0 = 𝜆/(𝑛(𝜆) − 𝑛𝑆);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' for our material at the operating wavelength 𝜆 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='6328 𝜇𝑚, 𝑛(𝜆) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='696 and 𝑛𝑆 = 1 (for air immersed kinoforms), the condition is satisfied for ℎ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='9092 𝜇𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' In this simple case if ℎ0 ∗ is the implemented modulation depth, the ratio 𝑚 = ℎ0 ∗/ℎ0 denotes a mismatch parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Under those conditions diffraction efficiency 𝜂𝛼 can be written as (60): 𝜂𝛼 = 𝑠𝑖𝑛𝑐2(𝑚 − 𝛼) (S4) The condition 𝑚 = 1 guarantees the maximum diffraction efficiency 𝜂1 = 1, ensuring that all the incident optical power is effectively shaped in the reconstructed holographic pattern [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' During the encoding process, more complex distortion effects can determine a non-linear form for g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' These certainly include quantization and pixelation effects and non-linear responses of the material to the structuring process that led to a kinoform in which phase mismatches are included providing 𝜂1 ≠ 1 even if target modulation depth is reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Supplementary figures Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S1: Theoretical diffraction efficiency from an ideal kinoform as function of the height mismatch error m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S2: Schematic of the experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Beam expander - lenses 𝐿1 (f1=-50 mm) and 𝐿2 (f2=250 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' SLM - Holoeye Pluto, LCOS spatial light modulator, phase only (reflective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 4f configuration - lenses 𝐿3, (f3=300 mm) and 𝐿4 (f2=175 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' QWP - quarter wave plate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' BS - 70/30 beam splitter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Objective - 50X Mitutoyo Plan Apo Infinity Corrected Long WD Objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' TL - tube lens (fTL=200 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fourier transforming lens - lens 𝐿5, (f5=300 mm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' CCD1/2 - “DCC3240M Thorlabs” camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S3: Azopolymer optical characterization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Molecular structural formula.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Absorbance in the UV-visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Probe wavelength 𝜆𝑝 = 633 𝑛𝑚 is chosen different from the writing beam and out of the absorption band of the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Refractive index at 𝜆𝑝 was measured via ellipsometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 2Ne=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='696 Assisting/erasing (405 nm) Wriling Probe (491 nm) (633 nm) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S4: Holographic setup spatial resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Holographic reconstruction of square shaped light pattern with lateral size Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Experimental squares size Δ as function of designed size Δ’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The slope 𝑏 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='376 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='002 𝜇𝑚 of the fitted line trend Δ = 𝑏Δ′ defines the calibration of physical dimensions of patterns in the polymer plane with respect to the analytically designed target images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C Contrast of the holographic reconstructed square as function of lateral size Δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Contrast is defined as 𝐶 = (𝐼𝑊 + 𝐼𝐵)/(𝐼𝑊 − 𝐼𝐵) with 𝐼𝑊 and 𝐼𝐵 representing the average experimental intensity levels corresponding to white and black areas of the target image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Resolution limit is reasonably set to Δ0 = 5𝑏 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='88 𝜇𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Kinoforms resulting from design step are scaled by a factor 5 before being encoded into the illumination pattern, ensuring the highest contrast for the holographic writing pattern maintaining a reasonably reduced pixel size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S5: Azopolymer response to an intensity structured field with designed lateral size Δ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='88 𝜇𝑚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Intensity Surface Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S6: Holographic setup intensity level modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Holographic reconstruction of square shaped light pattern with lateral size Δ0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='88 𝜇𝑚 and linearly spaced gray levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Implemented intensity levels as function of the addressed gray value in the target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' The line trend has a slope equal to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='077 (𝑎.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑢.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S7: Height distribution and height modulation depth estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Height distribution 𝑃𝑖 related to the AFM micrograph presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 3A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Full range dispersion, allowing for the experimental estimation of the total modulation depth ℎ0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='910 𝜇𝑚 is defined as ℎ0 = 𝑁 ∙ Δℎ, where 𝑁 = 91 is the number of occupied bins while ∆ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='01 𝜇𝑚 is the fixed bin width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Implemented phase depth is considered for a probe wavelength 𝜆𝑃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='6328 𝜇𝑚 assuming a refractive index equal to 𝑛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Integral function 𝐼(𝑛) = ∑ 𝑃𝑖 𝑛 𝑖=0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Experimental estimation of ℎ0(90%) represents the height range, uniformly distributed around the median of the distribution, where there is the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='90 of probability to find a corresponding height value in the AFM micrograph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='05 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S8: Comparison between: A phase distribution probability in the target phase map resulting from GS algorithm, B intensity distribution probability in the holographic pattern and C implemented phase distribution retrieved from the AFM image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Implemented phase depth is considered for a probe wavelength 𝜆𝑃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='6328 𝜇𝑚 assuming a refractive index equal to 𝑛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' For visual clarity, data are binned considering 𝑁 = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S9: Temporal characterization of structured surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Height distribution for six different exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Each dot represents the expected value ℎ̅ and relative variance 𝜎2 for the corresponding distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Root mean square error defined as RMSE = √∑ (𝑃𝑖 − 𝑃̅)2 𝑁 as function of the total exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' 𝑃̅ represents the target uniform distribution expected at different exposure times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C Full range modulation depth ℎ0 and 90% dispersion range as function of the total exposure time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Implemented phase depth is considered for a probe wavelength 𝜆𝑃 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='6328 𝜇𝑚 assuming a refractive index equal to 𝑛 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='696.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' t 20 s t=40 s t 60 s t=80 s t=100 s t=120 s -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='904 2元 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S10: Experimental determination of diffraction efficiency 𝜂𝑖 determined by integrating the CCD signal over the regions of interest delimited by the colored trace in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Green area corresponds to the holographic image efficiency while light blue and orange area correspond to the DC order and ghost image, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S11: Kinoform transmittance over exposure time determined by integrating the CCD signal over the full sensor size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S12: Experimental determination of pattern visibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Visibility is defined as 𝑉 = (𝐼𝑆𝑅 + 𝐼𝑁𝑅)/(𝐼𝑆𝑅 − 𝐼𝑁𝑅) where 𝐼𝑆𝑅 is the average intensity inside the signal region (green area) and 𝐼𝑁𝑅 is the average noise level outside the holographic image (orange area).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fologra t=40s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5mm GhostOff axis OnaxisNR SR 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5 mm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S13: Optimization of multiplexed spherical profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Axial shifting Δ𝑧 of the holographic image as function of the spherical phase profile parameter 𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Maximum visibility achieved with different spherical phase profile parameter 𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Best value for multiplexed focal length is 𝑓 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='450 𝑚𝑚, allowing for high visibility and reasonable orders separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S14: Speckle contrast reduction process by holograms time averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Average holographic pattern acquired after 10 writing/erasing cycles representing the on-axis image of the Greek letter pi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Speckle noise severity as function of the number of averaged frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Severity is defined as 𝜎/〈𝐼〉 where 〈𝐼〉 is the mean intensity and 𝜎 is its standard deviation measured in the image, see also (67).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C-D Average holographic pattern acquired after 10 writing/erasing cycles representing the on-axis image of a music note and a smile, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' DCorder noise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='owresolutionA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5mm C 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5mm 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5mm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S15 Speckle analysis of a time averaged grayscale pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' A Target image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Resulting average holographic pattern acquired after 10 writing/erasing cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' C Comparison between the mean intensity level of three cube faces for the single frame and the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' D Comparison between the speckle severity of three cube faces for the single frame and the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' B Top Top Side Side Front Front 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content='5mm Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' S16: Look up table for optical encryption and decryption of text messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} +page_content=' Trit Decimal Char +' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/v9AyT4oBgHgl3EQfaffQ/content/2301.00245v1.pdf'} diff --git a/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf b/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..5659005e4f6f9d747dc1408b2d0f34d71805e5eb --- /dev/null +++ b/v9FJT4oBgHgl3EQffixw/content/2301.11557v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:0c809a94bcc5e059fad237fce9893cc5e1fc0755d2227d1fd88bd4dd981c3a5f +size 2368111 diff --git a/v9FJT4oBgHgl3EQffixw/vector_store/index.faiss b/v9FJT4oBgHgl3EQffixw/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..75fd045ceaa69021fc78d2b2318e936fcf210cfc --- /dev/null +++ b/v9FJT4oBgHgl3EQffixw/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:801d268a2b3f4bc14c2fd345fcd9cc310d8a172434c2fdfeeebbab4142a74b08 +size 1835053 diff --git a/vNFAT4oBgHgl3EQfiR1K/content/tmp_files/2301.08598v1.pdf.txt b/vNFAT4oBgHgl3EQfiR1K/content/tmp_files/2301.08598v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..eccac37d8af946261e1c0b8ca95660dbbdb490f5 --- /dev/null +++ b/vNFAT4oBgHgl3EQfiR1K/content/tmp_files/2301.08598v1.pdf.txt @@ -0,0 +1,816 @@ +Pressure Ulcers and Dressings: A Strain Sensitivity Analy- +sis of the Boundary Conditions of a Finite Element Model +Nolwenn Fougeron 1, Isabelle Rivals 2,3, Nathanaël Connesson 1, Grégory Chagnon 1, Thierry Alonso 1, +Laurent Pasquinet 4, Stéphane Auguste 4, Antoine Perrier 1,5 and Yohan Payan 1 +1 Laboratoire TIMC-IMAG, Univ. Grenoble Alpes, CNRS, UMR 5525, VetAgro Sup, Grenoble INP, TIMC, +38000 Grenoble, France +2 UMR8256 Biological Adaptation and Ageing Research Group, Sorbonne Université, INSERM, UMRS1158, +Neurophysiologie Respiratoire Expérimentale et Clinique, 75013 Paris, France +3 Équipe de Statistique Appliquée, ESPCI Paris, PSL Research University, UMRS1158, 75005 Paris, France +4 Urgo Research, Innovation & Development, 21300 Chenôve, France +5 Département de Médecine de L’adolescent, Sorbonne Université Médecine, Assistance Publique Hôpitaux +de Paris (APHP), Service de Diabétologie, Hôpital Pitié-Salpêtrière, 75013 Paris, France +Abstract: Recently, a new bi-layer dressing was proposed by Urgo RID to reduce the healing time of pressure +ulcers (PU). This dressing was numerically evaluated in previously published work. In the current work, the +influence on the maximal shear strains of modelling parameters such as the dressing local geometry, the pres- +sure applied by the gauze inside the wound, the wound deepness, and the mattress stiffness, was assessed. A +sensitivity analysis was performed on these four parameters. Among all experiments, the mean maximal +Green–Lagrange shear strain was 0.29. The gauze pressure explained 60% of the model response in terms of +the volume of tissues under strains of 0.3, while the wound deepness explained 28%. The mattress had a +significant, but low impact, whereas the dressing local geometry had no significant impact. As expected, the +wound deepness was one of the most influential parameters. The gauze turned out to be more significant than +expected. This may be explained by the large range of values chosen for this study. The results should be +extended to more subjects, but still suggest that the gauze is a parameter that might not be neglected. Care +should also be taken in clinical practice when using gauze that could have either a positive or negative impact +on the soft tissues’ strains. This may also depend on the wound deepness. +Keywords: finite element analysis; pressure ulcers; dressing; soft tissues; internal strains; sen- +sitivity analysis + +1. Introduction +Pressure ulcers (PU) are injuries to the skin and underlying tissues that are common +adverse events in healthcare. For example, in intensive care units, PU prevalence reaches +almost 27% [1]. In any healthcare facility, the risk of developing a PU is increased for older +patients, patients with spinal cord injuries, or comorbidities [2]. PU have terrible conse- +quences on the quality of life of patients including longer hospitalisation time, social iso- +lation, and pain [3,4]. +PU are localised wounds that propagate in the soft tissues after a detrimental external loading. They are +classified from stage-1, for light wounds, to stage-4, for the most severe wounds. Short time (some minutes), but +intense, load application is sufficient to cause a PU, while reduced loads applied for an extended period (2 to 4 +h) can also lead to this kind of wound [5]. They have a multifactorial origin, but mechanical loads applied to the +tissues are considered to play a significant role in the onset of PU. Pressure or shear loads applied at the skin level +may lead to significant internal strains [6]. Strains and, more particularly, the Green–Lagrange maximal shear +strains, appeared to be a mechanical biomarker for the development of PU [7]. When these strains exceed the +cell’s ability to deform, in most cases under bony prominences, this eventually leads to cell death and the devel- +opment of a wound [7–9]. The sacrum is the most affected area of the human body. In this case, the recommended +procedure to treat PU consists of the unloading of the weakened tissues, which can be tedious to do continuously, +particularly if several PU are present. Dressings are common medical devices used to improve the healing process +of PU and a huge range of products are proposed to clinicians. Yet, the mean healing time of PU is estimated to +be 3 weeks and can sometimes exceed 10 weeks [10]. Recently, Urgo RID developed a new concept of dressing to +Citation: Fougeron, N.; Rivals, I.; +Connesson, N.; Chagnon, G.; +Alonso, T.; Pasquinet, L.; Auguste, +S.; Perrier, A.; Payan, Y. Pressure +Ulcers and Dressings: A Strain +Sensitivity Analysis of the Boundary +Conditions of a Finite Element +Model. Biomechanics 2023, 3, 1–12. +https://doi.org/10.3390/biomechan- +ics3010001 + + + +improve the healing of PU by reducing internal strains. This dressing consists of two layers, the first one being +the classic Urgo Start Plus Border dressing and the second one consisting of an unloading material. This material +is cut into alveoli that can be removed under the wound to relocate the loads outside of the wound region when +complete unloading of the PU is not temporally possible. The ability of this dressing to alleviate soft tissues has +already been studied previously [11]; however, question marks remain about the use of the dressing. The impact +of the dressing with different wound deepness, quantities of alveoli removed, or mattress stiffnesses still needs +to be estimated. Furthermore, the interaction with the gauze that is sometimes applied in the wound to absorb +part of the exudate has not been studied yet. +Finite element modelling is a common method applied to compute the soft tissues’ internal strains. Ceelen +et al. [12] proposed and validated this method on rat models for the estimation of the Green–Lagrange maximal +shear strains. Yet, few models of the sacrum region were proposed in the literature [13]. These models were +mainly proposed to compute internal and external stresses in soft tissues without PU. Some studies also applied +the finite element modelling method to evaluate penetrating ulcers in the cardiovascular domain [14], yet few +efforts were made for PU in soft tissues such as skin or adipose tissues. To the authors’ knowledge, the group of +Amit Gefen (Tel Aviv University) was the only one to propose a finite element model of the injured tissues with +a stage-4 PU. The authors showed that adding a multilayer dressing allowed the reduction of internal and exter- +nal stresses around the wound. Several other studies from that group also performed comparative analyses of +the finite element model of the sacrum region with various dressings or mattresses. They compared the use of +silicone foam dressing with various material parameters and showed that dressings anisotropy helped reduce +the internal and external stresses [15]. In another study, the authors from this group showed that the increase in +mattress stiffness induced an increase in internal stresses [16]. They also studied various soft cellulose fluff core +dressings with two mattress conditions and two moisture states of the dressings. Few differences could be noted +among the dressings, but better performances were obtained with the softest mattress [17,18]. These studies bring +interesting insights into how the change of boundary conditions may impact the response of the soft tissue and +potentially the onset or propagation of a PU. However, none of the previous studies reported statistical analysis +on the relative importance of the studied parameters [19]. Furthermore, the gauze inside the wound has still not +been modelled. +The current study aims to estimate the relative impact of the dressing geometry, mattress stiffness, use of +gauze, and PU deepness on the soft tissues’ maximal shear strains around the wound. A sensitivity analysis was +performed on these four parameters using a previously designed parametric model of the sacrum region. + + + + +2. Materials and Methods +2.1. Reference Finite Element Model +To reduce the computation time, a parametric approach was adopted. The model consisted of several layers: +the skin, adipose tissues, both dressing layers, and a mattress. The skin and adipose tissue thicknesses were set +to 1.30 mm and 22.30 mm, respectively [19,20]. To simulate a bony prominence on the median crest of the sacrum, +an imprint of the sacrum geometry was approximated by a portion of a sphere with an ellipsoidal volume on top +of it. The adipose tissue thickness was thus reduced to 13.30 mm under the bony prominence. The sacrum bone +was set as rigid with the pilot node at the centre of the area. A PU from stage-2 to stage-3 was added to the model, +with various depths defined after, while the radius was set to 15.00 mm. The dressing was modelled with two +layers referred to as dressing layer 1 and dressing layer 2 (Figure 1). Dressing layer 1 is the unloading material +cut into alveoli that is in contact with the mattress, and dressing layer 2 is the UrgoStart Plus Border dressing that +is in contact with the skin. Both layers were modelled as a cylindrical layer with a radius of 125.00 mm. The +thickness of dressing layer 2 was set to 3.50 mm, whereas the thickness of dressing layer 1 was set to 5.20 mm. A +mattress with a height of 50.00 mm was added to the model. The diameter of the model was 250.00 mm to avoid +boundary effects in the wound area. Symmetry in the sagittal plane was also considered so only half of the model +was used for the simulation (Figure 2). All components were meshed with SOLID185 linear hexahedral elements +ANSYS APDL (ANSYS 2020 R2 software, ANSYS Inc., Cannonsburg, PA, USA) with a mixed pressure-displace- +ment formulation for the soft tissues. The model was composed, at most, of 6088 elements. + + +Figure 1. The new dressing design developed by Urgo RID. + +The dressing layers were tied together. Tie constraints were also used between the soft tissue layers and +between the skin and dressing layer 2. A coefficient of friction of 0.62 was defined between dressing layer 2 and +the mattress. This value was computed from friction tests performed at Urgo RID. The dressing, glued on a cali- +brated weight, was positioned on a rigid plate cover with a clinical sheet. A gradually increasing force was ap- +plied to a cable attached to the dressing. The coefficient of friction was the ratio of the force that pulled the dress- +ing and the calibrated weight. Between the skin and the mattress, this coefficient was set to 0.43 [20]. A vertical +force of 217 N was applied to the pilot node of the sacrum area, as illustrated in Figure 2a. Considering the sym- +metry of the model, this corresponded to 47% of the bodyweight of a 94 kg subject [21]. The bottom nodes of the +mattress were fixed in position. Simulations were performed with ANSYS in a quasi-static analysis with an im- +plicit scheme. + +Dressing layer 1 +Dressing layer 2 + +Figure 2. Model geometry and boundary conditions. + +Soft tissues were modelled with non-linear hyperelastic isotropic constitutive equations. More particularly, +the skin was modelled with the law proposed by Isihara et al. [22]. The material parameters were optimised using +a curve-fitting method from the data of Ni Annaidh et al. [23]. The adipose tissues were modelled with the equa- +tion developed by Yeoh [24] with parameters fitted according to the data of Sommer et al. [25]. The soft tissue +stiffness was increased close to the wound region, as detailed in Fougeron et al. [11], to account for the stiffening +of the tissues surrounding a PU. The constitutive equation for the different tissues is: +������������ = � ������������������������0(������������1� − 3)������������ + +������������ +������������=1 +� 1 +������������������������ +(������������ − 1)2������������ +������������ +������������=1 + + +������������1 = ������������2 = ������������3 = 3(1 − 2������������) +2������������10(1 + ������������) + +where W is the strain energy density function, ������������������������0 the material parameters, ������������1� the first deviatoric invariant of the +right Cauchy–Green deformation tensor, J the Jacobian of the deformation gradient, and ������������������������ the nearly incom- +pressibility parameters expressed with the Poisson’s ratio ν by the formula provided by Mott et al. [26]. The +indices i and k are between 1 and 3 for the skin and between 1 and 2 for the adipose tissues. Soft tissue stiffening +was considered by multiplying the C10 parameters of the skin and the adipose tissue by a coefficient of 1.0, 1.5, +and 2.0 for the soft, medium, and stiff areas, respectively, as detailed in Figure 2. +The value of Poisson’s ratio was set to 0.4999 to account for the nearly incompressibility of the soft tissues. +Dressing layer 2 was modelled with a linear elastic orthotropic material, whereas layer 1 was defined as a com- +pressible material and modelled with a Blatz–Ko constitutive equation [27]. +������������ = ������������ +2 �������������2 +������������3 ++ �������������3 − 5� + +where ������������2 and ������������3 are the second and third invariants of the right Cauchy–Green deformation tensor, respectively, +and µ is the initial shear modulus. + +a) MODEL BOUNDARY CONDITIONS +ANSYS +Pilot node +Mattress +Nodes constrained by the pilot node +Vertical force +b) MODEL GEOMETRY +Soft adipose tissue +Medium adipose tissue +Rigid adipose tissue +Boundary with the sacrum +Boundary with the bony prominence +Adipose tissue +Skin +Stage-2/Stage-3 PU +Dressing layer 2 +Dressing layer 1 +Soft skin +Medium skin +Rigid skin +Mattress +The initial shear modulus µ of dressing layer 1 and Young’s moduli of dressing layer 2 were computed from +compression and tension tests. According to the literature data, the Poisson ratio of dressing layer 2 was set to +0.2560. The mattress was modelled as a linear elastic isotropic material with a Poisson ratio of 0.3000 and a refer- +ence Young modulus, E, of 0.23 MPa. The material parameters are detailed in Table 1. Further details about the +reference model are provided in Fougeron et al. [11]. + +Table 1. Material parameters of the model’s components. +Component +C10 (MPa) C20 (MPa) C30 (MPa) µ (MPa) EX (MPa) EY (MPa) EZ (MPa) d1 (MPa−1) +ν +Adipose tissue +1.3 × 10−4 +0.0 +12.2 × 10−3 +- +- +- +- +1.6 +0.4999 +Skin +2.7 × 10−1 +1.9 +- +- +- +- +- +- +0.4999 +Dressing layer 1 +- +- +- +1.0 × 10−3 +- +- +- +- +- +Dressing layer 2 +- +- +- +- +4.4 +1.8 +2.6 × 10−2 +- +0.2560 +Mattress +- +- +- +- +2.3 × 10−1 +- +- +- +0.3000 + +2.2. Sensitivity Analysis +Principal stretches λ1, λ2, and λ3 were extracted to compute the Green–Lagrange principal strains (Equation +(4)). The maximal shear strain, Eshear, was calculated as detailed in Equation (5). +������������������������ = 1 +2 (������������������������ 2 − 1), ������������ ∈ [1, 2, 3] + +������������������������ℎ������������������������������������ = 1 +2 max (|������������1 − ������������2|, |������������2 − ������������3|, |������������3 − ������������1|) + +Green–Lagrange maximal shear strains are recognised as potential mechanical biomarkers to study the onset +and development of PU [7]. In the current study, a region of interest (ROI) was defined for the computation of +the strains. The ROI included soft tissues under the wound and in the perilesional area within three times the +radius of the PU.Experiments performed on rats suggested the possibility to define a threshold of damage that +should be subject-specific [7]. Due to the lack of data on human subjects, the threshold was arbitrarily fixed to 0.3 +considering that Eshear was below this threshold for healthy tissues. +A sensitivity analysis was performed to assess the relative significance of the model parameters on the vol- +ume of healthy tissues. The finite element model was emulated with a polynomial model detailed after, following +the method described by Macron et al. [19], to investigate the impact of the following parameters on the volume +of healthy tissues: wound deepness, alveoli cutting size, mattress stiffness, and pressure applied by the gauze. +The parameters varied between their minimal and maximal values, as detailed in Table 2. After normalisation in +[−1, 1], experimental points were chosen according to a three-level full factorial design resulting in 34 combina- +tions (i.e., 81 simulations). Based on the knowledge of expert clinicians, the wound deepness extrema were set to +1.30 mm and 5.00 mm to respectively account for a stage-2 and a stage-3 PU. The recommendations from Urgo +about the use of the bi-layer dressing are to remove alveoli around the wound, so this was used as the mean level +of the parameter. Then, a layer of alveoli around the wound was added or removed to respectively define the +minimal and maximal levels (Figure 3). The mattress stiffness limits were defined according to literature values +[16,28]. In the clinical routine, the pressure applied by the gauze may significantly vary depending on its satura- +tion in fluid and on the person who inserts the gauze in the wound. As a consequence, it was chosen to model +the effect of the gauze by the pressure applied on the wound walls rather than that of the gauze itself. A finite +element preliminary study was performed to define the gauze pressure values. To this end, the volume of healthy +tissues was analysed with a 5.0 mm deep PU model for multiple values of pressure between 0.00 MPa and 0.08 +MPa. A local optimum was found at 0.02 MPa. As a consequence, the minimal and maximal values were set to +0.00 MPa (i.e., no gauze in the wound) and 0.04 MPa. + + + +Table 2. Parameters’ minimal, intermediate, and maximal values used as levels for the experimental points of the sensitivity +analysis. +Parameters +Minimal Level +Intermediate +Level +Maximal Level +Wound deep- +ness +1.30 mm +3.20 mm +5.00 mm +Alveoli cut +Recommended ++1 layer +Recom- +mended +Recommended +−1 layer +Mattress +stiffness +0.03 MPa +0.23 MPa +0.43 MPa +Gauze pres- +sure +0.00 MPa +0.02 MPa +0.04 MPa +Given that the local finite element model is rather a qualitative model, a full polynomial model of degree +two was considered sufficient to emulate it: +������������(������������) = ������������0 + � ������������������������������������������������ +������������ +������������=1 ++ � ������������������������������������������������������������ +������������ +������������=1 +2 ++ � � ������������������������������������������������������������������������������������ +������������>������������ +������������ +������������=1 + + +where y is the volume of healthy tissues, m the number of parameters, xi the value of the ith parameter, and θ the +vector of the adjustable coefficients, which was estimated with ordinary least squares. The value of two for the +degree will be further justified in the results section. The sensitivity of the model to each input (linear term, +square, order-two interaction) can be simply defined as the percentage of variance due to this input. Assuming, +for simplicity, the m = 4 parameters independent and uniformly distributed in [−1, 1] (i.e., with second- and +fourth-order moments of respectively 1/3 and 4/45), it becomes: +2 +2 +2 +2 +2 +2 +2 +2 +1 +1 +1 +1 +var( +) +var( ) +3 +4 +var( +) +var( +) +45 +1 +var( +) +var( ) var( +) +9 +var( ) +i +i +i +i +i +i +ii +ii +i +ii +i +ii +ij +ij +i +j +ij +i +j +ij +m +m +m +i +ii +ij +i +i +i +j i +s +x +x +s +x +x +s +x x +x +x +y +s +s +s +θ +θ +θ +θ +θ +θ +θ +θ +θ += += += +> + += += += +× + + + += += += +× + + += += += +× + + += ++ ++ + + +∑ +∑ +∑∑ + + +The sensitivities to the ith parameter and to its interaction with parameter j are hence given by the percent- +ages: + +(8) + +s. +s. +S. +s +var(y) +var(y) + +Figure 3. Minimal, intermediate, and maximal levels of the alveoli cutting (a) and +wound deepness (b) parameters. + +3. Results +This section may be divided by subheadings. It should provide a concise and precise 208 description of the +experimental results, their interpretation, as well as the experimental 209 conclusions that can be drawn in Table +3. + +Table 3. Parameter coefficients and polynomial model sensitivities (>1%) in decreasing order of magnitude. +Parameters +Coefficients θi and θii or θij +Sensitivities Si or Sij (%) +Gauze pressure +−3.9, −10.7 +60 +Wound deepness +−4.3, −3.3 +28 +Wound deepness∗Gauze pressure ++4.6 +10 +Mattress stiffness ++1.1, −0.9 +1 + +One may notice that approximately 99% of the model response y was explained by four parameters: the +gauze pressure, the wound deepness, the interaction of the wound deepness and the gauze pressure, and the +mattress stiffness. More particularly, the gauze pressure explained about 60% of the model response, as illus- +trated in Figure 4a. Considering dressing layer 1, this layer was shown to reduce the maximal shear strains on +one model of a stage-2 PU in a previous study. When close enough to the recommended (i.e., plus or minus one +layer of alveoli), the change in the volume of healthy tissues was not significant, as presented in Figure 4. On the +contrary, wound deepness was a significant parameter that explained 28% of the response (cf. Figure 4). As ex- +pected, the interaction of the wound deepness and the gauze pressure was also important, whereas the mattress +stiffness had a significant, but low impact on the volume of healthy tissues (cf. Figure 4). Extreme values of gauze +pressure seem to have a negative impact on the volume of healthy tissues (cf. Figure 4), suggesting that an optimal +value can be found. Tissues around deep PU tend to have more important strains (cf. Figure 4) and softer mat- +tresses may not be suitable in all cases, since the interquartile range of the volume of healthy tissues is larger than +for stiffer mattresses. Worst-case scenarios were defined as the 10% experiments with the highest peak maximal +shear strains. Among these nine experiments, the peak maximal shear strains were greater than 0.80 and all were +designed with the softest mattress and the maximal gauze pressure with various wound deepness and alveoli +cut. + +a)ALVEOLICUTS +b)WOUNDDEEPNESSES +Minimallevel +Intermediatelevel +Maximal level + +Figure 4. Effect of the four parameters on the volume of healthy tissues (i.e., tissues with strains lower than 0.3), +with the other three parameters being set to their intermediary value. + +To illustrate the results, Green–Lagrange maximal shear strains in the ROI were plotted for some experi- +ments in Figure 5. + +a) Effect of the Gauze pressure +100 +(%) : +tissues +06 +Volume of healthy +80 +70 +60 +50 +Pressure=0.00MPa +Pressure = 0.02 MPa +Pressure = 0.04 MPa +b) Effect of the Alveoli cut +100 +(%) +tissues +06 +80 +70 +60 +50 +Recommended cut +1 Layer +Recommended cut +Recommended cut -1 Layer +c) Effect of the Wound deepness +100 +(%) : + tissues +90 +Volume of healthy t +80 +70 +60 +50 +Deepness = 1.3 mm +Deepness = 3.2 mm +Deepness = 5.0 mm +d) Effect of the Mattress stiffness +100 +(%) +Volume of healthy tissues +06 +80 +70 +60 +50 +E= 0.03 MPa +E = 0.23 MPa +E= 0.43 MPa + +Figure 5. Green–Lagrange maximal shear strains in the ROI of some experiments. All parame- +ters were set to the intermediate values except for one that varied according to the defined lev- +els: (a) changes in the gauze pressure, (b) changes in the alveoli cut, (c) changes in the wound +deepness, and (d) changes in the mattress stiffness. The ROI appears in grey in (e). + +4. Discussion +A new bi-layer dressing has been proposed by Urgo RID to improve the healing of PU. This dressing has +previously been studied to evaluate its mechanical impact on the soft tissues in one specific scenario. In this case, +the use of the dressing allowed the reduction of internal strains around the wound. Yet, some factors may affect +the conclusions: the dressing alveoli cutting, the pressure applied by the gauze inside the wound, the deepness +of the wound, and/or the stiffness of the mattress. Thereby, the present study aimed to evaluate the relative im- +portance of these parameters regarding the maximal shear strains around the PU. A sensitivity analysis was per- +formed following a three-level full factorial design. +Among all experiments, the mean maximal shear strain was 0.29 and the peak value reached 0.97. The ex- +periments that reached the highest values of maximal shear strains were all designed with the softest mattress +and the maximal gauze pressure. The strain values are in range with the previously published results, but are + +a) GAUZE PRESSURES +b) ALVEOLI CUTS + Minimal level +Intermediate level + Maximal level +C)WOUNDDEEPNESSES +d)MATTRESS STIFFNESS +Minimal level +Intermediate level + Maximal level +0.28 +0.56 +0.14 +0.42 +0.7 +Maximal shear strains (Green-Lagrange) +e) REGION OF INTEREST +lower than those obtained by Macron et al., for whom peak values ranged between 1.42 and 4.14. Macron et al. +studied the strains under the ischial tuberosities in subjects in a sitting position, which may explain the differences +[19]. The computation of the peak maximal shear strain is also local and thus highly sensitive to mesh quality and +model non-linearities. Therefore, the volume of healthy tissues was preferred here as a discriminant measure for +the sensitivity analysis. The gauze pressure alone explained 60% of the model response, while the wound deep- +ness and the interaction between the gauze pressure and the wound deepness accounted for 28% and 10% of the +response, respectively. To the authors’ knowledge, this study is the first attempt to assess the impact of these two +parameters on the computation of the strains where they both significantly impacted the results. The mattress +also had a significant, but low impact. Contrary to the previous studies of Linder-Ganz and Gefen [16], the softest +mattress did not necessarily reduce the strains in the ROI. This may be due to the use of the bi-layer dressing in +this particular study, which adds a cushion layer between the soft tissues and the mattress. Furthermore, the local +approach proposed in this study may not be able to capture the impact of the mattress on a large scale, since +weight-bearing areas are limited here. It is worth noting that the results could be affected by the levels chosen for +the sensitivity analysis. Mattress stiffness is highly dependent on the brand and few data are provided by the +manufacturers. The mattress was modelled with linear elastic homogeneous isotropic material properties, which +may not be appropriate for all mattress technologies. The use of gauze was modelled as a homogeneous pressure +applied inside the PU. Various products are used by clinicians and the filling of the gauze inside the wound is +highly dependent on the operator and the exudate of the wound. The use of pressure allows one to model the +effect of the gauze without the need to model all types of commercialised products or operators’ protocols. The +wound deepness is a significant parameter with an important impact, but in the present study, PU 5.3 mm deep +at most were designed. Consequently, the conclusion might not be extrapolated to deeper PU. Other parameters +could also have been included in the sensitivity analysis. A geometrical description of the PU such as its diameter +or the interaction between the PU diameter and the dressing alveoli cutting could modify the strain distribution. +Subject-specific parameters were also not studied in this work. As detailed by Macron et al. [19], materials and +thicknesses of soft tissues as well as bone geometries may have a significant impact on strain computation [19]. +The material parameters of soft tissues were estimated from cadaveric tests of the literature. Therefore, the current +study does not account for the variability of the constitutive behaviours that are proposed in the literature +[13,29,30]. The Poisson ratio was also higher than in most literature studies, but this is in range with the recom- +mendation of Bonet and Wood [29] to be close to incompressibility. The soft tissue thicknesses were fixed in the +current study even though values from 4.0 mm to 33.5 mm were reported by Clark et al. [30]. Yet, considering all +of the parameters would have entailed too many experiments. As a result, it was decided for this study to focus +on one particular case for which the model was previously experimentally evaluated, and to evaluate the param- +eters relating to the use of the dressing in this particular environment: the alveoli cutting, the gauze pressure, the +wound deepness, and the mattress stiffness. The present study was not exhaustive on the studied parameters. +Further analyses are necessary to include subject-specific parameters obtained on healthy subjects, but also on +subjects with PU. The threshold of the strains used to define healthy tissues could also have an impact on the +results. Thus, the same sensitivity analysis was performed with a threshold of 0.65 as prescribed by Ceelen et al. +[7]. Small discrepancies, a few percent, were noted in terms of sensitivities, but the relative order of the parame- +ters remained the same. +Finally, the results presented here suggest that care should be taken when filling the wound with gauze. +Gauze is important to maintain an optimal environment in the wound, particularly in terms of moisture. How- +ever, gauze should not be crammed into the wound or filled with too much fluid at the risk of applying too much +pressure inside the wound and thus exacerbating the deformations of already weakened soft tissue. Furthermore, +as was expected, the deeper the wound, the more strains. Even though the unloading of soft tissues is always +prescribed for PU, special care should be taken when dealing with stage-2 and higher PU. To consolidate the +conclusion, future work will include the transfer of the proposed modelling on realistic subject-specific geome- +tries of the sacrum and the heel in several patients. This study is a first attempt to numerically evaluate the effect +of new dressing designs and to potentially propose guidelines to industrials and clinicians for the use of these +medical devices. + +Conflicts of Interest: This study was financially supported by Urgo RID. +References +1. +Labeau, S.O.; Afonso, E.; Benbenishty, J.; Blackwood, B.; Boulanger, C.; Brett, S.J.; Calvino-Gunther, S.; Chaboyer, W.; Coyer, F.; +Deschepper, M.; et al. Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: The +DecubICUs study. Intensiv. Care Med. 2020, 47, 160–169. https://doi.org/10.1007/s00134-020-06234-9. + + +2. +Demarré, L.; Van Lancker, A.; Van Hecke, A.; Verhaeghe, S.; Grypdonck, M.; Lemey, J.; Annemans, L.; Beeckman, D. The cost +of prevention and treatment of pressure ulcers: A systematic review. Int. J. Nurs. Stud. 2015, 52, 1754–1774. +https://doi.org/10.1016/j.ijnurstu.2015.06.006. +3. +Graves, N.; Birrell, F.; Whitby, M. Effect of Pressure Ulcers on Length of Hospital Stay. Infect. Control Hosp. Epidemiol. 2005, 26, +293–297. https://doi.org/10.1086/502542. +4. +Theisen, S.; Drabik, A.; Stock, S. Pressure ulcers in older hospitalised patients and its impact on length of stay: A retrospective +observational study. J. Clin. Nurs. 2011, 21, 380–387. https://doi.org/10.1111/j.1365-2702.2011.03915.x. +5. +Loerakker, S.; Stekelenburg, A.; Strijkers, G.; Rijpkema, J.J.M.; Baaijens, F.P.T.; Bader, D.L.; Nicolay, K.; Oomens, C.W.J. Tem- +poral effects of mechanical loading on deformation-induced damage in skeletal muscle tissue. Ann. Biomed. Eng. 2010, 38, 2577– +2587. https://doi.org/10.1007/s10439-010-0002-x. +6. +Gefen, A.; Brienza, D.M.; Cuddigan, J.; Haesler, E.; Kottner, J. Our contemporary understanding of the aetiology of pressure +ulcers/pressure injuries. Int. Wound J. 2021, 19, 692–704. https://doi.org/10.1111/iwj.13667. +7. +Ceelen, K.; Stekelenburg, A.; Loerakker, S.; Strijkers, G.; Bader, D.; Nicolay, K.; Baaijens, F.; Oomens, C. Compression-induced +damage and internal tissue strains are related. J. Biomech. 2008, 41, 3399–3404. https://doi.org/10.1016/j.jbiomech.2008.09.016. +8. +Bouten, C.V.C.; Knight, M.M.; Lee, D.A.; Bader, D.L. Compressive deformation and damage of muscle cell subpopulations in a +model system. Ann. Biomed. Eng. 2001, 29, 153–163. https://doi.org/10.1114/1.1349698. +9. +Oomens, C.W.J.; Bressers, O.F.J.T.; Bosboom, E.M.; Bouten, C.; Bader, D.D. Can loaded interface characteristics influence strain +distributions in muscle adjacent to bony prominences?. Comput. Methods Biomech. Biomed. Eng. 2003, 6, 171–180. +https://doi.org/10.1080/1025584031000121034. +10. +Palese, A.; Saiani, L.; Pota, I.; Laquintana, D.; Stinco, G.; Di Giulio, P. What is the healing time of stage II pressure ulcers? +Findings from a secondary analysis. Adv. Ski. Wound Care 2015, 28, 69–75. https://doi.org/10.1097/01.asw.0000459964.49436.ce. +11. +Fougeron, N.; Connesson, N.; Chagnon, G.; Alonso, T.; Pasquinet, L.; Bahuon, M.; Guillin, E.; Perrier, A.; Payan, Y. New pressure +ulcers dressings to alleviate human soft tissues: A finite element study. J. Tissue Viability 2022, 31, 506–513. +https://doi.org/10.1016/j.jtv.2022.05.007. +12. +Ceelen, K.K.; Stekelenburg, A.; Mulders, J.L.J.; Strijkers, G.J.; Baaijens, F.P.T.; Nicolay, K.; Oomens, C.W.J. Validation of a nu- +merical model of skeletal muscle compression with MR tagging: A contribution to pressure ulcer research. J. Biomech. Eng. 2008, +130, 061015. https://doi.org/10.1115/1.2987877. +13. +Savonnet, L.; Wang, X.; Duprey, S. Finite element models of the thigh-buttock complex for assessing static sitting discomfort +and +pressure +sore +risk: +A +literature +review. +Comput. +Methods +Biomech. +Biomed. +Eng. +2018, +21, +379–388. +https://doi.org/10.1080/10255842.2018.1466117. +14. +Gefen, A.; Krämer, M.; Brehm, M.; Burckardt, S. The biomechanical efficacy of a dressing with a soft cellulose fluff core in +prophylactic use. Int. Wound J. 2020, 17, 1968–1985. https://doi.org/10.1111/iwj.13489. +15. +Linder-Ganz, E.; Gefen, A. Stress analyses coupled with damage laws to determine biomechanical risk factors for deep tissue +injury during sitting. J. Biomech. Eng. 2008, 131, 011003. https://doi.org/10.1115/1.3005195. +16. +Schwartz, D.; Gefen, A. The biomechanical protective effects of a treatment dressing on the soft tissues surrounding a non- +offloaded sacral pressure ulcer. Int. Wound J. 2019, 16, 684–695. https://doi.org/10.1111/iwj.13082. +17. +Schwartz, D.; Levy, A.; Gefen, A. A Computer Modeling Study to Assess the Durability of Prophylactic Dressings Subjected to +Moisture +in +Biomechanical +Pressure +Injury +Prevention. +Ostomy +Wound +Manag. +2018, +64, +18–26. +https://doi.org/10.25270/oqm.2018.7.1826. +18. +Macron, A.; Pillet, H.; Doridam, J.; Rivals, I.; Sadeghinia, M.J.; Verney, A.; Rohan, P.-Y. Is a simplified Finite Element model of +the gluteus region able to capture the mechanical response of the internal soft tissues under compression?. J. Tissue Viability +2019, 217, 81–90. +19. +Call, E.; Pedersen, J.; Bill, B.; Black, J.; Alves, P.; Brindle, C.T.; Dealey, C.; Santamaria, N.; Clark, M. Enhancing pressure ulcer +prevention +using +wound +dressings: +What +are +the +modes +of +action?. +Int. +Wound +J. +2013, +12, +408–413. +https://doi.org/10.1111/iwj.12123. +20. +Plagenhoef, S.; Evans, F.G.; Abdelnour, T. Anatomical Data for Analyzing Human Motion University of Massachusetts -Am- +herst. Res. Q. Exerc. Sport 1983, 54, 169–178. +21. +Isihara, A.; Hashitsume, N.; Tatibana, M. Statistical theory of rubber-like elasticity. IV. (Two-dimensional stretching). J. Chem. +Phys. 1951, 19, 1508–1512. https://doi.org/10.1063/1.1748111. +22. +Annaidh, A.N.; Bruyère-Garnier, K.; Destrade, M.; Gilchrist, M.D.; Otténio, M. Characterization of the anisotropic mechanical +properties of excised human skin. J. Mech. Behav. Biomed. Mater. 2012, 5, 139–148. https://doi.org/10.1016/j.jmbbm.2011.08.016. +23. +Yeoh, O.H. Some forms of the strain energy function for rubber. Rubber Chem. Technol. 1993, 66, 754–771. +https://doi.org/10.5254/1.3538343. +24. +Sommer, G.; Eder, M.; Kovacs, L.; Pathak, H.; Bonitz, L.; Mueller, C.; Regitnig, P.; Holzapfel, G.A. Multiaxial mechanical prop- +erties and constitutive modeling of human adipose tissue: A basis for preoperative simulations in plastic and reconstructive +surgery. Acta Biomater. 2013, 9, 9036–9048. https://doi.org/10.1016/j.actbio.2013.06.011. +25. +Mott, P.; Dorgan, J.; Roland, C. The bulk modulus and Poisson’s ratio of “incompressible” materials. J. Sound Vib. 2008, 312, +572–575. https://doi.org/10.1016/j.jsv.2008.01.026. +26. +Lee, W.; Won, B.H.; Cho, S.W. Finite element modeling for predicting the contact pressure between a foam mattress and the +human +body +in +a +supine +position. +Comput. +Methods +Biomech. +Biomed. +Eng. +2016, +20, +104–117. +https://doi.org/10.1080/10255842.2016.1203421. + + +27. +Dickinson, A.; Steer, J.; Worsley, P. Finite element analysis of the amputated lower limb: A systematic review and recommen- +dations. Med. Eng. Phys. 2017, 43, 1–18. https://doi.org/10.1016/j.medengphy.2017.02.008. +28. +Keenan, B.E.; Evans, S.L.; Oomens, C.W. A review of foot finite element modelling for pressure ulcer prevention in bedrest: +Current perspectives and future recommendations. J. Tissue Viability 2021, 31, 73–83. https://doi.org/10.1016/j.jtv.2021.06.004. +29. +Bonet, J.; Wood, R.D. Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd ed.; Cambridge University Press: New York, +NY, USA, 2008. https://doi.org/10.1017/cbo9780511755446. +30. +Clark, M.; Rowland, L.; Wood, H.; Crow, R. Measurement of soft tissue thickness over the sacrum of elderly hospital patients +using B-mode ultrasound. J. Biomed. Eng. 1989, 11, 200–202. https://doi.org/10.1016/0141-5425(89)90141-6. + + diff --git a/vNFAT4oBgHgl3EQfiR1K/content/tmp_files/load_file.txt b/vNFAT4oBgHgl3EQfiR1K/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5a063e1af27be198699fd99e517119ad524e4869 --- /dev/null +++ b/vNFAT4oBgHgl3EQfiR1K/content/tmp_files/load_file.txt @@ -0,0 +1,1143 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf,len=1142 +page_content='Pressure Ulcers and Dressings: A Strain Sensitivity Analy- sis of the Boundary Conditions of a Finite Element Model Nolwenn Fougeron 1, Isabelle Rivals 2,3, Nathanaël Connesson 1, Grégory Chagnon 1, Thierry Alonso 1, Laurent Pasquinet 4, Stéphane Auguste 4, Antoine Perrier 1,5 and Yohan Payan 1 1 Laboratoire TIMC-IMAG, Univ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Grenoble Alpes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' UMR 5525,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' VetAgro Sup,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Grenoble INP,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' TIMC,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 38000 Grenoble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' France 2 UMR8256 Biological Adaptation and Ageing Research Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sorbonne Université,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' INSERM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' UMRS1158,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Neurophysiologie Respiratoire Expérimentale et Clinique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 75013 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' France 3 Équipe de Statistique Appliquée,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ESPCI Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' PSL Research University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' UMRS1158,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' France 4 Urgo Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Innovation & Development,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 21300 Chenôve,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' France 5 Département de Médecine de L’adolescent,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sorbonne Université Médecine,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Assistance Publique Hôpitaux de Paris (APHP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Service de Diabétologie,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Hôpital Pitié-Salpêtrière,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 75013 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' France Abstract: Recently,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' a new bi-layer dressing was proposed by Urgo RID to reduce the healing time of pressure ulcers (PU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This dressing was numerically evaluated in previously published work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In the current work, the influence on the maximal shear strains of modelling parameters such as the dressing local geometry, the pres- sure applied by the gauze inside the wound, the wound deepness, and the mattress stiffness, was assessed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A sensitivity analysis was performed on these four parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Among all experiments, the mean maximal Green–Lagrange shear strain was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The gauze pressure explained 60% of the model response in terms of the volume of tissues under strains of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3, while the wound deepness explained 28%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The mattress had a significant, but low impact, whereas the dressing local geometry had no significant impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' As expected, the wound deepness was one of the most influential parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The gauze turned out to be more significant than expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This may be explained by the large range of values chosen for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The results should be extended to more subjects, but still suggest that the gauze is a parameter that might not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Care should also be taken in clinical practice when using gauze that could have either a positive or negative impact on the soft tissues’ strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This may also depend on the wound deepness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Keywords: finite element analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' pressure ulcers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' dressing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' soft tissues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' internal strains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' sen- sitivity analysis 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Introduction Pressure ulcers (PU) are injuries to the skin and underlying tissues that are common adverse events in healthcare.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' For example, in intensive care units, PU prevalence reaches almost 27% [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In any healthcare facility, the risk of developing a PU is increased for older patients, patients with spinal cord injuries, or comorbidities [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' PU have terrible conse- quences on the quality of life of patients including longer hospitalisation time, social iso- lation, and pain [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' PU are localised wounds that propagate in the soft tissues after a detrimental external loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' They are classified from stage-1, for light wounds, to stage-4, for the most severe wounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Short time (some minutes), but intense, load application is sufficient to cause a PU, while reduced loads applied for an extended period (2 to 4 h) can also lead to this kind of wound [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' They have a multifactorial origin, but mechanical loads applied to the tissues are considered to play a significant role in the onset of PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pressure or shear loads applied at the skin level may lead to significant internal strains [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Strains and, more particularly, the Green–Lagrange maximal shear strains, appeared to be a mechanical biomarker for the development of PU [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' When these strains exceed the cell’s ability to deform, in most cases under bony prominences, this eventually leads to cell death and the devel- opment of a wound [7–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The sacrum is the most affected area of the human body.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In this case, the recommended procedure to treat PU consists of the unloading of the weakened tissues, which can be tedious to do continuously, particularly if several PU are present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dressings are common medical devices used to improve the healing process of PU and a huge range of products are proposed to clinicians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Yet, the mean healing time of PU is estimated to be 3 weeks and can sometimes exceed 10 weeks [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Recently, Urgo RID developed a new concept of dressing to Citation: Fougeron, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Rivals, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Connesson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Chagnon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Alonso, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pasquinet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Auguste, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Perrier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Payan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pressure Ulcers and Dressings: A Strain Sensitivity Analysis of the Boundary Conditions of a Finite Element Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomechanics 2023, 3, 1–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3390/biomechan- ics3010001 improve the healing of PU by reducing internal strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This dressing consists of two layers, the first one being the classic Urgo Start Plus Border dressing and the second one consisting of an unloading material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This material is cut into alveoli that can be removed under the wound to relocate the loads outside of the wound region when complete unloading of the PU is not temporally possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The ability of this dressing to alleviate soft tissues has already been studied previously [11];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' however, question marks remain about the use of the dressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The impact of the dressing with different wound deepness, quantities of alveoli removed, or mattress stiffnesses still needs to be estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Furthermore, the interaction with the gauze that is sometimes applied in the wound to absorb part of the exudate has not been studied yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Finite element modelling is a common method applied to compute the soft tissues’ internal strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ceelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [12] proposed and validated this method on rat models for the estimation of the Green–Lagrange maximal shear strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Yet, few models of the sacrum region were proposed in the literature [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' These models were mainly proposed to compute internal and external stresses in soft tissues without PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Some studies also applied the finite element modelling method to evaluate penetrating ulcers in the cardiovascular domain [14], yet few efforts were made for PU in soft tissues such as skin or adipose tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' To the authors’ knowledge, the group of Amit Gefen (Tel Aviv University) was the only one to propose a finite element model of the injured tissues with a stage-4 PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The authors showed that adding a multilayer dressing allowed the reduction of internal and exter- nal stresses around the wound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Several other studies from that group also performed comparative analyses of the finite element model of the sacrum region with various dressings or mattresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' They compared the use of silicone foam dressing with various material parameters and showed that dressings anisotropy helped reduce the internal and external stresses [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In another study, the authors from this group showed that the increase in mattress stiffness induced an increase in internal stresses [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' They also studied various soft cellulose fluff core dressings with two mattress conditions and two moisture states of the dressings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Few differences could be noted among the dressings, but better performances were obtained with the softest mattress [17,18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' These studies bring interesting insights into how the change of boundary conditions may impact the response of the soft tissue and potentially the onset or propagation of a PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' However, none of the previous studies reported statistical analysis on the relative importance of the studied parameters [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Furthermore, the gauze inside the wound has still not been modelled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The current study aims to estimate the relative impact of the dressing geometry, mattress stiffness, use of gauze, and PU deepness on the soft tissues’ maximal shear strains around the wound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A sensitivity analysis was performed on these four parameters using a previously designed parametric model of the sacrum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Materials and Methods 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Reference Finite Element Model To reduce the computation time, a parametric approach was adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The model consisted of several layers: the skin, adipose tissues, both dressing layers, and a mattress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The skin and adipose tissue thicknesses were set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='30 mm and 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='30 mm, respectively [19,20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' To simulate a bony prominence on the median crest of the sacrum, an imprint of the sacrum geometry was approximated by a portion of a sphere with an ellipsoidal volume on top of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The adipose tissue thickness was thus reduced to 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='30 mm under the bony prominence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The sacrum bone was set as rigid with the pilot node at the centre of the area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A PU from stage-2 to stage-3 was added to the model, with various depths defined after, while the radius was set to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The dressing was modelled with two layers referred to as dressing layer 1 and dressing layer 2 (Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dressing layer 1 is the unloading material cut into alveoli that is in contact with the mattress, and dressing layer 2 is the UrgoStart Plus Border dressing that is in contact with the skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Both layers were modelled as a cylindrical layer with a radius of 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The thickness of dressing layer 2 was set to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='50 mm, whereas the thickness of dressing layer 1 was set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='20 mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A mattress with a height of 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 mm was added to the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The diameter of the model was 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 mm to avoid boundary effects in the wound area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Symmetry in the sagittal plane was also considered so only half of the model was used for the simulation (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' All components were meshed with SOLID185 linear hexahedral elements ANSYS APDL (ANSYS 2020 R2 software, ANSYS Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=', Cannonsburg, PA, USA) with a mixed pressure-displace- ment formulation for the soft tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The model was composed, at most, of 6088 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The new dressing design developed by Urgo RID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The dressing layers were tied together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tie constraints were also used between the soft tissue layers and between the skin and dressing layer 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A coefficient of friction of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='62 was defined between dressing layer 2 and the mattress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This value was computed from friction tests performed at Urgo RID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The dressing, glued on a cali- brated weight, was positioned on a rigid plate cover with a clinical sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A gradually increasing force was ap- plied to a cable attached to the dressing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The coefficient of friction was the ratio of the force that pulled the dress- ing and the calibrated weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Between the skin and the mattress, this coefficient was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='43 [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A vertical force of 217 N was applied to the pilot node of the sacrum area, as illustrated in Figure 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Considering the sym- metry of the model, this corresponded to 47% of the bodyweight of a 94 kg subject [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The bottom nodes of the mattress were fixed in position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Simulations were performed with ANSYS in a quasi-static analysis with an im- plicit scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dressing layer 1 Dressing layer 2 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Model geometry and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Soft tissues were modelled with non-linear hyperelastic isotropic constitutive equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' More particularly, the skin was modelled with the law proposed by Isihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The material parameters were optimised using a curve-fitting method from the data of Ni Annaidh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The adipose tissues were modelled with the equa- tion developed by Yeoh [24] with parameters fitted according to the data of Sommer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The soft tissue stiffness was increased close to the wound region, as detailed in Fougeron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [11], to account for the stiffening of the tissues surrounding a PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The constitutive equation for the different tissues is: ������������ = � ������������������������0(������������1� − 3)������������ + ������������ ������������=1 � 1 ������������������������ (������������ − 1)2������������ ������������ ������������=1 ������������1 = ������������2 = ������������3 = 3(1 − 2������������) 2������������10(1 + ������������) where W is the strain energy density function,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ������������������������0 the material parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ������������1� the first deviatoric invariant of the right Cauchy–Green deformation tensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J the Jacobian of the deformation gradient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' and ������������������������ the nearly incom- pressibility parameters expressed with the Poisson’s ratio ν by the formula provided by Mott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The indices i and k are between 1 and 3 for the skin and between 1 and 2 for the adipose tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Soft tissue stiffening was considered by multiplying the C10 parameters of the skin and the adipose tissue by a coefficient of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='5, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0 for the soft, medium, and stiff areas, respectively, as detailed in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The value of Poisson’s ratio was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='4999 to account for the nearly incompressibility of the soft tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dressing layer 2 was modelled with a linear elastic orthotropic material, whereas layer 1 was defined as a com- pressible material and modelled with a Blatz–Ko constitutive equation [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ������������ = ������������ 2 �������������2 ������������3 + �������������3 − 5� where ������������2 and ������������3 are the second and third invariants of the right Cauchy–Green deformation tensor, respectively, and µ is the initial shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='a) MODEL BOUNDARY CONDITIONS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ANSYS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Pilot node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Mattress ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Nodes constrained by the pilot node ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Vertical force ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='b) MODEL GEOMETRY ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Soft adipose tissue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Medium adipose tissue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Rigid adipose tissue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Boundary with the sacrum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Boundary with the bony prominence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Adipose tissue ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Skin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Stage-2/Stage-3 PU ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Dressing layer 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Dressing layer 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Soft skin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Medium skin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Rigid skin ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Mattress ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='The initial shear modulus µ of dressing layer 1 and Young’s moduli of dressing layer 2 were computed from ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='compression and tension tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' According to the literature data, the Poisson ratio of dressing layer 2 was set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2560.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The mattress was modelled as a linear elastic isotropic material with a Poisson ratio of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3000 and a refer- ence Young modulus, E, of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='23 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The material parameters are detailed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Further details about the reference model are provided in Fougeron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Material parameters of the model’s components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Component C10 (MPa) C20 (MPa) C30 (MPa) µ (MPa) EX (MPa) EY (MPa) EZ (MPa) d1 (MPa−1) ν Adipose tissue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 × 10−4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 × 10−3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='4999 Skin 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='7 × 10−1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='4999 Dressing layer 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0 × 10−3 Dressing layer 2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='6 × 10−2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2560 Mattress 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 × 10−1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3000 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sensitivity Analysis Principal stretches λ1, λ2, and λ3 were extracted to compute the Green–Lagrange principal strains (Equation (4)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The maximal shear strain, Eshear, was calculated as detailed in Equation (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ������������������������ = 1 2 (������������������������ 2 − 1), ������������ ∈ [1, 2, 3] ������������������������ℎ������������������������������������ = 1 2 max (|������������1 − ������������2|, |������������2 − ������������3|, |������������3 − ������������1|) Green–Lagrange maximal shear strains are recognised as potential mechanical biomarkers to study the onset and development of PU [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In the current study, a region of interest (ROI) was defined for the computation of the strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The ROI included soft tissues under the wound and in the perilesional area within three times the radius of the PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='Experiments performed on rats suggested the possibility to define a threshold of damage that should be subject-specific [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Due to the lack of data on human subjects, the threshold was arbitrarily fixed to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 considering that Eshear was below this threshold for healthy tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A sensitivity analysis was performed to assess the relative significance of the model parameters on the vol- ume of healthy tissues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The finite element model was emulated with a polynomial model detailed after, following the method described by Macron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [19], to investigate the impact of the following parameters on the volume of healthy tissues: wound deepness, alveoli cutting size, mattress stiffness, and pressure applied by the gauze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The parameters varied between their minimal and maximal values, as detailed in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' After normalisation in [−1, 1], experimental points were chosen according to a three-level full factorial design resulting in 34 combina- tions (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=', 81 simulations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Based on the knowledge of expert clinicians, the wound deepness extrema were set to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='30 mm and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 mm to respectively account for a stage-2 and a stage-3 PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The recommendations from Urgo about the use of the bi-layer dressing are to remove alveoli around the wound, so this was used as the mean level of the parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Then, a layer of alveoli around the wound was added or removed to respectively define the minimal and maximal levels (Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The mattress stiffness limits were defined according to literature values [16,28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In the clinical routine, the pressure applied by the gauze may significantly vary depending on its satura- tion in fluid and on the person who inserts the gauze in the wound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' As a consequence, it was chosen to model the effect of the gauze by the pressure applied on the wound walls rather than that of the gauze itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A finite element preliminary study was performed to define the gauze pressure values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' To this end, the volume of healthy tissues was analysed with a 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0 mm deep PU model for multiple values of pressure between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 MPa and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='08 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A local optimum was found at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='02 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' As a consequence, the minimal and maximal values were set to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 MPa (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=', no gauze in the wound) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='04 MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Parameters’ minimal, intermediate, and maximal values used as levels for the experimental points of the sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Parameters Minimal Level Intermediate Level Maximal Level Wound deep- ness 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='30 mm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='20 mm 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 mm Alveoli cut Recommended +1 layer Recom- mended Recommended −1 layer Mattress stiffness 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='03 MPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='23 MPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='43 MPa Gauze pres- sure 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00 MPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='02 MPa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='04 MPa Given that the local finite element model is rather a qualitative model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' a full polynomial model of degree two was considered sufficient to emulate it: ������������(������������) = ������������0 + � ������������������������������������������������ ������������ ������������=1 + � ������������������������������������������������������������ ������������ ������������=1 2 + � � ������������������������������������������������������������������������������������ ������������>������������ ������������ ������������=1 where y is the volume of healthy tissues,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' m the number of parameters,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' xi the value of the ith parameter,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' and θ the vector of the adjustable coefficients,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' which was estimated with ordinary least squares.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The value of two for the degree will be further justified in the results section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The sensitivity of the model to each input (linear term, square, order-two interaction) can be simply defined as the percentage of variance due to this input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Assuming, for simplicity, the m = 4 parameters independent and uniformly distributed in [−1, 1] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' with second- and fourth-order moments of respectively 1/3 and 4/45),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' it becomes: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ) var( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=') ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='9 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='var( ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ii ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ii ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ii ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ii ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='j ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ii ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ij ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='j i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='θ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='> ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4\uf8f4\uf8f2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='\uf8f4\uf8f3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='∑∑ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='The sensitivities to the ith parameter and to its interaction with parameter j are hence given by the percent- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ages: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='(8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' +s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' s var(y) var(y) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Minimal, intermediate, and maximal levels of the alveoli cutting (a) and wound deepness (b) parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Results This section may be divided by subheadings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' It should provide a concise and precise 208 description of the experimental results, their interpretation, as well as the experimental 209 conclusions that can be drawn in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Parameter coefficients and polynomial model sensitivities (>1%) in decreasing order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Parameters Coefficients θi and θii or θij Sensitivities Si or Sij (%) Gauze pressure −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='9, −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='7 60 Wound deepness −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3, −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 28 Wound deepness∗Gauze pressure +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='6 10 Mattress stiffness +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1, −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='9 1 One may notice that approximately 99% of the model response y was explained by four parameters: the gauze pressure, the wound deepness, the interaction of the wound deepness and the gauze pressure, and the mattress stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' More particularly, the gauze pressure explained about 60% of the model response, as illus- trated in Figure 4a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Considering dressing layer 1, this layer was shown to reduce the maximal shear strains on one model of a stage-2 PU in a previous study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' When close enough to the recommended (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=', plus or minus one layer of alveoli), the change in the volume of healthy tissues was not significant, as presented in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' On the contrary, wound deepness was a significant parameter that explained 28% of the response (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' As ex- pected, the interaction of the wound deepness and the gauze pressure was also important, whereas the mattress stiffness had a significant, but low impact on the volume of healthy tissues (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Extreme values of gauze pressure seem to have a negative impact on the volume of healthy tissues (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Figure 4), suggesting that an optimal value can be found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tissues around deep PU tend to have more important strains (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Figure 4) and softer mat- tresses may not be suitable in all cases, since the interquartile range of the volume of healthy tissues is larger than for stiffer mattresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Worst-case scenarios were defined as the 10% experiments with the highest peak maximal shear strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Among these nine experiments, the peak maximal shear strains were greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='80 and all were designed with the softest mattress and the maximal gauze pressure with various wound deepness and alveoli cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' a)ALVEOLICUTS b)WOUNDDEEPNESSES Minimallevel Intermediatelevel Maximal level Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Effect of the four parameters on the volume of healthy tissues (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=', tissues with strains lower than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3), with the other three parameters being set to their intermediary value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' To illustrate the results, Green–Lagrange maximal shear strains in the ROI were plotted for some experi- ments in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' a) Effect of the Gauze pressure 100 (%) : tissues 06 Volume of healthy 80 70 60 50 Pressure=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='00MPa Pressure = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='02 MPa Pressure = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='04 MPa b) Effect of the Alveoli cut 100 (%) tissues 06 80 70 60 50 Recommended cut +1 Layer Recommended cut Recommended cut -1 Layer c) Effect of the Wound deepness 100 (%) : tissues 90 Volume of healthy t 80 70 60 50 Deepness = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 mm Deepness = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2 mm Deepness = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0 mm d) Effect of the Mattress stiffness 100 (%) Volume of healthy tissues 06 80 70 60 50 E= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='03 MPa E = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='23 MPa E= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='43 MPa Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Green–Lagrange maximal shear strains in the ROI of some experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' All parame- ters were set to the intermediate values except for one that varied according to the defined lev- els: (a) changes in the gauze pressure, (b) changes in the alveoli cut, (c) changes in the wound deepness, and (d) changes in the mattress stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The ROI appears in grey in (e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Discussion A new bi-layer dressing has been proposed by Urgo RID to improve the healing of PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This dressing has previously been studied to evaluate its mechanical impact on the soft tissues in one specific scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' In this case, the use of the dressing allowed the reduction of internal strains around the wound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Yet, some factors may affect the conclusions: the dressing alveoli cutting, the pressure applied by the gauze inside the wound, the deepness of the wound, and/or the stiffness of the mattress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Thereby, the present study aimed to evaluate the relative im- portance of these parameters regarding the maximal shear strains around the PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A sensitivity analysis was per- formed following a three-level full factorial design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Among all experiments, the mean maximal shear strain was 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='29 and the peak value reached 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The ex- periments that reached the highest values of maximal shear strains were all designed with the softest mattress and the maximal gauze pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The strain values are in range with the previously published results, but are a) GAUZE PRESSURES b) ALVEOLI CUTS Minimal level Intermediate level Maximal level C)WOUNDDEEPNESSES d)MATTRESS STIFFNESS Minimal level Intermediate level Maximal level 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='7 Maximal shear strains (Green-Lagrange) e) REGION OF INTEREST lower than those obtained by Macron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=', for whom peak values ranged between 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='42 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Macron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' studied the strains under the ischial tuberosities in subjects in a sitting position, which may explain the differences [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The computation of the peak maximal shear strain is also local and thus highly sensitive to mesh quality and model non-linearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Therefore, the volume of healthy tissues was preferred here as a discriminant measure for the sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The gauze pressure alone explained 60% of the model response, while the wound deep- ness and the interaction between the gauze pressure and the wound deepness accounted for 28% and 10% of the response, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' To the authors’ knowledge, this study is the first attempt to assess the impact of these two parameters on the computation of the strains where they both significantly impacted the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The mattress also had a significant, but low impact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Contrary to the previous studies of Linder-Ganz and Gefen [16], the softest mattress did not necessarily reduce the strains in the ROI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This may be due to the use of the bi-layer dressing in this particular study, which adds a cushion layer between the soft tissues and the mattress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Furthermore, the local approach proposed in this study may not be able to capture the impact of the mattress on a large scale, since weight-bearing areas are limited here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' It is worth noting that the results could be affected by the levels chosen for the sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Mattress stiffness is highly dependent on the brand and few data are provided by the manufacturers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The mattress was modelled with linear elastic homogeneous isotropic material properties, which may not be appropriate for all mattress technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The use of gauze was modelled as a homogeneous pressure applied inside the PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Various products are used by clinicians and the filling of the gauze inside the wound is highly dependent on the operator and the exudate of the wound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The use of pressure allows one to model the effect of the gauze without the need to model all types of commercialised products or operators’ protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The wound deepness is a significant parameter with an important impact, but in the present study, PU 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3 mm deep at most were designed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Consequently, the conclusion might not be extrapolated to deeper PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Other parameters could also have been included in the sensitivity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A geometrical description of the PU such as its diameter or the interaction between the PU diameter and the dressing alveoli cutting could modify the strain distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Subject-specific parameters were also not studied in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' As detailed by Macron et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [19], materials and thicknesses of soft tissues as well as bone geometries may have a significant impact on strain computation [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The material parameters of soft tissues were estimated from cadaveric tests of the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Therefore, the current study does not account for the variability of the constitutive behaviours that are proposed in the literature [13,29,30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The Poisson ratio was also higher than in most literature studies, but this is in range with the recom- mendation of Bonet and Wood [29] to be close to incompressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The soft tissue thicknesses were fixed in the current study even though values from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0 mm to 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='5 mm were reported by Clark et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Yet, considering all of the parameters would have entailed too many experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' As a result, it was decided for this study to focus on one particular case for which the model was previously experimentally evaluated, and to evaluate the param- eters relating to the use of the dressing in this particular environment: the alveoli cutting, the gauze pressure, the wound deepness, and the mattress stiffness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The present study was not exhaustive on the studied parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Further analyses are necessary to include subject-specific parameters obtained on healthy subjects, but also on subjects with PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The threshold of the strains used to define healthy tissues could also have an impact on the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Thus, the same sensitivity analysis was performed with a threshold of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='65 as prescribed by Ceelen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Small discrepancies, a few percent, were noted in terms of sensitivities, but the relative order of the parame- ters remained the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Finally, the results presented here suggest that care should be taken when filling the wound with gauze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gauze is important to maintain an optimal environment in the wound, particularly in terms of moisture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' How- ever, gauze should not be crammed into the wound or filled with too much fluid at the risk of applying too much pressure inside the wound and thus exacerbating the deformations of already weakened soft tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Furthermore, as was expected, the deeper the wound, the more strains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Even though the unloading of soft tissues is always prescribed for PU, special care should be taken when dealing with stage-2 and higher PU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' To consolidate the conclusion, future work will include the transfer of the proposed modelling on realistic subject-specific geome- tries of the sacrum and the heel in several patients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' This study is a first attempt to numerically evaluate the effect of new dressing designs and to potentially propose guidelines to industrials and clinicians for the use of these medical devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Conflicts of Interest: This study was financially supported by Urgo RID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Labeau, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Afonso, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Benbenishty, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Blackwood, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Boulanger, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Brett, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Calvino-Gunther, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Chaboyer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Coyer, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Deschepper, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Prevalence, associated factors and outcomes of pressure injuries in adult intensive care unit patients: The DecubICUs study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Intensiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Care Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2020, 47, 160–169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1007/s00134-020-06234-9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Demarré, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Van Lancker, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Van Hecke, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Verhaeghe, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Grypdonck, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Lemey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Annemans, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Beeckman, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The cost of prevention and treatment of pressure ulcers: A systematic review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Nurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2015, 52, 1754–1774.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ijnurstu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='006.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Graves, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Birrell, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Whitby, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Effect of Pressure Ulcers on Length of Hospital Stay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Infect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Control Hosp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Epidemiol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2005, 26, 293–297.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1086/502542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Theisen, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Drabik, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stock, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pressure ulcers in older hospitalised patients and its impact on length of stay: A retrospective observational study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Clin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Nurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2011, 21, 380–387.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1365-2702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='03915.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Loerakker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stekelenburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Strijkers, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Rijpkema, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Baaijens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bader, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Nicolay, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Oomens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tem- poral effects of mechanical loading on deformation-induced damage in skeletal muscle tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2010, 38, 2577– 2587.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1007/s10439-010-0002-x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gefen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Brienza, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Cuddigan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Haesler, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Kottner, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Our contemporary understanding of the aetiology of pressure ulcers/pressure injuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wound J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2021, 19, 692–704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1111/iwj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='13667.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ceelen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stekelenburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Loerakker, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Strijkers, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bader, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Nicolay, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Baaijens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Oomens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Compression-induced damage and internal tissue strains are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2008, 41, 3399–3404.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='jbiomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bouten, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Knight, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Lee, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bader, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Compressive deformation and damage of muscle cell subpopulations in a model system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2001, 29, 153–163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1114/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1349698.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Oomens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bressers, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bosboom, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bouten, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bader, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Can loaded interface characteristics influence strain distributions in muscle adjacent to bony prominences?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='. Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Methods Biomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2003, 6, 171–180.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1080/1025584031000121034.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Palese, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Saiani, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pota, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Laquintana, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stinco, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Di Giulio, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' What is the healing time of stage II pressure ulcers?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Findings from a secondary analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Adv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wound Care 2015, 28, 69–75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1097/01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='asw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='0000459964.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='49436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='ce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Fougeron, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Connesson, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Chagnon, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Alonso, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pasquinet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bahuon, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Guillin, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Perrier, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Payan, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' New pressure ulcers dressings to alleviate human soft tissues: A finite element study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tissue Viability 2022, 31, 506–513.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='jtv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='007.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ceelen, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stekelenburg, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Mulders, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Strijkers, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Baaijens, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Nicolay, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Oomens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Validation of a nu- merical model of skeletal muscle compression with MR tagging: A contribution to pressure ulcer research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2008, 130, 061015.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1115/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2987877.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Savonnet, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Duprey, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Finite element models of the thigh-buttock complex for assessing static sitting discomfort and pressure sore risk: A literature review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Methods Biomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2018, 21, 379–388.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1080/10255842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1466117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gefen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Krämer, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Brehm, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Burckardt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The biomechanical efficacy of a dressing with a soft cellulose fluff core in prophylactic use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wound J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2020, 17, 1968–1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1111/iwj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='13489.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Linder-Ganz, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gefen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Stress analyses coupled with damage laws to determine biomechanical risk factors for deep tissue injury during sitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2008, 131, 011003.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1115/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3005195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Schwartz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gefen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The biomechanical protective effects of a treatment dressing on the soft tissues surrounding a non- offloaded sacral pressure ulcer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wound J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2019, 16, 684–695.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1111/iwj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='13082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Schwartz, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Levy, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gefen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A Computer Modeling Study to Assess the Durability of Prophylactic Dressings Subjected to Moisture in Biomechanical Pressure Injury Prevention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Ostomy Wound Manag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2018, 64, 18–26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='25270/oqm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Macron, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pillet, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Doridam, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Rivals, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sadeghinia, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Verney, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Rohan, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='-Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Is a simplified Finite Element model of the gluteus region able to capture the mechanical response of the internal soft tissues under compression?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='. J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tissue Viability 2019, 217, 81–90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Call, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pedersen, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bill, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Black, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Alves, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Brindle, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dealey, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Santamaria, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Clark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Enhancing pressure ulcer prevention using wound dressings: What are the modes of action?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='. Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wound J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2013, 12, 408–413.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1111/iwj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='12123.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Plagenhoef, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Evans, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Abdelnour, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Anatomical Data for Analyzing Human Motion University of Massachusetts -Am- herst.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Exerc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sport 1983, 54, 169–178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Isihara, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Hashitsume, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tatibana, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Statistical theory of rubber-like elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' (Two-dimensional stretching).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 1951, 19, 1508–1512.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1063/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1748111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Annaidh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bruyère-Garnier, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Destrade, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Gilchrist, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Otténio, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Characterization of the anisotropic mechanical properties of excised human skin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Behav.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Mater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2012, 5, 139–148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='jmbbm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='08.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Yeoh, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Some forms of the strain energy function for rubber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Rubber Chem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Technol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 1993, 66, 754–771.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='5254/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='3538343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sommer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eder, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Kovacs, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Pathak, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bonitz, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Mueller, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Regitnig, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Holzapfel, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Multiaxial mechanical prop- erties and constitutive modeling of human adipose tissue: A basis for preoperative simulations in plastic and reconstructive surgery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Acta Biomater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2013, 9, 9036–9048.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='actbio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Mott, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dorgan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Roland, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' The bulk modulus and Poisson’s ratio of “incompressible” materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Sound Vib.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2008, 312, 572–575.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='jsv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='026.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Lee, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Won, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Cho, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Finite element modeling for predicting the contact pressure between a foam mattress and the human body in a supine position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Methods Biomech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2016, 20, 104–117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1080/10255842.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1203421.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Dickinson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Steer, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Worsley, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Finite element analysis of the amputated lower limb: A systematic review and recommen- dations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Med.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 2017, 43, 1–18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='medengphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='02.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Keenan, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Evans, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Oomens, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' A review of foot finite element modelling for pressure ulcer prevention in bedrest: Current perspectives and future recommendations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Tissue Viability 2021, 31, 73–83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='jtv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Bonet, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wood, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Nonlinear Continuum Mechanics for Finite Element Analysis, 2nd ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Cambridge University Press: New York, NY, USA, 2008.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1017/cbo9780511755446.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Clark, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Rowland, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Wood, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=';' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Crow, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Measurement of soft tissue thickness over the sacrum of elderly hospital patients using B-mode ultrasound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Biomed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' Eng.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' 1989, 11, 200–202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content=' https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} +page_content='1016/0141-5425(89)90141-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/vNFAT4oBgHgl3EQfiR1K/content/2301.08598v1.pdf'} diff --git a/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf b/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..d8ca2c10c0d0597a05e047195f93b748292e61c0 --- /dev/null +++ b/yNAzT4oBgHgl3EQftP0T/content/2301.01671v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:053ad3b3307ec394b59da0a58f864290dcc3dfc271481873f0977060464305c1 +size 476828 diff --git a/yNAzT4oBgHgl3EQftP0T/vector_store/index.faiss b/yNAzT4oBgHgl3EQftP0T/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..4a43feb859a2b2932ca7ba92d06fb1b4f73ea872 --- /dev/null +++ b/yNAzT4oBgHgl3EQftP0T/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:f38392e353dee168e32a0ba5c3a8dbbd770342d7f03780ade25f3e7bc1896d27 +size 7077933 diff --git a/yNAzT4oBgHgl3EQftP0T/vector_store/index.pkl b/yNAzT4oBgHgl3EQftP0T/vector_store/index.pkl new file mode 100644 index 0000000000000000000000000000000000000000..d302f40d202690ba8593b9ad186dc165dc14f361 --- /dev/null +++ b/yNAzT4oBgHgl3EQftP0T/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:403482b1f85e53c3f28d4f58646b7ca6b9ffe426cd90965d3072b38626d9d68c +size 269368 diff --git a/ytFKT4oBgHgl3EQfMC3E/content/tmp_files/2301.11749v1.pdf.txt b/ytFKT4oBgHgl3EQfMC3E/content/tmp_files/2301.11749v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3a3df0e451b5dd04299dca71eb70d100c539aece --- /dev/null +++ b/ytFKT4oBgHgl3EQfMC3E/content/tmp_files/2301.11749v1.pdf.txt @@ -0,0 +1,2599 @@ +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +1 +A Multi-task Multi-stage Transitional Training +Framework for Neural Chat Translation +Chulun Zhou∗, Yunlong Liang∗, Fandong Meng, Jie Zhou, Jinan Xu, +Hongji Wang, Min Zhang and Jinsong Su† +Abstract—Neural chat translation (NCT) aims to translate a cross-lingual chat between speakers of different languages. Existing +context-aware NMT models cannot achieve satisfactory performances due to the following inherent problems: 1) limited resources of +annotated bilingual dialogues; 2) the neglect of modelling conversational properties; 3) training discrepancy between different stages. +To address these issues, in this paper, we propose a multi-task multi-stage transitional (MMT) training framework, where an NCT +model is trained using the bilingual chat translation dataset and additional monolingual dialogues. We elaborately design two auxiliary +tasks, namely utterance discrimination and speaker discrimination, to introduce the modelling of dialogue coherence and speaker +characteristic into the NCT model. The training process consists of three stages: 1) sentence-level pre-training on large-scale parallel +corpus; 2) intermediate training with auxiliary tasks using additional monolingual dialogues; 3) context-aware fine-tuning with gradual +transition. Particularly, the second stage serves as an intermediate phase that alleviates the training discrepancy between the +pre-training and fine-tuning stages. Moreover, to make the stage transition smoother, we train the NCT model using a gradual transition +strategy, i.e., gradually transiting from using monolingual to bilingual dialogues. Extensive experiments on two language pairs +demonstrate the effectiveness and superiority of our proposed training framework. +Index Terms—Neural Chat Translation, Monolingual Dialogue, Dialogue Coherence, Speaker Characteristic, Gradual Transition. +! +1 +INTRODUCTION +N +EURAL Chat Translation (NCT) is to translate a cross- +lingual chat between speakers of different languages +into utterances of their individual mother tongue. Fig. 1 +depicts an example of cross-lingual chat where one speaks +in English and another in Chinese with their corresponding +translations. With more international communication and +cooperation all around the world, the chat translation task +becomes more important and has broader applications in +daily life. +In this task, sentence-level Neural Machine Translation +(NMT) models [1], [2], [3] can be directly used to translate +dialogue utterances sentence by sentence. In spite of its +practicability, sentence-level NMT models often generate +∗ +C. Zhou and Y. Liang equally contribute to this paper. +† +Jinsong Su is the corresponding author. +• +C. Zhou and H. Wang are with School of Informatics, Xiamen University, +Xiamen 361005, China. +E-mail: clzhou@stu.xmu.edu.cn, whj@xmu.edu.cn +• +Y. Liang and J. Xu are with Beijing Jiaotong University, Beijing 100044, +China. +E-mail: yunlongliang@bjtu.edu.cn, jaxu@bjtu.edu.cn +• +F. Meng and J. Zhou are with Pattern Recognition Center, WeChat AI, +Tencent Inc, China. +E-mail: fandongmeng@tencent.com, withtomzhou@tencent.com +• +M. Zhang is with Soochow University 215031, Suzhou, China. +E-mail: minzhang@suda.edu.cn. +• +J. Su is with School of Informatics and Institute of Artificial Intelligence, +Xiamen University 361005, Xiamen, China. Meanwhile, he is with +Laboratory of Digital Protection and Intelligent Processing of Intangible +Cultural Heritage of Fujian and Taiwan (Xiamen University), Ministry +of Culture and Tourism, China. He is also with Pengcheng Laboratory, +China. +E-mail: jssu@xmu.edu.cn +Manuscript received July 25, 2021; revised May 6, 2022; accepted Dec 18, +2022. +𝐲!"#: !"#$%&'(&)*+, +-./(rú guǒ nǐ kàn dào tā!jiào tā +gǎn jǐn shōu shí xíng lǐ") +𝐱!"#: Well, if you see him, tell him +to pack his bags. +s1 +𝐱#: Oh, hi Max! Hey, do you know +everybody? +𝐲#: 0123#456789:(mài +kè sī#nǐ hái rèn shí dà huǒ ma$) +s2 +𝐲$: ;56/#$<7=>9:(bù +rèn shí"nǐ kàn jiàn dà wèi le ma$) +𝐱$: No. Have you seen David? +𝐱%: No, no, he hasn’t been around. +𝐲%: ?@'&?AB/(méi yǒu! +tā méi zài zhè") +𝐱!: So when, when do you leave? +𝐲!: #CDEFGHI:(nǐ +men shén me shí hou dòng shēn$) +…… +…… +s2 +s1 +s1 +Speaker s1, is the translation of Speaker s1-specific utterance. +𝐲!: … +s1 +𝐱!: … +Speaker s2, is the translation of Speaker s2-specific utterance. +𝐱!: … +s2 +𝐲!: … +Source-side context 𝐶𝐱! +Target-side context 𝐶𝐲! +Fig. 1. An example of cross-lingual chat (En⇔Zh). The speaker s1- +specific utterance xu is being translated from English to Chinese with +corresponding dialogue history context. +unsatisfactory translations due to ignoring the contextual +information in dialogue history. To address this problem, +many researches [4], [5], [6], [7], [8], [9], [10], [11], [12], +[13], [14] adapt context-aware NMT models to make chat +translation through their capability of incorporating dia- +logue history context. Generally, these methods adopt a +pretrain-finetune paradigm, which first pre-train a sentence- +level NMT model on a large-scale parallel corpus and then +fine-tune it on the chat translation dataset in a context-aware +way. However, they still can not obtain satisfactory results in +the scenario of chat translation, mainly due to the following +aspects of limitations: 1) The resource of bilingual chat +arXiv:2301.11749v1 [cs.CL] 27 Jan 2023 + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +2 +translation corpus is usually limited, thus making an NCT +model insufficiently trained to fully exploit dialogue con- +text. 2) Conventional ways of incorporating dialogue con- +text neglect to explicitly model its conversational properties +such as dialogue coherence and speaker characteristic, re- +sulting in incoherent and speaker-inconsistent translations. +3) The abrupt transition from sentence-level pre-training to +context-aware fine-tuning breaks the consistency of model +training, which hurts the potential performance of the final +NCT model. Therefore, it is of great significance to train a +better NCT models by resolving the above three aspects of +limitations. +In this paper, we propose a multi-task multi-stage +transitional (MMT) training framework where an NCT +model is trained using the bilingual chat translation dataset +and additional monolingual dialogues. Specifically, our pro- +posed framework consists of three training stages, also +following the pretrain-finetune paradigm. The first stage +is still to pre-train the NCT model through sentence-level +translation on the large-scale parallel corpus, resulting in +the model M1. At the second stage, using M1 for model +initialization, we continue to train the model through the +previous sentence-level translation task along with two aux- +iliary dialogue-related tasks using additional monolingual +dialogues, obtaining the model M2. The auxiliary tasks are +related to dialogue coherence and speaker characteristic, +which are two important conversational properties of dia- +logue context. For the dialogue coherence, we design the +task of Utterance Discrimination (UD). The UD task is to +judge whether an utterance and a given section of contextual +utterances are within the same dialogue. For the speaker +characteristic, we design the Speaker Discrimination (SD) task. +The SD task is to discriminate whether a given utterance +and a piece of speaker-specific dialogue history contexts +are spoken by the same speaker. Finally, at the last stage, +initialized by M2, the model is fine-tuned using a gradual +transition strategy and eventually becomes a context-aware +NCT model M3. Concretely, the NCT model is trained +through the objective comprised of chat translation, UD and +SD tasks. During this process, we initially construct training +samples for the two auxiliary tasks from additional mono- +lingual dialogues and gradually transit to using bilingual +dialogues. +The MMT training framework enhances the NCT model +from the following aspects. Firstly, the relatively abundant +monolingual dialogues function as a supplement to the +scarce annotated bilingual dialogues, making the model +more sufficiently trained to exploit dialogue context. Sec- +ondly, the UD and SD tasks are directly related to dialogue +coherence and speaker characteristic, thus introducing the +modelling of these two conversational properties into the +NCT model. Thirdly, the second training stage serves as an +intermediate phase that alleviates the discrepancy between +sentence-level pre-training and context-aware fine-tuning. +Particularly, it endows the model with the preliminary +capability to capture dialogue context for the subsequent +NCT training. It is notable that the two dialogue-related +auxiliary tasks exist at both the second and third stages +with different training data, which maintains the training +consistency to some extent. Therefore, at the third stage, the +NCT model can be more effectively fine-tuned to leverage +dialogue context using the chat translation dataset with only +a small number of annotated bilingual dialogues. +In essence, the major contributions of our paper are as +follows: +• +In NCT, our work is the first attempt to use ad- +ditional relatively abundant monolingual dialogues +for training, which helps the model more sufficiently +trained to capture dialogue context for chat transla- +tion. +• +We elaborately design two dialogue-related auxiliary +tasks, namely utterance discrimination and speaker +discrimination. This makes the model more capable +of modelling dialogue coherence and speaker char- +acteristic, which are two important conversational +properties of dialogue context. +• +We propose to alleviate the training discrepancy be- +tween pre-training and fine-tuning by introducing an +intermediate stage (Stage 2) and adopting a gradual +transition strategy for the context-aware fine-tuning +(Stage 3). At the second stage, the model is simul- +taneously optimized with the two auxiliary tasks on +the additional monolingual dialogues. Moreover, at +the third stage, we train the NCT model by gradually +transiting from using monolingual to bilingual dia- +logues, making the stage transition smoother. Thus, +the NCT model can be more effectively fine-tuned on +the small-scale bilingual chat translation dataset. +• +We will release the code of this work on Github +https:// github.com/DeepLearnXMU. +The remainder of this paper is organized as follows. Sec- +tion 2 gives the NCT problem formalization, introduces the +basic architecture of our NCT model and describes the con- +ventional two-stage training including sentence-level pre- +training and context-aware fine-tuning. Section 3 elaborates +our proposed MMT training framework. In Section 4, we +report the experimental results and make in-depth analysis. +Section 5 summarizes the related work, mainly involving +several existing studies on NCT and context-aware NMT +models. Finally, in Section 6, we draw the conclusions of +this paper. +2 +BACKGROUND +In this section, we first give the NCT problem formalization +(Section 2.1). Then, we describe the Flat-NCT model, which +is the model architecture used in this work (Section 2.2). +Finally, we introduce the dominant approach of training an +NCT model, which consists of sentence-level pre-training +(Section 2.3.1) and context-aware fine-tuning (Section 2.3.2). +2.1 +Problem Formalization +In the scenario of this work, we denote the two speakers +involved in a dialogue as s1 and s2. For a cross-lingual chat, +as shown in the example in Fig. 1, the two speakers speak +in the source and target language, respectively. We assume + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +3 +[𝐶𝐱!; 𝐱"] +𝑦#$ +𝐡% +(') +𝐡),$ +(') +𝑦$ +Decoder +Input Representation Layer +emb +emb +emb +emb +emb +emb +emb +𝑐𝑙𝑠 +𝑥" +𝑥# +𝑥$ +𝑥% +𝑠𝑒𝑝 +𝑒𝑜𝑠 +𝐡&,( +($) +𝐡&,$ +($) +𝐡&,# +($) +𝐡&,% +($) +𝐡&,+ +($) +𝐡&,, +($) +𝐡&," +($) +𝐡&,, +(") +𝐡&,, +(-) +𝐶𝐱! +𝐱" +emb +emb +emb +𝐡&,. +($) +𝐡&,/ +($) +𝐡&,0 +($) +𝐡&,. +(") +𝐡&,/ +(") +𝐡&,0 +(") +𝐡&,. +(-) +𝐡&,/ +(-) +𝐡&,0 +(-) +𝑥, +𝑥/ +𝑥. +Encoder +Softmax +… +… +… +… +Self-attention +Self-attention +Fig. 2. The architecture of the Flat-NCT model used in this work. The left part depicts the attention mechanism inside Flat-NCT encoder. For +illustration, we assume the input sequence Cxu; xu is the concatenation of Cxu=x1,x2,x3,x4 and xu=x6,x7,x8,⟨eos⟩ separated by a special token +“⟨sep⟩”. Notably, words in Cxu can only be attended to by those in xu at the first encoder layer. At the other encoder layers, Cxu is masked and the +self-attention is only conducted within words of xu. +TABLE 1 +Definitions of Different Dialogue History Contexts +Symbol +Definition +Meaning +Cxu +x1, x2, x3, ..., xu−1 +Source-side context of xu +Cyu +y1, y2, y3, ..., yu−1 +Target-side context of yu +Cs1 +xu +x1, x3, ..., xu−2 +s1-specific context of xu +Cs2 +xu +x2, x4, ..., xu−1 +s2-specific context of xu +Cs1 +yu +y1, y3, ..., yu−2 +s1-specific context of yu +Cs2 +yu +y2, y4, ..., yu−1 +s2-specific context of yu +Cxu +x1, x2, x3, ..., xu−1 +Context of xu +Cyu +y1, y2, y3, ..., yu−1 +Context of yu +Cs1 +xu +x1, x3, ..., xu−2 +s1-specific context of xu +Cs2 +xu +x2, x4, ..., xu−1 +s2-specific context of xu +Cs1 +yu +y1, y3, ..., yu−2 +s1-specific context of yu +Cs2 +yu +y2, y4, ..., yu−1 +s2-specific context of yu +xu represents an utterance from the source-language monolingual +dialogue X and yu is from the target-language monolingual dia- +logue Y . +they have alternately given utterances in their own lan- +guages for u turns, resulting in the source-language utter- +ance sequence X=x1, x2, x3, x4, ..., xu−1, xu and the target- +language utterance sequence Y =y1, y2, y3, y4, ..., yu−1, yu. +Notably, X and Y contain both the utterances originally +spoken by one speaker and the translated utterances from +the other speaker. Specifically, among these utterances, +x1, x3, ..., xu are originally spoken by the source-language +speaker s1 and y1, y3, ..., yu are the corresponding trans- +lations in the target language. Analogously, y2, y4, ..., yu−1 +are originally spoken by the target-language speaker s2 and +x2, x4, ..., xu−1 are the translated utterances in the source +language. +Besides the bilingual dialogues, our proposed train- +ing framework uses additional monolingual dialogues DX +of the source language and DY of the target language. +Slightly different from the bilingual dialogue, the two +speakers (s1 and s2) in a monolingual dialogue speak +in the same language. We also assume a source-language +monolingual dialogue X∈DX and a target-language mono- +lingual Y ∈DY +proceed to the u-th turn, resulting in +x1, x2, x3, x4, ..., xu−1, xu and y1, y2, y3, y4, ..., yu−1, yu, +respectively. +Then, we give the necessary definitions in the remainder +of this paper. For clarity, we list all definitions1 in Table 1. +For a bilingual dialogue, we define the dialogue history +context of xu on the source side as Cxu=x1, x2, x3, ..., xu−1 +and that of yu on the target side as Cyu=y1, y2, y3, ..., yu−1. +According to original speakers, on the source side, we +define the speaker s1-specific dialogue history context of +xu as the partial sequence of its preceding utterances +Cs1 +xu=x1, x3, ..., xu−2 and the speaker s2-specific dialogue +history context of xu as Cs2 +xu=x2, x4, ..., xu−1. On the target +side, Cs1 +yu=y1, y3, ..., yu−2 and Cs2 +yu=y2, y4, ..., yu−1 denote +the speaker s1-specific and s2-specific dialogue history con- +texts of yu, respectively. When it comes to a monolingual +dialogue, we also formalize different types of dialogue +history contexts {Cxu, Cyu, Cs1 +xu, Cs2 +xu, Cs1 +yu, Cs2 +yu} in a similar +way. +2.2 +The NCT model +We use the Flat-Transformer introduced in [14] as our basic +NCT model, which we denote as Flat-NCT. Figure 2 shows +the architecture of the Flat-NCT, mainly including input +representation layer, encoder and decoder. +2.2.1 +Input Representation Layer +For each utterance xu=x1, x2,· · ·, x|xu| to be translated, +[Cxu; xu] is fed into the NCT model as input, where [; ] +1. For each item of {Cxu, Cyu, Cs1 +xu, Cs2 +xu, Cs1 +yu, Cs2 +yu, Cxu, Cyu, Cs1 +xu, +Cs2 +xu, Cs1 +yu, Cs2 +yu}, taking Cxu for instance, we prepend a special token +‘[cls]’ to it and use another special token ‘[sep]’ to delimit its included +utterances, as implemented in [15]. + +JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015 +4 +denotes the concatenation. Different from the conventional +embedding layer that only includes word embedding WE +and position embedding PE, we additionally add a speaker +embedding SE and a turn embedding TE. The final embed- +ding B(xi) of each input word xi can be written as +B(xi) = WE(xi) + PE(xi) + SE(xi) + TE(xi), +(1) +where WE ∈ R|V |×d, SE ∈ R2×d and TE ∈ R|U|×d. +Here, |V |, |U| and d denote the size of shared vocabulary, +maximum dialogue turns, and the hidden size, respectively. +2.2.2 +Encoder +The encoder of our NCT model has L identical layers, +each of which is composed of a self-attention (SelfAtt) sub- +layer and a feed-forward network (FFN) sub-layer.2 Let +h(l) +e +denote the hidden states of the l-th encoder layer, it +is calculated using the following equations: +z(l) +e += SelfAtt(h(l−1) +e +) + h(l−1) +e +, +h(l) +e += FFN(z(l) +e ) + z(l) +e , +(2) +where h(0) +e +is initialized as the embedding of input words. +Particularly, words in Cxu can only be attended to by those +in xu at the first encoder layer while Cxu is masked at the +other layers, as implemented in [14]. +2.2.3 +Decoder +The decoder also consists of L identical layers, each of +which additionally has a cross-attention (CrossAtt) sub- +layer compared to the encoder. Let h(l) +d +denote the hidden +states of the l-th decoder layer, it is computed as +z(l) +d = SelfAtt(h(l−1) +d +) + h(l−1) +d +, +c(l) +d = CrossAtt(z(l) +d , h(L) +e +) + z(l) +d , +h(l) +d = FFN(c(l) +d ) + c(l) +d , +(3) +where h(L) +e +corresponds to the top-layer encoder hidden +states. +At each decoding time step t, the t-th decoder hidden +state h(L) +d,t is fed into a linear transformation layer and a +softmax layer to predict the probability distribution of the +next target token: +p(yt|y